D_{jj}=\frac{\prod_{t=1}^{k}(\lambda^{(k+1)}_{j}-\lambda^{(k)}_{t})}{\prod_{t=1}^{k}\Gamma(1+\lambda^{(k+1)}_{j}-\lambda^{(k+1)}_{t})},\
\ \ 1\leq j\leq k,$ $\displaystyle
D_{k+1,k+1}=\frac{1}{\prod_{t=1}^{k}\Gamma(1+\lambda^{(k+1)}_{j}-\lambda^{(k+1)}_{t})}.$

Thus the normalized connection matrix $C^{(k+1)}(A_{k})L^{(k+1)}$, what we are
computing, differs from the connection factor $\widetilde{\Omega_{0}}$ by $D$.
One checks that multiplying the formula [6, Formula 7.3-7.4] for
$\widetilde{\Omega_{0}}$ by $D$ gives rise to the formula in Proposition 3.22
(provided that the replacement of the matrix $A$ by $-A$ and the irregular
term $E_{k+1}$ by $\iota E_{k+1}$ are accounted for).

####  3.5.4 Entries of Stokes matrices

For any $1\leq k\leq n-1$, we define a map $\widetilde{C}_{k}\colon{\rm
Herm}(k)\rightarrow U(k)$ by

$\widetilde{C}_{k}(B)=\widetilde{C}(E_{1},\delta_{1}(B))\cdots\widetilde{C}(E_{k},\delta_{k}(B)),\hskip
8.53581ptfor\ B\in{\rm Herm}(k).$

Then we define upper and lower $k$ by $k$ matrices $S_{k\pm}(B)$ via

$\displaystyle\widetilde{C}_{k}(B)e^{B_{k}}\widetilde{C}_{k}(B)^{-1}=S_{k-}(B)S_{k+}(B),$
(81)

with the diagonal part $[S_{k+}(B)]=[S_{k-}(B)]=e^{\frac{[B_{k}]}{2}}$. By
definition $S_{k\pm}(B)$ are just the $k\times k$ Stokes matrices at $u_{\rm
cat}$.

Recall that the $n\times n$ Stokes matrix $S_{+}(u_{\rm cat},A)$ at $u_{\rm
cat}$ is upper triangular. For any $1\leq k\leq n-1$, let us denote by
$b_{k+1}(A)$ the column vector consisting of the first $k$ elements of the
$k+1$-th column of $S_{+}(A)$.

######  Lemma 3.25.

The column vector $b_{k+1}(A)$ is equal to the product of matrices

$\displaystyle
b_{k+1}(A)=S_{k+}(A^{(k)}_{{k-1}})\widetilde{C}_{k}(A^{(k)})e^{-\frac{A_{k}^{(k)}}{2}}b^{(k+1)}(A^{(k+1)}_{k}),$
(82)

where $b^{(k+1)}(A^{(k+1)}_{k})$ is the column vector consisting of the first
$k$ entries of the $k+1$-th column of the Stokes matrix
$S_{+}\big{(}E_{k+1},\delta_{k+1}(A_{k})\big{)}$ of equation (76).

Proof. It follows from the expression of the monodromy relation (66) that the
upper left $k+1$ submatrix of $S_{+}(u_{\rm cat},A)$ coincides with
$S_{k+1}(A^{(k+1)})$. So it is enough to prove the case $k=n-1$. Using the
monodromy relation

$\widetilde{C}(E_{n},A)e^{A_{n}}\widetilde{C}(E_{n},A)^{-1}=S_{-}(E_{n},A_{n-1})S_{+}(E_{n},A_{n-1}),$

and the expression

$S_{-}(E_{n},A_{n-1})=\left(\begin{array}[]{cc}e^{\frac{1}{2}A_{n-1}^{(n-1)}}&0\\\
b^{(n)}_{-}(A_{n-1})&\ast\end{array}\right),\ \ \
S_{+}(E_{n},A_{n-1})=\left(\begin{array}[]{cc}e^{\frac{1}{2}A_{n-1}^{(n-1)}}&b^{(n)}(A_{n-1})\\\
0&\ast\end{array}\right),$

we get

$\displaystyle S_{n-}(A)S_{n+}(A)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\widetilde{C}_{n-1}(A^{(n-1)})&0\\\
0&1\end{array}\right)\widetilde{C}(E_{n},A)e^{A_{n}}\widetilde{C}(E_{n},A)^{-1}\left(\begin{array}[]{cc}\widetilde{C}_{n-1}(A^{(n-1)})^{-1}&0\\\
0&1\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\widetilde{C}_{n-1}(A^{(n-1)})&0\\\
0&1\end{array}\right)\left(\begin{array}[]{cc}e^{\frac{1}{2}A_{n-1}^{(n-1)}}&0\\\
b^{(n)}_{-}&\ast\end{array}\right)\left(\begin{array}[]{cc}e^{\frac{1}{2}A_{n-1}^{(n-1)}}&b^{(n)}\\\
0&\ast\end{array}\right)\left(\begin{array}[]{cc}\widetilde{C}_{n-1}(A^{(n-1)})^{-1}&0\\\
0&1\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\widetilde{C}_{n-1}(A^{(n-1)})e^{A_{n-1}^{(n-1)}}\widetilde{C}_{n-1}(A^{(n-1)})^{-1}&\widetilde{C}_{n-1}(A^{(n-1)})e^{\frac{1}{2}A_{n-1}^{(n-1)}}b^{(n)}\\\
b^{(n)}_{-}e^{\frac{1}{2}A_{n-1}^{(n-1)}}\widetilde{C}_{n-1}(A^{(n-1)})^{-1}&\ast\end{array}\right)$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cc}S_{n-1,-}&0\\\
b^{(n)}_{-}e^{\frac{1}{2}A_{n-1}^{(n-1)}}(S_{n-1,+}\widetilde{C}_{n-1})^{-1}&\ast\end{array}\right)\left(\begin{array}[]{cc}S_{n-1,+}&S_{n-1,+}\widetilde{C}_{n-1}e^{-\frac{1}{2}A_{n-1}^{(n-1)}}b^{(n)}\\\
0&\ast\end{array}\right).$

Here in the last equality, we use again the monodromy relation

$\widetilde{C}_{n-1}(A^{(n-1)})e^{A_{n-1}^{(n-1)}}\widetilde{C}_{n-1}(A^{(n-1)})^{-1}=S_{n-1,-}\big{(}A_{n-2}^{(n-1)}\big{)}S_{n-1,+}\big{(}A_{n-2}^{(n-1)}\big{)}.$

Then the proof is finished for $k=n-1$.

Proof of Theorem 1.2. Note that $(S_{+})_{k,k+1}$ is the $k$-th entry of the
column vector $b_{k+1}(A)$. Lemma 3.25 says that

$b_{k+1}(A)=S_{k+}(A^{(k)}_{k-1})\widetilde{C}_{k}(A^{(k)})e^{-\frac{A_{k}^{(k)}}{2}}b^{(k+1)}(A^{(k+1)}_{k}).$

On the one hand, the $k$-th row of $S_{k+}(A^{(k)}_{k-1})$ is simply
$(0,...,0,e^{\frac{1}{2}\lambda^{(k-1)}_{k}})$. Here recall that
$\lambda^{(k-1)}_{k}=A_{kk}$. On the other hand, by the definition of
$\widetilde{C}_{k}$ for $k=1,...,n$, we have

$\widetilde{C}_{k}(A^{(k)})=\left(\begin{array}[]{cc}\widetilde{C}_{k-1}(A^{(k-1)})&0\\\
0&1\end{array}\right)\cdot\widetilde{C}(E_{k},A^{(k)}).$

Thus the $k$-th row of $S_{k+}(A^{(k)}_{k-1})\widetilde{C}_{k}(A^{(k)})$ is
just the scalar multiplication of $e^{\frac{1}{2}\lambda^{(k-1)}_{k}}$ with
the $k$-th row of $\widetilde{C}^{(k)}(A^{(k)})$.

Therefore, the $k$-th entry of $b_{k+1}(A)$ is given by multiplying the $k$-th
row of $\widetilde{C}^{(k)}(A^{(k)})$ with the $k+1$-th column of
$e^{\frac{-1}{2}A_{k}^{(k)}}b^{(k)}(A_{k}^{(k+1)})$. By Proposition 3.22, that
is

$\displaystyle(S_{+})_{k,k+1}$ $\displaystyle=$ $\displaystyle
e^{\frac{\small{\lambda^{(k-1)}_{k}-\lambda^{(k)}_{k+1}}}{4}}\sum_{i=1}^{k}\Big{(}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+\frac{\lambda^{(k)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1+\frac{\lambda^{(k+1)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+\frac{\lambda^{(k)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1+\frac{\lambda^{(k-1)}_{l}-\lambda^{(k)}_{i}}{2\pi\iota})}\Big{)}\cdot\frac{a^{(k)}_{i}}{N_{i}^{(k+1)}}.$

Then the expression of $(S_{+})_{k,k+1}$ in the theorem follows immediately
from the identity $a^{(k)}_{i}/{N_{i}^{(k+1)}}=m^{(k)}_{i}$. The expression of
$(S_{-})_{k+1,k}$ follows in a similar way.

Since the Riemann-Hilbert-Birkhoff map $\nu(u_{\rm cat})$ (at the caterpillar
point) is a Poisson map (see the last section), other entries of Stokes
matrices are therefore determined by sub-diagonal ones, via the classical RLL
formulation, see Section 6.6. Thus the other entries are given by explicit
algebraic combinations of the sub-diagonal ones.

######  Remark 3.26.

Since each $C(E_{k},\delta_{k}(A))$ smoothly depend on $A$, the Stokes
matrices $S_{\pm}(u_{\rm cat},\cdot)$ are smoothly defined on ${\rm Herm}(n)$.
However, in Definition 3.20 we introduce the normalized connection matrix
$\widetilde{C}(E_{k},\delta_{k}(A))$, associated to a systematically defined
chain of matrices $P_{k}(A)$ diagonalizing $A^{(k)}$. The matrix
$\widetilde{C}(E_{k},\delta_{k}(A))$ is only defined for $A\in{\rm
Herm}_{0}(n)$, simply because $P_{k}$ is only defined there (a family of
matrices $P(t)$, which diagonalizes a family of $A(t)$, is generally singular
when $t$ approaches to the point where eigenvalues of $A(t)$ coincides). In
this way, we can only derive the explicit expression of the Stokes matrices on
${\rm Herm}_{0}(n)\subset{\rm Herm}(n)$. But one checks that the expression
smoothly (actually real analytic) extends to ${\rm Herm}(n)$. We also remark
that it is interesting to study the behaviour of the expression in Theorem 1.2
under the toric degeneration of Gelfand-Tsetlin systems, see e.g., [44].

###  3.6 The diffeomorphism $\Phi_{u_{\rm cat}}(u)$

For any fixed $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$, the Riemann-Hilbert-
Birkhoff map (also known as the dual exponential map) is

$\displaystyle\nu(u)\colon{\rm Herm}(n)\cong\mathfrak{u}(n)^{*}\rightarrow{\rm
Herm}^{+}(n)\cong{\rm U}(n)^{*};\ A\mapsto S_{-}(u,A)S_{+}(u,A).$ (87)

It follows from [10, Theorem 2] that $\nu(u)$ is a diffeomorphism. Similarly,

######  Definition 3.27.

The Riemann-Hilbert-Birkhoff map at $u_{\rm cat}$ is

$\displaystyle\nu(u_{\rm cat})\colon{\rm Herm}(n)\rightarrow{\rm
Herm}^{+}(n);\ A\mapsto S_{-}(u_{\rm cat},A)S_{+}(u_{\rm cat},A).$ (88)

The smooth map $\nu(u_{\rm cat})$ intertwines the Gelfand-Tsetlin system
defined on the open dense subset, see Section 7.1, thus is a diffeomorphism.

######  Proposition 3.28.

Given any $A\in{\rm Herm}(n)$, there exists a solution $\Phi(u;A)$ of the
isomonodromy equation (27) with the prescribed asymptotics $A$ at $u_{\rm
cat}$.

Proof. Since $\nu(u)$ and $\nu(u_{\rm cat})$ are diffeomorphisms, for any
fixed $A\in{\rm Herm}(n)$, there exists a unique function $Y(u)\in{\rm
Herm}(n)$ of $u\in U_{\rm id}$ such that $S_{\pm}(u,Y(u))=S_{\pm}(u_{\rm
cat},A)$. Because $S_{\pm}(u_{\rm cat},A)$ is independent of $u$, the function
$Y(u)$ is a solution of isomonodromy equation (27). By Theorem 1.1 and
Corollary 3.18, there exists $A^{\prime}$, as the asymptotics of the solution
$Y(u)$, such that

$S_{\pm}(u_{\rm cat},A)=S_{\pm}(u,Y(u))=S_{\pm}(u_{\rm cat},A^{\prime}).$

It follows that $A=A^{\prime}$, i.e., the asymptotics of the solution $Y(u)$
is $A$.

######  Definition 3.29.

We introduce the map

$\Phi_{u_{\rm cat}}(u):{\rm Herm}(n)\rightarrow{\rm
Herm}(n)~{};~{}A\mapsto\Phi(u;A),$

where $\Phi(u;A)$ is the solution of (27) with the prescribed asymptotics $A$.
Then by Corollary 3.18 it is the diffeomorphism such that
$\nu(u)\circ\Phi_{u_{\rm cat}}(u)=\nu(u_{\rm cat})$.

###  3.7 Proof of Theorem 1.5: regularized limits of Stokes matrices as
approaching to caterpillar points

This subsection is a proof of Theorem 1.5.

Suppose we are given the solution $\Phi(u;B)$ of the isomonodromy equation
(27) on $U_{\rm id}$ with the asymptotics $B\in{\rm Herm}(n)$. Following the
proof of Theorem 1.1, we have as $0<u_{1}\ll u_{2}\ll...\ll u_{n}$,

$\displaystyle\Big{(}\overrightarrow{\prod_{k=1,...,n}}(\frac{u_{k-1}}{u_{k}})^{\frac{\delta_{k}(\Phi)}{2\pi\iota}}\Big{)}\cdot\Phi(u;B)\cdot\Big{(}\overrightarrow{\prod_{k=1,...,n}}(\frac{u_{k-1}}{u_{k}})^{\frac{\delta_{k}(\Phi)}{2\pi\iota}}\Big{)}^{-1}=B+O({\rm
In}(u_{2})/u_{2}),$ (89)

uniformly for all $B$ satisfying the norm $|B|<d$ for a fixed constant $d>0$.
For simplicity, for any $B\in{\rm Herm}(n)$ and $B^{\prime}\in{\rm
Herm}^{+}(n)$, let us denote by

$g(u;B)=\overrightarrow{\prod_{k=1,...,n}}({u_{k-1}}/{u_{k}})^{\frac{\delta_{k}(B)}{2\pi\iota}},\hskip
14.22636ptand\hskip
14.22636ptG(u;B^{\prime})=\overrightarrow{\prod_{k=1,...,n}}({u_{k-1}}/{u_{k}})^{\frac{{\rm
log}(\delta_{k}(B^{\prime}))}{2\pi\iota}}.$

Here the $\delta_{k}(B^{\prime})$ is positive definition Hermitian matrix, and
${\rm log}$ takes its logarithm.

######  Lemma 3.30.

For any $u$ with $0<u_{1}<\cdots<u_{n}$, $B\in{\rm Herm}(n)$ and
$B^{\prime}\in{\rm Herm}^{+}(n)$, if

$W=g(u;B)\cdot B\cdot g(u;B)^{-1},\hskip
14.22636ptW^{\prime}=G(u;B^{\prime})\cdot B^{\prime}\cdot
g(u;B^{\prime})^{-1},$

then

$B=g(u;W)^{-1}\cdot W\cdot g(u;W),\hskip
14.22636ptB^{\prime}=G(u;W^{\prime})\cdot W^{\prime}\cdot
g(u;W^{\prime})^{-1}.$

Proof. It follows directly from the definition of the operator $\delta_{k}$,
the chosen order in the product $\overrightarrow{\prod}$.

Note that $|g(u;B)|=1$ for any $B\in{\rm Herm}(n)$, the above Lemma and (89)
imply

$\displaystyle\Phi(u,B)=g(u;B)^{-1}\cdot B\cdot g(u;B)+O({\rm
In}(u_{2})/u_{2}).$ (90)

It further implies as $0<u_{1}\ll u_{2}\ll...\ll u_{n}$,

$\displaystyle S_{\pm}\Big{(}u,~{}g(u;B)^{-1}Bg(u;B)+O({\rm
In}(u_{2})/u_{2})\Big{)}=S_{\pm}(u,\Phi(u;B))=S_{\pm}(u_{\rm cat},B).$ (91)

Here the last identity follows from the definition of Stokes matrices at
$u_{\rm cat}$. Setting $A:=g(u;B)^{-1}\cdot B\cdot g(u;B)$, it arrives at

$\displaystyle S_{\pm}\Big{(}u,~{}A+O({\rm
In}(u_{2})/u_{2})\Big{)}=S_{\pm}\Big{(}u_{\rm cat},~{}g(u;A)\cdot A\cdot
g(u;A)^{-1}\Big{)}.$ (92)

A manipulation of Gauss decomposition and monodromy relation (201) shows the
following equivariant property of Riemann-Hilbert-Birkhoff map $\nu(u_{\rm
cat})$ (see Section 7.1 for a stronger equivariant condition of $\nu(u_{\rm
cat})$ with respect to the Thimm action)

$\displaystyle\nu\Big{(}u_{\rm cat},g(u;A)\cdot A\cdot
g(u;A)^{-1}\Big{)}=G\Big{(}u;\nu(u_{\rm cat},A)\Big{)}\cdot\nu(u_{\rm
cat},A)\cdot G\Big{(}u;\nu(u_{\rm cat},A)\Big{)}^{-1}.$ (93)

Therefore, the identities (92) and (93) give

$\displaystyle\nu\Big{(}u,A+O({\rm
In}(u_{2})/u_{2})\Big{)}=G\Big{(}u;\nu(u_{\rm cat},A)\Big{)}\cdot\nu(u_{\rm
cat},A)\cdot G\Big{(}u;\nu(u_{\rm cat},A)\Big{)}^{-1}.$

Since the Riemann-Hilbert-Birkhoff map $\nu(u,A)$ is analytic on $A$, applying
Lemma 3.30 to the above identity proves Theorem 1.5. Up to a Gauss
decomposition, it states that $S_{\pm}(u,A)$ exist properly regularized limits
as $0<u_{1}\ll u_{2}\ll...\ll u_{n}$.

######  Example 3.31 (2 by 2 cases).

Let us consider the rank two case, that is

$\displaystyle\frac{dF}{dz}=\left(\left(\begin{array}[]{cc}\iota u_{1}&0\\\
0&\iota u_{2}\end{array}\right)-\frac{1}{2\pi\iota
z}{\left(\begin{array}[]{cc}t_{1}&a\\\
\bar{a}&t_{2}\end{array}\right)}\right)\cdot F.\quad$

Following [[7] Proposition 8], the Stokes matrices (with respect to the chosen
branch of ${\rm log}(z)$) are

$S_{-}(u,A)=\left(\begin{array}[]{cc}e^{\frac{t_{1}}{2}}&0\\\
\frac{\bar{a}\cdot
e^{\frac{t_{1}-t_{2}}{4}}({u_{2}-u_{1}})^{\frac{t_{2}-t_{1}}{2\pi\iota}}}{\Gamma(1-\frac{\lambda_{1}-t_{1}}{2\pi\iota})\Gamma(1-\frac{\lambda_{2}-t_{1}}{2\pi\iota})}&e^{\frac{t_{2}}{2}}\end{array}\right),\
\ S_{+}(u,A)=\left(\begin{array}[]{cc}e^{\frac{t_{1}}{2}}&\frac{a\cdot
e^{\frac{t_{1}-t_{2}}{4}}({u_{2}-u_{1}})^{\frac{t_{1}-t_{2}}{2\pi\iota}}}{\Gamma(1+\frac{\lambda_{1}-t_{1}}{2\pi\iota})\Gamma(1+\frac{\lambda_{2}-t_{1}}{2\pi\iota})}\\\
0&e^{\frac{t_{2}}{2}}\end{array}\right).$

By the definition of $\delta_{k}$ given in (32), we have that
$\delta_{0}(S_{\pm}(u,A))=\delta_{1}(S_{\pm}(u,A))$ is the diagonal part of
$S_{\pm}(u,A)$. Then ${\frac{{\rm
log}(\delta_{1}(S_{-})\delta_{1}(S_{+}))}{2\pi\iota}}={{\rm
diag}(\frac{t_{1}}{2\pi\iota},\frac{t_{1}}{2\pi\iota})}$, and we have

$\left(\begin{array}[]{cc}u_{2}^{-\frac{t_{1}}{2\pi\iota}}&0\\\
0&u_{2}^{-\frac{t_{2}}{2\pi\iota}}\end{array}\right)S_{-}(u,A)S_{+}(u,A)\left(\begin{array}[]{cc}u_{2}^{\frac{t_{1}}{2\pi\iota}}&0\\\
0&u_{2}^{\frac{t_{2}}{2\pi\iota}}\end{array}\right)\rightarrow S_{-}(u_{\rm
cat},A)S_{+}(u_{\rm cat},A),\ \ as\ u_{1}\ll u_{2},$

where

$S_{-}(u_{\rm cat},A)^{\dagger}=S_{+}(u_{\rm
cat},A)=\left(\begin{array}[]{cc}e^{\frac{t_{1}}{2}}&\frac{a\cdot
e^{\frac{t_{1}-t_{2}}{4}}}{\Gamma(1+\frac{\lambda_{1}-t_{1}}{2\pi\iota})\Gamma(1+\frac{\lambda_{2}-t_{1}}{2\pi\iota})}\\\
0&e^{\frac{t_{2}}{2}}\end{array}\right).$

###  3.8 The leading terms of Stokes matrices via the Gelfand-Tsetlin systems

Recall that in Section 3.7, we introduce the unitary matrix

$g(u;A)=\overrightarrow{\prod_{k=0,...,n-1}}({u_{k}}/{u_{k+1}})^{\frac{\delta_{k}(A)}{2\pi\iota}}.$

Setting $A=g(u;B)^{-1}\cdot B\cdot g(u;B)++O({\rm In}(u_{2})/u_{2})$ as the
argument in the left hand side of the identity (91), it arrives at

######  Proposition 3.32.

For any fixed $A\in{\rm Herm}(n)$, we have, as $0<u_{1}\ll u_{2}\ll...\ll
u_{n}$,

$\displaystyle S_{\pm}(u,A)=S_{\pm}\Big{(}u_{\rm cat},\hskip
5.69054ptg(u;A)\cdot A\cdot g(u;A)^{-1}+O({\rm In}(u_{2})/u_{2})\Big{)}.$ (95)

As a corollary, we can describe explicitly the leading terms of $S_{\pm}(u,A)$
as $u\rightarrow u_{\rm cat}$.

######  Corollary 3.33.

For any fixed $A\in{\rm Herm}(n)$, the subdiagonal entries of Stokes matrices
$S_{\pm}(u,A)$ satisfy

$\displaystyle(S_{+}(u,A))_{k,k+1}={l}^{(\pm)}_{k,k+1}(u,A)+o(l^{(+)}_{k,k+1}(u,A)),\hskip
8.53581ptas\ 0<u_{1}\ll u_{2}\ll...\ll u_{n},$ (96)

where ${l}^{(+)}_{k,k+1}(u,A)$ coincides with the $(S_{+})_{k,k+1}$ given by
the formula in Theorem 1.2, provided we replace the term $m_{i}^{(k)}(A)$ by

$\displaystyle m_{i}^{(k)}(A)\cdot
u_{k}^{-\frac{A_{kk}}{2\pi\iota}}({u_{k}}/{u_{k+1}})^{\frac{\lambda^{(k)}_{i}(A)}{2\pi\iota}}.$
(97)

Proof of Corollary 3.33. First, under the Gelfand-Tsetlin action and angle
coordinates, the Thimm action (see Section 3.5.1) of an element

${\rm diag}(e^{\iota\theta^{(k)}_{1}},...,e^{\iota\theta^{(k)}_{k}})\in T(k)$

is described by

$\displaystyle\lambda^{(i)}_{j}\mapsto\lambda^{(i)}_{j},\hskip
14.22636pt\psi^{(i)}_{j}=\psi^{(i)}_{j}+\delta_{ki}\theta^{(k)}_{j}.$ (98)

For any $A$ and $u$, let us introduce an element in the torus

$\prod_{k=1,...,n}({u_{k-1}}/{u_{k}})^{\frac{\lambda^{(k-1)}(A)}{2\pi\iota}}\in
T(1)\times\cdots\times T(n-1)$

where $\lambda^{(k-1)}(A):={\rm
diag}(\lambda^{(k-1)}_{1},...,\lambda^{(k-1)}_{k-1})$. Let us introduce the
diagonal matrix

$D(u;A)={\rm
diag}(u_{1}^{{A_{11}}/{2\pi\iota}},...,u_{n}^{{A_{nn}}/{2\pi\iota}})\in T(n).$

Then one checks

$\displaystyle
g(u;A)Ag(u;A)^{-1}=D(u;A)^{-1}\cdot\Big{(}\prod_{k=1,...,n}({u_{k-1}}/{u_{k}})^{\frac{\lambda^{(k-1)}(A)}{2\pi\iota}}\bullet
A\Big{)}\cdot D(u;A).$ (99)

Therefore, we have

$\displaystyle\lambda^{(k)}_{i}(g(u;A)Ag(u;A)^{-1})$ $\displaystyle=$
$\displaystyle\lambda^{(k)}_{i}(A),$ $\displaystyle
a^{(k)}_{i}(g(u;A)Ag(u;A)^{-1})$ $\displaystyle=$ $\displaystyle
a^{(k)}_{i}(A)\cdot
u_{i}^{-\frac{A_{ii}}{2\pi\iota}}(u_{k}/u_{k+1})^{\frac{\lambda^{(k)}_{i}(A)}{2\pi\iota}},$

where the $u_{i}^{-{A_{ii}}/{2\pi\iota}}$ term in the second identity comes
from the action of $D(u;A)$ in (99). Furthermore, using the identity of
functions $a^{(k)}_{i}$ and ${m^{(k)}_{i}}$ on ${\rm Herm}_{0}(n)$,

$a^{(k)}_{i}={m^{(k)}_{i}}\cdot\sqrt{\frac{\prod_{v=1,v\neq
i}^{k+1}(\lambda^{(k+1)}_{v}-\lambda^{(k+1)}_{i})}{\prod_{v=1}^{k}(\lambda^{(k)}_{v}-\lambda^{(k+1)}_{i})}},$

we get that the values of the function $m_{i}^{(k)}$ at the two points $A$ and
$g(u;A)Ag(u;A)^{-1}$ are related by

$\displaystyle m_{i}^{(k)}(g(u;A)Ag(u;A)^{-1})=m_{i}^{(k)}(A)\cdot
u_{k}^{-\frac{A_{kk}}{2\pi\iota}}({u_{k}}/{u_{k+1}})^{\frac{\lambda^{(k)}_{i}(A)}{2\pi\iota}},$

that is the identity (97). In the end, the corollary follows from the real
analyticity of $S_{\pm}(u_{\rm cat},A)$ with respect to $A$.

One can prove similar results for the other entries of $S_{\pm}(u,A)$. Since
the functions $m^{(k)}_{i}$ are expressed by
$a^{(k)}_{i}=e^{\iota\psi^{(k)}_{i}}$, $\lambda^{(k)}_{l}$ and
$\lambda^{(k+1)}_{l}$, Corollary 3.33 gives the expression of the leading
terms of Stokes matrices as $u\rightarrow u_{\rm cat}$, in terms of the
Gelfand-Tsetlin action and angle coordinates. Furthermore, we see that the
leading terms include a fast spin on the corresponding Liouville torus of the
integrable systems, which is cancelled out as far as the regularized limit is
considered.

###  3.9 Regularized limits of Stokes matrices with degenerate irregular terms

Although the results in the previous sections are obtained under the
assumption that $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$, a generalization of
the construction to the degenerate case, i.e.,
$u\in\mathfrak{h}(\mathbb{R})\setminus\mathfrak{h}_{\rm reg}(\mathbb{R})$, is
direct.

Given any partition $\underline{d}$ of $n$, i.e., a set of integers
$\\{d_{j}\\}_{j=1,...,m}$ such that $n=d_{1}+\cdots+d_{m}$, let us consider
the subspace of $\mathfrak{h}(\mathbb{R})$

$U_{\underline{d}}=\\{u={\rm
diag}(\underbrace{u_{1},...,u_{1}}_{d_{1}},\underbrace{u_{2},...,u_{2}}_{d_{2}},....,\underbrace{u_{m},...,u_{m}}_{d_{m}})\in\mathfrak{h}(\mathbb{R})~{}|~{}u_{i}\neq
u_{j}\ \ if\ \ i\neq j\\}.$

Any $n\times n$ matrix $A$ can be seen as a block matrix $A=(A_{ij})_{1\leq
i,j\leq m}$ according to the tuple $\underline{d}$, where each $A_{ij}$
denotes the $d_{i}\times d_{j}$ block/submatrix formed by the corresponding
$d_{i}$ rows and $d_{j}$ columns of $A$. Then for each $k=0,...,m-1$, we
denote by $\delta_{k}(A)$ the blocked matrix

$\displaystyle\delta_{k}(A)_{ij}=\left\\{\begin{array}[]{lr}A_{ij},&if\ \
1\leq i,j\leq k,\ or\ i=j\\\ 0,&otherwise.\end{array}\right.$ (102)

######  Remark 3.34.

The identity (102) depends on the partition $d_{1},...,d_{m}$, which
generalizes the definition of $\delta_{k}(A)$ given in (32) corresponding to
the case $d_{1}=\cdots=d_{n}=1$. Although this more general notation can lead
to confusion with (32), the context typically eliminates any ambiguity.

Associated to degenerate irregular terms, the isomonodromy equation becomes
the differential equation for a matrix valued function
$\Phi(u):U_{\underline{d}}\rightarrow{\rm Herm}(n)$

$\displaystyle\frac{\partial\Phi}{\partial
u_{k}}=\frac{1}{2\pi\iota}[\Phi,{\rm ad}^{-1}_{u}{\rm
ad}_{E(\underline{d})_{k}}\Phi],\ for\ all\ k=1,...,n.$ (103)

Here

$E(\underline{d})_{k}={\rm diag}(0,...,0,{\rm Id}_{d_{k}},0,...,0)$

is the block diagonal matrix whose $k-$th block diagonal entry is the rank
$d_{k}$ identity matrix ${\rm Id}_{d_{k}}$. Note that ${\rm
ad}_{E(\underline{d})_{k}}\Phi$ takes values in the space
${\mathfrak{gl}}_{n}^{od}(\underline{d})$ of block off diagonal matrices with
respect to the tuple $(\underline{d})$, and that ${\rm ad}_{u}$ is invertible
when restricted to ${\mathfrak{gl}}_{n}^{od}(\underline{d})$. Then we have

######  Proposition 3.35.

For any solution $\Phi(u)$ of the equation 103 on $U_{\underline{d}}$, the
canonical solutions $F_{\pm}$ of the system

$\displaystyle\frac{dF}{dz}=\Big{(}\iota
u-\frac{1}{2\pi\iota}\frac{\Phi(u)}{z}\Big{)}\cdot F,$ (104)

satisfy

$\displaystyle\frac{\partial F}{\partial u_{k}}=\Big{(}\iota
E_{k}z-\frac{1}{2\pi\iota}{\rm ad}^{-1}_{u}{\rm
ad}_{E(\underline{d})_{k}}\Phi(u)\Big{)}\cdot F.$ (105)

Thus the Stokes matrices $S_{\pm}(\Phi(u),u)$ of (104) are locally constants
(independent of $u\in U_{\underline{d}}$). Furthermore the isomonodromy
equation (103) is the compatibility condition of the systems (104) and (105).

Now in a similar way, we can prove results analog to Theorem 3.3 and Theorem
3.15 for the systems (104) and (105).

######  Theorem 3.36.

For any fixed $A\in{\rm Herm}(n)$, the limit of the matrix valued function of
$u\in U_{\underline{d}}$

$\displaystyle\Big{(}\overrightarrow{\prod_{k=0,...,m-1}}(\frac{u_{k}}{u_{k+1}})^{\frac{{\rm
log}\big{(}\delta_{k}(S_{-})\delta_{k}(S_{+})\big{)}}{2\pi\iota}}\Big{)}\cdot
S_{-}(u,A)S_{+}(u,A)\cdot\Big{(}\overrightarrow{\prod_{k=0,...,m-1}}(\frac{u_{k}}{u_{k+1}})^{\frac{{\rm
log}\big{(}\delta_{k}(S_{-})\delta_{k}(S_{+})\big{)}}{2\pi\iota}}\Big{)}^{-1},$
(106)

as $0<u_{1}\ll u_{2}\ll...\ll u_{m}$, equals to (here $u_{0}:=1$)

$\displaystyle\Big{(}\overrightarrow{\prod_{k=1,...,m}}C\big{(}E(\underline{d})_{k},\delta_{k}(A)\big{)}\Big{)}\cdot
e^{A}\cdot\Big{(}\overrightarrow{\prod_{k=1,...,m}}C\big{(}E(\underline{d})_{k},\delta_{k}(A)\big{)}\Big{)}^{-1}.$
(107)

Here the product $\overrightarrow{\prod}$ is taken with the index $i$ to the
right of $j$ if $j<i$, and each
$C\big{(}E(\underline{d})_{k},\delta_{k}(A)\big{)}$ denotes the connection
matrix of the system

$\displaystyle\frac{dF}{dz}=\Big{(}\iota
E(\underline{d})_{k}-\frac{1}{2\pi\iota}\frac{\delta_{k}(A)}{z}\Big{)}F,$
(108)

where $\delta_{k}(A)$ is given in (102), and

$E(\underline{d})_{k}={\rm diag}(0,...,0,{\rm Id}_{d_{k}},0,...,0)$

is the block diagonal matrix whose $k-$th block diagonal entry is the rank
$d_{k}$ identity matrix ${\rm Id}_{d_{k}}$.

This theorem will be used to derive the expression of quantum Stokes matrices,
see Section 5.2.

##  4 Closure of Stokes matrices in the De Concini-Procesi space

The infinite point $u_{\rm cat}$ is actually a special point in the De
Concini-Procesi space $\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$. In this
section, we generalize the result in Section 3 from $u_{\rm cat}$ to general
points on $\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$. First, in Section
4.1, we recall the definition of the space $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})$. In Section 4.2, we recall the Deligne-Mumford space of
stable rational curves with marked points, its stratification and coordinate
charts. Then in Sections 4.3 and 4.4, we introduce respectively the connection
and Stokes matrices at the boundary in $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})$, which depend on some discrete choices. In Section 4.5, we
how that how the connection and Stokes matrices explicitly depend on the
discrete choices. In Section 4.6, we generalize the results in Section 3 from
the caterpillar point $u_{\rm cat}$ to an arbitrary boundary point, and prove
Theorem 1.6.

###  4.1 The De Concini-Procesi space

In this subsection, let us fix a rank $r$ simple Lie algebra $\mathfrak{g}$
with Cartan subalgebra $\mathfrak{t}$, set of roots
$\Delta\subset\mathfrak{t}^{*}$, positive roots $\Delta_{+}$, and simple roots
$\\{\alpha_{i}:i\in I\\}$.

We let ${\mathcal{G}}$ denote the _minimal building set_ associated to the set
of roots. To define ${\mathcal{G}}$, let ${\mathcal{G}}^{\prime}$ denote the
set of all non-zero subspaces of $\mathfrak{t}^{*}$ which are spanned by a
subset of $\Delta$. Let $V\in{\mathcal{G}}^{\prime}$. We say that
$V=V_{1}\oplus\cdots\oplus V_{k}$ is a _decomposition_ of $V$ if
$V_{1},\dots,V_{k}\in{\mathcal{G}}^{\prime}$, and if whenever
$\alpha\in\Delta$ and $\alpha\in V$, then $\alpha\in V_{i}$ for some $i$. From
Section 2.1 of [15], every element of ${\mathcal{G}}^{\prime}$ admits a unique
decomposition. Then we define ${\mathcal{G}}$ be the set of indecomposable
elements of ${\mathcal{G}}^{\prime}$. This set can be described as follows.
There is an action of the Weyl group $W$ on $\mathfrak{t}$. This action
preserves $\Delta$. Thus, we get actions of $W$ on ${\mathcal{G}}$ and
${\mathcal{G}}^{\prime}$. If $J\subseteq I$ is a non-empty, connected subset
of the Dynkin diagram $I$ of $\mathfrak{g}$, we can form $V_{J}={\rm
span}(\alpha_{j}:j\in J)$. Then $V_{J}\in{\mathcal{G}}$. In fact, every
$V\in{\mathcal{G}}$ is of the form $w(V_{J})$ for some $w\in W$ and $J$ as
above.

Let $\mathfrak{t}_{\rm reg}=\\{\chi\in\mathfrak{t}:\alpha(\chi)\neq 0,\text{
for all }\alpha\in\Delta\\}$. For any $V\in{\mathcal{G}}$, we have a map
$\mathfrak{t}_{\rm reg}\rightarrow\mathbb{P}(\mathfrak{g}/V^{\perp})$.

######  Definition 4.1.

The De Concini-Procesi space $\widetilde{\mathfrak{t}_{\rm
reg}}\subset\mathfrak{t}\times\prod_{V\in{\mathcal{G}}}\mathbb{P}(\mathfrak{t}/V^{\perp})$
is the closure of the image of the map $\mathfrak{t}_{\rm
reg}\rightarrow\mathfrak{t}\times\prod_{V\in{\mathcal{G}}}\mathbb{P}(\mathfrak{t}/V^{\perp})$.

Let us consider its projective analog. Note that the multiplicative group
$\mathbb{C}^{\times}$ acts on $\mathfrak{t}_{\rm reg}$, and the regular
morphism $\mathfrak{t}_{\rm
reg}\subset\prod_{V\in{\mathcal{G}}}\mathbb{P}(\mathfrak{t}/V^{\perp})$ is
constant on the $\mathbb{C}^{\times}$ orbits. Thus we get a map
$\mathfrak{t}_{\rm
reg}/\mathbb{C}^{\times}\rightarrow\prod_{V\in{\mathcal{G}}}\mathbb{P}(\mathfrak{t}/V^{\perp})$.

######  Definition 4.2.

The De Concini-Procesi space $\overline{\mathfrak{t}_{\rm
reg}}\subset\prod_{V\in{\mathcal{G}}}\mathbb{P}(\mathfrak{t}/V^{\perp})$ is
the closure of the image of the map
$\mathfrak{t}^{reg}/\mathbb{C}^{\times}\rightarrow\prod_{V\in{\mathcal{G}}}\mathbb{P}(\mathfrak{t}/V^{\perp})$.

Thus a point of $\overline{\mathfrak{t}_{\rm reg}}$ consists of a collection
$(L_{V})_{V\in{\mathcal{G}}}$ where $L_{V}\in\mathfrak{t}/V^{\perp}$. We will
think of $L_{V}$ as a subspace of $\mathfrak{t}$ containing $V^{\perp}$ as a
hyperplane. Note that if $\chi\in\mathfrak{t}_{\rm reg}$, then $\chi\notin
V^{\perp}$ for all $V\in{\mathcal{G}}$ and the image of $\chi$ in
$\overline{\mathfrak{t}_{\rm reg}}$ is the collection
$L_{V}=V^{\perp}+\mathbb{C}\chi$.

By [15, Theorem 4.1], $\widetilde{\mathfrak{t}_{\rm reg}}$ is the total space
of a tautological line bundle on $\overline{\mathfrak{t}_{\rm reg}}$.
Furthermore, following [15, Theorem 3.1 and 3.2], the boundary $D$ of
$\mathfrak{t}_{\rm reg}$ in $\widetilde{\mathfrak{t}_{\rm reg}}$ is a divisor
with normal crossings, and is the union of smooth irreducible divisors
$D_{\omega(V_{J})}$ indexed by the elements ${\omega(V_{J})}$ in
${\mathcal{G}}$. Then by [15, Theorem 4.1], $\overline{\mathfrak{t}_{\rm
reg}}$ is isomorphic to the boundary divisor $D_{\omega(V_{I})}$.

Since the root system $\Delta$ is defined over $\mathbb{R}$, the variety is
defined over $\mathbb{R}$ and so it make sense to consider the real points
$\overline{\mathfrak{t}_{\rm reg}(\mathbb{R})}$.

####  4.1.1 Type A cases and the parameter space of Stokes matrices

Recall that we denote by $\mathfrak{h}_{\rm reg}(\mathbb{R})$ the space of
$n\times n$ diagonal matrices with distinct real eigenvalues, and in this
paper we consider the meromorphic linear system

$\displaystyle\frac{dF}{dz}=\Big{(}\iota
u-\frac{1}{2\pi\iota}\frac{A}{z}\Big{)}\cdot F,$

with $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$ and $A\in{{\rm Herm}}(n)$. By
definition, the Stokes matrices $S_{\pm}(u,A)$ are invariant under the
translation action on $\mathfrak{h}_{\rm reg}(\mathbb{R})$. Thus let
$\mathbb{R}$ act on $\mathfrak{h}_{\rm reg}(\mathbb{R})$ by translation, then
for any fixed $A$, $S_{\pm}(u,A)$ are parameterized by $\mathfrak{h}_{\rm
reg}(\mathbb{R})/\mathbb{R}$.

Now let $\mathfrak{t}(\mathfrak{sl}_{n})$ denote the Cartan subalgebra of
$\mathfrak{sl}_{n}$. The space $\widetilde{\mathfrak{t}_{\rm
reg}(\mathfrak{sl}_{n})}$ is obtained by succesive blowups along the
intersections of the diagonal hyperplanes $u_{i}=u_{j}$. Our goal is to extend
the definition of Stokes matrices from $\mathfrak{h}_{\rm
reg}(\mathbb{R})/\mathbb{\mathbb{R}}\cong\mathfrak{t}_{\rm
reg}(\mathfrak{sl}_{n})(\mathbb{R})$ to $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R}):=\widetilde{\mathfrak{t}_{\rm
reg}(\mathfrak{sl}_{n})}(\mathbb{R})$. But in the rest of this section, we
will only define Stokes matrices on $\overline{\mathfrak{t}_{\rm
reg}(\mathfrak{sl}_{n})}(\mathbb{R})\subset\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})$. This is because, on the one hand, for the other boundary
points, the construction is rather similar, and on the other hand, extension
to $\overline{\mathfrak{t}_{\rm reg}(\mathfrak{sl}_{n})}(\mathbb{R})$ is
enough for the applications in Poisson geometry, integrable systems and
representation theory.

######  Remark 4.3.

$G$-valued Stokes matrices have been introduced in [11] for any complex
reductive Lie group $G$, beyond the classical ${\rm GL}_{n}$ case.
Accordingly, one can extend $G$-Stokes matrices from $\mathfrak{t}_{\rm
reg}(\mathbb{R})$ to $\overline{\mathfrak{t}_{\rm reg}}(\mathbb{R})$. Here
$\mathfrak{t}$ is a Cartan subalgebra of $\mathfrak{g}={\rm Lie}(G)$.

###  4.2 The space $\overline{\mathcal{M}_{0,n+1}}(\mathbb{R})$

Since in this paper we only consider the type $A$ case, it is more convenient
to identify $\overline{\mathfrak{t}_{\rm reg}(\mathfrak{sl}_{n})}$ with the
Deligne-Mumford space $\overline{\mathcal{M}_{0,n+1}}$ of stable rational
curves with $n+1$ marked points. Here a rational curve with $n+1$ marked
points is a finite union $\Sigma$ of projective lines
$\Sigma_{1},...,\Sigma_{m}$ together with marked distinct points
$z_{1},...,z_{n}\in\Sigma$ such that

  * •

Each marked point belongs to a unique $\Sigma_{j}$;

  * •

The intersection of projective lines $\Sigma_{i}\cap\Sigma_{j}$ is either
empty or consists of one point, and in the latter case the intersection is
transversal;

  * •

The graph of components (whose vertices are the lines $C_{i}$ and whose edges
correspond to pairs of intersecting lines) is a tree;

  * •

The total number of special points (i.e. marked points or intersection points)
that belong to a given component $\Sigma_{i}$ is at least 3.

A point of $\overline{\mathcal{M}_{0,n+1}}$ is then an isomorphism classes of
such stable curves. The space $\overline{\mathcal{M}_{0,n+1}}$ contains a
dense open subset corresponding to curves with one component. This open subset
is isomorphic to
$\Big{(}(\mathbb{P}^{1})^{n+1}\setminus\Delta\Big{)}/PSL_{2}(\mathbb{C})$.
Here $PSL_{2}(\mathbb{C})$ is the automorphism group of $\mathbb{P}^{1}$. We
have that

$\Big{(}(\mathbb{P}^{1})^{n+1}\setminus\Delta\Big{)}/PSL_{2}(\mathbb{C})\cong(\mathbb{C}^{n}\setminus\Delta)/B\cong\mathfrak{h}_{\rm
reg}/B\cong\mathfrak{t}_{\rm reg}(\mathfrak{sl}_{n})/\mathbb{C}^{\times}$

where $\Delta$ denotes the thick diagonal, $B\subset PSL_{2}$ is the Borel
subgroup, and $B\cong\mathbb{C}^{\times}\ltimes\mathbb{C}$ acting on
$\mathbb{C}^{n}$ by scaling and translation. It follows from [15, Section 4.3]
that the above isomorphism can be extended to an identification
$\overline{\mathcal{M}_{0,n+1}}\cong\overline{\mathfrak{t}_{\rm
reg}(\mathfrak{sl}_{n})}$. Taking the real points, we have

$\overline{\mathcal{M}_{0,n+1}}(\mathbb{R})\cong\overline{\mathfrak{t}_{\rm
reg}(\mathfrak{sl}_{n})}(\mathbb{R}).$

Therefore we can identity these two spaces, and work with
$\overline{\mathcal{M}_{0,n+1}}(\mathbb{R})$.

The space $\overline{\mathcal{M}_{0,n+1}}(\mathbb{R})$ is a connected and
compact smooth manifold of dimension $n-2$. It can be regarded as a
compactification of the configuration space $\mathcal{M}_{0,n+1}(\mathbb{R})$
of ordered $(n+1)$-tuples $(u_{1},...,u_{n+1})$ of pairwise distinct points on
$\mathbb{RP}^{1}$ modulo the automorphism group $PSL_{2}(\mathbb{R})$. Since
it consists of equivalence classes of stable rational curves over
$\mathbb{R}$, points on $\overline{\mathcal{M}_{0,n+1}}(\mathbb{R})$ can be
thought as “cactus-like” pictures-trees of circles with marked points on them.
Thus it has a stratification, with the strata indexed by the combinatorial
types of stable rational curves, i.e. by rooted trees with $n$ leaves colored
by the marked points $u_{1},...,u_{n}$ (the root vertex is colored by
$u_{n+1}$). Here the marked points (resp. double points) of a stable curve
$\Sigma$ become external (resp. internal) edges, and components $\Sigma_{i}$
of $\Sigma$ become vertices of the corresponding tree. Two edges share a
common vertex if and only if the corresponding points are on the same line
$\Sigma_{i}$.

Let $T$ be such a tree, then the corresponding stratum $\mathcal{M}_{T}$ is
the product of $\mathcal{M}_{0,d(I)}$ over all internal vertices $I$ of $T$
with $d(I)$ the index of $I$. In particular, $0$-dimensional strata correspond
to binary rooted trees with $n$ ordered leaves, while $1$-dimensional strata
correspond to almost binary trees (with exactly one $4$-valent internal
vertex). The stratum corresponding to a tree $T$ lies in the closure of the
one corresponding to another tree $T^{\prime}$ if and only if $T^{\prime}$ is
obtained from $T$ by contracting some edges.

We denote by $T(u)$ a planar rooted tree $T$ with $n$ leaves colored by the
components $u_{1},...,u_{n}$ of $u$. We say that it is compatible with
$\sigma\in S_{n}$ if all internal vertices of the tree are in the lower half
plane, all leaves are on the horizontal line $y=0$ and are colored by
$u_{\sigma(1)},...,u_{\sigma(n)}$ from left to right.

######  Definition 4.4.

We denote by $T_{\sigma}(u)$ a planar rooted tree with $n$ leaves colored by
$u$ and compatible with $\sigma\in S_{n}$.

$u_{3}$$u_{5}$$u_{6}$$u_{2}$$u_{1}$$u_{4}$ Figure 1: A planar binary rooted
tree with $6$ leaves colored by $u_{1},...,u_{6}$.

Any $T_{\sigma}(u)$ represents a point in
$\overline{\mathcal{M}_{0,n+1}}(\mathbb{R})$ as follows. First, to any planar
binary rooted tree $T^{\prime}_{\sigma}(u)$ compatible with $\sigma$, one can
assign a set of coordinates $x_{I}$, indexed by internal vertices $I$ of
$T^{\prime}_{\sigma}$, in an appropriate neighborhood
$U_{T^{\prime}_{\sigma}}$ of the corresponding 0-dimensional stratum. Then for
any planar rooted tree $T_{\sigma}$ which can be obtained from
$T^{\prime}_{\sigma}$ by contracting some edges, the coloring
$u=(u_{1},...,u_{n})$ of the leaves of $T_{\sigma}(u)$ determines a point in
the chart $U_{T^{\prime}_{\sigma}}$: in the coordinates $\\{x_{I}\\}$, the
point is given by

$\displaystyle
x_{I}=\frac{u_{r(I)}-u_{l(I)}}{u_{r(I^{\prime})}-u_{l(I^{\prime})}},\ if\ I\
and\ I^{\prime}\ are\ contracted\ in\ T,$ (109) $\displaystyle x_{I}=0,\ \ \ \
\ \ otherwise.$ (110)

where $I^{\prime}$ is the preceding vertex of $I$ in $T^{\prime}$, i.e.
$I^{\prime}:={\rm max}\\{J\in T~{}|~{}J<I\\}$ in the partial ordering $<$ of
the vertices of $T_{\sigma}(u)$ with the root being the minimal element. Here
for any vertex $I$, $l(I)\in[1,...,n]$ is such that $\sigma(l(I))$ is the
maximal index of the $u_{i}\textquoteright s$ in the left branch at $I$, and
analogously, $r(I)\in[1,...,n]$ is such that $\sigma(r(I))$ the minimal index
of the $u_{i}^{\prime}s$ in the right branch at $I$.

On the one hand, for different $u$, $T_{\sigma}(u)$ can represent the same
point. In particular, if we have a binary tree $T_{\sigma}$, then for all
$u\in U_{\sigma}$, $T_{\sigma}(u)$ represents a same point. On the other hand,
if the ordering $\sigma$ is obtained from $\sigma^{\prime}$ by reversing the
order of the descendants of any internal vertex of $T(u)$, then
$T_{\sigma}(u)$ and $T_{\sigma^{\prime}}(u)$ represent the same point.

It is clear that the stratum
$\mathcal{M}_{T}\subset\overline{\mathcal{M}_{0,n+1}}(\mathbb{R})$
corresponding to a rooted tree $T$ in the local coordinates determined by a
binary rooted tree $T^{\prime}$ can be described as follows.

######  Proposition 4.5.

The stratum $\mathcal{M}_{T}$ has a nonempty intersection with
$U_{T^{\prime}_{\sigma}}$ if and only if $T$ is obtained from $T^{\prime}$ by
contracting some edges. In the latter case, $\mathcal{M}_{T}$ is a subset of
$U_{T^{\prime}_{\sigma}}$ defined as follows: $x_{I}\neq 0$ if the (unique)
edge of $T^{\prime}$ which ends at $I$ is contracted in $T$, and $x_{I}=0$
otherwise.

###  4.3 Connection matrices parameterized by $\overline{\mathfrak{t}_{\rm
reg}(\mathfrak{sl}_{n})}(\mathbb{R})\cong\overline{\mathcal{M}_{0,n+1}}(\mathbb{R})$

For any $\sigma\in S_{n}$, let us take the component $U_{\sigma}$ of
$\mathfrak{t}_{\rm reg}(\mathfrak{sl}_{n})(\mathbb{R})$, and denote by
$\overline{U_{\sigma}}$ its closure in $\overline{\mathfrak{t}_{\rm
reg}(\mathfrak{sl}_{n})}(\mathbb{R})$. Note that $\overline{U_{\sigma}}$ is
consisting of the points which can be represented by certain planar tree
$T_{\sigma}(u)$ compatible with $\sigma$.

Now given any $u_{0}\in\overline{U_{\sigma}}$, let $T_{\sigma}(u)$ be a
representative of $u_{0}$. For any internal vertex $I\in T_{\sigma}(u)$ with
$E(I)$ outgoing edges, let us assign the subsets $I_{j}\subset\\{1,...,n\\}$
formed by all leaves of $T_{\sigma}(u)$ on the $j$-th branch (counting from
left to right) of $T_{\sigma}(u)$ at $I$ for all $1\leq j\leq E(I)$. Let
$u_{I_{j,0}}$ and $u_{I_{j,1}}$ be the coloring of the first and the last
leaves (counting from left) of $T_{\sigma}(u)$ in the subset $I_{j}$. the
coloring of the first leave (counting from left) of $T_{\sigma}(u)$ in the
subset $I_{j}$. Let $\\{t_{I_{j}}\\}_{j=1,...,i}$ be a collection of numbers
assigned to the branches of $I$ such that

$\displaystyle t_{I_{1}}=0,\ t_{I_{E(I)}}=1\hskip 8.53581ptand\hskip
8.53581pt\frac{t_{I_{j}}-t_{I_{j-1}}}{t_{I_{j+1}}-t_{I_{j}}}=\frac{u_{I_{j,0}}-u_{I_{j-1,1}}}{u_{I_{j+1,0}}-u_{I_{j,1}}},\
for\ 1<j<E(I).$ (111)

Then let us introduce an $n\times n$ diagonal matrix $\tilde{u}^{I}={\rm
diag}(\tilde{u}^{I}_{1},...,\tilde{u}^{I}_{n})$ with

$\displaystyle\tilde{u}^{I}_{k}=t_{I_{j}},\ \ \ if\ k\in I_{j}\ \ for\
certain\ 1\leq j\leq E(I),$ (112) $\displaystyle\tilde{u}^{I}_{k}=0,\hskip
34.14322ptotherwise,$ (113)

Besides, for any $A\in{\rm Herm}(n)$ we denote by $A^{I}\in{\rm Herm}(n)$ the
matrix

$\displaystyle A^{I}_{kj}=\left\\{\begin{array}[]{lr}A^{I}_{kj},&if\ \ k,j\in
B(I),\ or\ k=j\\\ 0,&otherwise.\end{array}\right.$ (116)

where $B(I)$ is the subset of $\\{1,...,n\\}$ formed by all leaves of
$T_{\sigma}(u)$ at $I$.

Now given any internal vertex $I\in T_{\sigma}(u)$ and $A\in{\rm Herm}(n)$,
let us consider the linear system

$\displaystyle\frac{dF}{dz}=\Big{(}\iota\tilde{u}^{I}-\frac{1}{2\pi\iota}\frac{A^{I}}{z}\Big{)}F.$
(117)

The system has only two anti-Stokes directions, the two halves of the
imaginary axis. We will choose the right half plane ${\rm
Sect}_{+}:=\\{z\in\mathbb{C}~{}|~{}{\rm Re}(z)>0\\}$ as the initial Stokes
sector, and take the branch of ${\rm log}(z)$ which is real on
$\mathbb{R}_{>0}$. Let us denote by $C^{I}(\tilde{u}^{I},A^{I})$ and
$S_{\pm}(\tilde{u}^{I},A^{I})$ the associated connection and Stokes matrices.
Let $U(n)^{I_{d}}\subset U(n)$ be the stabilizer subgroup of
$u\in\mathfrak{h}_{\rm reg}(\mathbb{R})\subset{\rm Herm}(n)$ under the
conjugation action of $U(n)$ on ${\rm Herm}(n)$. Then the connection map

$C^{I}(\tilde{u}^{I}):{\rm Herm}(n)\rightarrow U(n);~{}A\mapsto
C^{I}(\tilde{u}^{I},A^{I})$

is equivariant with respect to the group $U(n)^{I_{d}}$, and (similar to the
$u_{\rm cat}$ case given in (200)) the Stokes matrices
$S_{\pm}(\tilde{u}^{I},A^{I})$ are blocked according to the index set
$I_{1},...,I_{E(I)}$. Furthermore, we have the monodromy relation

$\displaystyle
C^{I}(\tilde{u}^{I},A^{I})e^{A^{I}}C^{I}(\tilde{u}^{I},A^{I})^{-1}=S_{-}(\tilde{u}^{I},A^{I})S_{+}(\tilde{u}^{I},A^{I}).$
(118)

######  Definition 4.6.

For any $u_{0}\in\overline{U_{\sigma}}$ and $A\in{\rm Herm}(n)$, the
connection matrix $C^{\sigma}(u_{0},A)$ is

$\displaystyle C^{\sigma}(u_{0},A):=\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}C^{I}(\tilde{u}^{I},A^{I}),$ (119)

where $T_{\sigma}(u)$ is a representative of $u_{0}$, the product is taken
over all internal vertices $I$ of $T_{\sigma}(u)$, and is taken with $C^{I}$
to the right of $C^{I^{\prime}}$ if $I^{\prime}<I$ in the partial ordering.

Since each factor $C^{I}(\tilde{u}^{I},A^{I})$ in the product is unitary, we
have

######  Proposition 4.7.

The connection matrix $C^{\sigma}(u_{0},A)$ is unitary.

By comparing identities (109) and (110) with (111), we see that the irregular
term $\tilde{u}^{I}$ in the equation (117) only depends on the coordinates of
$u_{0}$ in certain fixed chart $U_{T^{\prime}_{\sigma}}$. Thus the right hand
side of (119) is independent of the different choices of the coloring $u$, as
long as $T_{\sigma}(u)$ is compatible with $\sigma$ and represents the same
point $u_{0}$. Therefore, $C^{\sigma}(u_{0},A)$ is well defined. In
particular, if $u_{0}$ is in the $0$-dimensional strata, then any coloring
$u\in U_{\sigma}$ of the binary tree $T_{\sigma}(u)$ will define the same
$C^{\sigma}(u_{0},A).$ However, the definition depends on the choice of
$\sigma$. That is for
$u_{0}\in\overline{U_{\sigma}}\cap\overline{U_{\sigma^{\prime}}}$, there
exists different $\sigma,\sigma^{\prime}\in S_{n}$ such that $u_{0}$ can be
represented by certain $T_{\sigma}(u)$ and $T_{\sigma^{\prime}}(u)$. Then
$C^{\sigma}(u_{0},A)$ and $C^{\sigma^{\prime}}(u_{0},A)$ will be in general
different, see Section 4.5.

###  4.4 Stokes matrices at the boundary

Given any $\sigma\in S_{n}$, denote by $P_{\sigma}\in{\rm GL}_{n}$ the
corresponding permutation matrix.

######  Definition 4.8.

For any $u_{0}\in\overline{U_{\sigma}}$ and $A\in{\rm Herm}(n)$, the Stokes
matrices $S^{\sigma}_{\pm}(u_{0},A)$ are the unique triangular matrices, with
${P_{\sigma}}^{-1}e^{\frac{[A]}{2}}P_{\sigma}$ as diagonal part, determined by
the identity (Gauss decomposition)

$\displaystyle{C^{\sigma}(u_{0},A)}e^{A}{C^{\sigma}(u_{0},A)}^{-1}=P_{\sigma}S_{+}^{\sigma}(u_{0},A)S_{-}^{\sigma}(u_{0},A){P_{\sigma}}^{-1}.$
(120)

Here $[A]$ is the diagonal part of $A$.

######  Example 4.9.

Let us take the caterpillar point $u_{\rm cat}$, i,e, the "infinite point"
$u_{1}\ll\cdots\ll u_{n}$. It has a representative

$u_{n}$$u_{1}$$u_{2}$$u_{3}$$\ddots$$u_{n-1}$ Figure 2: A caterpillar point
with a chosen planar embedding

Then one checks that for $u_{0}=u_{\rm cat}$, Definitions 4.6 and 4.8 coincide
with the definition of connection and Stokes matrices introduced in Section
3.4.

###  4.5 Different choices of planar embeddings and wall-crossing formula

In this subsection, we study how connection and Stokes matrices associated to
planar trees depend on the choices of planar embeddings.

Suppose that a given point $u_{0}$ lies at the intersection of
$\overline{U_{\sigma}}$ and $\overline{U_{\sigma^{\prime}}}$, for different
$\sigma,\sigma^{\prime}\in S_{n}$. Then there are two equivalent
representatives of $u_{0}$, i.e., planar embeddings $T_{\sigma}(u)$ and
$T_{\sigma^{\prime}}(u)$ of a same rooted tree with coloring $u$ compatible
with $\sigma$ and $\sigma^{\prime}$ respectively. Here equivalence means that
$T_{\sigma^{\prime}}(u)$ is obtained from $T_{\sigma}(u)$ by reversing the
order of the descendants of some internal vertex of $T_{\sigma}(u)$. Let us
assume that $\sigma$ and $\sigma^{\prime}$ differ by reversing only one cycle.
Then $T_{\sigma^{\prime}}(u)$ is given by reversing the order of the
descendants of some vertex $I$ of $T_{\sigma}(u)$.

Let us denote by $C^{J}(\tilde{u}^{J},A^{J})$ and
$C_{op}^{J}(\tilde{u}^{J},A^{J})$ the connection matrices associated to the
vertices $J$ of $T_{\sigma}(u)$ and $T_{\sigma^{\prime}}(u)$ respectively.
Then by the assumption and the definition of the connection matrices, we have

$\displaystyle
C^{J}_{op}(\tilde{u}^{J},A^{J})=\left\\{\begin{array}[]{lr}C^{J}(\tilde{u}^{J},A^{J}),&if\
J\neq I,\\\ C^{I}_{-}(\tilde{u}^{I},A^{I}),&J=I,\end{array}\right.$

where $C^{I}_{-}(\tilde{u}^{I},A^{I})$ is the connection matrix of the system
(117) but with respect to the left half plane (the Stokes sector ${\rm
Sect}_{-}$). To be more precise, similar to Definition 2.10, the connection
matrix $C^{I}_{-}(\tilde{u}^{I},A^{I})$ of the system (11) is determined by
$F_{0}(z)=F_{-}(z)\cdot C^{I}_{-}(\tilde{u}^{I},A^{I})$, where $F_{-}(z)$ is
the canonical solution on ${\rm Sect}_{-}$. Furthermore, since a negative half
cycle (i.e., in clockwise direction) around $0$ is a positive half cycle
around $\infty$, we get the monodromy relation

$\displaystyle
C^{I}_{-}(\tilde{u}^{I},A)=(S^{I}_{+}(\tilde{u}^{I},A)S^{I}_{-}(\tilde{u}^{I},A))^{-\frac{1}{2}}S^{I}_{+}(\tilde{u}^{I},A)C^{I}(\tilde{u}^{I},A),$
(122)

where $S^{I}_{\pm}(\tilde{u}^{I},A)$ are the Stokes matrices of the system
(117). In summary, we have the following "wall-crossing" formula of the
connections matrices as $u$ crosses the common face of $U_{\sigma}\cup
U_{\sigma^{\prime}}$.

######  Proposition 4.10.

Given any point $u_{0}$ in
$\overline{U_{\sigma}}\cap\overline{U_{\sigma^{\prime}}}$, where $\sigma$
differs with $\sigma^{\prime}$ by reversing some segments, let $T_{\sigma}(u)$
and $T_{\sigma^{\prime}}(u)$ be its two representatives. Then for any
$A\in{\rm Herm}(n)$, $C^{\sigma^{\prime}}(u_{0},A)$ equals to
$C^{\sigma}(u_{0},A)=\overrightarrow{\prod}_{J\in
T_{\sigma}(u)}C^{J}(\tilde{u}^{J},A)$ after replacing the corresponding
factors $C^{I}(\tilde{u}^{J},A)$ by $C^{I}_{-}(\tilde{u}^{J},A)$ in the
product.

Following Definition 4.8, we also get a "wall-crossing" formula of the Stokes
matrices as $u$ crosses the common face of $U_{\sigma}\cup
U_{\sigma^{\prime}}$.

######  Remark 4.11.

From the above discussion, we see that the (discrete) choice of planar
embeddings in the definition of connection and Stokes matrices at boundary is
a natural generation of the choice of initial Stokes sector in the definition
of the ordinary connection/Stokes matrices. It is a new phenomenon naturally
arising from the closure of Stokes matrices.

######  Example 4.12 (Caterpillar points).

For each $1\leq i\leq n$, let $\tau_{i}\in S_{n}$ be the permutation reversing
the segment $[1,...,i]$. Then associated to $\tau_{i}$, there is a
representative of the caterpillar point $u_{\rm cat}$ as in Figure 3.

$u_{n}$$u_{i}$$u_{i-1}$$\ddots$$u_{1}$$u_{i+1}$ $\ddots$ Figure 3: A
caterpillar point with a planar embedding given by $\tau_{i}$

For any $A\in{\rm Herm}(n)$, we denote by $S_{+}^{\tau_{i}}(u_{\rm cat},A)$
the Stokes matrix associated to the planar tree. For $\tau_{1}={\rm id}\in
S_{n}$, it is just the Stokes matrix $S_{+}(u_{\rm cat},A)$. For any other
$\tau_{i}$, $S_{+}^{\tau_{i}}(u_{\rm cat},A)$ is the Stokes matrix at the
caterpillar point but with respect to a different choice of planar embedding.
Thus we can use Proposition 4.10 to express $S_{+}^{\tau_{i}}(u_{\rm cat},A)$
by $S_{+}(u_{\rm cat},A)$.

######  Proposition 4.13.

Under the change of the planar embedding, the associated Stokes matrices at
$u_{\rm cat}$ change as

$\displaystyle S_{+}(u_{\rm cat},A)=\left(\begin{array}[]{cc}S_{i+}&B\\\
0&C\end{array}\right)\rightarrow S_{+}^{\tau_{i}}(u_{\rm
cat},A)=\left(\begin{array}[]{cc}P_{i}S_{i+}^{\dagger}P_{i}^{-1}&P_{i}(S_{i+}S_{i+}^{\dagger})^{-\frac{1}{2}}S_{i+}B\\\
0&C\end{array}\right),$ (127)

where $S_{i+}$ are the left-top $i$-th principal submatrices of $S_{+}(u_{\rm
cat},A)$, and $S_{i+}^{\dagger}$ is the complex conjugate of $S_{i+}$.

Proof. By Definition 4.8, the Stokes matrix $S^{\tau_{i}}_{+}(u_{\rm cat},A)$
associated to $\tau_{i}$, is given by

$\displaystyle{C^{\tau_{i}}(u_{\rm cat},A)}e^{A}{C^{\tau_{i}}(u_{\rm
cat},A)}^{-1}=P_{i}S_{-}^{\tau_{i}}(u_{\rm cat},A)S_{+}^{\tau_{i}}(u_{\rm
cat},A){P_{i}}^{-1},$ (128)

where $C^{\tau_{i}}=C_{-}^{(1)}\cdot\cdot\cdot
C_{-}^{(i)}C^{(i+1)}\cdot\cdot\cdot C^{(n)}$ and $P_{i}$ is the permutation
matrix associated to $\tau_{i}$. Here for $1\leq k\leq i$, $C_{-}^{(k)}(A)$ is
the connection matrix of $\frac{dF}{dz}=\Big{(}-\iota
E_{k}-\frac{1}{2\pi\iota}\frac{A^{(k)}}{z}\Big{)}F,$ and recall that
$C^{(k)}(A)$, $S_{\pm}^{(k)}(A)$ are the connection and Stokes matrices of
$\frac{dF}{dz}=\Big{(}\iota
E_{k}-\frac{1}{2\pi\iota}\frac{A^{(k)}}{z}\Big{)}F.$

######  Lemma 4.14.

For any $A\in{\rm Herm}(n)$, we have

$\displaystyle C_{-}^{(1)}\cdot\cdot\cdot
C_{-}^{(i)}=(S_{i+}S_{i-})^{-\frac{1}{2}}S_{i+}C^{(1)}\cdot\cdot\cdot
C^{(i)}.$ (129)

Proof. Consider a rank $i$ linear system $\frac{dF(z)}{dz}=\left(\iota
u-\frac{1}{2\pi\iota}\frac{\Phi(u)}{z}\right)\cdot F(z)$. Since a negative
half cycle (i.e., in clockwise direction) around $0$ is a positive half cycle
around $\infty$, we get the monodromy relation

$\displaystyle
C_{-}(u,\Phi(u))=(S_{+}(u,\Phi(u))S_{-}(u,\Phi(u)))^{-\frac{1}{2}}S_{+}(u,\Phi(u))C_{-}(u,\Phi(u)).$

If we take $\Phi(u)$ the solution of the corresponding isomonodromy equation
with the asymptotics $A$, then the above relation implies the identity (129).

The lemma implies

$C^{\tau_{i}}(u_{\rm cat},A)=(S_{i+}S_{i-})^{-\frac{1}{2}}S_{i+}\cdot C(u_{\rm
cat},A).$

Then by a direct blocked matrix manipulation, the proposition follows from the
defining relations (66) and (128) of $S_{+}(u,\Phi(u))$ and
$S_{+}^{\tau_{i}}(u,\Phi(u))$, and the relation
$S_{i-}(u,\Phi(u))=S_{i+}(u,\Phi(u))^{\dagger}$.

######  Remark 4.15.

The Cactus group $Cact_{n}$ is a group generated by elements $\tau_{i}$,
$1\leq i\leq n$, subject to some relations. See e.g., [29]. Proposition 4.13
can be used to interpret the cactus group actions on the Gelfand-Tsetlin cones
[9] from the perspective of WKB approximation. See our next papers.

###  4.6 Closure of Stokes matrices

In this subsection, we generalize the results in Section 3, particularly the
Theorem 3.15, from the caterpillar points to any boundary points.

We use the same notation as in Section 4.3. Let $T_{\sigma}(u)$ be a planar
labelled rooted tree with $n$ leaves and compatible with $\sigma$. If it has
$k$ internal vertex, then any labelling $u$ represents a point
$u_{0}\in\overline{U_{\sigma}}$ in the codimension $k$ strata. For any vertex
$I$ of $T_{\sigma}(u)$, let us introduce a parameter $r_{I}\in\mathbb{R}$ such
that $u_{I_{j,0}}-u_{I_{j-1,1}}=r_{I}\cdot(t_{I_{j}}-t_{I_{j-1}})$ for all
$1\leq j\leq E(I)$. The labelling $u$ depends on $\\{r_{I}\\}$ and
$\\{t_{I_{j}}\\}$ for all vertices $I$, and $u_{0}$ can be understood as the
infinite point as $r_{I}\gg r_{I^{\prime}}$ in $u$, whenever $I<I^{\prime}$ in
the partial order. The collection of numbers $\\{t_{I_{j}}\\}$ can be seen as
a coordinate of $u_{0}$, and we write $u_{0}(t)$ to stress the dependence of
$u_{0}$ on them.

Now associated to each vertex $I$, denote by
$\mathfrak{gl}(n)^{d_{I}}\subset\mathfrak{gl}(n)$ the space of matrices whose
$(i,j)$ entry is zero unless $i$ and $j$ are both in the subset
$I_{k}\subset\\{1,...,n\\}$ for some $k$. Here recall that $I_{k}$ is formed
by all leaves of on the $k$-th branch of $T_{\sigma}(u)$ at $I$. Thus
$\mathfrak{gl}(n)^{d_{I}}$ is just the subspace of the block diagonal
matrices, blocked by the index sets $\\{I_{k}\\}_{k=1,...,E(I)}$. For any
$A\in\mathfrak{gl}(n)$, we denote by $A^{d_{I}}\in\mathfrak{gl}(n)^{I}$ the
corresponding projection.

Then the analogs of Theorem 1.1 and Theorem 3.15 are respectively

######  Theorem 4.16 (Asymptotics of solutions of the isomonodromy equations).

For any solution $\Phi(u)$ of the isomonodromy equation (27) on $U_{\sigma}$,
there exists a function $A(u_{0})\in{\rm Herm}(n)$ of $t$ such that

$\displaystyle\Big{(}\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}({r_{I}}/{r_{I^{\prime}}})^{\frac{\Phi^{d_{I}}}{2\pi\iota}}\Big{)}\cdot\Phi(u)\cdot\Big{(}\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}({r_{I}}/{r_{I^{\prime}}})^{\frac{\Phi^{d_{I}}}{2\pi\iota}}\Big{)}^{-1}\rightarrow
A(u_{0}),\hskip 8.53581ptas\ r_{I}\gg r_{I^{\prime}}\ whenever\ I<I^{\prime}\
in\ T_{\sigma}(u),$

where $I^{\prime}$ is the preceding vertex of $I$, the product
$\overrightarrow{\prod}$ is taken over all internal vertices $I$ of
$T_{\sigma}(u)$, and is taken with $I$ to the right of ${J}$ if $I<J$ in the
partial order. Here $\Phi^{I}$ is defined as (116) associated to the vertex
$I$. We call $A(u_{0})$ the asymptotics of $\Phi(u)$ at
$u_{0}\in\overline{U_{\sigma}}$, and denote the solution by
$\Phi_{u_{0}}^{\sigma}(u,A)$.

######  Theorem 4.17 (Stokes matrices at the boundary via the isomonodromy
deformation).

Let us denote by $\Phi(u;A(u_{0}))$ the solution of (27) with the asymptotics
$A(u_{0})$ at $u_{0}\in\overline{U_{\sigma}}$, then we have

$\displaystyle
S_{\pm}(u,\Phi_{u_{0}}^{\sigma}(u;A))=S_{\pm}^{\sigma}(u_{0},A(u_{0})).$

######  Theorem 4.18 (Regularized limits of Stokes matrices at the boundary).

For any $A\in{\rm Herm}(n)$, we have

$\displaystyle\Big{(}\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}({r_{I}}/{r_{I^{\prime}}})^{\frac{{\rm
log}(S_{-}^{d_{I}}S_{+}^{d_{I}})}{2\pi\iota}}\Big{)}^{-1}\cdot
S_{-}(u,A)S_{+}(u,A)\cdot\Big{(}\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}({r_{I}}/{r_{I^{\prime}}})^{\frac{{\rm
log}(S_{-}^{d_{I}}S_{+}^{d_{I}})}{2\pi\iota}}\Big{)}\rightarrow
S_{-}^{\sigma}(u_{0},A)S_{+}^{\sigma}(u_{0},A),$

as $r_{I}\gg r_{I^{\prime}}$ whenever $I<I^{\prime}$ in $T_{\sigma}(u)$. Here
the none zero blocks of $S_{-}(u,A)^{d_{I}}S_{+}(u,A)^{d_{I}}$ is a set of
positive definition Hermitian matrices, and ${\rm log}$ takes the logarithm of
the corresponding blocks and leaves other blocks zero.

For $u_{0}=u_{\rm cat}$, they recover Theorem 1.1 and Theorem 3.15. For
general $u_{0}$, in order to prove them, we can repeat the recursive
procedures in Sections 3.2–3.4. The only difference is that now we need to
block the (1) in a different way, i.e., study the set of intermediate linear
systems determined by $u_{0}$ and the associated isomonodromy equations
restricted to each variable $r_{I}$ (instead of the variable $u_{i}$ as in the
caterpillar point case). The complete proofs will be omitted here.

Theorem 4.18 can be proved in the same way to the proof of Theorem 1.5 given
in Section 3.7, except that we need the following equivariance of the Riemann-
Hilbert-Birkhoff map $\nu(u_{0})$ at a general boundary point $u_{0}$,
generalizing the identity 93 at $u_{\rm cat}$,

$\displaystyle S_{-}^{\sigma}\Big{(}u_{0},\Big{(}\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}(\frac{r_{I}}{r_{I^{\prime}}})^{\frac{A^{I}}{2\pi\iota}}\Big{)}A\Big{(}\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}(\frac{r_{I}}{r_{I^{\prime}}})^{\frac{A^{I}}{2\pi\iota}}\Big{)}^{-1}\Big{)}\cdot
S_{+}^{\sigma}\Big{(}u_{0},\Big{(}\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}(\frac{r_{I}}{r_{I^{\prime}}})^{\frac{A^{I}}{2\pi\iota}}\Big{)}A\Big{(}\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}(\frac{r_{I}}{r_{I^{\prime}}})^{\frac{A^{I}}{2\pi\iota}}\Big{)}^{-1}\Big{)}$
$\displaystyle=\Big{(}\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}({r_{I}}/{r_{I^{\prime}}})^{\frac{{\rm
log}(S_{-}^{I}S_{+}^{I})}{2\pi\iota}}\Big{)}\cdot
S^{\sigma}_{-}(u_{0},A)S^{\sigma}_{+}(u_{0},A)\cdot\Big{(}\overrightarrow{\prod_{I\in
T_{\sigma}(u)}}({r_{I}}/{r_{I^{\prime}}})^{\frac{{\rm
log}(S_{-}^{I}S_{+}^{I})}{2\pi\iota}}\Big{)}^{-1}.$

The above identity follows from a manipulation of the equivariance of the
connection maps $C^{I}$ with respect to $U(n)^{I_{d}}$ for each vertex $I$,
and the (blocked Gauss decomposition) monodromy relation (118).

######  Remark 4.19.

Theorem 4.17 can be understood as a branching rule of the Stokes matrices
$S(u,A)$ as $u$ degenerates to $u_{0}$ according to the associated planar tree
$T_{\sigma}(u)$. It relates the Stokes matrices at $u\in{\mathfrak{h}_{\rm
reg}}(\mathbb{R})$ to the analytic data of a set of linear systems of lower
ranks via the solutions of isomonodromy equation with prescribed asymptotics,
as some of components $u_{i}$ of $u$ collapse (in a comparable speed).

##  5 Quantum Stokes matrices and their regularized limits at caterpillar
points

This section gives the quantum analog of the results in Section 3, i.e., the
expression of the regularized limits, as well as the leading terms, of quantum
Stokes matrices as $u\rightarrow u_{\rm cat}$, in terms of the Gelfand-Tsetlin
basis. In Section 5.1, we introduce the quantum Stokes matrices of the linear
system (1). In Section 5.2, we recall the regularized limits of quantum Stokes
matrices at a caterpillar point, and their explicit expressions. In Section
5.3 and 5.5, we we introduce the Gelfand-Tsetlin basis, and describe the
leading terms of quantum Stokes matrices as $u\rightarrow u_{\rm cat}$ in
terms of the basis.

###  5.1 Quantum Stokes matrices

In this subsection, we recall the Stokes matrices of the linear system (1)
associated to a representation $L(\lambda)$

$\displaystyle\frac{dF}{dz}=h\Big{(}\iota
u+\frac{1}{2\pi\iota}\frac{T}{z}\Big{)}\cdot F.$ (130)

Under a choice of basis in $L(\lambda)$, it becomes a special case of the
equation (11) with rank $n\times{\rm dim}(L(\lambda))$. But it is more
conveniently seen as a $n\times n$ system with coefficients valued in a non-
commutative space ${\rm End}(L(\lambda))$. The two Stokes sectors are still
the right/left half planes ${\rm Sect}_{\pm}=\\{z\in\mathbb{C}~{}|~{}\pm{\rm
Re}(z)>0\\}$. And again, let us choose the branch of ${\rm log}(z)$, which is
real on the positive real axis, with a cut along $i\mathbb{R}_{\geq 0}$. As a
special case of Theorem 2.2, we have that

######  Corollary 5.1.

For any $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$, on ${\rm Sect}_{\pm}$ there
is a unique (therefore canonical) holomorphic fundamental solution
$F_{h\pm}(z;u)\in{\rm End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$ of
(130) such that $F_{h\pm}\cdot e^{-\iota huz}z^{\frac{-h[T]}{2\pi\iota}}$ can
be analytically continued to $S(-\pi,\pi)$ and $S(-2\pi,0)$ respectively, and
for every small $\varepsilon>0$,

$\displaystyle\lim_{z\rightarrow\infty}F_{h+}(z;u)\cdot e^{-\iota huz}\cdot
z^{\frac{-h[T]}{2\pi\iota}}$ $\displaystyle=$ $\displaystyle 1,\ \ \ as\ \ \
z\in\overline{S}(-\pi+\varepsilon,\pi-\varepsilon),$
$\displaystyle\lim_{z\rightarrow\infty}F_{h-}(z;u)\cdot e^{-\iota huz}\cdot
z^{\frac{-h[T]}{2\pi\iota}}$ $\displaystyle=$ $\displaystyle 1,\ \ \ as\ \ \
z\in\overline{S}(-2\pi+\varepsilon,-\varepsilon),$

where $[T]={\rm diag}(E_{11},...,E_{nn})\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$.

######  Definition 5.2.

The quantum Stokes matrices of (130) (with respect to ${\rm Sect}_{+}$ and the
chosen branch of ${\rm log}(z)$) are the elements $S_{h\pm}(u)\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$ determined by

$\displaystyle F_{h+}(z)=F_{h-}(z)\cdot e^{\frac{h[T]}{2}}S_{h+}(u),\ \ \ \ \
F_{h-}(ze^{-2\pi\iota})=F_{h+}(z)\cdot S_{h-}(u)e^{\frac{-h[T]}{2}},$ (131)

where the first (resp. second) identity is understood to hold in ${\rm
Sect}_{-}$ (resp. ${\rm Sect}_{+}$) after $F_{h+}$ (resp. $F_{h-}$) has been
analytically continued anticlockwise around $z=\infty$.

Let us assume $u\in U_{\rm id}\subset\mathfrak{h}_{\rm reg}(\mathbb{R})$, then
the asymptotics in Theorem 5.1 ensures that $S_{h+}=(s_{ij}^{(+)})$ is a upper
triangular matrix, and $S_{h-}=(s_{ij}^{(-)})$ is lower triangular with
entries $s_{ij}^{(\pm)}$ in ${\rm End}(L(\lambda))$, see e.g., [49, Chapter
9.1].

###  5.2 Quantum Stokes matrices at a caterpillar point

Since (130) can be seen as a special form of the general system (11), we have

$S_{h\pm}(u)=S_{\pm}(hu,-hT),$

where the right hand side are the (classical) Stokes matrices of the equation
(11) with $u$ and $A$ replaced by $hu$ and $-hT$. Here the irregular term
$\iota hu={\rm diag}(\iota hu_{1},...,\iota hu_{n})\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$ is degenerate, i.e., has
repeated eigenvalues. One can still study the regularized limits of
$S_{h\pm}(u)$ as the components $u_{i}$ of $u\in\mathfrak{h}_{\rm
reg}(\mathbb{R})$ collapse in a comparable speed, and introduce the quantum
Stokes matrices for any boundary point in $\widehat{\mathfrak{h}_{\rm
reg}}(\mathbb{R})\setminus{\mathfrak{h}_{\rm reg}}(\mathbb{R})$. In
particular, the construction in Section 3.9 enables us to define the quantum
Stokes matrices at $u_{\rm cat}$ as follows.

First, the system (130) has rank $n\times{\rm dim}(L(\lambda))$. In terms of
the notation in Section 3.9, let us take the partition $\underline{d}$ of
$n\times{\rm dim}(L(\lambda))$ with $d_{1}=\cdots=d_{n}={\rm
dim}(L(\lambda))$, then the irregular term in the Stokes matrices
$S_{\pm}(hu,-hT)$ of (130) lives in $U_{\underline{d}}$. Following Theorem
3.36, the matrix $S_{-}(hu,-hT)\cdot S_{+}(hu,-hT)$ has a regularized limit.

######  Definition 5.3.

The quantum Stokes matrices at $u_{\rm cat}$ are the upper and lower $n\times
n$ triangular matrices $S_{h\pm}(u_{\rm cat})$ (having the same diagonal
part), with entries valued in ${\rm End}(L(\lambda))$, such that (the blocked
Gauss decomposition)

$S_{h-}(u_{\rm cat})S_{h+}(u_{\rm cat})$

equals to the regularized limit of the function $S_{-}(hu,-hT)\cdot
S_{+}(hu,-hT)$ on $U_{\underline{d}}$, i.e., the limit of

$\displaystyle\Big{(}\overrightarrow{\prod_{k=0,...,n-1}}(\frac{u_{k}}{u_{k+1}})^{\frac{{\rm
log}\big{(}\delta_{k}(S_{-})\delta_{k}(S_{+})\big{)}}{2\pi\iota}}\Big{)}\cdot
S_{-}S_{+}\cdot\Big{(}\overrightarrow{\prod_{k=0,...,n-1}}(\frac{u_{k}}{u_{k+1}})^{\frac{{\rm
log}(\big{(}\delta_{k}(S_{-})\delta_{k}(S_{+})\big{)}}{2\pi\iota}}\Big{)}^{-1}$
(132)

as $0<u_{1}\ll u_{2}\ll...\ll u_{n}$, where $\delta_{k}(S_{\pm}(hu,-hT))$
takes the ${\rm End}(\lambda)$ valued entries of the matrices
$S_{\pm}(hu,-hT)$ as in (102).

To write down the explicit expression of $S_{h\pm}(u_{\rm cat})$, as a non-
commutative version of the formula in Theorem 1.2, let us define
$T(\zeta):=\zeta{\rm Id}-T$. Then for any $1\leq m\leq n$, elements
$(a_{1},...,a_{m})$ and $(b_{1},...,b_{m})$ of $\\{1,...,n\\}$, let us
consider the $m$ by $m$ quantum minor

$\displaystyle\Delta^{a_{1},...,a_{m}}_{b_{1},...,b_{m}}(T(\zeta)):=\sum_{\sigma\in
S_{n}}(-1)^{\sigma}T_{a_{\sigma(1)},b_{1}}(\zeta_{1})\cdots
T_{a_{\sigma(m)},b_{m}}(\zeta_{m})\in U({\mathfrak{gl}}_{n})[\zeta],$

where $\zeta_{j}=\zeta+(j-1)$. For any $1\leq k\leq n$, let

$\displaystyle
M_{k}(\zeta):=\Delta^{1,...,k}_{1,...,k}\Big{(}T(\zeta-\frac{1}{2}(k-1))\Big{)}$
(133)

be the upper left $k\times k$ quantum-minor of $T$. It is known that the
subalgebra, generated in $U({\mathfrak{gl}}_{n})$ by the coefficients in all
$M_{k}(\zeta)$ for $1\leq k\leq n$, is commutative and is called the Gelfand-
Tsetlin subalgebra.

Let $\\{\zeta^{(k)}_{i}\\}_{1\leq i\leq k\leq n}$ denote the roots of
$M_{k}(\zeta)=0$ for all $k=1,...,n$ (in an appropriate splitting extension),
that are commutative with each other. Let us consider the equation (1)
associated to $L(\lambda)$, then

######  Theorem 5.4.

For any $1\leq k\leq n-1$, the $(k,k+1)$-entry of $S_{h\pm}(u_{\rm cat})$, as
an element in ${\rm End}(L(\lambda))$ is given by

$\displaystyle s_{k,k+1}^{(+)}=-(\iota
h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1})}{2\pi\iota}}\sum_{i=1}^{k}\Big{(}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2}}{2\pi\iota})}\Big{)}\cdot
h\alpha^{(k)}_{i},$ $\displaystyle s_{k+1,k}^{(+)}=-(\iota
h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}+1)}}{2\pi\iota}}\sum_{i=1}^{k}\Big{(}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1-h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1-h\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}+\frac{1}{2}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1-h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1-h\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}+\frac{1}{2}}{2\pi\iota})}\Big{)}\cdot
h\beta^{(k)}_{i},$

with

$\displaystyle\alpha^{(k)}_{j}=\sum_{j=1}^{k}\frac{(-1)^{k-j}\Delta^{1,...,\hat{j},...,k}_{1,...,k-1}\Big{(}T(\zeta^{(k)}_{i}-\frac{(k-1)}{2})\Big{)}}{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}E_{j,k+1}$ (134)
$\displaystyle\beta^{(k)}_{j}=\sum_{j=1}^{k}\frac{(-1)^{k-j}\Delta_{1,...,\hat{j},...,k}^{1,...,k-1}\Big{(}T(\zeta^{(k)}_{i}-\frac{(k-3)}{2})\Big{)}}{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}E_{k+1,j}.$ (135)

######  Remark 5.5.

A universal version of the above formula of quantum Stokes matrices at $u_{\rm
cat}$ is given in [52], and is used to derive an explicit Drinfeld
isomorphism.

Theorem 5.4 is better to be understood in terms of the Gelfand-Tsetlin basis
in $L(\lambda)$. Before giving a proof of the theorem in Section 5.4, let us
first introduce the Gelfand-Tsetlin orthonormal basis.

###  5.3 Gelfand-Tsetlin basis for representations of ${\mathfrak{gl}}_{n}$

Recall that $\\{E_{ij}\\}_{i,j=1,...,n}$ is the standard basis of the Lie
algebra ${\mathfrak{gl}}_{n}$. Denote by ${\mathfrak{gl}}_{n-1}$ the
subalgebra spanned by the elements $\\{E_{ij}\\}_{i,j=1,...,n-1}$. Finite
dimensional irreducible representations of ${\mathfrak{gl}}_{n}$ are
parameterized by the highest weight, i.e., $n$-tuples of numbers
$\lambda=(\lambda_{1},...,\lambda_{n})$ with

$\lambda_{i}-\lambda_{i+1}\in\mathbb{Z}_{+},\hskip 14.22636pt,\ \forall\
i=1,...,n-1.$

We denote by $L(\lambda)$ the corresponding representation. It has a lowest
vector $\xi_{0}$ such that $E_{ii}\xi=\lambda_{i}\xi$ for $i=1,...,n$, and
$E_{ij}\xi=0$ for $1\leq j<i\leq n$. Then the simple branching rule for the
reduction from ${\mathfrak{gl}}_{n}$ to ${\mathfrak{gl}}_{n-1}$ states that
the restriction of $L(\lambda)$ to the subalgebra ${\mathfrak{gl}}_{n-1}$ is
isomorphic to the direct sum of pairwise inequivalent irreducible
representations

$L(\lambda)|_{{\mathfrak{gl}}_{n-1}}\cong\underset{\lambda^{(n-1)}}{\bigoplus}L^{\prime}(\lambda^{(n-1)}),$

where the summation is over the highest weights $\lambda^{(n-1)}$ satisfying
the interlacing conditions

$\displaystyle\lambda_{i}^{(n)}-\lambda^{(n-1)}_{i}\in\mathbb{Z}_{\geq
0},\hskip 14.22636pt\lambda^{(n-1)}_{i}-\lambda^{(n)}_{i+1}\in\mathbb{Z}_{\geq
0},\hskip 14.22636pt\forall\ i=1,...,n-1.$ (136)

Thus a chain of subalgebras

${\mathfrak{gl}}_{1}\subset\cdots\subset{\mathfrak{gl}}_{n-1}\subset{\mathfrak{gl}}_{n}$

produces a basis in $L(\lambda)$, parameterized by the Gelfand-Tsetlin
patterns. Such a pattern $\Lambda$ (for fixed $\lambda^{(n)}$) is a collection
of numbers $\\{\lambda^{(i)}_{j}\\}_{1\leq i\leq j\leq n-1}$ satisfying the
interlacing conditions

$\displaystyle\lambda_{j}^{(i)}-\lambda^{(i-1)}_{j}\in\mathbb{Z}_{\geq
0},\hskip 14.22636pt\lambda^{(i-1)}_{j}-\lambda^{(i)}_{j+1}\in\mathbb{Z}_{\geq
0},\hskip 14.22636pt\forall\ i=1,...,n-1.$

######  Definition 5.6.

We denote by $P_{GZ}(\lambda;\mathbb{Z})$ the set of Gelfand-Tsetlin patterns
in $L(\lambda)$, seen as the set of integer points in the real Gelfand-Tsetlin
polytope $P_{GZ}(\lambda;\mathbb{R})$.

The structure of the basis obtained in this way is summarized in the following
proposition.

######  Proposition 5.7.

There exists an orthonormal basis $\xi_{\Lambda}$ of $L(\lambda)$, called the
Gelfand-Tsetlin basis, parameterized by all patterns $\Lambda\in
P_{GZ}(\lambda;\mathbb{Z})$, such that for any $1\leq i\leq k\leq n$, the
actions of $E_{kk}$, $\zeta^{(k)}_{i}$ and $\alpha^{(k)}_{i}$ on the basis
${\xi_{\Lambda}}$ of $L(\lambda)$ are given by

$\displaystyle E_{kk}\cdot\xi_{\Lambda}=$
$\displaystyle(\sum_{i=1}^{k}\lambda^{(k)}_{i}-\sum_{i=1}^{k-1}\lambda^{(k-1)}_{i})\xi_{\Lambda},$
(137) $\displaystyle\zeta^{(k)}_{i}\cdot\xi_{\Lambda}=$
$\displaystyle(\lambda^{(k)}_{i}(\Lambda)-i+1+(k-1)/2)\xi_{\Lambda},$ (138)
$\displaystyle\alpha^{(k)}_{i}\cdot\xi_{\Lambda}=$
$\displaystyle\sqrt{-\frac{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}{\prod_{l=1}^{k+1}(\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2})}\frac{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1)}{\prod_{l=1}^{k-1}(\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2})}}\cdot\xi_{\Lambda+\delta^{k}_{i}}$
(139)

where the pattern $\Lambda+\delta^{(k)}_{i}$ is obtained from $\Lambda$ by
replacing $\lambda^{(k)}_{i}$ by $\lambda^{(k)}_{i}+1$. It is supposed that
$\xi_{\Lambda}$ is zero if $\Lambda$ is not a pattern.

Proof. We refer to [43, Sectioin 2] for the existence of the orthogonal
Gelfand-Tsetlin basis, and the norms of the elements in the basis [43,
Proposition 2.4]. We then perform the explicit normalization to get an
orthonormal basis. The actions of $E_{kk}$ and $\zeta^{(k)}_{i}$ on the
orthogonal basis given in [43, Theorem 2.3] imply the identities (137) and
(138).

By the Laplace expansion, the $\alpha^{(k)}_{i}$ in (134) equals to the
normalized quantum minor

$\displaystyle\alpha^{(k)}_{i}=\frac{-1}{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}\Delta^{1,...,k}_{1,...,k-1,k+1}\Big{(}T(\zeta-\frac{1}{2}(k-1))\Big{)}.$

Then the identity (139) follows from the action of
$\Delta^{1,...,k}_{1,...,k-1,k+1}$ on the orthogonal basis given in [43,
Section 2.5], provided the normalization, from the orthogonal to the
orthonormal basis, is accounted for.

###  5.4 Proof of Theorem 5.4

The explicit expressions in Theorem 5.4 can be derived in the same way as the
ones in Theorem 1.2, except that in the non-commutative setting we should use
Proposition 5.8 to exchange the orders in the product of the ${\rm
End}(L(\lambda))$ valued functions $\zeta^{(k)}_{i}$ and $\alpha^{(k)}_{i}$.

Following Definition 5.3 and Theorem 3.36, we have

$\displaystyle S_{h-}(u_{\rm cat})S_{h+}(u_{\rm
cat})=\Big{(}\overrightarrow{\prod_{k=1,...,n}}C^{(k)}_{h}(T)\Big{)}\cdot
e^{hT}\cdot\Big{(}\overrightarrow{\prod_{k=1,...,n}}C^{(k)}_{h}(T)\Big{)}^{-1},$
(140)

where each $C^{(k+1)}_{h}(T)$ denotes the connection matrix of

$\displaystyle\frac{dF}{dz}=h\Big{(}\iota
E_{k+1,k+1}+\frac{1}{2\pi\iota}\frac{\delta_{k+1}(T)}{z}\Big{)}\cdot F.$ (141)

Here $E_{k+1}\in{\rm End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$ is the
matrix whose $(k+1,k+1)$ entry is $1\in{\rm End}(L(\lambda))$ and other
entries are zero. Therefore, to get the explicit expressions of
$S_{h\pm}(u_{\rm cat})$, we need to compute each $C^{(k+1)}_{h}(T)$.

Diagonalization in stages. To simplify the equation (141), let us diagonalize
the upper left $k$-th submatrix of its coefficient matrix. Recall that
$\zeta^{(k)}_{1},...,\zeta^{(k)}_{k}$ of $M_{k}(\zeta)$ denote the roots of
the quantum minor
$\Delta^{1,...,k}_{1,...,k}\Big{(}T(\zeta-\frac{h}{2}(k-1))\Big{)}$, which act
diagonally on the Gelfand-Zeitlin basis in $L(\lambda)$.

######  Proposition 5.8.

$(a).$ For any $2\leq k\leq n$, the $n\times n$ matrix $P_{k}$ with entries in
${\rm End}(L(\lambda))$,

$\displaystyle(P_{k})_{ij}:=\frac{(-1)^{k-j}}{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}\Delta^{1,...,\hat{j},...,k}_{1,...,k-1}\Big{(}T(\zeta^{(k)}_{i}-\frac{1}{2}(k-1))\Big{)},$
(142) $\displaystyle(P_{k})_{ii}:=1,\ \ if\ i>k,\ \ \ \ \ \ (P_{k})_{ij}:=0,\
\ otherwise,$

diagonalizes the upper left $k$-th principal submatrix $T^{(k)}$ of $T$, and
is such that $T_{k}:=P_{k}\cdot T\cdot P_{k}^{-1}$ takes the form

$T_{k}=\begin{pmatrix}\zeta^{(k)}_{1}+\frac{1}{2}(k-1)&&&&\alpha^{(k)}_{1}&\cdots\\\
&\ddots&&&\vdots&\cdots\\\
&&\zeta^{(k)}_{k}+\frac{1}{2}(k-1)&&\alpha^{(k)}_{k}&\cdots\\\ &&&&\\\
\beta_{1}^{(k)}&\cdots&\beta_{k}^{(k)}&&E_{k+1,k+1}&\cdots\\\
\vdots&\cdots&\vdots&&\vdots&\ddots\end{pmatrix}.$

$(b).$ Furthermore, for any $1\leq k\leq n-1$, and $1\leq i,j\leq k$, we have
the commutators

$\displaystyle[\zeta^{(k)}_{i},\alpha^{(k)}_{j}]=\delta_{ij}\alpha^{(k)}_{j}.$
(143)

Proof. The Laplace expansion, of the quantum minor $M_{k}(\zeta)$ as a
weighted sum of $k$ quantum minors of size $k-1$, shows that for any tuples
$(a_{1},...,a_{k})$ and $(b_{1},...,b_{k})$,

$\Delta^{a_{1},...,a_{k}}_{b_{1},...,b_{k}}(T)(\zeta)=\sum_{j=1}^{k}(-1)^{k-j}\Delta^{a_{1},...,\widehat{a_{j}},...,a_{k}}_{b_{1},...,b_{k-1}}(T)(\zeta)\cdot
T_{a_{j}b_{k}}(\zeta+(k-1)).$

In the following, let us fix $(a_{1},...,a_{k})=(1,...,k)$ and
$(b_{1},...,b_{k-1})=(1,...,k-1)$. If we take $b_{k}=k$ and for each
$i=1,...,k$ plug in $\zeta=\zeta_{i}^{(k)}-\frac{1}{2}(k-1)$, we get the
identity

$\displaystyle
0=\Delta^{1,...,k}_{1,...,k}(T)\Big{(}\zeta^{(k)}_{i}-\frac{1}{2}(k-1)\Big{)}=\sum_{j=1}^{k}(-1)^{k-j}\Delta^{1,...,\hat{j},...,k}_{1,...,k-1}(T)\Big{(}\zeta^{(k)}_{i}-\frac{1}{2}(k-1)\Big{)}\cdot
T_{jk}\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k-1)\Big{)}.$

Similarly, taking $b_{k}=1,...,k-1$ and
$\zeta=\zeta_{i}^{(k)}-\frac{1}{2}(k-1)$ eventually implies that for each $i$,
the row vector

$v_{i}=\frac{1}{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}\Big{(}(P_{k})_{i1},...,(P_{k})_{ik}\Big{)}$

satisfies

$v_{i}\cdot T^{(k)}=\Big{(}\zeta^{(k)}_{i}+\frac{h}{2}(k-1)\Big{)}v_{i}.$

Here we denote by $T^{(k)}$ the upper left $k$-th principal submatrix of $T$.
It verifies $(a)$.

The relation in $(b)$ follows from Proposition 5.7.

One can define $P_{k}$ recursively. That is for any $1\leq k\leq n-1$, we
introduce the $n\times n$ matrix $L^{(k+1)}$ with entries in ${\rm
End}(L(\lambda))$

$\displaystyle
L^{(k+1)}_{ij}=\alpha^{(k)}_{i}\cdot\frac{1}{\zeta^{(k+1)}_{j}-\zeta^{(k)}_{i}-\frac{1}{2}},\
\ \ 1\leq i\leq k,\ \ 1\leq j\leq k+1,$ (144) $\displaystyle
L^{(k+1)}_{k+1,j}=1,\ \ \ \ \ 1\leq j\leq k+1.$ (145) $\displaystyle
L^{(k+1)}_{ii}=1,\ \ if\ i>k+1,\ \ \ L^{(k+1)}_{ij}=0,\ otherwise.$

Then ${L^{(k+1)}}^{-1}\cdot T_{k}\cdot{L^{(k+1)}}=T_{k+1}$, and
$P_{k}=L^{(k+1)}P_{k+1}$.

Confluent hypergeometric functions in representation spaces. Diagonalizing the
upper left $k$-th submatrix of the coefficient of (141), we are left with the
equation

$\displaystyle\frac{dF}{dz}=h\Big{(}\iota
E_{k+1,k+1}+\frac{1}{2\pi\iota}\frac{\delta_{k+1}(T_{k})}{z}\Big{)}\cdot F.$
(146)

A solution of this equation, as a function valued in ${\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$, is described using the
generalized confluent hypergeometric functions as follows. First recall that
the confluent hypergeometric functions associated to any
$\alpha_{j}\in\mathbb{C}$ and
$\beta_{j}\in\mathbb{C}\setminus\\{0,-1,-2,...\\},1\leq j\leq m$, are

${}_{k}F_{k}(\alpha_{1},...,\alpha_{k},\beta_{1},...,\beta_{k};z)=\sum_{n=0}^{\infty}\frac{(\alpha_{1})_{n}\cdots(\alpha_{k})_{n}}{(\beta_{1})_{n}\cdots(\beta_{k})_{n}}\frac{z^{n}}{n!},$
(147)

where $(\alpha)_{0}=1$, $(\alpha)_{n}=\alpha\cdots(\alpha+n-1),$ $n\geq 1$.
Since $\\{\zeta^{(k)}_{i}\\}_{1\leq i\leq k\leq n}$ are commutative elements
in ${\rm End}(L(\lambda))$, we can introduce the following confluent
hypergeometric functions valued in ${\rm End}(L(\lambda))$,

$\displaystyle
H(z)_{ij}=\scriptsize{\frac{1}{h(\zeta^{(k+1)}_{j}-\zeta^{(k)}_{i}-\frac{1}{2})}}\cdot{{}_{k}F_{k}}(\alpha_{ij,1},...,\alpha_{ij,k},\beta_{ij,1},...,\widehat{\beta_{ij,j}},...,\beta_{ij,k+1};\iota
z),\ \ \ 1\leq i\leq k,1\leq j\leq k+1,$ $\displaystyle
H(z)_{k+1,j}={{}_{k}F_{k}}(\alpha_{k+1j,1},...,\alpha_{k+1j,k},\beta_{k+1j,1},...,\widehat{\beta_{k+1j,j}},...,\beta_{k+1j,k+1};\iota
z),\ \ i=k+1,\ 1\leq j\leq k+1,$

with the variables $\\{\alpha_{ij,l}\\}$ and $\\{\beta_{ij,l}\\}$ are given by

$\displaystyle\alpha_{ij,i}=\frac{h}{2\pi\iota}\Big{(}\zeta^{(k+1)}_{j}-\zeta^{(k)}_{i}-\frac{1}{2}\Big{)},\
\ \ 1\leq i\leq k,\ 1\leq j\leq k+1,$
$\displaystyle\alpha_{(k+1,j),i}=1+\frac{h}{2\pi\iota}\Big{(}\zeta^{(k+1)}_{j}-\zeta^{(k)}_{i}-\frac{1}{2}\Big{)},\
\ \ 1\leq j\leq k+1,$
$\displaystyle\alpha_{ij,l}=1-\frac{h}{2\pi\iota}\Big{(}\zeta^{(k+1)}_{j}-\zeta^{(k)}_{l}-\frac{1}{2}\Big{)},\
\ \ \ l\neq i,\ 1\leq l\leq k,\ 1\leq i,j\leq k+1,$
$\displaystyle\beta_{ij,l}=1+\frac{h}{2\pi\iota}\Big{(}\zeta^{(k+1)}_{j}-\zeta^{(k+1)}_{l}\Big{)},\
\ \ \ l\neq j,\ 1\leq l\leq k+1,\ \ 1\leq i,j\leq k+1.$

######  Proposition 5.9.

The equation (146) has a fundamental solution $F(z)\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(L(\lambda))$ taking the form

$\displaystyle F(z)=\left(\begin{array}[]{cc}{\rm
diag}(ha^{(k)}_{1},...,ha^{(k)}_{k},1)&0\\\ 0&{\rm
Id}_{n-k-1}\end{array}\right)\left(\begin{array}[]{cc}\big{(}H_{ij}(z)\big{)}_{i,j=1}^{k+1}&0\\\
0&{\rm Id}_{n-k-1}\end{array}\right)\cdot z^{\frac{h[T_{k+1}]}{2\pi\iota}},$
(152)

where $[T_{k+1}]$ is the diagonal part of $T_{k+1}$, i.e.,

${[T_{k+1}]}=\left(\begin{array}[]{cc}{\rm
diag}({\zeta^{(k+1)}_{1}+\frac{k}{2}},...,{\zeta^{(k+1)}_{k+1}+\frac{k}{2}})&0\\\
0&{\rm diag}({E_{k+2,k+2}},...,{E_{n,n}})\end{array}\right).$

Proof. We only need to verify the identities

$\displaystyle
z\frac{d}{dz}(F(z)_{ij})=\frac{h}{2\pi\iota}\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k-1)\Big{)}F(z)_{ij}+\frac{h}{2\pi\iota}\alpha^{(k)}_{i}F(z)_{k+1,j},\
\ \ for\ 1\leq i\leq k,$ (153) $\displaystyle
z\frac{d}{dz}(F(z)_{k+1,j})=h\iota
zF(z)_{k+1,j}+\frac{h}{2\pi\iota}\sum_{i=1}^{k}\beta^{(k)}_{i}F(z)_{ij}+\frac{h}{2\pi\iota}\zeta^{(k)}_{k+1}F(z)_{k+1,j},$
(154)

for the functions

$\displaystyle F(z)_{ij}=h\alpha^{(k)}_{i}\cdot H_{ij}(z)\cdot
z^{\frac{h}{2\pi\iota}(\zeta^{(k+1)}_{j}+\frac{k}{2})},\ \ \ for\ 1\leq i\leq
k,\ 1\leq j\leq k+1$ $\displaystyle F(z)_{k+1,j}=H_{k+1,j}(z)\cdot
z^{\frac{h}{2\pi\iota}(\zeta^{(k+1)}_{j}+\frac{k}{2})},\ \ \ for\ 1\leq j\leq
k+1.$

Since $\alpha^{(k)}_{i}$ and $\zeta^{(k)}_{i}$ do not commute, to compare the
left and right hand sides of (153), we need to keep the terms
$\alpha^{(k)}_{i}$ in the front. That is to pull the term $\alpha^{(k)}_{i}$
in

$\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k-1)\Big{)}F(z)_{ij}=\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k-1)\Big{)}\cdot\alpha^{(k)}_{i}H_{ij}(z)z^{\frac{1}{2\pi\iota}(\zeta^{(k+1)}_{j}+\frac{k}{2})}$

in front, via the commutator relation

$\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k-1)\Big{)}\cdot\alpha^{(k)}_{i}=\alpha^{(k)}_{i}\cdot\Big{(}\zeta^{(k)}_{i}+\frac{1}{2}(k+1)\Big{)}.$

Then the equation (153) is equivalent to the differential equation

$\displaystyle
z\frac{dH_{ij}}{dz}=\frac{h({\small\zeta^{(k)}_{i}-\zeta^{(k+1)}_{j}}+\frac{1}{2})}{2\pi\iota}H_{ij}-\frac{1}{2\pi\iota}H_{k+1,j},\
\ \ for\ 1\leq i\leq k,$

which in turn can be verified by the definition (147) of the confluent
hypergeometric functions. Using the similar differential equation of confluent
hypergeometric functions, as well as the characteristic polynomials of
$\delta_{k+1}(T)$, one can verify (153).

We denote by $C^{(k+1)}_{h}(T_{k})$ the connection matrix of the equation
(146). Then the Stokes matrices $S_{h+}^{(k+1)}(T)$ and the normalized
connection matrix $\widetilde{C}_{h}^{(k+1)}(T):=C^{(k+1)}_{h}(T_{k})\cdot
L^{(k+1)}$ of the linear system (146) is described by

######  Proposition 5.10.

$(1)$ The entries $s_{ij}$ of $S_{h+}^{(k+1)}(T)$ are

$\displaystyle s_{j,k+1}=-\frac{(\iota
h)^{\frac{h(\zeta^{(k)}_{j}-E_{k+1,k+1}-1)}{2\pi\iota}}e^{\frac{-h\zeta^{(k)}_{j}}{2}}\prod_{l=1,l\neq
j}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{j}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k)}_{j}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi\iota})}\cdot
h\alpha^{(k)}_{j},\ \ \ for\ j=1,...,k;$

and

$s_{ii}=e^{\frac{E_{ii}}{2}},\ \ \ i=1,...,n;\hskip 14.22636pts_{ij}=0,\ \
otherwise.$

$(2)$. The entries $c_{ij}$ of the matrix $\widetilde{C}_{h}^{(k+1)}(T)$ are

$\displaystyle c_{ij}=-\frac{(\iota
h)^{\frac{h(\zeta^{(k+1)}_{j}-\zeta^{(k)}_{i})}{2\pi\iota}}\prod\limits_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k+1)}_{j}-\zeta^{(k+1)}_{l}}{2\pi\iota})\prod\limits_{l=1}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod\limits_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k+1)}_{j}-\zeta^{(k)}_{l}}{2\pi\iota})\prod\limits_{l=1,l\neq
j}^{k+1}\Gamma(1+h\frac{\zeta^{(k+1)}_{l}-\zeta^{(k)}_{i}+\frac{1}{2}}{2\pi\iota})}\cdot
h\alpha^{(k)}_{i},$

for $1\leq j\leq k+1,1\leq i\leq k$;

$\displaystyle c_{k+1,j}=\frac{(\iota
h)^{\frac{hE_{k+1,k+1}-h\zeta^{(k+1)}_{j}}{2\pi\iota}}\prod_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k+1)}_{j}-\zeta^{(k+1)}_{l}}{2\pi\iota})}{\prod_{l=1}^{k}\Gamma(1+h\frac{\zeta^{(k+1)}_{j}-\zeta^{(k)}_{l}-\frac{1}{2}}{2\pi\iota})},\
\ \ for\ 1\leq j\leq k+1;$

and

$\ c_{ii}=1\ \ for\ \ k+2\leq i\leq n;\hskip 14.22636ptc_{ij}=0\ \ otherwise.$

Proof. It follows from Proposition 5.9 and the known asymptotics of the
functions ${}_{k}F_{k}(\alpha_{1},...,\alpha_{k},\beta_{1},...,\beta_{k};z)$:

$\displaystyle\frac{\prod_{l=1}^{k}\Gamma(\alpha_{l})}{\prod_{l=1}^{k}\Gamma(\beta_{l})}\cdot{{}_{k}F_{k}}(\alpha_{1},...,\alpha_{k},\beta_{1},...,\beta_{k};z)$
(155) $\displaystyle\sim$
$\displaystyle\sum_{m=1}^{k}\Gamma(\alpha_{m})\frac{\prod_{l=1,l\neq
m}^{k}\Gamma(\alpha_{l}-\alpha_{m})}{\prod_{l=1}^{k}\Gamma(\beta_{l}-\alpha_{m})}(\mp
z)^{-\alpha_{m}}(1+O(\frac{1}{z}))+e^{z}z^{\sum_{l=1}^{k}(\alpha_{l}-\beta_{l})}(1+O(\frac{1}{z})).$

where upper or lower signs are chosen according as $z$ lies in the upper or
lower half-plane. To be more precise, if we take the solution $F(z)$ in
Proposition 5.9, then

$\displaystyle F(z)\sim\hat{F}(z)\cdot Y\cdot U_{+},\ \ \ as\ z\rightarrow\
\infty\ in\ \overline{S}(-\pi+\varepsilon,\pi-\varepsilon)$ $\displaystyle
F(z)\sim\hat{F}(z)\cdot Y\cdot U_{-},\ \ \ as\ z\rightarrow\ \infty\ in\
\overline{S}(-2\pi+\varepsilon,-\varepsilon),$

for small $\varepsilon$, where

$\hat{F}(z)=(1+O(z^{-1}))e^{\iota
hzE_{k+1,k+1}}z^{\frac{h}{2\pi\iota}[T_{k}]}$

is the unique formal solution, and $U_{\pm}$ are explicit matrices determined
by the asymptotics (155). Note that the matrices $U_{\pm}$ are certain
functions of the commutative variables $\\{\zeta^{(k)}_{j}\\}_{j=1,...,k}$ and
$\\{\zeta^{(k+1)}_{j}\\}_{j=1,...,k+1}$. Then by the uniqueness in Theorem
2.2, we know that the canonical solutions

$\displaystyle F(z)=F_{+}(z)YU_{+}\hskip 8.53581ptin\ \ {\rm Sect}_{+},$
$\displaystyle F(z)=F_{-}(z)YU_{-}\hskip 8.53581ptin\ \ {\rm Sect}_{-}.$

Then by definition, the Stokes matrices are given by (here to derive the
second formula, the change in choice of ${\rm log}(z)$ is accounted for)

$\displaystyle S^{(k+1)}_{h+}(T)=YU_{-}U_{+}^{-1}Y^{-1},\ \ \ \ \ \
S^{(k+1)}_{h-}(T)=YU_{-}e^{h[T_{k+1}]}U_{+}^{-1}Y^{-1}.$

Then to get the expression of $b^{(k+1)}_{j}$ in the proposition, we only need
to use the commutative relation (143) to commute $a^{(k)}_{j}$ in the first
factor of $Y$ in $YU_{-}U_{+}^{-1}Y^{-1}$ with the arguments $\zeta^{(k)}_{j}$
in $U_{-}U_{+}^{-1}$. Similarly, one can get the expression of the normalized
connection matrix $\widetilde{C}^{(k+1)}(T)$ by considering the asymptotics of
$F(z)$ at $z=0$. The involved computation using the asymptotics of
${}_{k}F_{k}$ is straight-forward but lengthy. Here we only outline the
computational process, and refer the reader to [37] for the full details
including the explicit expressions of $U_{\pm}$.

Subdiagonal entries of quantum Stokes matrices. The connection matrices of the
equations (141) and (146) are related by

$C_{h}^{(k+1)}(T)=P_{k}^{-1}\cdot C_{h}^{(k+1)}(T_{k})\cdot P_{k},$

where $P_{k}$ is given in Proposition 5.8. By the relation
$\widetilde{C}_{h}^{(k+1)}(T)=C^{(k+1)}_{h}(T_{k})\cdot L^{(k+1)}$ and
$P_{k}=L^{(k+1)}P_{k+1}$, we get

$\displaystyle
C_{h}^{(k+1)}(T)=P_{k}^{-1}\cdot\widetilde{C}_{h}^{(k+1)}(T_{k})\cdot
P_{k+1}.$ (156)

Plugging (156) for all $k=1,...,n-1$ into (140) gives

$\displaystyle S_{h-}(u_{\rm cat})S_{h+}(u_{\rm
cat})=\Big{(}\overrightarrow{\prod_{k=1,...,n}}\widetilde{C}_{h}^{(k)}(T)\Big{)}\cdot
e^{h[T]}\cdot\Big{(}\overrightarrow{\prod_{k=1,...,n}}\widetilde{C}_{h}^{(k)}(T)\Big{)}^{-1}.$

A manipulation of (blocked) Gauss decomposition, as well as the explicit
formula of $\widetilde{C}_{h}^{(k+1)}(T)$ and $S_{h+}^{(k+1)}(T)$ given in
Proposition 5.10, gives rise to the formula of $s^{(+)}_{k,k+1}$ in Theorem
5.4. We refer the reader to [37] for the full details.

###  5.5 The leading terms of quantum Stokes matrices via the Gelfand-Tsetlin
basis

Given any $L(\lambda)$, let us think of the associated quantum Stokes matrices
as blocked matrices. Let us take a proper norm on the vector space ${\rm
End}(\lambda)$, then up to a slight modification, Proposition 3.32 and
Corollary 3.33 can be applied to the quantum/blocked cases. That is if we
introduce the unitary matrix

$g(u;T)=\overrightarrow{\prod_{k=0,...,n-1}}({u_{k}}/{u_{k+1}})^{\frac{\delta_{k}(T)}{2\pi\iota}},$

then

######  Corollary 5.11.

For any fixed none zero real number $h$, we have, as $0<u_{1}\ll
u_{2}\ll...\ll u_{n}$,

$\displaystyle S_{h\pm}(u)=S_{\pm}\Big{(}u_{\rm cat},\hskip
5.69054ptg(hu;hT)\cdot hT\cdot g(hu;hT)^{-1}+O((u_{2}-u_{1})^{-1})\Big{)}.$
(157)

Furthermore, the subdiagonal entries of $S_{h\pm}(u)$ satisfy

$\displaystyle
s^{(\pm)}_{k,k+1}(u)={l}^{(\pm)}_{k,k+1}(u)+o(l^{(\pm)}_{k,k+1}(u)),\hskip
8.53581ptas\ 0<u_{1}\ll u_{2}\ll...\ll u_{n},$ (158)

where the leading terms ${l}^{(\pm)}_{k,k+1}(u)$ is given by the expressions
$s^{(\pm)}_{k,k+1}(u_{\rm cat})$ of in Theorem 5.4 provided we replace
respectively $\alpha^{(k)}_{i}$ and $\beta^{(k)}_{i}$ by

$\displaystyle
u_{k}^{-\frac{E_{kk}}{2\pi\iota}}({u_{k}}/{u_{k+1}})^{\frac{h(\zeta^{(k)}_{i}-\frac{k-1}{2})}{2\pi\iota}}\cdot\alpha^{(k)}_{i}\hskip
8.53581ptand\hskip
8.53581ptu_{k}^{-\frac{E_{kk}}{2\pi\iota}}({u_{k}}/{u_{k+1}})^{\frac{h(\zeta^{(k)}_{i}-\frac{k-3}{2})}{2\pi\iota}}\cdot\beta^{(k)}_{i}.$
(159)

Here $\frac{k-1}{2}$ in $(\zeta^{(k)}_{i}-\frac{k-1}{2})$ comes from the
Capelli correction in the quantum minors, see the computation in Appendix.

By Proposition 5.7 and Corollary 5.11, the leading terms of entries of quantum
Stokes matrices, as elements in ${\rm End}(L(\lambda))$, can be written down
explicitly in terms of the Gelfand-Tsetlin basis.

######  Remark 5.12.

In the next section, we will study the WKB approximation of $S_{h\pm}(u_{\rm
cat})$. By the definition given in the introduction and the replacement of
$\alpha^{(k)}_{i}$ and $\beta^{(k)}_{i}$ given in (159), we see that
${l}^{(\pm)}_{k,k+1}(u)$ and $s^{(\pm)}_{k,k+1}(u_{\rm cat})$ have the same
WKB leading term as $h\rightarrow-\infty.$ The relation in (158) is for any
fixed $h>0$ (the proof of Proposition 3.32 given in [51] makes the assumption
that the norm of $A$ is fixed), therefore, the WKB datum at $u_{\rm cat}$
encodes the "repeated limit" of $S_{h\pm}(u)$ as $u\rightarrow u_{\rm cat}$
and then $h\rightarrow-\infty$.

##  6 Applications in representation theory: crystals arising from the WKB
approximation

In the previous section, we introduce the relation between the regularized
limits of quantum Stokes matrices at $u_{\rm cat}$ and Gelfand-Tsetlin basis.
In this section, we deepen the relation between Stokes phenomenon and
representation theory by showing that the WKB approximation of quantum Stokes
matrices at a caterpillar point gives rise to $\mathfrak{gl}_{n}$-crystals.
Section 6.1 computes explicitly the WKB approximation. Section 6.2 unveils
that how to recover the Gelfand-Tsetlin basis from the WKB approximation. Then
Section 6.3 introduces the notion of WKB datum of quantum Stokes matrices. In
the end, Section 6.6 recalls the relation between quantum Stokes matrices and
the RLL relations in quantum groups.

###  6.1 WKB approximation of quantum Stokes matrices at caterpillar points

Since the derivative in equation (1) is multiplied by a small parameter $1/h$,
we will call the leading term, as $h\rightarrow-\infty$, of Stokes matrices of
(1) as the WKB approximation. In this subsection, we study the WKB
approximation of Stokes matrices given in (5.4).

First, for any Gelfand-Tsetlin pattern $\Lambda$ of $L(\lambda)$ given in
Section 5.3, set

$\displaystyle
x^{(k)}_{j}(\Lambda):=-\lambda^{(k)}_{j}+\lambda^{(k-1)}_{j-1}-\lambda^{(k)}_{j-1}+\lambda^{(k+1)}_{j},\hskip
8.53581pt1\leq j\leq k+1,$ (160) $\displaystyle
y^{(k)}_{j}(\Lambda):=\lambda^{(k)}_{j}-\lambda^{(k-1)}_{j}+\lambda^{(k)}_{j+1}-\lambda^{(k+1)}_{j+1},\hskip
8.53581pt0\leq j\leq k,$ (161)

and

$\displaystyle X^{(k)}_{j}(\Lambda):=\sum_{i=1}^{j}x^{(k)}_{i}(\Lambda),\hskip
8.53581pt1\leq j\leq k+1,$ (162) $\displaystyle
Y^{(k)}_{j}(\Lambda):=\sum_{i=j}^{k}y^{(k)}_{i}(\Lambda),\hskip 8.53581pt0\leq
j\leq k.$ (163)

Note that $Y^{(k)}_{0}(\Lambda)=Y^{(k)}_{j}(\Lambda)-X^{(k)}_{j}(\Lambda)$ for
any $j=0,...,k+1$. Furthermore, we define

$\displaystyle
wt_{k}(\Lambda)=\sum_{i=1}^{k}\lambda^{(k)}_{i}(\Lambda)-\sum_{i=1}^{k-1}\lambda^{(k-1)}_{i}(\Lambda),$
(164) $\displaystyle\varepsilon_{k}(\Lambda)={\rm
max}\\{X_{1}^{(k)}(\Lambda),X_{2}^{(k)}(\Lambda),...,X_{k}^{(k)}(\Lambda)\\},$
(165) $\displaystyle\phi_{i}(\Lambda)={\rm
max}\\{Y_{1}^{(k)}(\Lambda),Y_{2}^{(k)}(\Lambda),...,Y_{k}^{(k)}(\Lambda)\\},$
(166)

and define the functions $l_{1}(\Lambda)<\cdots<l_{m_{k}}(\Lambda)$ of
$\Lambda\in P_{GZ}(\lambda;\mathbb{Z})$ be those ordered labels such that

$\displaystyle X^{(k)}_{l_{j}}(\Lambda)=\varepsilon_{k}(\Lambda).$ (167)

Let us denote by

$P^{k}_{GZ}(\lambda;\mathbb{Z}):=\\{\Lambda\in
P_{GZ}(\lambda;\mathbb{Z})~{}|~{}\Lambda+\delta^{(k)}_{l_{1}}\in
P_{GZ}(\lambda;\mathbb{Z})\ and\ \Lambda+\delta^{(k)}_{l_{i}}\notin
P_{GZ}(\lambda;\mathbb{Z})\ for\ i=2,...,m_{k}\\}.$

In particular, all $\Lambda$ satisfying $m_{k}(\Lambda)=1$ and
$\Lambda+\delta^{(k)}_{l_{i}}\notin P_{GZ}(\lambda;\mathbb{Z})$ are in
$P^{k}_{GZ}(\lambda;\mathbb{Z})$. From (160)–(167), the function
$m_{k}(\Lambda)$ is defined for any point $\Lambda$ in the real polytope
$P_{GZ}(\lambda;\mathbb{R})$. Thus
$P^{k}_{GZ}(\lambda;\mathbb{R}):=P^{k}_{GZ}(\lambda;\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{R}$
is an open dense part of $P_{GZ}(\lambda;\mathbb{R})$, whose complements are
cut out by various equalities between $\lambda^{(j)}_{i}(\Lambda)$ for
$j=k-1,k,k+1$. By this reason, the elements in the subset
$P^{k}_{GZ}(\lambda;\mathbb{Z})$ are called generic (it gives precise meaning
of the word generic in Theorem 1.16).

Let $\xi_{\Lambda}$ be the Gelfand-Tsetlin orthonormal basis given in Section
5.3. Then we have

######  Proposition 6.1.

For $k=1,...,n-1$ and for any $\Lambda\in P^{k}_{GZ}(\lambda;\mathbb{Z})$,
there exists a real number $\theta_{k}\in[0,2\pi)$ independent of $q=e^{h/2}$
such that

$\displaystyle s_{k,k+1}^{(+)}\cdot\xi_{\Lambda}\sim
q^{-\varepsilon_{k}\big{(}\Lambda+\delta^{(k)}_{l_{1}(\Lambda)}\big{)}+\iota\theta_{k}}\xi_{\Lambda+\delta^{(k)}_{l_{1}(\Lambda)}},\hskip
8.53581ptas\ \ q\rightarrow 0.$ (168)

Here
$\varepsilon_{k}\big{(}\Lambda+\delta^{(k)}_{l_{1}(\Lambda)}\big{)}=\varepsilon_{k}(\Lambda)+1.$

Proof. By the asymptotics of gamma function

${\rm In}(\Gamma(1+z))\sim z{\rm In}(z)-z+\frac{1}{2}{\rm
In}(z)+\frac{1}{2}{\rm In}(2\pi)+\frac{1}{12z}+o(\frac{1}{z}),\ \ as\
|z|\rightarrow\infty\ at\ constant\ |{\rm arg}(z)|<\pi,$

For $r$ a real number and $h\rightarrow-\infty$, we have

$\displaystyle{\rm
In}(\Gamma(1+\frac{rh}{2\pi\iota}))\sim\left\\{\begin{array}[]{lr}\frac{rh{\rm
In}(-h)}{2\pi\iota}+\frac{rh}{2\pi\iota}{\rm
In}(\frac{r}{2\pi})+\frac{rh}{4}-\frac{rh}{2\pi\iota}+\frac{1}{2}{\rm
In}(\frac{rh}{\iota}),&r>0,\\\ \frac{rh{\rm
In}(-h)}{2\pi\iota}+\frac{rh}{2\pi\iota}{\rm
In}(\frac{-r}{2\pi})-\frac{rh}{4}-\frac{rh}{2\pi\iota}+\frac{1}{2}{\rm
In}(\frac{rh}{\iota}),&r<0.\end{array}\right.$ (171)

Here we use ${\rm In}(\frac{\pm 1}{\iota})=\mp\frac{\pi\iota}{2}$ to separate
the real and imaginary part of the $h$ linear terms. By (138),

$\zeta^{(k)}_{i}\cdot\xi_{\Lambda}=(\lambda^{(k)}_{i}(\Lambda)-i+1+(k-1)/2)\cdot\xi_{\Lambda},$

by abuse of notation, we will take $\zeta^{(k)}_{i}$ as the number
$\lambda^{(k)}_{i}(\Lambda)-i+1+(k-1)/2$ when the vector $\xi_{\Lambda}$ is
specified. Then by (171) and the interlacing inequalities between
$\lambda^{(i)}_{j}$ for $i=k-1,k,k+1$,

$\displaystyle{\rm In}\Big{(}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2}}{2\pi\iota})}\Big{)}$
$\displaystyle\sim$ $\displaystyle\frac{h{\rm
In}(-h)}{2\pi\iota}A^{(k)}_{i}+\iota
h\theta^{(k)}_{i}+\frac{h}{4}B^{(k)}_{i}+C^{(k)}_{i},\hskip 14.22636ptas\
h\rightarrow-\infty,$

where

$\displaystyle
A^{(k)}_{i}=\sum_{l=1}^{k-1}\zeta^{(k-1)}_{l}+\sum_{l=1}^{k+1}\zeta^{(k+1)}_{l}-2\sum_{l=1}^{k}\zeta^{(k)}_{l}+1,$
$\displaystyle
B^{(k)}_{i}=-2\sum_{l=1}^{i-1}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})+2\sum_{l=i+1}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})+\sum_{l=1}^{i-1}(\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l})$
$\displaystyle\hskip
34.1433pt-\sum_{l=i}^{k-1}(\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l})+\sum_{l=1}^{i}(\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l})-\sum_{l=i+1}^{k+1}(\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}),$
$\displaystyle C^{(k)}_{i}=\frac{1}{2}{\rm
In}\Big{(}-\frac{1}{h^{2}}\frac{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l})}{\prod_{l=1}^{k+1}(\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2})}\frac{\prod_{l=1,l\neq
i}^{k}(\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1)}{\prod_{l=1}^{k-1}(\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2})}\Big{)},$

and

$\displaystyle\theta^{(k)}_{i}=\frac{h}{2\pi}A^{(k)}_{i}-\sum_{l=1}^{i-1}\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi}{\rm
In}(\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi})+\sum_{l=i+1}^{k}(\frac{\zeta^{(k)}_{l}-\zeta^{(k)}_{i}}{2\pi}){\rm
In}(\frac{\zeta^{(k)}_{l}-\zeta^{(k)}_{i}}{2\pi})$ $\displaystyle\hskip
34.1433pt-\sum_{l=1}^{i-1}\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi}{\rm
In}(\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi})+\sum_{l=i+1}^{k}(\frac{\zeta^{(k)}_{l}-\zeta^{(k)}_{i}-1}{2\pi}){\rm
In}(\frac{\zeta^{(k)}_{l}-\zeta^{(k)}_{i}-1}{2\pi})$ $\displaystyle\hskip
34.1433pt+\sum_{l=1}^{i-1}(\frac{\zeta^{(k-1)}_{l}-\zeta^{(k)}_{i}-\frac{1}{2}}{2\pi}){\rm
In}(\frac{\zeta^{(k-1)}_{l}-\zeta^{(k)}_{i}-\frac{1}{2}}{2\pi})-\sum_{l=i}^{k-1}(\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2}}{2\pi}){\rm
In}(\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2}}{2\pi})$
$\displaystyle\hskip
34.1433pt+\sum_{l=1}^{i}(\frac{\zeta^{(k+1)}_{l}-\zeta^{(k)}_{i}-\frac{1}{2}}{2\pi}){\rm
In}(\frac{\zeta^{(k+1)}_{l}-\zeta^{(k)}_{i}-\frac{1}{2}}{2\pi})-\sum_{l=i+1}^{k+1}(\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi}){\rm
In}(\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi}).$

Finally, the action of $s^{(+)}_{k,k+1}$ on $\xi_{\Lambda}$ is given by

$\displaystyle(\iota
h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1)}}{2\pi\iota}}\sum_{i=1}^{k}\Big{(}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}}{2\pi\iota})}{\prod_{l=1}^{k+1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k+1)}_{l}-\frac{1}{2}}{2\pi\iota})}\frac{\prod_{l=1,l\neq
i}^{k}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k)}_{l}-1}{2\pi\iota})}{\prod_{l=1}^{k-1}\Gamma(1+h\frac{\zeta^{(k)}_{i}-\zeta^{(k-1)}_{l}-\frac{1}{2}}{2\pi\iota})}\Big{)}\cdot(-ha^{(k)}_{i})\cdot\xi_{\Lambda}$
$\displaystyle\sim$
$\displaystyle(-h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1)}}{2\pi\iota}}(-\iota)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1)}}{2\pi\iota}}\sum_{i=1}^{k}e^{\frac{h{\rm
In}(-h)}{2\pi\iota}A^{(k)}_{i}}e^{\iota
h\theta^{(k)}_{i}+\frac{h}{4}B^{(k)}_{i}}e^{C^{(k)}_{i}}(-h\alpha^{(k)}_{i})\cdot\xi_{\Lambda}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}\Big{(}(-h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1)}}{2\pi\iota}}e^{\frac{h{\rm
In}(-h)}{2\pi\iota}A^{(k)}_{i}}\Big{)}\Big{(}e^{\frac{h\small{(E_{k+1,k+1}-E_{kk}+1)}}{4}}e^{\iota
h\theta^{(k)}_{i}+\frac{h}{4}B^{(k)}_{i}}\Big{)}\Big{(}-he^{C^{(k)}_{i}}\alpha^{(k)}_{i}\cdot\xi_{\Lambda}\Big{)}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}q^{-X^{(k)}_{i}(\Lambda+\delta^{(k)}_{i})+2\iota
h\theta^{(k)}_{i}}\cdot\xi_{\Lambda+\delta^{(k)}_{i}}.$

Here in the last equality, we use the identities (137), (138), (160) and (162)
to get

$\displaystyle(-h)^{\frac{h\small{(E_{kk}-E_{k+1,k+1}-1)}}{2\pi\iota}}e^{\frac{h{\rm
In}(-h)}{2\pi\iota}A^{(k)}_{i}}$ $\displaystyle=$ $\displaystyle 1,$
$\displaystyle
e^{\frac{h\small{(E_{k+1,k+1}-E_{kk}+1)}}{4}}e^{\frac{h}{4}B^{(k)}_{i}+\iota
h\theta^{(k)}_{i}}\cdot\xi_{\Lambda+\delta^{(k)}_{i}}$ $\displaystyle=$
$\displaystyle q^{-X^{(k)}_{i}(\Lambda+\delta^{(k)}_{i})+2\iota
h\theta^{(k)}_{i}}\xi_{\Lambda+\delta^{(k)}_{i}},$

and use the expression (139) of $\alpha^{(k)}_{i}\cdot\xi_{\Lambda}$ and the
expression (6.1) of $C^{(k)}_{i}$ to get

$-he^{C^{(k)}_{i}}\alpha^{(k)}_{i}\cdot\xi_{\Lambda}=\xi_{\Lambda+\delta^{(k)}_{i}}.$

Then the proposition follows from the definitions (165) and (167) of
$\varepsilon_{k}(\Lambda)$ and $l_{i}(\Lambda)$ with $i=1,...,m_{k}$, and the
assumption $m_{k}(\Lambda)=1.$ The constant $\theta_{k}$ in (168) is just
$2\theta^{(k)}_{l_{1}}$.

###  6.2 Gelfand-Tsetlin basis form WKB approximation

It particularly implies the following construction of the set $E(\lambda,u)$
from Stokes matrices themselves.

For any integers $i,j$ with $1\leq n-i\leq j\leq n-1$, we introduce the
monomial $Q_{ij}(u_{\rm cat})=s_{j,j+1}^{(+)}s_{j-1,j}^{(+)}\cdots
s_{n-i,n-i+1}^{(+)}$ of the entries of quantum Stokes matrix at $u_{\rm cat}$.
For any tuple of nonnegative integers $\underline{d}=(d^{(k)}_{j})_{1\leq
j\leq k\leq n-1}$, we define the operator

$\displaystyle Q(u_{\rm cat})^{\underline{d}}=\overrightarrow{\prod_{i+j\geq
n}}(Q_{ij})^{d^{(i)}_{j}}\in{\rm End}(L(\lambda)),$ (172)

where the product is taken with $Q_{ij}$ to the right of
$Q_{i^{\prime}j^{\prime}}$ if $j>j^{\prime}$; or $j=j^{\prime}$ and
$i>i^{\prime}$. Let $\xi_{\Lambda_{0}}$ be the lowest vector in $L(\lambda)$,
which corresponds to the pattern $\Lambda_{0}$ satisfying
$\lambda^{(k)}_{i}(\Lambda_{0})=\lambda^{(k+1)}_{i+1}(\Lambda_{0})$ for all
$i,k$. Following the definition of $Q^{\underline{d}}$ and the proof of
Proposition 6.1, we have

######  Proposition 6.2.

For any $\underline{d}$, there exists a constant $c(\underline{d})$
independent of $q$ such that

$q^{c(\underline{d})}Q(u_{\rm
cat})^{\underline{d}}\cdot\xi_{\Lambda_{0}}\rightarrow\xi_{\Lambda},\hskip
5.69054ptas\hskip 5.69054ptq\rightarrow 0,$

where $\Lambda\in P_{GZ}(\lambda;\mathbb{Z})$ is uniquely determined by the
linear equations

$\displaystyle\lambda^{(i)}_{j}(\Lambda)-\lambda^{(i+1)}_{j+1}(\Lambda)=d^{(i)}_{j},\hskip
8.53581ptfor\ all\ 1\leq j\leq i\leq n-1.$ (173)

Proof. The ordering in the definition of $Q^{\underline{d}}$, together with
the formula (139) and the expressions in Theorem 5.4, enforces

$Q(u_{\rm cat})^{\underline{d}}\cdot\xi_{\Lambda_{0}}=f(q)\xi_{\Lambda},$

for certain function of $q$, where $\Lambda$ is determined by (173). Take
$Q^{\underline{d}}=s_{34}(s_{34}s_{23})(s_{34}s_{23}s_{12})^{2}$ as an
example, by (5.4) and (139) we have

$\displaystyle
s_{12}\cdot\xi_{\Lambda_{0}}=f_{1}\xi_{\Lambda_{0}+\delta^{(1)}_{1}},\hskip
8.53581pts_{23}s_{12}\cdot\xi_{\Lambda_{0}}=f_{2}\xi_{\Lambda_{0}+\delta^{(1)}_{1}+\delta^{(2)}_{2}},\hskip
8.53581pts_{34}s_{23}s_{12}\cdot\xi_{\Lambda_{0}}=f_{3}\xi_{\Lambda_{0}+\delta^{(1)}_{1}+\delta^{(2)}_{2}+\delta^{(3)}_{3}},$
$\displaystyle
s_{12}s_{34}s_{23}s_{12}\cdot\xi_{\Lambda_{0}}=f_{4}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+\delta^{(2)}_{2}+\delta^{(3)}_{3}},\hskip
14.22636pts_{23}s_{12}s_{34}s_{23}s_{12}\cdot\xi_{\Lambda_{0}}=f_{5}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+2\delta^{(2)}_{2}+\delta^{(3)}_{3}},$
$\displaystyle(s_{34}s_{23}s_{12})^{2}\cdot\xi_{\Lambda_{0}}=f_{6}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+2\delta^{(2)}_{2}+2\delta^{(3)}_{3}},\hskip
14.22636pts_{23}(s_{34}s_{23}s_{12})^{2}\cdot\xi_{\Lambda_{0}}=f_{7}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+3\delta^{(2)}_{2}+2\delta^{(3)}_{3}},$
$\displaystyle
s_{34}s_{23}(s_{34}s_{23}s_{12})^{2}\cdot\xi_{\Lambda_{0}}=f_{8}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+3\delta^{(2)}_{2}+3\delta^{(3)}_{3}},\hskip
5.69054pts_{34}s_{34}s_{23}(s_{34}s_{23}s_{12})^{2}\cdot\xi_{\Lambda_{0}}=f_{9}\xi_{\Lambda_{0}+2\delta^{(1)}_{1}+3\delta^{(2)}_{2}+4\delta^{(3)}_{3}},$

for certain coefficients $\\{f_{i}(q)\\}_{i=1,...,9}$. Thus $Q(u_{\rm
cat})^{\underline{d}}\cdot\xi_{\Lambda_{0}}$ corresponds to a chain of vectors
$f_{i}(q)v_{i}$ as shown in the example, and the ordering in the product
$Q^{\underline{d}}$ guarantees that in each immediate step the assumption in
Proposition 6.1 is satisfied. By Proposition 6.1, for the $i$-th vector in the
chain, there exists a constant $c_{i}$ such that
$q^{c_{i}}\frac{f_{i}(q)}{f_{i-1}(q)}\rightarrow 1$ as $q\rightarrow 0$. Here
we assume $f_{0}(q)=1.$ Since $f(q)=\prod_{i}\frac{f_{i}(q)}{f_{i-1}(q)}$, it
follows from that there exists constants a $c$ such that $q^{c}f(q)\rightarrow
1$ as $q\rightarrow 0$. It proves the proposition.

######  Corollary 6.3.

The set $B(\lambda,u_{\rm cat})$ of the WKB approximation of all nonzero
vectors $Q(u_{\rm cat})^{\underline{d}}\cdot\xi_{0}$ is a base of
$L(\lambda)$, with a natural parameterization: the element corresponding to
$Q(u_{\rm cat})^{\underline{d}}\cdot\xi_{0}$ is labelled by the integer point
$\Lambda\in P_{GZ}(\lambda;\mathbb{Z})$ satisfying (173). Actually
$B(\lambda,u_{\rm cat})$ is isomorphic to the set of Gelfand-Tsetlin basis.

###  6.3 WKB operators

In this subsection, we introduce a combinatorial structure to encode the WKB
leading terms (of entries) of quantum Stokes matrices at a caterpillar point.

As we have seen from the above sections, the $q$ leading term of the action of
monomials $Q^{d}$ for all possible $d$ on the lowest vector $\xi_{0}$ produces
a basis $B(\lambda,u_{\rm cat})$ of $L(\lambda)$, parameterized by
$P_{GZ}(\lambda;\mathbb{Z})$. It coincides with the Gelfand-Tsetlin basis. If
we denote by $B^{k}(\lambda,u_{\rm cat})\subset B(\lambda,u_{\rm cat})$ the
subset consisting of the elements parametrized by
$P_{GZ}^{k}(\lambda;\mathbb{Z})\subset P_{GZ}(\lambda;\mathbb{Z})$, then the
formula (168) shows that taking the $q$ leading term of $s_{k,k+1}^{(+)}$
naturally induces a map

$\displaystyle\tilde{e}_{k}:\ B^{k}(\lambda,u_{\rm cat})\rightarrow
B(\lambda,u_{\rm
cat})~{};~{}\tilde{e}_{k}(\xi_{\Lambda})=\xi_{\Lambda+\delta^{(k)}_{l_{1}}}.$

Equivalently, as we identify $B(\lambda,u_{\rm cat})$ with
$P_{GZ}(\lambda;\mathbb{Z})$ by mapping $\xi_{\Lambda}$ to $\Lambda$,
$\tilde{e}_{k}$ can be seen as a map

$\displaystyle\tilde{e}_{k}:\ P_{GZ}^{k}(\lambda;\mathbb{Z})\rightarrow
P_{GZ}(\lambda;\mathbb{Z})~{};~{}\tilde{e}_{k}({\Lambda})={\Lambda+\delta^{(k)}_{l_{1}}}.$

The map has a canonical extension to the whole $P_{GZ}(\lambda;\mathbb{Z})$ as
follows.

First, the expression of $\tilde{e}_{k}$ is universal, i.e., doesn’t depend on
the choice of $\lambda$. Thus for any positive integer $N$, if we set
$N\lambda=(N\lambda^{(n)}_{1},...,N\lambda^{(n)}_{n})$, then $\tilde{e}_{k}$
can be equivalently seen as a map from
$P^{k}_{GZ}(\lambda;\frac{\mathbb{Z}}{N}):=\\{\Lambda\in
P^{k}_{GZ}(\lambda;\mathbb{R})~{}|~{}\lambda^{(i)}_{j}(\Lambda)\in\frac{\mathbb{Z}}{N}\\}$
to $P_{GZ}(\lambda;\frac{\mathbb{Z}}{N}):=\\{\Lambda\in
P_{GZ}(\lambda;\mathbb{R})~{}|~{}\lambda^{(i)}_{j}(\Lambda)\in\frac{\mathbb{Z}}{N}\\}$

$\displaystyle\tilde{e}_{k}^{\frac{1}{N}}(\Lambda):=\Lambda+\frac{1}{N}\delta^{(k)}_{l},$
(174)

where
$\lambda^{(i)}_{j}(\Lambda+\frac{1}{N}\delta^{(k)}_{l})=\lambda^{(i)}_{j}+\frac{\delta_{ik}\delta_{jl}}{N}$.
Now as $N\rightarrow\infty$, the discrete "dynamical system" (174) approaches
to a unique continuous system in the inner part of the whole real polytope
$P_{GZ}(\lambda;\mathbb{R})$:

$\displaystyle\tilde{e}_{k}^{t}(\Lambda):=\Lambda+t_{1}\delta^{(k)}_{r_{1}}+\cdots+t_{m}\delta^{(k)}_{r_{m}},$
(175)

where the time

$t\in\Big{[}0,\hskip 5.69054pt\sum_{i=1}^{k}({\rm
min}(\lambda^{(k+1)}_{i}(\Lambda),\lambda^{(k-1)}_{i-1}(\Lambda))-\lambda^{(k)}_{i}(\Lambda))\Big{)},$

and $t_{1},...,t_{m}$, $r_{1},...,r_{m}$ are determined by

$\displaystyle{r_{1}}=l_{1}(\Lambda+s\delta^{(k)}_{r_{1}}),\hskip
8.53581pt\text{for all}\ 0\leq s<t_{1},$ $\displaystyle
r_{i}=l_{1}(\Lambda+t_{1}\delta^{(k)}_{r_{1}}+\cdots
t_{i-1}\delta^{(k)}_{r_{i-1}}+s\delta^{(k)}_{r_{i}}),\hskip 8.53581pt\text{for
all}\ 0\leq s<t_{i},\ for\ i=2,...,m,$ $\displaystyle t=t_{1}+\cdots+t_{m}.$

Here the function $l_{1}(\Lambda)$ is defined for any point $\Lambda$ in
$P_{GZ}(\lambda;\mathbb{R})$ just as (160)–(167).

This continuation is unique, thus canonically determines an extension of
$\tilde{e}_{k}$ to the complements of the "generic part"
$P^{k}_{GZ}(\lambda;\mathbb{Z})$ in $P_{GZ}(\lambda;\mathbb{Z})$. It gives
rise to

######  Definition 6.4.

For each $k$, the WKB operator $\tilde{e}_{k}$ from $B(\lambda,u_{\rm
cat})\cong P_{GZ}(\lambda;\mathbb{Z})$ to $B(\lambda,u_{\rm cat})\cup\\{0\\}$
is given by

$\displaystyle\tilde{e}_{k}\cdot\xi_{\Lambda}:=\xi_{\Lambda+\delta^{(k)}_{l_{1}(\Lambda)}},\hskip
14.22636pt\forall\Lambda\in P_{GZ}(\lambda;\mathbb{Z})$ (176)

Here recall that $l_{1}(\Lambda)$ is the integer given by (167), and it is
supposed that $\tilde{e}_{k}\cdot\xi_{\Lambda}$ is zero if
$\Lambda+\delta^{(k)}_{l_{1}(\Lambda)}$ doesn’t belong to
$P_{GZ}(\lambda;{\mathbb{Z}})$.

The above computation and discussion carry to the WKB approximation of the
entries $s^{(-)}_{k+1,k}$ of lower triangular Stokes matrix $S_{\hbar-}(u_{\rm
cat})$. It produces the same set $B(\lambda,u_{\rm cat})$, and induces
operators $\tilde{f}_{k}$ on $B(\lambda,u_{\rm cat})$

$\displaystyle\tilde{f}_{k}\cdot\xi_{\Lambda}=\xi_{\Lambda-\delta^{(k)}_{l_{m_{k}}(\Lambda)}},\hskip
14.22636pt\forall\Lambda\in P_{GZ}(\lambda;\mathbb{Z}).$ (177)

One checks that $\tilde{e}_{k}$ and $\tilde{f}_{k}$ satisfy that for all
$\Lambda,\Lambda^{\prime}\in P_{GZ}(\lambda;{\mathbb{Z}})$,

$\displaystyle\varepsilon_{k}(\Lambda)={\rm
max}\\{j:\tilde{e}_{k}^{j}(\xi_{\Lambda})\neq 0\\},$
$\displaystyle\phi_{k}(\Lambda)={\rm
max}\\{j:\tilde{f}_{k}^{j}(\xi_{\Lambda})\neq 0\\},$

and

$\tilde{e}_{k}\cdot\xi_{\Lambda}=\xi_{\Lambda}^{\prime}\hskip 8.53581ptif\
and\ only\ if\hskip
8.53581pt\tilde{f}_{k}\cdot\xi_{\Lambda}^{\prime}=\xi_{\Lambda}.$

######  Definition 6.5.

We call $(B(\lambda,u_{\rm
cat}),\tilde{e}_{k},\tilde{f}_{k},\varepsilon_{k},\phi_{k})$ the WKB datum of
the Stokes matrices $S_{h\pm}(u_{\rm cat})$ at the caterpillar point
associated to the representation $L(\lambda)$.

###  6.4 WKB datum are crystals

In Section 6.4.1 we recall the notion of crystals and their tensor products.
In Section 6.4.2 we prove that the WKB datum of quantum Stokes matrices at
$u_{\rm cat}$ are $\mathfrak{gl}_{n}$-crystals. Furthermore, in Section 6.4.3
we also realize the tensor products of $\mathfrak{gl}_{n}$-crystals by WKB
analysis.

####  6.4.1 Crystals and tensor products

Let $\mathfrak{g}$ be a semisimple Lie algebra with a Cartan datum
$(A,\Delta_{+}=\\{\alpha_{i}\\}_{i\in
I},\Delta^{\vee}_{+}=\\{\alpha_{i}^{\vee}\\}_{i\in I},P,P^{\vee})$ be a Cartan
datum, where $P\subset\mathfrak{h}^{*}$ denotes the weight lattice, $I$
denotes the set of vertices of its Dynkin diagram, $\alpha_{i}\in I$ denote
its simple roots, and $\alpha^{\vee}_{i}$ the simple coroots.

######  Definition 6.6.

A $\mathfrak{g}$-crystal is a finite set $B$ along with maps

$\displaystyle wt$ $\displaystyle:$ $\displaystyle B\rightarrow P,$
$\displaystyle\tilde{e}_{i},\tilde{f}_{i}$ $\displaystyle:$ $\displaystyle
B\rightarrow B\cup\\{0\\},\hskip 5.69054pti\in I,$
$\displaystyle\varepsilon_{i},\phi_{i}$ $\displaystyle:$ $\displaystyle
B\rightarrow\mathbb{Z}\cup\\{-\infty\\},\hskip 5.69054pti\in I,$

satisfying for all $b,b^{\prime}\in B$, and $i\in I$,

  * •

$\tilde{f}_{i}(b)=b^{\prime}$ if and if $b=\tilde{e}_{i}(b^{\prime})$, in
which case

$wt(b^{\prime})=wt(b)-\alpha_{i},\hskip
8.53581pt\varepsilon_{i}(b^{\prime})=\varepsilon_{i}(b)+1,\hskip
8.53581pt\phi_{i}(b^{\prime})=\phi_{i}(b)-1.$

  * •

$\phi_{i}(b)=\varepsilon_{i}(b)+\langle wt(b),\alpha_{i}^{\vee}\rangle$, and
if $\phi_{i}(b)=\varepsilon_{i}(b)=-\infty$, then
$\tilde{e}_{i}(b)=\tilde{f}_{i}(b)=0.$

The map $wt$ is called the weight map, $\tilde{e}_{i}$ and $\tilde{f}_{i}$ are
called Kashiwara operators or crystal operators.

######  Definition 6.7.

Let $B_{1}$ and $B_{2}$ be two crystals. The tensor product $B_{1}\otimes
B_{2}$ is the crystal with the underlying set $B_{1}\times B_{2}$ (the
Cartesian product) and structure maps

$\displaystyle wt(b_{1},b_{2})$ $\displaystyle=$ $\displaystyle
wt(b_{1})+wt(b_{2}),$ $\displaystyle\tilde{e}_{i}(b_{1},b_{2})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lr}(\tilde{e}_{i}(b_{1}),b_{2}),&if\
\varepsilon_{i}(b_{1})>\phi(b_{2})\\\
(b_{1},\tilde{e}_{i}(b_{2})),&otherwise\end{array}\right.$
$\displaystyle\tilde{f}_{i}(b_{1},b_{2})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{lr}(\tilde{f}_{i}(b_{1}),b_{2}),&if\
\varepsilon_{i}(b_{1})\geq\phi(b_{2})\\\
(b_{1},\tilde{f}_{i}(b_{2})),&otherwise.\end{array}\right.$

####  6.4.2 ${\mathfrak{gl}}_{n}$-crystals from WKB approximation

Let us take the Cartan datum of type $A_{n-1}$, where $I=\\{1,2,...,n-1\\}$,
the weight lattice $P=\mathbb{C}\\{v_{1},...,v_{n}\\}$, and
$\alpha_{i}=v_{i}-v_{i+1}$, $\alpha_{i}^{\vee}$ is given by $\langle
v_{i},\alpha_{j}^{\vee}\rangle=\delta_{ij}-\delta_{i,j+1}.$ Let us consider
the WKB datum, given in Definition 6.6, of the quantum Stokes matrices at the
caterpillar point $u_{\rm cat}$ associated to the representation $L(\lambda)$.
Let us define a weight map

$wt:B(\lambda,u_{\rm cat})\rightarrow
P~{};~{}\xi_{\Lambda}\rightarrow\sum_{k=1}^{n}wt_{k}(\Lambda)v_{k}=\sum_{k=1}^{n}(\sum_{i=1}^{k}\lambda^{(k)}_{i}(\Lambda)-\sum_{i=1}^{k-1}\lambda^{(k-1)}_{i}(\Lambda))v_{k}.$

The following theorem implies Theorem 1.16 in the introduction.

######  Theorem 6.8.

The WKB datum $(B(\lambda,u_{\rm
cat}),\tilde{e}_{k},\tilde{f}_{k},\varepsilon_{k},\phi_{k})$ with the weight
map $wt$ is a ${\mathfrak{gl}}_{n}$-crystal.

Proof. By the formula (176) and (177), the WKB operators on the finite set
$B(\lambda,u_{\rm cat})$ are:

$\displaystyle\tilde{e}_{k}\cdot\xi_{\Lambda}=\left\\{\begin{array}[]{lr}\xi_{\Lambda+\delta^{(k)}_{l}},&if\
\ \varepsilon_{k}(\Lambda)>0,\\\ 0,&if\ \
\varepsilon_{k}(\Lambda)=0,\end{array}\right.where\ l={\rm
min}\\{j=1,...,k~{}|~{}X_{j}^{(k)}(\Lambda)=\varepsilon_{k}(\Lambda)\\},$
(182)
$\displaystyle\tilde{f}_{k}\cdot\xi_{\Lambda}=\left\\{\begin{array}[]{lr}\xi_{\Lambda-\delta^{(k)}_{l}},&if\
\ \phi_{k}(\Lambda)>0,\\\ 0,&if\ \
\phi_{k}(\Lambda)=0,\end{array}\right.where\ l={\rm
max}\\{j=1,...,k~{}|~{}Y_{j}^{(k)}(\Lambda)=\phi_{k}(\Lambda)\\},$ (185)

The explicit realization (182) and (185) of the WKB operators $\tilde{e}_{k}$,
$\tilde{f}_{k}$ coincides with the known crystal operators of
${\mathfrak{gl}}_{n}$-crystal realized on the Gelfand-Tsetlin basis, see e.g.,
[30].

####  6.4.3 Tensor products from WKB approximation

Recall that given any representation $L(\lambda)$, the quantum Stokes matrices
at $u_{\rm cat}$ produce operators $s_{ij}^{(\pm)}\in{\rm End}(L(\lambda)).$
Now given two representations $L(\lambda_{1})$ and $L(\lambda_{2})$, let us
consider the actions of

$\displaystyle s_{k,k}^{(+)}\otimes s_{k,k+1}^{(+)}+s_{k,k+1}^{(+)}\otimes
s_{k+1,k+1}^{(+)}\ \ and\ \ s_{k+1,k}^{(+)}\otimes
s_{k,k}^{(+)}+s_{k+1,k+1}^{(-)}\otimes s_{k+1,k}^{(-)},\ \ k=1,...,n-1,$ (186)

on the tensor product $L(\lambda_{1})\otimes L(\lambda_{2})$. Let
$\\{\xi_{\Lambda_{1}}\\}$ and $\\{\xi_{\Lambda_{2}}\\}$ be the basis of
$L(\lambda_{1})$ and $L(\lambda_{2})$ respectively. Similar to Proposition
6.1, one can compute the WKB leading term of the operators in (186) under the
basis $\xi_{\Lambda_{1}}\otimes\xi_{\Lambda_{2}}$. (Since the diagonal
elements $s_{k,k}^{(\pm)}$ of quantum Stokes matrices have a rather simple
expression, the computation is direct and is omitted here.) Then following the
same argument as previous sections, one verifies

######  Proposition 6.9.

The WKB approximation of the operators in (186) induces the crystal operators
$\tilde{e}_{k}$ and $\tilde{f}_{k}$ on the tensor product
$E(\lambda_{1},u_{\rm cat})\otimes E(\lambda_{2},u_{\rm cat})$ of
$\mathfrak{gl}_{n}-$crystals.

###  6.5 The structure of the WKB approximation

In a summary, the WKB approximation of quantum Stokes matrices at caterpillar
points naturally produces Gelfand-Tsetlin basis in any $L(\lambda)$: the WKB
approximation of monomials $Q^{d}$ of entries of Stokes matrices produces the
basis $B(\lambda,u_{\rm cat})$ of $L(\lambda)$. Then for any generic $v\in
B(\lambda,u_{\rm cat})$, the WKB approximation of the vector
$s_{k,k+1}^{(+)}\cdot v$ takes the form

$\displaystyle
q^{-\varepsilon_{k}(\tilde{e}_{k}(v))+\iota\theta_{k}(\tilde{e}_{k}(v))}\tilde{e}_{k}(v),$
(187)

where $\tilde{e}_{k}$ is the WKB/crystal operator on $B(\lambda,u_{\rm cat})$.
The expression (187) can be understood as a structure theorem of quantum
Stokes matrices at $u_{\rm cat}$ under the Gelfand-Tsetlin basis. In a next
paper, we would like to use the method of isomonodromy deformation to obtain
the same expression (187) for quantum Stokes matrices at general $u$, with
$B(\lambda,u_{\rm cat})$ replaced by the eigenbasis $B(\lambda,u)$ of the
shift of argument subalgebra $\mathcal{A}(u)$ of $U(\mathfrak{gl}_{n})$.

###  6.6 The RLL relations and semiclassical limits

This appendix explains the reason why the WKB approximation in differential
equations should be related to the theory of crystal bases in quantum groups.
It recalls the quantum and classical RLL relations satisfied by quantum and
classical Stokes matrices respectively.

In [52], we showed that $S_{h\pm}(u)=(s_{ij}^{(\pm)})$ satisfy the RLL
relation, i.e., are $L_{\pm}$-operators, in the Faddeev-Reshetikhin-Takhtajan
realization [46] of the quantum group $U_{q}(\mathfrak{gl}_{n})$ of
${\mathfrak{gl}}_{n}$. To be more precise, let us take the R-matrix $R\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(\mathbb{C}^{n})$ given by

$\displaystyle R=\sum_{i\neq j,i,j=1}^{n}E_{ii}\otimes
E_{jj}+e^{\frac{h}{2}}\sum_{i=1}^{n}E_{ii}\otimes
E_{ii}+(e^{\frac{h}{2}}-e^{-\frac{h}{2}})\sum_{1\leq j<i\leq n}E_{ij}\otimes
E_{ji}.$ (188)

Set $\mathcal{S}_{h+}(u):={S}_{h+}(u)$ and
$\mathcal{S}_{h-}(u):={S}_{h+}(u)^{-1}\in{\rm End}(\mathbb{C}^{n})\otimes{\rm
End}(L(\lambda))$. The following theorem can be found in [52, 53] in various
generalities.

######  Theorem 6.10.

For any $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$ and $L(\lambda)$, the quantum
Stokes matrices $\mathcal{S}_{h\pm}(u)$ satisfy

$\displaystyle
R^{12}\mathcal{S}^{13}_{h\pm}\mathcal{S}^{23}_{h\pm}=\mathcal{S}^{23}_{h\pm}\mathcal{S}^{13}_{h\pm}R^{12}\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(\mathbb{C}^{n})\otimes{\rm
End}(\lambda),$ (189) $\displaystyle
R^{12}\mathcal{S}^{13}_{h+}\mathcal{S}^{23}_{h-}=\mathcal{S}^{23}_{h+}\mathcal{S}^{13}_{h-}R^{12}\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(\mathbb{C}^{n})\otimes{\rm
End}(\lambda),$ (190)

and

$(\mathcal{S}_{h+})_{ii}\cdot(\mathcal{S}_{h-})_{ii}=(\mathcal{S}_{h-})_{ii}\cdot(\mathcal{S}_{h+})_{ii}=1,\
\ \ for\ i=1,...,n.$

Thus the Stokes matrices $S_{h\pm}(u_{\rm cat})$ can be understood as the
$L_{\pm}$ operators in the FRT realization of the quantum group
$U_{q}(\mathfrak{gl}_{n})$ [46] in a representation. The associated coproduct
takes the form

$\Delta(s_{ij}^{(\pm)})=\sum_{k=1}^{n}s_{ik}^{(\pm)}\otimes s_{kj}^{(\pm)},\
i,j=1,...,n.$

Thus the operators in (186) are just the coproduct of the sub-diagonal
elements of quantum Stokes matrices.

One checks that the "gauge transformation" appearing in (132) preserves the
RLL relations, that is if we write the term in (132) as
$\mathcal{G}_{h-}(u)^{-1}\mathcal{G}_{h+}(u)$ via the Gauss decomposition,
then $\mathcal{G}_{\pm}(u)$ satisfy the identities (189) and (190). Since
$S_{h\pm}(u_{\rm cat})$ are the limits of $\mathcal{G}_{h\pm}(u)$, we get

######  Corollary 6.11.

The quantum Stokes matrices $S_{h\pm}(u_{\rm cat})$ at $u_{\rm cat}$ satisfy
the RLL relations (189) and (190).

Similarly, one can show that the RLL relations are preserved under the
regularized limits of quantum Stokes matrices at a general boundary point in
$\widehat{\mathfrak{h}_{\rm reg}}(\mathbb{R})$. We refer to [51] for the
classical analog, where the classical RLL relation (191) is shown to be
preserved under the closure of (classical) Stokes matrices. We see that the
Stokes matrices at a caterpillar point share many properties with the generic
ones, thus we can first study these properties using the closed formula at
$u_{\rm cat}$, and then use the method of isomonodromy/isospectral deformation
to understand them at generic $u$.

Semicalssical limits. Let us consider the Lie algebra ${\mathfrak{u}}(n)$ of
the unitary group ${\rm U}(n)$, consisting of skew-Hermitian matrices, and
identify ${\rm Herm}(n)\cong{\mathfrak{u}}(n)^{*}$ via the pairing $\langle
A,\xi\rangle=2{\rm Im}({\rm tr}A\xi)$. Thus ${\rm Herm}(n)$ inherits a Poisson
structure from the canonical linear (Kostant-Kirillov-Souriau) Poisson
structure on ${\mathfrak{u}}(n)^{*}$.

For any fixed $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$, by varying $A\in{\rm
Herm}(n)$, we can think of the Stokes matrices $S_{\pm}(u;A)$ of equation (11)
as matrix valued functions $S_{\pm}(u)$ on ${\rm
Herm}(n)\cong\mathfrak{u}(n)^{*}$. Let us introduce the function
$M(u)={S}_{-}(u)^{-1}{S}_{+}(u)$, and let $r_{\pm}\in{\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(\mathbb{C}^{n})$ be the standard
classical r-matrices, that is the semiclassical limit of the $R$-matrix in
(188), see e.g., [4, Formula (212) and (213)]. Then a reformulation of
Boalch’s remarkable theorem states that

######  Theorem 6.12.

[10] For any fixed $u\in\mathfrak{h}_{\rm reg}(\mathbb{R})$, the matrix valued
function $M(u)$ satisfies

$\displaystyle\\{M^{1},M^{2}\\}=r_{+}M^{1}M^{2}+M^{1}M^{2}r_{+}-M^{1}r_{+}M^{2}-M^{2}r_{-}M^{1}.$
(191)

Here the tensor notation is used: both sides are ${\rm
End}(\mathbb{C}^{n})\otimes{\rm End}(\mathbb{C}^{n})$ valued functions on
${\rm Herm}(n)$, $M^{1}:=M\otimes 1$, $M^{2}:=1\otimes M$, and the $ij,kl$
coefficient of the matrix $\\{M^{1},M^{2}\\}$ is defined as the Poisson
bracket $\\{M_{ij},M_{kl}\\}$ of the functions on ${\rm
Herm}(n)\cong\mathfrak{u}(n)^{*}$.

The right hand side of (191) coincides with the classical RLL formulation of
the Poisson brackets on the standard dual Poisson Lie group (see [4, Formula
(235)]). In particular, the relation (191) can be seen as a semiclassical
limit of the relations (189)-(190). This is why the Stokes matrices of the
equation (130) are called quantum.

##  7 Applications in Poisson geometry

In this section, we prove Theorem 1.15 and Theorem 1.11, and give a new proof
of Theorem 1.10. Section 7.1 shows that the Riemann-Hilbert-Birkhoff maps at
caterpillar points intertwine Gelfand-Tsetlin systems and their multiplicative
analogs. Then Section 7.2 uses the phase transformation of Gelfand-Tsetlin
systems to derive the explicit formula of the map $\Gamma_{AM}$. In the end,
Section 7.3 shows the Poisson geometric nature of taking closure of Stokes
matrices, and provides a new proof to Theorem 1.10.

###  7.1 Riemann-Hilbert-Birkhoff maps at caterpillar points are compatible
with Gelfand-Tsetlin systems

Recall that in Section 3.5.1, we have introduced the Gelfand-Tsetlin maps and
Thimm torus actions. In this subsection, we first introduce their
multiplicative analogs [22] on ${\rm Herm}^{+}(n)$, and then prove that the
map $\nu(u_{\rm cat})$ is compatible with them.

Logarithmic Gelfand-Tsetlin maps. Let ${\rm Herm}^{+}(n)\subset{\rm Herm}(n)$
denote the subset of positive definite Hermitian matrices, and define a
logarithmic Gelfand-Tsetlin map

$\mu\colon{\rm Herm}^{+}(n)\to\mathbb{R}^{\frac{n(n+1)}{2}},$ (192)

taking $A$ to the collection of numbers
$\mu^{(k)}_{i}(A)=\log(\lambda^{(k)}_{i}(A))$. Here recall that
$\lambda^{(k)}_{i}(A)^{\prime}s$ are the ordered set of eigenvalues of the
upper left $k$-th submatrix $A^{(k)}$ of $A\in{\rm Herm}(n)$. Then $\mu$ is a
continuous map from ${\rm Herm}^{+}(n)$ onto the Gelfand-Tsetlin cone
$\mathcal{C}(n)$.

Thimm torus actions. Let $\mathcal{C}_{0}(n)\subset\mathcal{C}(n)$ denote the
subset where all of the eigenvalue inequalities (68) are strict. Let ${\rm
Herm}_{0}^{+}(n)$ denote the intersection of ${\rm Herm}_{0}(n)$ and ${\rm
Herm}^{+}(n)$, i.e., ${\rm Herm}_{0}^{+}(n)=\mu^{-1}(\mathcal{C}_{0}(n))$.
Then the actions of Thimm torus on ${\rm Herm}_{0}(n)$ defined in (69)
restrict to a torus action on ${\rm Herm}_{0}^{+}(n)$. The action preserves
the logarithmic Gelfand-Tsetlin map $\mu$.

Recall from Definition 3.27 that the Riemann-Hilbert-Birkhoff map at $u_{\rm
cat}$ is

$\nu(u_{\rm cat})\colon{\rm Herm}(n)\rightarrow{\rm Herm}^{+}(n);\ A\mapsto
S_{-}(u_{\rm cat},A)S_{+}(u_{\rm cat},A).$

######  Proposition 7.1.

[51] The dual exponential map

$\nu(u_{\rm cat}):{\rm Herm}(n)\rightarrow{\rm Herm}^{+}(n)$

is a diffeomorphism compatible with the Gelfand-Tsetlin systems. That is

  * (a).

$\nu(u_{\rm cat})$ intertwines the Gelfand-Tsetlin maps: $\mu\circ\nu(u_{\rm
cat})=\lambda$;

  * (b).

$\nu(u_{\rm cat})$ intertwines the Thimm’s torus actions on ${\rm
Herm}_{0}(n)$ and ${\rm Herm}_{0}^{+}(n)$.

The proof of the proposition relies on the following linear algebra fact.

######  Lemma 7.2.

For each $0<k\leq n$, let $\Psi^{(k)}:{\rm Herm}(k)\rightarrow{\rm U}(k)$ be a
smooth map satisfying the conditions

  1. (1).

$\Psi^{(k)}$ is a ${\rm U}(k-1)$-equivariant map, i.e.,
$\Psi^{(k)}(gAg^{-1})={\rm Ad}_{g}\Psi^{(k)}(A)$, for any $g\in{\rm
U}(k-1)\subset U(k)$;

  2. (2).

for any $A\in{\rm Herm}(k)$, there is a block decomposition of
$\Psi^{(k)}(A)e^{A}\Psi^{(k)}(A)^{-1}$ taking the form

$\Psi^{(k)}(A)e^{A}\Psi^{(k)}(A)^{-1}=\left(\begin{array}[]{cc}{\rm
Id_{k-1}}&0\\\
{B^{(k)}}^{\dagger}&1\end{array}\right)\left(\begin{array}[]{cc}e^{A^{(k-1)}}&0\\\
0&\star\end{array}\right)\left(\begin{array}[]{cc}{\rm Id_{k-1}}&B^{(k)}\\\
0&1\end{array}\right),$

where $B^{(k)}$ is a column with $k-1$ elements, and ${B_{+}^{(k)}}^{\dagger}$
the conjugate transpose.

Let us extend $\Psi^{(k)}:{\rm Herm}(k)\rightarrow{\rm U}(k)$ as a map from
${\rm Herm}(n)$ to ${\rm U}(n)$, using the projection of ${\rm Herm}(n)$ onto
${\rm Herm}(k)$ and the natural inclusion ${\rm U}(k)\subset{\rm U}(n)$. Let
$\Psi:=\Psi^{(1)}\cdot\cdot\cdot\Psi^{(n)}$ be the map from ${\rm Herm}(n)$ to
${\rm U}(n)$ given by the pointwise multiplication, then the map

$\Gamma_{\Psi}:={\rm Ad}_{\Psi}\circ{\rm exp}\colon{\rm Herm}(n)\to{\rm
Herm}^{+}(n);\ A\mapsto\Psi(A)e^{A}\Psi(A)^{-1}$

is a diffeomorphism compatible with the Gelfand-Tsetlin systems.

Proof. We will prove this theorem inductively on $n$. When $n=1$, the result
is obvious. For the inductive step $n>1$, we first assume that the map
$\Psi_{n-1}:=\Psi^{(1)}\cdot\cdot\cdot\Psi^{(n-1)}\colon{\rm
Herm}(n-1)\rightarrow{\rm Herm}^{+}(n-1)$ is such that ${\rm
Ad}_{\Psi_{n-1}}\circ{\rm exp}$ satisfies the conditions $(a)$ and $(b)$
(provided replacing $\nu(u_{\rm cat})$ by $\Gamma_{\Psi}$).

Now we try to prove that the map ${\rm Ad}_{\Psi_{n-1}\Psi^{(n)}}\circ{\rm
exp}$ also satisfy $(a)$ and $(b)$. First, using the identity in the
assumption $(2)$

$\Psi^{(n)}(A)e^{A}\Psi^{(n)}(A)^{-1}=\left(\begin{array}[]{cc}{\rm
Id}_{n-1}&0\\\
{B^{(n)}}^{\dagger}&1\end{array}\right)\left(\begin{array}[]{cc}e^{A^{(n-1)}}&0\\\
0&\star\end{array}\right)\left(\begin{array}[]{cc}{\rm Id}_{n-1}&B^{(n)}\\\
0&1\end{array}\right),$

we have

$\displaystyle{\rm
Ad}_{\Psi_{n-1}(A)\Psi^{(n)}(A)}e^{A}=\left(\begin{array}[]{cc}\Psi_{n-1}e^{A^{(n-1)}}\Psi_{n-1}^{-1}&\Psi_{n-1}e^{A^{(n-1)}}B^{(n)}\\\
{B^{(n)}}^{\dagger}e^{A^{(n-1)}}\Psi_{n-1}^{-1}&\star\end{array}\right).$

Hence by the assumption about the map ${\rm Ad}_{\Psi_{n-1}}\circ{\rm exp}$,
we observe that ${\rm Ad}_{\Psi_{n-1}\Psi^{(n)}}\circ{\rm exp}$ intertwines
the Gelfand-Tsetlin maps.

For the Thimm’s torus action, by the assumption for $n-1$, the map ${\rm
Ad}_{\Psi_{n-1}}\circ{\rm exp}$ intertwines the $T(k)\subset U(k)$ actions on
${\rm Herm}_{0}(n-1)$ and ${\rm Herm}^{+}_{0}(n-1)$ for $1\leq k<n-1$. That is

$\displaystyle\Psi_{n-1}(t\bullet A)e^{t\bullet A}\Psi_{n-1}(t\bullet
A)^{-1}=t\bullet\left(\Psi_{n-1}(A)e^{A}\Psi_{n-1}(A)^{-1}\right).$ (194)

Here recall that the $k$-torus $T(k)\subset{\rm U}(k)\subset U(n)$ acts on
${\rm Herm}_{0}(n)$ as in (69). Furthermore, using the assumption $(1)$ on the
equivariance of $\Psi^{(n)}$, we have

${\rm Ad}_{\Psi_{n-1}(A)\Psi^{(n)}(A)}e^{A}={\rm
Ad}_{\Psi^{(n)}(\Psi_{n-1}(A)A\Psi_{n-1}(A)^{-1})}e^{\Psi_{n-1}(A)A\Psi_{n-1}(A)^{-1}}.$

Together with (194) and the definition of the torus action, we obtain

$\displaystyle\Gamma_{\Psi}(t\bullet A)$ $\displaystyle=$ $\displaystyle{\rm
Ad}_{\Psi_{n-1}(t\bullet A)\Psi^{(n)}(t\bullet A)}e^{t\bullet A}$
$\displaystyle=$ $\displaystyle{\rm
Ad}_{\Psi^{(n)}(t\bullet(\Psi_{n-1}(A)A\Psi_{n-1}(A)^{-1}))}e^{t\bullet(\Psi_{n-1}(A)A\Psi_{n-1}(A)^{-1})}$
$\displaystyle=$ $\displaystyle t\bullet\left({\rm
Ad}_{\Psi^{(n)}(\Psi_{n-1}(A)A\Psi_{n-1}(A)^{-1})}e^{\Psi_{n-1}(A)A\Psi_{n-1}(A)^{-1}}\right)$
$\displaystyle=$ $\displaystyle t\bullet\Gamma_{\Psi}(A),$

for any $t\in T(k)$ and $1\leq k\leq n$. It finishes the proof.

Proof of Proposition 7.1. By the definition of $\nu(u_{\rm cat})$, we only
need to prove that the connection map

$C^{(k)}:{\rm Herm}(k)\rightarrow{\rm U}(k)~{};~{}B\mapsto C(E_{k},B)$

of the $k\times k$ system

$\displaystyle\frac{dF}{dz}=\Big{(}\iota
E_{k}-\frac{1}{2\pi\iota}\frac{B}{z}\Big{)}F,$ (195)

(or equivalently the system (65)) satisfies the assumption $(1)$ and $(2)$ in
Lemma 7.2. To see that, let $S^{(k)}_{\pm}(B)\in{\rm GL}(k)$ denote the two
Stokes matrices of (65). By Definition 2.7 and the asymptotics of the
canonical solutions $F_{\pm}$ of (195), we have (here to derive the second
formula, the change in choice of ${\rm log}(z)$ is accounted for)

$\displaystyle F_{+}(z)F_{-}^{-1}(z)=e^{\iota
E_{k}z}z^{-\frac{\delta_{k-1}(B)}{2\pi\iota}}e^{\frac{-\delta_{k-1}(B)}{2}}S^{(k)}_{+}z^{\frac{\delta_{k-1}(B)}{2\pi\iota}}e^{-\iota
E_{k}z}\sim{\rm Id}_{k},\ \ \ as\ z\rightarrow\infty\ \ in\ the\ third\
quadrant,$ $\displaystyle F_{-}(ze^{-2\pi\iota})F_{+}^{-1}(z)=e^{\iota
E_{k}z}z^{-\frac{\delta_{k-1}(B)}{2\pi\iota}}S^{(k)}_{-}e^{\frac{-\delta_{k-1}(B)}{2}}z^{\frac{\delta_{k-1}(B)}{2\pi\iota}}e^{-\iota
E_{k}z}\sim{\rm Id}_{k},\ \ \ as\ z\rightarrow\infty\ \ in\ the\ first\
quadrant,$

where $\delta_{k-1}(B)$ is the projection of $B$ to the centralizer of $E_{k}$
in ${{\rm Herm}}(k)$. It follows that the Stokes matrices take the form

$\displaystyle
S^{(k)}_{-}(B)=\left(\begin{array}[]{cc}e^{\frac{B^{(k-1)}}{2}}&0\\\
{b_{-}^{(k)}}&\ast\end{array}\right),\ \ \
S^{(k)}_{+}(B)=\left(\begin{array}[]{cc}e^{\frac{B^{(k-1)}}{2}}&b^{(k)}\\\
0&\ast\end{array}\right),$ (200)

where $b^{(k)}$ is a column vector. Furthermore, the monodromy relation (26)
gives rise to

$\displaystyle
C^{(k)}(B)e^{B}C^{(k)}(B)^{-1}=S_{-}^{(k)}(B)S_{+}^{(k)}(B),\hskip
8.53581ptfor\ B\in{\rm Herm}(k).$ (201)

It follows from (200) and (201) that the connection map $C^{(k)}:{\rm