# Automorphisms of the Quantum Cohomology of the Springer Resolution and
Applications

Changzheng Li  School of Mathematics, Sun Yat-sen University, Guangzhou
510275, P.R. China<EMAIL_ADDRESS>,  Changjian Su  Yau
Mathematical Sciences Center, Tsinghua University, Beijing, China
<EMAIL_ADDRESS>and  Rui Xiong  Department of Mathematics and
Statistics, University of Ottawa, 150 Louis-Pasteur, Ottawa, ON, K1N 6N5,
Canada<EMAIL_ADDRESS>
###### Abstract.

In this paper, we introduce quantum Demazure–Lusztig operators acting by ring
automorphisms on the equivariant quantum cohomology of the Springer
resolution. Our main application is a presentation of the torus-equivariant
quantum cohomology in terms of generators and relations. We provide explicit
descriptions for the classical types. We also recover Kim’s earlier results
for the complete flag varieties by taking the Toda limit.

###### Key words and phrases:

Keywords: Quantum Demazure–Lusztig operators, Springer resolution, Quantum
cohomology

##  1\. Introduction

The quantum cohomology ring $QH^{*}(X)$ of a complex projective manifold $X$
is a deformation of the classical cohomology ring
$H^{*}(X)=H^{*}(X,\mathbb{C})$ by incorporating the Gromov–Witten invariants.
When $X$ is a non-compact algebraic variety equipped with a nice reductive
group $\mathbb{G}$ action, the equivariant quantum product can still be
defined via localizations (see e.g. [Liu13]). An interesting example of such
varieties is the symplectic resolution, which usually comes in pairs called
symplectic duals [Kam22]. The symplectic duality is a 3-dimensional mirror
symmetry, generalizing well-known structures in geometric representation
theory [BLPW16]. Okounkov and his collaborators initiated the study of the
symplectic duality via the enumerative geometry of the symplectic resolution,
which is deeply related to various structures in geometry, representation
theory, and mathematical physics [Oko18]. There have been studies of
$QH^{*}_{\mathbb{G}}(X)$ for various examples of symplectic resolutions [OP10,
BMO11, Su16, MS13, MO19, Dan22, MP15, KMP21].

The Springer resolution, which is the cotangent bundle of the complete flag
variety $\mathcal{B}$ of a complex semisimple, simply-connected Lie group $G$,
is the most classical example of the symplectic resolutions. It is a
resolution of the nilpotent cone in $\mathfrak{g}=\operatorname{Lie}(G)$ and
admits a natural action by $\mathbb{G}=G\times\mathbb{C}^{*}$, where
$\mathbb{C}^{*}$ dilates the cotangent fibers. Let $T$ be a maximal torus of
$G$ with Lie algebra $\mathfrak{t}$, and denote
$\mathbb{T}:=T\times\mathbb{C}^{*}$. The $\mathbb{T}$-equivariant quantum
cohomology ring $QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$ can be defined as the
deformation

$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})=\big{(}H^{*}_{\mathbb{T}}(T^{*}\mathcal{B})\otimes\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}),\,*\,\big{)},$

due to the equivariant quantum Chevalley formula proved by Braverman, Maulik,
and Okounkov [BMO11, Theorem 3.2]. Here
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}$ is the
complement of the union of root hyperplanes in the complex dual torus
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}$ (see Section 2).

Our first main theorem reveals a surprising symmetry in the equivariant
quantum cohomology ring of $T^{*}\mathcal{B}$. To explain this symmetry, let
us start with the classical Springer theory. Recall that the degenerate affine
Hecke algebra $\mathcal{H}_{\hbar}$ is a $\mathbb{C}[\hbar]$-algebra generated
by $\operatorname{Sym}(\mathfrak{t}^{*})$ and the group algebra
$\mathbb{C}[W]$, subject to interacting relations between them. As will be
reviewed in Section 2.1, $\mathcal{H}_{\hbar}$ acts on the
$\mathbb{T}$-equivariant cohomology $H^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$,
through an isomorphism of convolution algebras [CG10, Lus88]. The operators
corresponding to $w\in W\subset\mathcal{H}_{\hbar}$ are usually referred to as
the Demazure–Lusztig operators [Gin98]. On the other hand, there is a natural
Weyl group action on the coefficient ring
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$
induced by the Weyl group action on
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}$. The
_quantum Demazure–Lusztig operators_ $T_{w}$ over
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})=H^{*}_{\mathbb{T}}(T^{*}\mathcal{B})\otimes\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$
is the diagonal tensor product of these two actions, c.f. Definition 3.1. Our
first main theorem is the following unexpected result, for which we refer to
Example 3.3 for a quick illustration with $T^{*}\mathbb{P}^{1}$.

######  Theorem A (Theorem 3.2).

Every quantum Demazure–Lusztig operator is a ring automorphism of
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$.

Observe that the usual Demazure–Lusztig operator interacts with the
equivariant cup product via Leibniz-type formulas [Gin98, §12], which is never
a ring automorphism except for the trivial case. As the quantum
Demazure–Lusztig operators that we have defined create a pole at the origin of
the quantum variables, they cannot be projected down to the equivariant
cohomology. The key ingredient in our proof is the stable basis introduced by
Maulik and Okounkov [MO19]. We achieve this aim by direct computations using
an explicit formula about the multiplication of a divisor class with the
stable basis due to the second named author [Su16]. The proof uses some
crucial structures appearing in the cotangent bundle situation.

As an application of the above theorem, we obtain a ring presentation for
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$. For any dominant weight $\lambda$, we
define in (4.3) a square matrix $\Theta(\lambda)$ with values in
$QH^{2}_{\mathbb{T}}(T^{*}\mathcal{B})$, inspired from the quantum Chevalley
formula. It leads to an effective way of finding relations in the equivariant
quantum cohomology, due to an observation in communicative algebra as in Lemma
4.9. Via a Deformation Principle in Propositon 4.11, we obtain our second main
result.

######  Theorem B (Theorem 4.1).

The $\mathbb{T}$-equivariant quantum cohomology
$QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$ is generated by divisors with the
relations $\operatorname{tr}\big{(}\Theta(\lambda)^{k}\big{)}=\sum_{\mu\in
W\\!\lambda}\mu^{k}$ for all dominant weights $\lambda$ and all $k\geq 1$.

The above ring representation has an application in the study of the
Calogero–Moser system, which is an integrable system arising from the one-
dimensional many-body problems (see e.g. [Eti07, BH18] and references
therein). Combining Theorem B and [BMO11, Theorem 3.2], we obtain an explicit
description of the classical trigonometric Calogero–Moser map in Corollary
4.13.

We are going to provide an explicit ring presentation of
$QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$ in all classical types. Instead of
using the relations in Theorem B directly, we compute the characteristic
polynomial of $\Theta(\lambda)$. For instance, in type $A_{n-1}$, the flag
variety $\mathcal{B}=\mathcal{F}\ell_{n}$ parameterizes complete flags in
$\mathbb{C}^{n}$ and the Springer resolution $T^{*}\mathcal{F}\ell_{n}$
parameterizes pairs $(A,\phi)$ for $A$ an $n\times n$ nilpotent matrix and
$\phi\in\mathcal{F}\ell_{n}$ a complete flag of $A$-invariant subspaces. Among
the family of $\Theta(\lambda)$, We can find one matrix of the form

$\left[\begin{matrix}\chi_{1}&\frac{\hbar}{1-q_{1}/q_{2}}&\frac{\hbar}{1-q_{1}/q_{3}}&\cdots&\frac{\hbar}{1-q_{1}/q_{n}}\\\\[4.30554pt]
\frac{\hbar}{1-q_{2}/q_{1}}&\chi_{2}&\frac{\hbar}{1-q_{2}/q_{3}}&\cdots&\frac{\hbar}{1-q_{2}/q_{n}}\\\\[4.30554pt]
\frac{\hbar}{1-q_{3}/q_{1}}&\frac{\hbar}{1-q_{3}/q_{2}}&\chi_{3}&\cdots&\frac{\hbar}{1-q_{3}/q_{n}}\\\\[4.30554pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
\frac{\hbar}{1-q_{n}/q_{1}}&\frac{\hbar}{1-q_{n}/q_{2}}&\frac{\hbar}{1-q_{n}/q_{3}}&\cdots&\chi_{n}\end{matrix}\right],$
(1.1)

where
$\chi_{i}=x_{i}+\hbar\sum_{a<i}\frac{q_{a}/q_{i}}{1-q_{a}/q_{i}}-\hbar\sum_{i<b}\frac{q_{i}/q_{b}}{1-q_{i}/q_{b}}$.
The matrix (1.1) happens to coincide with the one in [Eti07, §2.8] The
presentation for the $\mathbb{T}$-equivariant quantum cohomology of
$T^{*}\mathcal{B}$ can be stated as follows,

######  Theorem C (Theorem 5.1).

The torus-equivariant quantum cohomology of $T^{*}\mathcal{F}\ell_{n}$ is
generated by divisors $x_{1},\ldots,x_{n}$ with relations
$\mathcal{E}_{k}(\chi)-e_{k}(t)$ for $k=1,\ldots,n$. Here
$\mathcal{E}_{k}(\chi)$ are the coefficients of the characteristic polynomial
of the matrix (1.1).

We will give a combinatorial description of $\mathcal{E}_{k}(\chi)$ in Theorem
5.2 via matching over $k$-subsets of $\\{1,\ldots,n\\}$, which coincides with
Feynman’s elementary computation [Pol19]. The computations for other classical
types are given in Theorem 5.4 (type $B$ and type $C$) and Theorem 5.7 (type
$D$).

There is another application of our explicit description of
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$. As explained in [BMO11, §8], we can
recover $QH^{*}_{T}(\mathcal{B})$ by taking the Toda limit. In the context of
quantum Schubert calculus and mirror symmetry, it is an important problem to
provide a precise description of the ring presentation of the
($T$-equivariant) quantum cohomology of $\mathcal{B}$ or, more generally, of a
partial flag variety. This problem has been solved for partial flag varieties
of type $A$ [GK95, AS95, CF99] and for the complete flag variety $\mathcal{B}$
of general Lie type by Kim [Kim99]. However, a precise description for partial
flag varieties of non-$A$ type in general is not yet available. By taking the
Toda limit, we are able to reobtain the presentation of
$QH^{*}_{T}(\mathcal{B})$ for classical Lie types and discuss its connection
with Kim’s description. For type $A_{n-1}$, the matrix (1.1) (after
conjugation) converges, under the Toda limit, to the well-known tridiagonal
matrix in quantum Schubert calculus:

$\left[\begin{matrix}x_{1}&-1&0&\cdots&0\\\\[4.30554pt]
q_{1}/q_{2}&x_{2}&-1&\cdots&0\\\\[4.30554pt]
0&q_{2}/q_{3}&x_{3}&\cdots&0\\\\[4.30554pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
0&0&0&\cdots&x_{n}\end{matrix}\right].$ (1.2)

This recovers the presentation obtained by Givental and Kim [GK95]. It would
be interesting to establish a direct (type-free) connection between both
descriptions. Our approach to finding the ring presentation of
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$ may be generalized to the cotangent
bundle of a partial flag variety. Hence, it is potentially useful in solving
the aforementioned problem for the partial flag varieties.

The paper is organized as follows. In Section 2, we review the background of
equivariant quantum cohomology of the Springer resolution $T^{*}\mathcal{B}$.
In Section 3, we define quantum Demazure–Lusztig operators and show that the
operators act on the equivariant quantum cohomology of $T^{*}\mathcal{B}$ by
ring automorphisms. In Section 4, we give the ring presentation of
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$, as well as an explicit description of
the classical trigonometric Calogero–Moser system. In Section 5, we carry out
computations of the ring presentation of
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$ for all classical Lie types. Finally,
in Appendix A and Appendix B, we discuss the ring presentation of
$QH^{*}_{T}(\mathcal{B})$ obtained by taking the Toda limit and provide some
details for the computations of $QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$ in
type $A_{n-1}$.

### Acknowledgement

The authors would like to thank Pavel Etingof, Michael Finkelberg, Michael
McBreen, Leonardo C. Mihalcea, Andrei Okounkov, Nicholas Proudfoot, Peng Shan,
Kirill Zainouline, and Yehao Zhou for helpful discussions and valuable
comments. C. Li is partially supported by NSFC Grants 12271529 and 11831017.
C. Su is partially supported by a startup grant from YMSC, Tsinghua
University. R. Xiong is partially supported by the NSERC Discovery grant
RGPIN-2015-04469, Canada.

##  2\. Preliminaries

In this section, we review some basic results about the quantum cohomology of
$T^{*}\mathcal{B}$.

### Notations

Let $G$ be a complex, semi-simple, simply connected group with Lie algebra
$\mathfrak{g}$, and $T\subset G$ be a maximal torus with Lie algebra
$\mathfrak{t}$. Let $R\subset\mathfrak{t}^{*}$ be the root system of
$(\mathfrak{g},\mathfrak{t})$. Fix a base
$\Pi=\\{\alpha_{i}\\}_{i=1}^{r}\subset R$ of simple roots, and let
$\\{\omega_{i}\\}_{i=1}^{r}\subset\mathfrak{t}^{*}$ denote the fundamental
weights where $r$ is the rank of $G$. Let
$R^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}=\\{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\mid\alpha\in
R\\}\subset\mathfrak{t}$ denote the set of coroots. Since $G$ is simply
connected, we can canonically identify the (co)character lattice with the
weight (resp. coroot) lattice:

$X^{*}(T)=\operatorname{Hom}(T,\mathbb{C}^{*})\cong\operatorname{Span}_{\mathbb{Z}}\\{\omega_{i}\\}_{i=1}^{r},\quad\operatorname{Hom}(\mathbb{C}^{*},T)\cong\operatorname{Span}_{\mathbb{Z}}R^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}.$

Unless otherwise stated, by $\alpha>0$ we will always mean an element $\alpha$
in the set $R^{+}$ of positive roots. The associated reflection
$s_{\alpha}:\mathfrak{t}^{*}\to\mathfrak{t}^{*}$ is defined by
$s_{\alpha}(\lambda)=\lambda-\langle\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\rangle\alpha$,
where
$\langle\cdot,\cdot\rangle:\mathfrak{t}^{*}\times\mathfrak{t}\to\mathbb{C}$ is
the natural pairing. We will also denote by $s_{\alpha}$ the induced
reflection on $\mathfrak{t}$. The Weyl group $W$ is generated by simple
reflections $s_{i}=s_{\alpha_{i}}$, for $i=1,\ldots,r$. It is equipped with
the Bruhat order $\geq$, and admits a standard length function
$\ell:W\to\mathbb{Z}_{\geq 0}$. Let $X^{*}(T)_{+}$ denote the set of dominant
weights in $X^{*}(T)$. For any $\pm\lambda\in X^{*}(T)_{+}$ we denote by
$W_{\lambda}$ the subgroup of $W$ that stabilizes $\lambda$, which is
generated by simple reflections $s_{i}$ with
$\langle\lambda,\alpha_{i}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\rangle=0$.
Each coset $uW_{\lambda}$ in $W/W_{\lambda}$ has a unique minimal length
representative, denoted as $\bar{u}$. Therefore $W/W_{\lambda}$ can be
identified with the subset $W^{\lambda}\subset W$ of minimal length
representatives.

The _degenerate affine Hecke algebra_ $\mathcal{H}_{\hbar}$ is a graded
$\mathbb{C}[\hbar]$-algebra generated by
$\\{x_{\lambda}\\}_{\lambda\in\mathfrak{t}^{*}}$ and $W$ such that

  1. (1)

$x_{c\lambda+\mu}=cx_{\lambda}+x_{\mu}$,
$x_{\lambda}x_{\mu}=x_{\mu}x_{\lambda}$,
$\forall\lambda,\mu\in\mathfrak{t}^{*},\forall c\in\mathbb{C}$;

  2. (2)

the elements $w\in W$ form the Weyl group inside $\mathcal{H}_{\hbar}$.

  3. (3)

$s_{i}x_{\lambda}-x_{s_{i}(\lambda)}s_{i}=\hbar\langle\alpha_{i}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}},\lambda\rangle$,
$\forall\lambda\in\mathfrak{t}^{*},\forall i=1,\ldots,r$;

The grading on $\mathcal{H}_{\hbar}$ is defined by $\deg x_{\lambda}=2,\deg
w=0,\deg\hbar=2$.

Let
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}:=\mathbb{C}^{*}\otimes_{\mathbb{Z}}X^{*}(T)$
be the complex dual torus, and
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}$ be the
complement of all the root hyperplanes
$\\{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}=1\\}_{\alpha\in
R^{+}}$. Here $q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}$ is
viewed as a function over $T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}$, and
$q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}=\prod_{i}(q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i}})^{a_{i}}$
for
$\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}=\sum_{i}a_{i}\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i}$.
In the present paper, we will mainly take coefficients in

$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})=\mathbb{C}\big{[}\,q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}},\tfrac{1}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\,\big{|}\,\alpha\in
R\,\big{]}.$ (2.1)

For any weight $\lambda\in X^{*}(T)$, we will consider the linear function
${p}_{\lambda}$ in $\mathcal{O}(\mathfrak{t})$ and the differential operator
$\partial_{\lambda}$ on
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$,
defined respectively by

${p}_{\lambda}(\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}})=\langle\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\rangle,\qquad\partial_{\lambda}q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}=\langle\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\rangle
q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}.$ (2.2)

We will also consider actions by the groups

$\mathbb{G}:=G\times\mathbb{C}^{*},\quad\mathbb{T}:=T\times\mathbb{C}^{*}.$
(2.3)

###  2.1. Equivariant cohomology of Springer resolution

The complete flag variety $\mathcal{B}$ of $G$ parameterizes Borel subalgebras
$\mathfrak{b}$ of $\mathfrak{g}$. Let $\mathcal{N}$ denote the cone of
nilpotent elements in $\mathfrak{g}$, i.e. those $x\in\mathfrak{g}$ with
$\mbox{ad}_{x}\in\mbox{End}(\mathfrak{g})$ being nilpotent. The cotangent
bundle $T^{*}\mathcal{B}$ is isomorphic to the subvariety

$\tilde{\mathcal{N}}=\\{(A,\mathfrak{b})\mid
A\in\mathcal{N},A\in\mathfrak{b}\\}\subset\mathcal{N}\times\mathcal{B}$

via the identification
$T^{*}_{\mathfrak{b}}\mathcal{B}\cong[\mathfrak{b},\mathfrak{b}]=\mathfrak{n}$.
The natural projection $\tilde{\mathcal{N}}\to\mathcal{N}$, or equivalently
the composition
$T^{*}\mathcal{B}\mathbin{\cong}\tilde{\mathcal{N}}\to\mathcal{N}$, is a
resolution of singularities, called the _Springer resolution_.

Let $B$ denote the standard Borel subgroup of $G$, whose Lie algebra is
spanned by $\mathfrak{t}$ and those root spaces parameterized by positive
roots. With the identification $\mathcal{B}=G/B$, the cotangent bundle
$T^{*}\mathcal{B}$ admits a natural $\mathbb{G}$-action with $G$-action on
$\mathcal{B}$ by left multiplication and $\mathbb{C}^{*}$-action given by
dilation. Precisely, for any $(g,z)\in\mathbb{G}$, the action is given by

$(g,z)\cdot(xB,\xi)=(gxB,z^{-1}g^{*}\xi),$

where $\xi$ is a cotangent vector at $gB\in G/B$ and $g^{*}\xi$ is pull back
of it to cotangent vector at $gxB$ induced by the left translation of $g$.

Let $Y$ denote $T^{*}\mathcal{B},\mathcal{B}$, or a point pt, and
$A\leq\mathbb{G}$ be a complex reductive subgroup. Let us describe the
$A$-equivariant cohomology $H_{A}^{*}(Y)=H_{A}^{*}(Y,\mathbb{C})$ of $Y$.
Denote also by $\hbar$ the standard generator of
$H_{\mathbb{C}^{*}}^{*}(\mbox{pt})$. We have

$\displaystyle H_{T}^{*}(\operatorname{pt})$
$\displaystyle=\operatorname{Sym}(\mathfrak{t}^{*}),$ $\displaystyle
H_{\mathbb{T}}^{*}(\operatorname{pt})$
$\displaystyle=\operatorname{Sym}(\mathfrak{t}^{*})[\hbar],$ $\displaystyle
H_{G}^{*}(\operatorname{pt})$
$\displaystyle=\operatorname{Sym}(\mathfrak{t}^{*})^{W},$ $\displaystyle
H_{\mathbb{G}}^{*}(\operatorname{pt})$
$\displaystyle=\operatorname{Sym}(\mathfrak{t}^{*})^{W}[\hbar].$

Note that $T^{*}\mathcal{B}$ is a vector bundle over $\mathcal{B}$, thus

$\displaystyle H^{*}_{\mathbb{G}}(T^{*}\mathcal{B})$
$\displaystyle=H^{*}_{\mathbb{G}}(\mathcal{B})=H^{*}_{G}(\mathcal{B})[\hbar],$
(2.4) $\displaystyle\text{and \quad}H^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$
$\displaystyle=H^{*}_{\mathbb{T}}(\mathcal{B})=H^{*}_{T}(\mathcal{B})[\hbar].$
(2.5)

Every $\lambda\in X^{*}(T)$ defines a line bundle
$\mathcal{L}_{\lambda}:=G\times_{B}\mathbb{C}_{\lambda}$ over $\mathcal{B}$.
The equivariant first Chern classes of $\mathcal{L}_{\lambda}$ are denoted as:

$D_{\lambda}^{G,\mathcal{B}}=c_{1}^{G}(\mathcal{L}_{\lambda})\in
H_{G}^{2}(\mathcal{B}),\quad
D_{\lambda}^{T,\mathcal{B}}=c_{1}^{T}(\mathcal{L}_{\lambda})\in
H_{T}^{2}(\mathcal{B}).$ (2.6)

Similarly, for $A\leq\mathbb{G}$, we denote by
$D_{\lambda}^{A,T^{*}\mathcal{B}}$ the $A$-equivariant first Chern class of
the line bundle over $T^{*}\mathcal{B}$ obtained via pullback of
$\mathcal{L}_{\lambda}$. Whenever there is no confusion, we will simply denote
all classes as $D_{\lambda}$.

Let $\operatorname{St}:=T^{*}\mathcal{B}\times_{\mathcal{N}}T^{*}\mathcal{B}$
denote the _Steinberg variety_ , which inherits a $\mathbb{G}$ action. Let
$H_{*}^{\mathbb{G}}(\operatorname{St})$ denote the $\mathbb{G}$-equivariant
Borel–Moore homology of the Steinberg variety, which has an associative
algebra structure via convolution, see [CG10]. The equivariant cohomology
$H^{*}_{\mathbb{G}}(T^{*}\mathcal{B})$ admits a canonical action by
$H_{*}^{\mathbb{G}}(\operatorname{St})$, and hence by the affine Hecke algebra
$\mathcal{H}_{\hbar}$, due to an isomorphism $\mathcal{H}_{\hbar}\simeq
H_{*}^{\mathbb{G}}(\operatorname{St})$ of $\mathbb{C}[\hbar]$-algebras proved
by Lusztig [Lus88]. The isomorphism can be explicitly described as follows.

  1. (1)

For any $\lambda\in X^{*}(T)$, $x_{\lambda}\in\mathcal{H}_{\hbar}$ is sent to
the the diagonal push forward of $D_{\lambda}$ from $T^{*}\mathcal{B}$ to
$\operatorname{St}$. In particular, for any $\gamma\in
H_{\mathbb{G}}(T^{*}\mathcal{B})$, the action of $x_{\lambda}$ on $\gamma$ is
given by the equivariant product with $D_{\lambda}$, i.e.

$x_{\lambda}(\gamma)=D_{\lambda}\cdot\gamma.$

  2. (2)

Let $\mathcal{P}_{i}:=G/P_{i}$, where $P_{i}:=B\cup Bs_{i}B$ is the minimal
parabolic subgroup containing $B$. The fundamental class of the conormal
bundle of $\mathcal{B}\times_{\mathcal{P}_{i}}\mathcal{B}$ inside
$\mathcal{B}\times\mathcal{B}$ is a well defined element in
$H_{*}^{\mathbb{G}}(\operatorname{St})$, and it corresponds to $s_{i}-1$.

The affine Hecke algebra $\mathcal{H}_{\hbar}$ also acts on the
$\mathbb{T}$-equivariant cohomology
$H^{*}_{\mathbb{T}}(T^{*}\mathcal{B})=H^{*}_{\mathbb{T}}(\operatorname{pt})\otimes_{H^{*}_{\mathbb{G}}(\operatorname{pt})}H^{*}_{\mathbb{G}}(T^{*}\mathcal{B})$.
The operators $x_{\lambda}$, $w$ are all
$H^{*}_{\mathbb{T}}(\operatorname{pt})$-linear. Moreover, the operators $w\in
W\subset\mathcal{H}_{\hbar}$ are also referred to as the Demazure–Lusztig
operators, see [Gin98].

Due to Borel, there is an isomorphism,

$\mathcal{O}(\mathfrak{t})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H_{G}^{*}(\mathcal{B})$
(2.7)

which sends ${p}_{\lambda}$ to $D_{\lambda}$; moreover, the following holds.

######  Proposition 2.1 ([Bor53], see also [AF]).

We have the following ring presentation

$\displaystyle H_{T}^{*}(\mathcal{B})$
$\displaystyle=\mathbb{C}[\lambda,D_{\lambda}]_{\lambda\in
X(T)}\big{/}\big{<}f(D_{\lambda})-f(\lambda)\mid
f(\lambda)\in\operatorname{Sym}(\mathfrak{t}^{*})^{W}\big{>},$

where $f(D_{\lambda})$ is obtained by substituting $\lambda$ by $D_{\lambda}$.

###  2.2. Calogero–Moser system

The Calogero–Moser system is a Hamiltonian system in the study of one-
dimensional many-body problems. The system was originally written down in its
rational version and was generalized to the trigonometric case and the
elliptic case by Sutherland and Krichever, respectively (see e.g. [Eti07,
BH18] and references therein).

For our purpose, we need the _trigonometric Calogero–Moser system_ mostly and
it is an integrable system over
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar]$
with Hamiltonian defined as follows. For any non-degenerate $W$-invariant
quadratic form $C\in\operatorname{Sym}^{2}(\mathfrak{t}^{*})^{W}$, we define
the Hamiltonian to be

$C(p)-\hbar^{2}\sum_{\alpha>0}\frac{C(\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}},\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}})}{(q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}/2}-q^{-\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}/2})^{2}}\in\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar],$
(2.8)

where ${p}_{\lambda}$ is defined (2.2) and naturally viewed as a function in
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})$.
The trigonometric Calogero–Moser system can be described by a
$\mathbb{C}[\hbar]$-linear embedding

$\eta_{\operatorname{CM},\hbar}^{\operatorname{cl}}:\operatorname{Sym}(\mathfrak{t}^{*})^{W}[\hbar]\hookrightarrow\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar],$
(2.9)

whose image consists of Possion commuting functions characterized by

  1. (1)

for any $f\in\operatorname{Sym}(\mathfrak{t}^{*})^{W}$, $f(p)$ appears as the
leading term of $\eta^{\operatorname{cl}}_{\operatorname{CM},\hbar}(f)$;

  2. (2)

for any $C\in\operatorname{Sym}^{2}(\mathfrak{t}^{*})^{W}$,
$\eta_{\operatorname{CM},\hbar}^{\operatorname{cl}}(C)$ is the Hamiltonian
defined in (2.8).

The existence of such a map can be ensured by the construction of Dunkl
operators. For any $\lambda\in X^{*}(T)$, the _classical trigonometric Dunkl
operator_ is

$\displaystyle\operatorname{Dun}_{\lambda}$
$\displaystyle:=p_{\lambda}-\hbar\sum_{\alpha>0}\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}s_{\alpha}\in\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar]\rtimes\mathbb{C}[W].$
(2.10)

For any polynomial $f(\lambda)\in\operatorname{Sym}(\mathfrak{t}^{*})$,
writing

$f(\operatorname{Dun}_{\lambda})=\sum_{w\in W}D(w,f,\hbar)\cdot
w\in\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar]\rtimes\mathbb{C}[W],$

where
$D(w,f,\hbar)\in\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar]$,
we define

$D(f,\hbar):=\sum_{w}D(w,f,\hbar)\in\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar].$

Then $\eta_{\operatorname{CM},\hbar}^{\operatorname{cl}}(f)=D(f,\hbar)$ for
$f\in\operatorname{Sym}(\mathfrak{t}^{*})^{W}$.

Let
$\mathcal{D}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$
denote the $\mathbb{C}$-algebra of differential operators on
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}$. The
quantum version of the trigonometric Calogero–Moser system can be similarly
described by

$\displaystyle\eta_{\operatorname{CM},\hbar}:\operatorname{Sym}(\mathfrak{t}^{*})^{W}[\hbar]\hookrightarrow\mathcal{D}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})[\hbar]$

characterized by similar conditions and can be constructed similarly using
(quantum) trigonometric Dunkl operators [Hec97, EFMV11].

For our purpose, we need a gauge transformation. Let

$\delta=\prod_{\alpha>0}(q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}-1).$
(2.11)

This defines an automorphism $\vartheta$ over
$\mathcal{D}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})[\hbar]$
given by $\vartheta(x)=\delta^{\hbar}x\delta^{-\hbar}$. By direct computation,
we have

$\displaystyle\vartheta(q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}})=q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}\quad\text{and}\quad\vartheta(\partial_{\lambda})=\partial_{\lambda}+\hbar\sum_{\alpha>0}\frac{\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}.$

This induces an automorphism $\vartheta^{\operatorname{cl}}$ of the Poisson
algebra
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar]$
such that

$\displaystyle\vartheta^{\operatorname{cl}}(q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}})=q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}\quad\text{and}\quad\vartheta^{\operatorname{cl}}({p}_{\lambda})={p}_{\lambda}+\hbar\sum_{\alpha>0}\frac{\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}.$

Let us denote

$\displaystyle\overline{\eta}_{\operatorname{CM},\hbar}$
$\displaystyle=\vartheta\circ\eta_{\operatorname{CM},\hbar},\qquad\overline{\eta}^{\operatorname{cl}}_{\operatorname{CM},\hbar}=\vartheta^{\operatorname{cl}}\circ\eta^{\operatorname{cl}}_{\operatorname{CM},\hbar}.$
(2.12)

###  2.3. Quantum cohomology

We will study the $A$-equivariant quantum cohomology $QH_{A}^{*}(X)$ of a
space $X\in\\{\mathcal{B},T^{*}\mathcal{B}\\}$. Here $A\leq\mathbb{G}$ is a
complex reductive subgroup. In particular for $X=T^{*}\mathcal{B}$, we require
that $A$ contains $\\{e\\}\times\mathbb{C}^{*}$, so that the $A$-fixed locus
$X^{A}$ is always compact.

Note that $G$ is simply connected. We have the identifications

$\mbox{Span}_{\mathbb{Z}}R^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}=\operatorname{Hom}(\mathbb{C}^{*},T)\simeq
H_{2}(X,\mathbb{Z}).$

Every positive coroot $\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}$
defines an $\operatorname{SL}_{2}$-subgroup
$G_{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}\subset G$, and then
defines a rational curve
$G_{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}B/B\subset\mathcal{B}\subset
T^{*}\mathcal{B}$. Therefore the cone $\operatorname{Eff}(X)$ of effective
classes in $H_{2}(X,\mathbb{Z})$ is generated by the positive coroots. For
each effective curve class $\beta\in\operatorname{Eff}(X)\subset
H_{2}(X,\mathbb{Z})$, and $\gamma_{1},\ldots,\gamma_{k}\in H_{A}(X)$, the
$k$-point, genus $0$, equivariant _Gromov–Witten invariants_ are given by

$\left<\gamma_{1},\ldots,\gamma_{k}\right>_{0,k,\beta}^{X}=\int_{[\overline{M}_{0,k}(X,\beta)]^{\mathrm{vir}}}\operatorname{ev}^{*}(\gamma_{1}\boxtimes\cdots\boxtimes\gamma_{k}),$

where $[\overline{M}_{0,k}(X,\beta)]^{\mathrm{vir}}$ is the virtual
fundamental class of the moduli space of $k$-pointed stable maps to $X$, see
[CK99] and [BMO11, section 4]. For $X=T^{*}\mathcal{B}$ which is noncompact,
the integral is defined via equivariant residue, see [Liu13]. The _quantum
cohomology_

$QH_{A}^{*}(X)=(H_{A}^{*}(X)\otimes_{\mathbb{C}}\mathbb{C}[\\![q^{\beta}]\\!]_{\beta\in\operatorname{Eff}(X)},\,*\,)$

is a deformation of the equivariant cohomology $H_{A}^{*}(X)$. The quantum
product $*$ is a
$\mathbb{C}[\\![q^{\beta}]\\!]_{\beta\in\operatorname{Eff}(X)}$-bilinear
product over $QH_{A}^{*}(X)$ such that for any
$\gamma_{1},\gamma_{2},\gamma_{3}\in H_{A}^{*}(X)$, we have

$(\gamma_{1}*\gamma_{2},\gamma_{3})=\sum_{\beta\in\operatorname{Eff}(X)\cup\\{0\\}}q^{\beta}\langle\gamma_{1},\gamma_{2},\gamma_{3}\rangle_{0,3,\beta}^{X}.$

Here $(\cdot,\cdot)$ is the Poincaré pairing, which can be evaluated using
equivariant residue in the case of $T^{*}\mathcal{B}$.

It is well-known that $QH_{A}^{*}(X)$ forms a graded commutative
$H_{A}^{*}(\operatorname{pt})$-algebra with identity $1$. Here, the degree of
$q^{\beta}$ is defined to be

$\deg q^{\beta}=2\left<-K_{X},\beta\right>,$

where $K_{X}$ is the canonical bundle of $X$. In particular, the degree of
$q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}$ is zero in
$QH^{*}_{A}(T^{*}\mathcal{B})$, since $K_{T^{*}\mathcal{B}}$ is trivial.

On the trivial vector bundle on $H^{2}(X)$ with fiber $H_{A}^{*}(X)$, there is
a flat connection $\nabla$, called the _Dubrovin connection_ and defined by

$\nabla_{\lambda}:=\partial_{\lambda}-\lambda*,\quad\forall\lambda\in
H^{2}(X),$

where $\partial_{\lambda}$ is as given in (2.2), by identifying $H^{2}(X)$
with $\mathfrak{t}^{*}$. The equivariant cohomology $QH_{A}^{*}(X)$ is then
equipped with an $H_{A}^{*}(\operatorname{pt})$-$\mathcal{D}$-module
structure, known as the _$A$ -equivariant quantum $\mathcal{D}$-module_ of
$X$. The main results of [BMO11] are the following.

######  Proposition 2.2 ([BMO11, Theorem 3.2]).

  1. (1)

The quantum multiplication by $D_{\lambda}$ is given by

$x_{\lambda}+\hbar\sum_{\alpha>0}\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}(s_{\alpha}-1).$

  2. (2)

The quantum connection of $T^{*}\mathcal{B}$ can be analytically extended to
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}$ and is
given by

$\nabla_{\lambda}=\partial_{\lambda}-x_{\lambda}-\hbar\sum_{\alpha>0}\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}(s_{\alpha}-1).$

Moreover, the $H_{\mathbb{G}}^{*}(\operatorname{pt})$-structure is given by
the quantum Calogero–Moser map $\overline{\eta}_{\operatorname{CM},\hbar}$.

  3. (3)

We have an
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$-algebra
isomorphism

$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar]\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}QH_{\mathbb{G}}^{*}(T^{*}\mathcal{B})$
(2.13)

sending ${p}_{\lambda}$ to $D_{\lambda}$. Moreover, the
$H_{\mathbb{G}}^{*}(\operatorname{pt})$-structure is given by the classical
Calogero–Moser map
$\overline{\eta}^{\operatorname{cl}}_{\operatorname{CM},\hbar}$.

######  Remark 2.3.

Part (3) of the above proposition admits an equivalent version as follows. By
introducing

$\Delta_{\lambda}=D_{\lambda}+\hbar\sum_{\alpha>0}\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}},$
(2.14)

we have an
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$-algebra
isomorphism as (2.13), which sends ${p}_{\lambda}$ to $\Delta_{\lambda}$. In
this case, the $H_{\mathbb{G}}^{*}(\operatorname{pt})$-structure is given by
the classical Calogero–Moser map
${\eta}^{\operatorname{cl}}_{\operatorname{CM},\hbar}$ (without gauge
transformation). The relation is summarized in the following diagram

---
$\textstyle{\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t}^{*})[\hbar]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{p}_{\lambda}\mapsto
D_{\lambda}}$$\textstyle{H^{*}_{\mathbb{G}}(\operatorname{pt})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{\operatorname{Sym}(\mathfrak{t}^{*})^{W}[\hbar]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\eta}_{\operatorname{CM},\hbar}^{\operatorname{cl}}}$$\scriptstyle{\eta_{\operatorname{CM},\hbar}^{\operatorname{cl}}}$$\textstyle{QH^{*}_{\mathbb{G}}(T^{*}\mathcal{B})}$$\textstyle{\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t}^{*})[\hbar]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{p}_{\lambda}\mapsto\Delta_{\lambda}}$$\scriptstyle{\vartheta^{\operatorname{cl}}}$

######  Remark 2.4.

Note that Proposition 2.2 (3) shows that
$QH^{*}_{\mathbb{G}}(T^{*}\mathcal{B})$ is generated by divisors $D_{\lambda}$
(or equivalently classes $\Delta_{\lambda}$). The isomorphism of (2.13) is
obtained after refining the equivariant quantum cohomology

$QH^{*}_{\mathbb{G}}(T^{*}\mathcal{B})=H^{*}_{\mathbb{G}}(T^{*}\mathcal{B})\otimes_{\mathbb{C}}\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$
(2.15)

by replacing the formal power series $\mathbb{C}[\\![q^{\beta}]\\!]_{\beta}$
with the function ring
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$.
In other words, the quantum product can be analytically extended to the ring
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}$.

The same argument also applies to $QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$.
That is, $QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$ is generated by divisors
$D_{\lambda}$ (or equivalently $\Delta_{\lambda}$) and we can realize it over

$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})=H^{*}_{\mathbb{T}}(T^{*}\mathcal{B})\otimes_{\mathbb{C}}\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}).$
(2.16)

These facts are crucial when defining Quantum Demazure–Lusztig operators.

##  3\. Automorhisms of $QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$

Let us define the main objects of this paper.

######  Definition 3.1 (Quantum Demazure–Lusztig operators).

For any $u\in W$, we define the _quantum Demazure–Lusztig operator_ $T_{u}$ to
be the operator on
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})=H^{*}_{\mathbb{T}}(T^{*}\mathcal{B})\otimes\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$
given by

$T_{u}=u\otimes u,$ (3.1)

where the first $u$ on the right-hand-side is an element of
$\mathcal{H}_{\hbar}$, i.e. a usual Demazure–Lusztig operator, and the second
$u$ denotes the induced action on
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$
from that on
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}$, namely
$u(q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}):=q^{u\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}$.

This section is devoted to the following surprising theorem, which will play
an important role in finding the presentation of
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$ in Section 4.

######  Theorem 3.2.

For any $u\in W$ and $\gamma_{1},\gamma_{2}\in
QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$,

$T_{u}(\gamma_{1}*\gamma_{2})=T_{u}(\gamma_{1})*T_{u}(\gamma_{2}).$ (3.2)

As a result, $T_{u}$ is a ring automorphism with respect to the quantum
product.

######  Example 3.3.

Consider the case $G=SL(2,\mathbb{C})$. Let $s:=s_{\alpha}\in W$ be the only
simple reflection, $\varpi$ be the fundamental weight, and
$q:=q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}$ denote the unique
primitive effective curve class. Note that $-D_{\varpi}=D_{-\varpi}\in
H_{\mathbb{T}}^{*}(T^{*}\mathbb{P}^{1})$ is the pullback of the equivariant
first Chern class $c_{1}^{\mathbb{T}}(\mathcal{O}(1))\in
H_{\mathbb{T}}^{*}(\mathbb{P}^{1})$. By Proposition 2.2 (1),

$D_{\varpi}*D_{-\varpi}=D_{\varpi}D_{-\varpi}+\hbar\frac{q}{1-q}(D_{\varpi}-D_{-\varpi}-\hbar).$

We have

$\displaystyle T_{s}(D_{\varpi})*T_{s}(D_{-\varpi})$
$\displaystyle=(D_{-\varpi}+\hbar)*(D_{\varpi}-\hbar)$
$\displaystyle=D_{\varpi}D_{-\varpi}+\hbar\frac{1}{1-q}(D_{\varpi}-D_{-\varpi}-\hbar),$
$\displaystyle T_{s}(D_{\varpi}*D_{-\varpi})$
$\displaystyle=D_{\varpi}D_{-\varpi}+\hbar\frac{q^{-1}}{1-q^{-1}}(D_{-\varpi}-D_{\varpi}+\hbar)$
$\displaystyle=D_{\varpi}D_{-\varpi}+\hbar\frac{1}{1-q}(-D_{-\varpi}+D_{\varpi}-\hbar).$

Thus

$T_{s}(D_{\varpi}*D_{-\varpi})=T_{s}(D_{\varpi})*T_{s}(D_{-\varpi}).$

###  3.1. Stable basis

The _stable basis_ was introduced by Maulik and Okounkov in their seminal work
on quantum cohomology of Nakajima quiver varieties [MO19]. It plays a crucial
role in our proof of Theorem 3.2. Let us briefly recall the definition of the
stable basis for the Springer resolution, see [MO19, Chapter 3] or [Su17] for
more details.

Recall that the torus $\mathbb{T}$ acts on $T^{*}\mathcal{B}$, and the fixed
points are $(wB,0)\in T^{*}\mathcal{B}$ for $w\in W$. For any $w\in W$ and
$\gamma\in H_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$, we denote by $\gamma|_{w}$
the restriction of $\gamma$ to the fixed point $(wB,0)$. The definition of
stable envelopes depends on a choice of a Weyl chamber in $\mathfrak{t}$. For
our purpose, we need to choose the negative Weyl chamber, for which we use $-$
in the subscript to indicate. Let $B^{-}$ denote the opposite Borel subgroup.
The stable basis is characterized by the following proposition.

######  Proposition 3.4 ([MO19, Su17]).

There exists a unique family of classes

$\big{\\{}\operatorname{Stab}_{-}(w)\in
H_{\mathbb{T}}^{2\dim\mathcal{B}}(T^{*}\mathcal{B})\,\big{|}\,w\in W\big{\\}}$

which satisfies the following properties:

  1. (1)

$\operatorname{Stab}_{-}(w)$ is supported on the union of conormal bundle of
opposite Schubert varieties $B^{-}uB/B$ for $u\geq w$, i.e.,
$\operatorname{Stab}_{-}(w)|_{u}=0$ unless $u\geq w$;

  2. (2)

$\operatorname{Stab}_{-}(w)|_{w}=\prod\limits_{\alpha>0,w\alpha>0}(w\alpha-\hbar)\prod\limits_{\alpha>0,w\alpha<0}w\alpha$;

  3. (3)

$\operatorname{Stab}_{-}(w)|_{u}$ is divisible by $\hbar$, for any $u>w$.

Moreover, the stable basis forms a basis of
$H_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$ after inverting the equivariant
parameters.

######  Proposition 3.5 ([Su16, Theorem 3.14]).

For $w\in W$ and $\lambda\in X^{*}(T)$, the following holds in
$QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$

$\displaystyle D_{\lambda}*\operatorname{Stab}_{-}(w)$
$\displaystyle=w(\lambda)\operatorname{Stab}_{-}(w)-\hbar\sum_{\alpha>0,w\alpha>0}\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\operatorname{Stab}_{-}(ws_{\alpha})$
$\displaystyle\quad-\hbar\sum_{\alpha>0}\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\big{(}\operatorname{Stab}_{-}(w)+\operatorname{Stab}_{-}(ws_{\alpha})\big{)}.$

######  Remark 3.6.

The above formula can also be deduced from Propositions 2.2 (1) as well as the
Chevalley formula [Su16, Theorem 3.7] of the stable basis for cup product.

Recall the class $\Delta_{\lambda}$ defined in (2.14). Proposition 3.5 can be
rewritten as

$\displaystyle\Delta_{\lambda}*\operatorname{Stab}_{-}(w)$
$\displaystyle=w(\lambda)\operatorname{Stab}_{-}(w)-\hbar\sum_{\alpha>0,w\alpha>0}\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{1}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(ws_{\alpha})$
$\displaystyle-\hbar\sum_{\alpha>0,w\alpha<0}\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(ws_{\alpha}).$
(3.3)

###  3.2. Proof of Theorem 3.2

Let us first prove the following lemma.

######  Lemma 3.7.

For any $w,u\in W$ and $\lambda\in X^{*}(T)$, we have

  1. (1)

$T_{u}\big{(}\operatorname{Stab}_{-}(w)\big{)}=(-1)^{\ell(u)}\operatorname{Stab}_{-}(wu^{-1})$;

  2. (2)

$T_{u}(\Delta_{\lambda})=\Delta_{u\lambda}$.

###### Proof.

Since $\operatorname{Stab}_{-}(w)$ has no quantum component, we have
$T_{u}(\operatorname{Stab}_{-}(w))=u(\operatorname{Stab}_{-}(w))$. We know
from [Su17, Lemma 3.2] that
$u\,\big{(}\operatorname{Stab}_{-}(w)\big{)}=(-1)^{\ell(u)}\operatorname{Stab}_{-}(wu^{-1})$.
This proves the first part (1).

For the second part (2), it suffices to show the case when $w=s_{i}$ is a
simple reflection:

$\displaystyle T_{s_{i}}(\Delta_{\lambda})$
$\displaystyle=s_{i}(D_{\lambda})+\hbar\sum_{\alpha>0}\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{s_{i}\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{s_{i}\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}$
$\displaystyle=D_{s_{i}\lambda}+\hbar\left<\lambda,\alpha_{i}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>+\hbar\sum_{\alpha>0,\alpha\neq\alpha_{i}}\left<s_{i}\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}+\hbar\left<\lambda,\alpha_{i}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{-\alpha_{i}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{-\alpha_{i}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}$
$\displaystyle=\Delta_{s_{i}\lambda}.$

Here in the second equality, we have used the facts
$s_{i}(R^{+}\setminus\\{\alpha_{i}\\})=R^{+}\setminus\\{\alpha_{i}\\}$ and
$s_{i}(D_{\lambda})=s_{i}x_{\lambda}\cdot
1=(x_{s_{i}\lambda}+\hbar\langle\lambda,\alpha_{i}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\rangle)\cdot
1=D_{s_{i}\lambda}+\hbar\langle\lambda,\alpha_{i}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\rangle$.
∎

With this lemma, we can prove the following special case of Theorem 3.2.

######  Proposition 3.8.

For any $w,u\in W$ and $\lambda\in X^{*}(T)$,

$T_{u}(\Delta_{\lambda}*\operatorname{Stab}_{-}(w))=T_{u}(\Delta_{\lambda})*T_{u}(\operatorname{Stab}_{-}(w)).$
(3.4)

###### Proof.

The proof is given by direct computation using the definition of $T_{u}$,
Lemma 3.7, and (3.3). On one hand, we have

$\displaystyle\quad(-1)^{\ell(u)}T_{u}(\Delta_{\lambda})*T_{u}(\operatorname{Stab}_{-}(w))=\Delta_{u\lambda}*\operatorname{Stab}_{-}(wu^{-1})$
$\displaystyle=w(\lambda)\operatorname{Stab}_{-}(wu^{-1})-\hbar\sum_{\alpha>0,wu^{-1}\alpha>0}\left<u\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{1}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(wu^{-1}s_{\alpha})$
$\displaystyle\quad-\hbar\sum_{\alpha>0,wu^{-1}\alpha<0}\left<u\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(wu^{-1}s_{\alpha}).$

On the other hand, we have

$\displaystyle\quad(-1)^{\ell(u)}T_{u}(\Delta_{\lambda}*\operatorname{Stab}_{-}(w))$
$\displaystyle=w(\lambda)\operatorname{Stab}_{-}(wu^{-1})-\hbar\sum_{\beta>0,w\beta>0}\left<\lambda,\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{1}{1-q^{u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(ws_{\beta}u^{-1})$
$\displaystyle\quad-\hbar\sum_{\beta>0,w\beta<0}\left<\lambda,\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\frac{q^{u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(ws_{\beta}u^{-1})$

We finish the proof by matching the coefficients in the two expressions above.

Case A. The coefficients of $\operatorname{Stab}_{-}(wu^{-1})$ are both
$w(\lambda)$.

Case B. Let $\alpha>0$ such that $u^{-1}\alpha=\beta>0$. If $wu^{-1}\alpha>0$,
i.e. $w\beta>0$, we have

$\displaystyle\left<u\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\dfrac{1}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(wu^{-1}s_{\alpha})$
$\displaystyle=\left<\lambda,\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\dfrac{1}{1-q^{u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(ws_{\beta}u^{-1}).$

Similarly, if $wu^{-1}\alpha<0$, i.e. $w\beta<0$, we have

$\displaystyle\left<u\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\dfrac{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(wu^{-1}s_{\alpha})$
$\displaystyle=\left<\lambda,\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\dfrac{q^{u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(ws_{\beta}u^{-1}).$

Case C. Let $\alpha>0$ such that $u^{-1}\alpha=-\beta>0$. If
$wu^{-1}\alpha>0$, i.e. $w\beta<0$, we have

$\displaystyle\left<u\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\dfrac{1}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(wu^{-1}s_{\alpha})$
$\displaystyle=\left<\lambda,-\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\dfrac{1}{1-q^{-u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(ws_{\beta}u^{-1})$
$\displaystyle=\left<\lambda,\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\dfrac{q^{u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(ws_{\beta}u^{-1}).$

Similarly, if $wu^{-1}\alpha<0$, i.e. $w\beta>0$, we have

$\displaystyle\left<u\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\dfrac{q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(wu^{-1}s_{\alpha})$
$\displaystyle=\left<\lambda,-\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\dfrac{q^{-u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}{1-q^{-u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(ws_{\beta}u^{-1})$
$\displaystyle=\left<\lambda,\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>\dfrac{1}{1-q^{u\beta^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\operatorname{Stab}_{-}(ws_{\beta}u^{-1}).\qed$

We also need the following easy lemma.

######  Lemma 3.9.

Let $\gamma_{1},\gamma_{2},\gamma_{3}\in
QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$.

  1. (1)

Assume $T_{u}(\gamma_{1}*\gamma_{2})=T_{u}(\gamma_{1})*T_{u}(\gamma_{2})$ and
$T_{u}(\gamma_{1}*\gamma_{3})=T_{u}(\gamma_{1})*T_{u}(\gamma_{3})$. Then for
any $f_{2},f_{3}\in
H^{*}_{\mathbb{T}}(\operatorname{pt})\otimes\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$,

$T_{u}(\gamma_{1}*(f_{2}\gamma_{2}+f_{3}\gamma_{3}))=T_{u}(\gamma_{1})*T_{u}(f_{2}\gamma_{2}+f_{3}\gamma_{3}).$

  2. (2)

Assume $T_{u}(\gamma_{i}*\gamma)=T_{u}(\gamma_{i})*T_{u}(\gamma)$ for $i=1,2$
and any $\gamma\in QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$. Then

$T_{u}(\gamma_{1}*\gamma_{2}*\gamma)=T_{u}(\gamma_{1}*\gamma_{2})*T_{u}(\gamma).$

###### Proof.

The first one follows from the definition of $T_{u}$, while the second one
follows from the associativity of the quantum product. ∎

Now we can prove Theorem 3.2.

###### Proof of Theorem 3.2.

Recall that by Proposition 3.4 the stable basis forms a basis of
$H_{\mathbb{T}}(T^{*}\mathcal{B})$ after inverting the equivariant parameters.
Thus by Proposition 3.8 and Lemma 3.9 (1), (3.2) is true for
$\gamma_{1}=\Delta_{\lambda}$ with $\lambda\in X^{*}(T)$ and arbitrary
$\gamma_{2}\in QH_{\mathbb{T}}(T^{*}\mathcal{B})$. Since
$QH_{\mathbb{T}}(T^{*}\mathcal{B})$ is generated by classes
$\Delta_{\lambda}$, the theorem follows from Lemma 3.9 (2). ∎

##  4\. Presentation of Quantum Cohomology

The equivariant cohomology of $T^{*}\mathcal{B}$ is generated by the divisors
$D_{\mu}$. Thus there is a natural surjective morphism $\pi$ of algebras over
$\mathcal{O}(T_{\operatorname{reg}}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}})[\hbar]$,

$\pi:\mathcal{O}(T_{\operatorname{reg}}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}})[\hbar][\mu,D_{\mu}]_{\mu\in
X^{*}(T)}\longrightarrow QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B}).$ (4.1)

In this section, we will determine the kernel $\ker\pi$. More precisely, for
any (anti-)dominant weight $\lambda\in X^{*}(T)_{\pm}$ and positive integer
$k$, we let

$\mathcal{R}_{\lambda}^{k}=\operatorname{tr}\big{(}\Theta(\lambda)^{k}\big{)}-\sum_{u\in
W^{\lambda}}(u\lambda)^{k}$ (4.2)

where $\Theta(\lambda)=\big{(}\Theta_{{u},{v}}(\lambda)\big{)}_{{u},{v}\in
W^{\lambda}}$ is a square matrix of order $|W^{\lambda}|=|W\\!\lambda|$
defined by

$\Theta_{{u},{v}}(\lambda)=\begin{cases}\Delta_{u\lambda},&\mbox{if
}{u}={v},\\\\[4.30554pt]
\frac{-\hbar\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>}{1-q^{u\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}},&\text{if
$\hskip 1.00006pt\overline{us_{\alpha}}\hskip 1.00006pt=v$ for some $\alpha\in
R^{+}\setminus R_{\lambda}^{+}$},\\\\[4.30554pt]
0,&\text{otherwise}.\end{cases}$ (4.3)

Here $R_{\lambda}^{+}:=R^{+}\bigcap(\sum_{s_{i}\in
W_{\lambda}}\mathbb{Z}\alpha_{i})$, and we recall that $\hskip
1.00006pt\overline{us_{\alpha}}\hskip 1.00006pt\in W^{\lambda}$ denotes the
minimal length representative of the coset $us_{\alpha}W_{\lambda}\in
W/W_{\lambda}$. The root $\alpha$ in the second case is unique [FW04]. The
following theorem is the main result of this section.

######  Theorem 4.1.

We have the following presentation:

$\displaystyle QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$
$\displaystyle\cong\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})[\hbar][\mu,D_{\mu}]_{\mu\in
X^{*}(T)}\bigg{/}\big{<}\mathcal{R}_{\lambda}^{k}\mid\lambda\in
X^{*}(T)_{+},k>0\big{>}.$

######  Example 4.2.

Let $G=SL(2,\mathbb{C})$, and use the same notation as in Example 3.3. Then

$\Theta(\varpi)=\bigg{[}\\!\\!\begin{subarray}{c}\phantom{\frac{1}{2}}D_{\varpi}+\hbar\frac{q}{1-q}\phantom{\frac{1}{2}}\\\
\frac{-\hbar}{1-q^{{-}1}}\end{subarray}\\!\\!\\!\\!\begin{subarray}{c}\frac{-\hbar}{1-q}\\\
\phantom{\frac{1}{2}}D_{{-}\varpi}-\hbar\frac{q}{1-q}\phantom{\frac{1}{2}}\end{subarray}\\!\\!\bigg{]},$

and

$\displaystyle\tfrac{1}{2}\operatorname{tr}\big{(}\Theta(\varpi)^{2}\big{)}$
$\displaystyle=\big{(}D_{\varpi}+\hbar\tfrac{q}{1-q}\big{)}^{2}+\tfrac{\hbar^{2}}{(1-q)(1-q^{-1})}$
$\displaystyle=D_{\varpi}*D_{\varpi}+2\hbar\tfrac{q}{1-q}D_{\varpi}-\hbar^{2}\tfrac{q}{1-q}.$

Theorem 4.1 gives

$\displaystyle
D_{\varpi}*D_{\varpi}+2\hbar\frac{q}{1-q}D_{\varpi}-\hbar^{2}\frac{q}{1-q}=\varpi^{2}.$
(4.4)

Thus $QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$ is generated by $D_{\varpi}$ with
relation (4.4).

On the other hand, $T^{*}\mathcal{B}=T^{*}\mathbb{P}^{1}$ is also a hypertoric
variety. Following [MS13], we denote by $x$ and $y$ the fundamental class of
the cotangent fibers at $0$ and $\infty$, respectively. We denote $h=-\hbar$.
Then

$x=-\varpi-D_{\varpi}\quad\text{and}\quad y=\varpi-D_{\varpi}.$

We thus get the following relation, compatible with [MS13, Theorem 1.1],

$x*y=q(h-x)*(h-y).$

######  Remark 4.3.

Note that the $\mathbb{G}$-equivariant (quantum) cohomology can be viewed as
the invariants ring under the left Weyl group action recalled at the beginning
of Section 4.1 below. By Lemma 4.4, it is not hard to obtain that

$QH_{\mathbb{G}}^{*}(T^{*}\mathcal{B})\cong\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})[\hbar][D_{\mu}]_{\mu\in
X^{*}(T)}\cong\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t}^{*})[\hbar].$

This recovers the first part of Proposition 2.2 (3) by Braverman, Maulik, and
Okounkov. Using the second part of Proposition 2.2 (3), our result provides an
application to the Calogero–Moser system, see Corollary 4.13.

###  4.1. Construction of Relations

In this subsection, we will show that the equalities
$\mathcal{R}_{\lambda}^{k}=0$ hold in $QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$.

Recall that the left Weyl group action on
$H^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$ is induced by the left multiplication
action of $G$ on $T^{*}\mathcal{B}$. By extending it linearly over
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$,
we get a left action of $W$ on $QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$. For
$w\in W$, we use $w^{L}$ to denote this action. By noting that $w^{L}$
preserves line bundles and using [MNS22, Equation (12)] 111The formula in
_loc. cit._ was stated for $QH^{*}_{T}(G/P)$, while the whole set-up works for
all $G$-varieties., we have

######  Lemma 4.4.

The following holds for any $f\in H^{*}_{\mathbb{T}}(\operatorname{pt})$,
$g\in\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$,
$\mu\in X^{*}(T)$ and $\gamma_{1},\gamma_{2}\in
QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$:

  1. (1)

$w^{L}(fgD_{\mu})=w(f)gD_{\mu}$;

  2. (2)

$w^{L}(\gamma_{1}*\gamma_{2})=(w^{L}\gamma_{1})*(w^{L}\gamma_{2})$.

Thus every $w^{L}$ is a ring automorphism of
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$. Moreover, the above action of $W$
commutes with the action by the affine Hecke algebra $\mathcal{H}_{\hbar}$.

######  Lemma 4.5.

Let $u,v,w\in W$ and $\alpha\in R^{+}\setminus R^{+}_{\lambda}$. We have

  1. (1)

$\Theta_{1,\hskip 0.70004pt\overline{s_{\alpha}}\hskip
0.70004pt}(\lambda)=\frac{-\hbar\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}$;

  2. (2)

$T_{u}(\Theta_{{1},{w}})=\Theta_{\hskip 0.70004pt\overline{u}\hskip
0.70004pt,\hskip 0.70004pt\overline{uw}\hskip 0.70004pt}$;

  3. (3)

$v^{L}\circ T_{u}(\Theta_{{1},{w}})=\Theta_{\hskip 0.70004pt\overline{u}\hskip
0.70004pt,\hskip 0.70004pt\overline{uw}\hskip 0.70004pt}$.

###### Proof.

Statement (1) follows directly from the definition of $\Theta(\lambda)$. Next,
we need to verify Statement (2) in each case of the definition of
$\Theta(\lambda)$ (4.3). In the first case, we have
$T_{u}(\Theta_{1,1})=T_{u}(\Delta_{\lambda})=\Delta_{u\lambda}=\Theta_{u,\hskip
0.70004pt\overline{u\cdot 1}\hskip 0.70004pt}$ by Lemma 3.7 (2); the second
case follows from the definition of $T_{u}$; if the third case occurs for
$\Theta_{1,w}$ (i.e. $\Theta_{1,w}=0$) then so is $\Theta_{u,\hskip
0.70004pt\overline{uw}\hskip 0.70004pt}$. Lastly, since every entry in
$\Theta(\lambda)$ is fixed by the left Weyl group action $v^{L}$ by Lemma 4.4
(1), Statement (3) follows. ∎

Let

$\varsigma_{\lambda}=\sum_{w\in
W_{\lambda}}(-1)^{\ell(w)}\operatorname{Stab}_{-}(w),$ (4.5)

denote the _averaging class_. Define

$\Sigma(\lambda)=\big{(}v^{L}\circ
T_{u}(\varsigma_{\lambda})\big{)}_{{u},{v}\in W^{\lambda}}.$ (4.6)

Note that $T_{u}(\varsigma_{\lambda})$ is not fixed222Using [MNS22, Theorem
4.3] and [AMSS17, Proposition 9.7], we can show that
$s_{i}^{L}(\varsigma_{\lambda})=\frac{\hbar+\alpha_{i}}{\hbar-\alpha_{i}}\varsigma_{\lambda}$
for any $s_{i}\in W_{\lambda}$.  by $W_{\lambda}$.

######  Lemma 4.6.

For any $w\in W_{\lambda}$ and $u,v\in W^{\lambda}$, we have

  1. (1)

$T_{w}(\varsigma_{\lambda})=\varsigma_{\lambda}$;

  2. (2)

$\Sigma(\lambda)_{{u},{v}}=v^{L}\left(\sum_{w\in
uW_{\lambda}}(-1)^{\ell(w)}\operatorname{Stab}_{-}(w^{-1})\right)$.

###### Proof.

Both statements follow immediately from Lemma 3.7 and the fact that
$\ell(wu^{-1})\equiv\ell(w)+\ell(u)\bmod 2$ for any $u,w\in W$. ∎

Define the diagonal matrix

$\Xi(\lambda)=\operatorname{diag}\big{(}u\lambda\big{)}_{{u}\in W^{\lambda}}.$
(4.7)

As an application of Theorem 3.2, we have the following key proposition.

######  Proposition 4.7.

For any $\lambda\in X^{*}(T)_{\pm}$, we have

$\Theta(\lambda)*\Sigma(\lambda)=\Sigma(\lambda)*\Xi(\lambda),$ (4.8)

where the product of matrices is evaluated by the quantum product.

###### Proof.

Let us first compare the $(1,1)$-entry of both sides. By definition,

$\displaystyle\quad(\Theta(\lambda)*\Sigma(\lambda))_{1,1}$
$\displaystyle=\sum_{w\in
W^{\lambda}}\Theta(\lambda)_{1,w}*\Sigma(\lambda)_{w,1}$
$\displaystyle=\Theta(\lambda)_{1,1}*\Sigma(\lambda)_{1,1}+\sum_{\alpha\in
R^{+}\setminus R^{+}_{\lambda}}\Theta(\lambda)_{1,\hskip
0.70004pt\overline{s_{\alpha}}\hskip 0.70004pt}*\Sigma(\lambda)_{\hskip
0.70004pt\overline{s_{\alpha}}\hskip 0.70004pt,1}$ $\displaystyle=\sum_{w\in
W_{\lambda}}(-1)^{\ell(w)}\Delta_{\lambda}*\operatorname{Stab}_{-}(w)+\hbar\\!\\!\\!\\!\sum_{\alpha\in
R^{+}\setminus
R^{+}_{\lambda}}\frac{\langle\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\rangle}{1-q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}}\sum_{w\in
W_{\lambda}}(-1)^{\ell(w)}\operatorname{Stab}_{-}(ws_{\alpha})$
$\displaystyle=\lambda\cdot\varsigma_{\lambda}=\lambda\cdot\Sigma(\lambda)_{1,1}.$

Here the third equality uses the equality $T_{\hskip
0.70004pt\overline{s_{\alpha}}\hskip
0.70004pt}(\varsigma_{\lambda})=T_{{s_{\alpha}}}(\varsigma_{\lambda})$ by
Lemma 4.6 (1) as well as the fact $(-1)^{\ell(s_{\alpha})}=-1$. The forth
equality follows from (3.3), and the facts that (i) $w(\lambda)=\lambda$ for
any $w\in W_{\lambda}$, (ii)
$\langle\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\rangle=0$
for every $\alpha\in R^{+}_{\lambda}$, and (iii) $w\alpha>0$ for every $w\in
W_{\lambda}$ and $\alpha\in R^{+}\setminus R^{+}_{\lambda}$.

For general $u,v\in W^{\lambda}$, by applying $v^{L}\circ T_{u}$ to the above
equalities, we get

$\displaystyle v(\lambda)\cdot\Sigma(\lambda)_{u,v}=v^{L}\circ
T_{u}(\lambda\cdot\Sigma(\lambda)_{1,1})$ $\displaystyle=v^{L}\circ
T_{u}\bigg{(}\sum_{w\in
W^{\lambda}}\Theta(\lambda)_{1,w}*\Sigma(\lambda)_{w,1}\bigg{)}$
$\displaystyle=\sum_{w\in W^{\lambda}}\Theta(\lambda)_{u,\hskip
0.70004pt\overline{uw}\hskip 0.70004pt}*\Sigma(\lambda)_{\hskip
0.70004pt\overline{uw}\hskip 0.70004pt,v}$ $\displaystyle=\sum_{w\in
W^{\lambda}}\Theta_{u,w}(\lambda)*\Sigma(\lambda)_{w,v}$
$\displaystyle=(\Theta(\lambda)*\Sigma(\lambda))_{u,v}.$

Here the second equality follows from Lemmas 4.4, 4.5, 4.6 and Theorem 3.2.
This finishes the proof of the proposition. ∎

######  Example 4.8.

Let $G=SL(2,\mathbb{C})$, and use the same notation as in Example 3.3. Then
Proposition 4.7 gives

$\bigg{[}\,\begin{subarray}{c}\phantom{\frac{1}{2}}\Delta_{\varpi}\phantom{\frac{1}{2}}\\\
\frac{-\hbar}{1-q^{{-}1}}\end{subarray}\begin{subarray}{c}\frac{-\hbar}{1-q}\\\
\phantom{\frac{1}{2}}\Delta_{-\varpi}\phantom{\frac{1}{2}}\end{subarray}\\!\\!\bigg{]}*\bigg{[}\begin{subarray}{c}\operatorname{Stab}_{-}(\operatorname{id})\\\\[2.15277pt]
-\operatorname{Stab}_{-}(s)\\\
\end{subarray}\,\,\begin{subarray}{c}s^{L}\operatorname{Stab}_{-}(\operatorname{id})\\\\[2.15277pt]
-s^{L}\operatorname{Stab}_{-}(s)\end{subarray}\bigg{]}=\bigg{[}\begin{subarray}{c}\operatorname{Stab}_{-}(\operatorname{id})\\\\[2.15277pt]
-\operatorname{Stab}_{-}(s)\\\
\end{subarray}\,\,\begin{subarray}{c}s^{L}\operatorname{Stab}_{-}(\operatorname{id})\\\\[2.15277pt]
-s^{L}\operatorname{Stab}_{-}(s)\end{subarray}\bigg{]}*\bigg{[}\begin{subarray}{c}\varpi\\\\[4.30554pt]
\phantom{0}\\\
\end{subarray}\,\,\,\,\begin{subarray}{c}\phantom{0}\\\\[4.30554pt]
-\varpi\end{subarray}\bigg{]}$

The next lemma works for any commutative algebra over an arbitrary field
$\mathbb{F}$.

######  Lemma 4.9.

Let $\Theta,\Sigma$ be square matrices of order $m$ with entries in an
$\mathbb{F}$-algebra $\mathcal{A}$, and
$\Xi=\operatorname{diag}(a_{1},\ldots,a_{m})$ with $a_{1},\ldots,a_{m}$
distinct elements in $\mathbb{F}$. Assume

$\Theta\cdot\Sigma=\Sigma\cdot\Xi.$

If there is at least one unit in each column of $\Sigma$, then $\det(\Sigma)$
is a unit in $\mathcal{A}$.

###### Proof.

Assume $\det(\Sigma)$ is not a unit in $\mathcal{A}$. Then there is a maximal
ideal $\mathfrak{m}$ containing $\det(\Sigma)$, so that
$\overline{\mathcal{A}}:=\mathcal{A}/\mathfrak{m}$ is a field and
$\overline{\det(\Sigma)}=0\in\overline{\mathcal{A}}$. Note that the columns of
$\overline{\Sigma}$ are all nonzero vectors by our assumption, and hence are
eigenvectors of $\overline{\Theta}$ of distinct eigenvalues by the hypothesis.
Therefore they form a basis of $\overline{\mathcal{A}}^{m}$. In other words,
$\overline{\det(\Sigma)}=\det(\overline{\Sigma})$ is invertible, resulting in
a contradiction. ∎

Let us denote
$\mathbb{F}=\mbox{Frac}H^{*}_{\mathbb{T}}(\operatorname{pt})(\\!(\hbar^{-1})\\!)$,
and consider the $\mathbb{F}$-algebra

$\mathcal{A}=QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})\otimes_{H_{\mathbb{T}}^{*}(\operatorname{pt})}\mathbb{F}.$
(4.9)

######  Proposition 4.10.

For any $\lambda\in X^{*}(T)_{\pm}$, the determinant $\det\Sigma(\lambda)$ is
a unit in $\mathcal{A}$.

###### Proof.

For any $v\in W^{\lambda}$, by [AMSS17, Proposition 9.7 and Lemma 7.3], we
have

$\Sigma(\lambda)_{1,{v}}=(-1)^{\dim\mathcal{B}}\hbar^{\dim\mathcal{B}}+\big{(}\text{terms
of lower degrees in $\hbar$}\big{)},$

Hence, it is a unit in $\mathcal{A}$. Thus every column of $\Sigma(\lambda)$
contains a unit in $\mathcal{A}$ (at the first row). Moreover, the diagonal
terms of $\Xi(\lambda)$ are distinct elements in $\mathbb{F}$. By Lemma 4.9,
$\det\Sigma(\lambda)$ is a unit in $\mathcal{A}$. ∎

######  Proposition 4.11.

For any $\lambda\in X^{*}(T)_{\pm}$ and $k\in\mathbb{Z}_{>0}$, in
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$, we have

$\operatorname{tr}\big{(}\Theta(\lambda)^{k}\big{)}=\sum_{{u}\in
W^{\lambda}}(u\lambda)^{k}.$ (4.10)

###### Proof.

By Proposition 4.10, $\det\Sigma(\lambda)$ is a unit in $\mathcal{A}$, so that
$\Sigma(\lambda)$ is invertible. Therefore by Proposition 4.7, we have the
following relation in $\mathcal{A}$:

$\operatorname{tr}\big{(}\Theta(\lambda)^{k}\big{)}=\operatorname{tr}\big{(}\Sigma(\lambda)^{-1}*\Theta(\lambda)^{k}*\Sigma(\lambda)\big{)}=\operatorname{tr}\big{(}\Xi(\lambda)^{k}\big{)}=\sum\nolimits_{u\in
W^{\lambda}}(u\lambda)^{k}.$

Notice that there is a natural embedding
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})\hookrightarrow
QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})\otimes_{H_{\mathbb{T}}(\operatorname{pt})}\mathbb{F}=\mathcal{A}$,
and both sides of the equality (4.10) take values in
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$, the statement follows. ∎

###  4.2. Proof of Theorem 4.1

By Proposition 4.11, all $\mathcal{R}_{\lambda}^{k}$ are in the kernel of
$\pi$. Once we prove that they generate the entire kernel, this will finish
our proof. The following Deformation Principle is a key ingredient in our
argument, the proof of which is similar to that of a lemma by Siebert and Tian
[ST97].

######  Proposition 4.12 (Deformation Principle).

Let $\\{\mathcal{R}_{i}\\}_{i\in I}$ be a family of elements in $\ker\pi$.
Assume for all $i\in I$

$\mathcal{R}_{i}=f_{i}(D_{\mu})-f_{i}(\mu)\mod\hbar,$

where $\\{f_{i}\\}_{i\in I}$ generates
$\operatorname{Sym}(\mathfrak{t}^{*})^{W}$. Then $\ker\pi$ is generated by
$\\{\mathcal{R}_{i}\\}_{i\in I}$.

###### Proof.

We have the following diagram

$\textstyle{\mathcal{O}(T_{\operatorname{reg}}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}})[\hbar][\mu,D_{\mu}]_{\mu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\pi}$$\textstyle{QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}(T_{\operatorname{reg}}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}})[\mu,D_{\mu}]_{\mu}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{QH_{T}^{*}(T^{*}\mathcal{B}),}$

where the vertical maps are modulo $\hbar$. Since $T^{*}\mathcal{B}$ is a
homomorphic symplectic variety, the quantum product and the equivariant
product coincide modulo $\hbar$ [BMO11]. Hence, by Proposition 2.1 and
isomorphism (2.5), we see that

$\ker\varphi=\big{<}f(D_{\mu})-f(\mu)\mid
f(\mu)\in\operatorname{Sym}(\mathfrak{t}^{*})^{W}\big{>}.$

Let $J$ be the ideal generated by $\\{\mathcal{R}_{i}\\}_{i\in I}$. Assume
$J\neq\ker\pi$. Let $\gamma$ be an element of minimal $\hbar$-degree such that
$\gamma\in\ker\pi\setminus J$. By the commutativity of the above diagram,

$\varphi(\gamma\bmod\hbar)=\pi(\gamma)\bmod\hbar=0.$

Note that $J/\left<\hbar\right>=\ker\varphi$ by our assumption. We can write

$\gamma=\gamma_{1}+\hbar\gamma_{2}$

for $\gamma_{1}\in J$ and some $\gamma_{2}\notin J$ of $\hbar$-degree less
than $\gamma$. Since $J\subset\ker\pi$, we have $\hbar\pi(\gamma_{2})=0$,
which implies $\pi(\gamma_{2})=0$. Hence, $\gamma_{2}\in\ker\pi\setminus J$,
and the $\hbar$-degree of $\gamma_{2}$ is less than that of $\gamma$, which
contradicts the choice of $\gamma$. ∎

###### Proof of Theorem 4.1.

First of all, we have $\mathcal{R}_{\lambda}^{k}\in\ker\pi$ by Proposition
4.11. Modulo $\hbar$, the matrix $\Theta(\lambda)$ becomes a diagonal matrix
with entries $D_{u\lambda}$ for ${u}\in W^{\lambda}$. Hence,

$\mathcal{R}_{\lambda}^{k}=\sum_{{u}\in
W^{\lambda}}(D_{u\lambda})^{k}-\sum_{{u}\in
W^{\lambda}}(u\lambda)^{k}\mod\hbar.$

By [Hum78, Section 23.1], the set of polynomials $\sum_{u\in
W^{\lambda}}(u\lambda)^{k}$ for all dominant weights $\lambda$ and $k\geq 1$
spans the space $\operatorname{Sym}(\mathfrak{t}^{*})^{W}$. Therefore the
family of all $\mathcal{R}_{\lambda}^{k}$ satisfies the condition of the
Deformation Principle 4.12. We can conclude that all
$\mathcal{R}_{\lambda}^{k}$ generate the kernel of $\pi$. ∎

###  4.3. Explicit formulae for the classical Calogero–Moser map

In this subsection, we will give an explicit formula for the classical
trigonometric Calogero–Moser map (2.8). Recall that
$q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}$ is naturally a
function over
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}$ for any
coroot $\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}$ and
${p}_{\lambda}$ defined in (2.2) is the linear function over $\mathfrak{t}$.
Both of them can be viewed as functions over
$T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t}$
naturally.

Let $Y(\lambda)$ be the matrix obtained from $\Theta(\lambda)$ by replacing
$\Delta_{u\lambda}$ by $p_{u\lambda}$ in (4.3). By Proposition 2.2 (3), we get
immediately

######  Corollary 4.13.

We have

$\eta_{\operatorname{CM},\hbar}^{\operatorname{cl}}\bigg{(}\sum_{u\in
W^{\lambda}}(u\lambda)^{k}\bigg{)}=\operatorname{tr}(Y(\lambda)^{k})\in\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar].$

######  Remark 4.14.

P. Etingof shared with us in private communication that our matrix
$Y(\lambda)$ can be viewed as the matrix of the Dunkl operator under the basis
$W$ of
$\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}}\times\mathfrak{t})[\hbar]\rtimes\mathbb{C}[W]$,
provided that $\lambda$ is strictly dominant (i.e., $W_{\lambda}$ is trivial).
In the rational case (analogous to our case), this method yields the known
explicit formula for the rational Calogero–Moser map in both classical and
quantum cases, obtained based on Lax pairs [SS93]. In the trigonometric case
(our case), Corollary 4.13 can be translated to the following trace-free
formula:

$\operatorname{tr}\left(\sum_{w\in
W}\operatorname{Dun}_{w\lambda}^{k}-\sum_{w\in
W}w\operatorname{Dun}_{\lambda}^{k}w^{-1}\right)=0.$ (4.11)

Recall that $\operatorname{Dun}_{\lambda}$ is the classical trigonometric
Dunkl operator defined in (2.10). It is worth noting that the trigonometric
Dunkl operators and Weyl group elements generate the degenerate Hecke algebra
$\mathcal{H}_{\hbar}$, and thus, unlike the rational case, the two sums in
(4.11) are not equal. To relate our result to the above formula, we need to
note that

$\displaystyle\operatorname{tr}(Y(\lambda)^{k})$
$\displaystyle=\operatorname{tr}(\operatorname{Dun}_{\lambda}^{k})=\frac{1}{|W|}\operatorname{tr}\left(\sum_{w\in
W}w\operatorname{Dun}_{\lambda}^{k}w^{-1}\right).$

On the other hand, by extending [EFMV11, Lemma 2.2] to the trigonometric case,
we can conclude that $\sum_{w\in W}\operatorname{Dun}_{w\lambda}^{k}$ is the
scalar matrix of
$\eta_{\operatorname{CM},\hbar}^{\operatorname{cl}}\left(\sum_{w\in
W}(w\lambda)^{k}\right)$. Hence,

$\displaystyle\eta_{\operatorname{CM},\hbar}^{\operatorname{cl}}\left(\sum_{w\in
W}(w\lambda)^{k}\right)$
$\displaystyle=\frac{1}{|W|}\operatorname{tr}\left(\sum_{w\in
W}\operatorname{Dun}_{w\lambda}^{k}\right).$

Corollary 4.13 is thus equivalent to (4.11). It is interesting to ask for a
direct proof of (4.11). Additionally, it should be noted that the trace-free
formula (4.11) is not valid for the quantum case.

In practical computation, rather than substituting square matrices of size
$|W|$ into a symmetric polynomial, our matrix is small and involves only
matrix powers. This reduces the computational complexity significantly.

######  Example 4.15 (Hamiltonian).

Let $\lambda\in X^{*}(T)_{+}$ be strictly dominant such that $W_{\lambda}$ is
trivial. By direct computations, we have

$\displaystyle\operatorname{tr}\big{(}Y(\lambda)^{2}\big{)}$
$\displaystyle=\sum_{w\in W}{p}_{w\lambda}^{2}+\hbar^{2}\sum_{w\in
W}\sum_{\alpha>0}\left<\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>^{2}\frac{1}{\big{(}1-q^{w\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}\big{)}\big{(}1-q^{-w\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}\big{)}}$
$\displaystyle=\sum_{w\in
W}{p}_{w\lambda}^{2}-\hbar^{2}\sum_{\alpha>0}\bigg{(}\sum_{w\in
W}\left<w\lambda,\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\right>^{2}\bigg{)}\frac{1}{\big{(}q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}/2}-q^{-\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}/2}\big{)}^{2}}.$

Let $C\in\operatorname{Sym}^{2}(\mathfrak{t}^{*})$ be a non-degenerate
quadratic form, which is unique up to a scalar. Thus,

$0\neq\sum_{w\in W}(w\lambda)^{2}=c_{\lambda}\cdot
C\in\operatorname{Sym}^{2}(\mathfrak{t}^{*}),$

for some constant $c_{\lambda}$. Then we can rewrite the above by

$c_{\lambda}\cdot\left(C({p})-\hbar^{2}\sum_{\alpha>0}\frac{C(\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}},\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}})}{\big{(}q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}/2}-q^{-\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}/2}\big{)}^{2}}\right).$

Corollary 4.13 shows

$\eta_{\operatorname{CM},\hbar}^{\operatorname{cl}}(C)=C({p})-\hbar^{2}\sum_{\alpha>0}\frac{C(\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}},\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}})}{\big{(}q^{\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}/2}-q^{-\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}/2}\big{)}^{2}},$

which is the Hamiltonian of the classical Calogero–Moser system (2.8).

##  5\. Computation for Classical Types

In this section, we compute $QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$ for groups
$G$ of classical types.

###  5.1. Type $A_{n-1}$

Recall that $G=SL(n,\mathbb{C})$ in type $A_{n-1}$. While for the convenience
of our computation, we prefer to use $GL$-notations. Let $\hat{T}$ be the
subgroup of diagonal matrices of $\hat{G}:=GL(n,\mathbb{C})$ and
$\hat{\mathfrak{t}}$ be its Lie algebra. We denote
$\hat{\mathbb{G}}=\hat{G}\times\mathbb{C}^{*}$ and
$\hat{\mathbb{T}}=\hat{T}\times\mathbb{C}^{*}$. By carefully examining the
proof, all the argument in this paper works for $GL$ with minor modification
after replacing $\mathfrak{t}$ by $\hat{\mathfrak{t}}$ in all the arguments.

Let us denote $\mathbf{e}_{i}\in X^{*}(\hat{T})$ the projection of the $i$-th
diagonal entry. Thus

$X^{*}(\hat{T})=\mathbb{Z}\mathbf{e}_{1}\oplus\cdots\oplus\mathbb{Z}\mathbf{e}_{n}.$

Denote by $\\{\mathbf{e}_{j}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}\\}$
the dual basis of $\\{\mathbf{e}_{i}\\}$. The simple coroots are given by

$\alpha_{i}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}=\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i}-\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i+1}\quad(i=1,\ldots,n-1).$

We denote
$q_{i}=q^{\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i}}$. Note
that
$q_{i}/q_{j}=q^{\mathbf{e}_{i}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}-\mathbf{e}_{j}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}}\in\operatorname{Eff}(\mathcal{B})$
for $i<j$. Following the notation from classical Schubert calculus (e.g.
[AF]), we denote

$x_{i}=D_{-\mathbf{e}_{i}},\qquad t_{i}=-\mathbf{e}_{i}.$

We denote also

$\chi_{i}=\Delta_{-\mathbf{e}_{i}}=x_{i}+\hbar\sum_{a<i}\frac{q_{a}/q_{i}}{1-q_{a}/q_{i}}-\hbar\sum_{i<b}\frac{q_{i}/q_{b}}{1-q_{i}/q_{b}}.$

By direct computation with respect to the ordered set
$W^{-\mathbf{e}_{1}}=\\{\operatorname{id},s_{1},\ldots,s_{n-1}\cdots
s_{1}\\}$, we have

$M(\chi):=\Theta(-\mathbf{e}_{1})=\left[\begin{matrix}\chi_{1}&\frac{\hbar}{1-q_{1}/q_{2}}&\frac{\hbar}{1-q_{1}/q_{3}}&\cdots&\frac{\hbar}{1-q_{1}/q_{n}}\\\\[4.30554pt]
\frac{\hbar}{1-q_{2}/q_{1}}&\chi_{2}&\frac{\hbar}{1-q_{2}/q_{3}}&\cdots&\frac{\hbar}{1-q_{2}/q_{n}}\\\\[4.30554pt]
\frac{\hbar}{1-q_{3}/q_{1}}&\frac{\hbar}{1-q_{3}/q_{2}}&\chi_{3}&\cdots&\frac{\hbar}{1-q_{3}/q_{n}}\\\\[4.30554pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
\frac{\hbar}{1-q_{n}/q_{1}}&\frac{\hbar}{1-q_{n}/q_{2}}&\frac{\hbar}{1-q_{n}/q_{3}}&\cdots&\chi_{n}\end{matrix}\right].$
(5.1)

From the proof of Proposition 4.11, we have the following identity over
$\mathcal{A}$

$\Sigma(-\mathbf{e}_{1})^{-1}*M(\chi)*\Sigma(-\mathbf{e}_{1})=\Xi(-\mathbf{e}_{1})=\operatorname{diag}(t_{1},\ldots,t_{n}).$

Let $y$ be an indeterminant. We thus have

$\det\bigg{(}y1_{n}+M(\chi)\bigg{)}=\prod_{i=1}^{n}(y+t_{i}).$

Expand the characteristic polynomial

$\det\bigg{(}y1_{n}+M(\chi)\bigg{)}=y^{n}+\mathcal{E}_{1}(\chi)y^{n-1}+\cdots+\mathcal{E}_{n-1}(\chi)y+\mathcal{E}_{n}(\chi).$

Thus the following relation holds in
$QH_{\hat{\mathbb{T}}}^{*}(T^{*}\mathcal{B})$,

$\mathcal{E}_{k}(\chi)=e_{k}(t)$

where $e_{k}$ is the $k$-th elementary symmetric polynomial. Since the algebra
of $W$-invariant polynomials is generated by the $k$-th elementary symmetric
polynomials. By Proposition 4.12, we can conclude the following presentation
of quantum cohomology.

######  Theorem 5.1.

For type $A_{n-1}$, we have

$QH^{*}_{\hat{\mathbb{T}}}(T^{*}\mathcal{B})=\frac{\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})[\hbar,x_{1},\ldots,x_{n},t_{1},\ldots,t_{n}]}{\left<\mathcal{E}_{k}(\chi)-e_{k}(t),k=1,\ldots,n\right>}.$
(5.2)

Consequently, since $T$ is cut from $\hat{T}$ by the relation
$\mathbf{e}_{1}+\cdots+\mathbf{e}_{n}=0$, we can conclude that

$QH^{*}_{{\mathbb{T}}}(T^{*}\mathcal{B})=QH^{*}_{\hat{\mathbb{T}}}(T^{*}\mathcal{B})/\left<e_{1}(t)\right>.$

Next, we will give a combinatorial formula for $\mathcal{E}_{k}(\chi)$. Let
$K$ be a subset of $\\{1,\ldots,n\\}$ of order $k$. A _matching_ of $K$ is a
map $\pi:K\to K$ with $\pi^{2}=\operatorname{id}$, i.e. an involution over
$K$. A matching $\pi$ is called perfect, if the involution $\pi$ has no fixed
points. Let us denote

$J(\pi)=\prod_{\pi(i)=i}\chi_{i}\quad\text{and}\quad
V(\pi)=\prod_{\pi(i)=j>i}\frac{\hbar^{2}q_{i}q_{j}}{(q_{i}-q_{j})^{2}}.$

Note that
$\frac{q_{i}q_{j}}{(q_{i}-q_{j})^{2}}=\frac{q_{i}/q_{j}}{(1-q_{i}/q_{j})^{2}}\in\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})$.

######  Theorem 5.2.

For any $1\leq k\leq n$,

$\mathcal{E}_{k}(\chi)=\sum_{K}\sum_{\pi}J(\pi)V(\pi),$ (5.3)

where the sum over all $k$-subsets $K$ of $\\{1,\ldots,n\\}$, and matchings
$\pi$ of $K$.

The proof of the above theorem involves some interesting equalities from the
viewpoint of combinatorics. We leave the details in Appendix B.

######  Example 5.3.

For $n=3$,

$\displaystyle\mathcal{E}_{1}(\chi)$
$\displaystyle=\chi_{1}+\chi_{2}+\chi_{3}$
$\displaystyle\mathcal{E}_{2}(\chi)$
$\displaystyle=\chi_{1}\chi_{2}+\chi_{1}\chi_{3}+\chi_{2}\chi_{3}+\frac{\hbar^{2}q_{1}q_{2}}{(q_{1}-q_{2})^{2}}+\frac{\hbar^{2}q_{1}q_{3}}{(q_{1}-q_{3})^{2}}+\frac{\hbar^{2}q_{2}q_{3}}{(q_{2}-q_{3})^{2}}$
$\displaystyle\mathcal{E}_{3}(\chi)$
$\displaystyle=\chi_{1}\chi_{2}\chi_{3}++\chi_{3}\frac{\hbar^{2}q_{1}q_{2}}{(q_{1}-q_{2})^{2}}+\chi_{2}\frac{\hbar^{2}q_{1}q_{3}}{(q_{1}-q_{3})^{2}}+\chi_{1}\frac{\hbar^{2}q_{2}q_{3}}{(q_{2}-q_{3})^{2}}.$

We can illustrate each term above by drawing the matching as follows

$\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(0.5,0.5)(0.0,-0.2)(0.5,-0.2)\qbezier(0.5,0.5)(1.0,-0.2)(0.5,-0.2)\end{picture}\\\
\chi_{1}\end{array}\quad\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(1.5,0.5)(1.0,-0.2)(1.5,-0.2)\qbezier(1.5,0.5)(2.0,-0.2)(1.5,-0.2)\end{picture}\\\
\chi_{2}\end{array}\quad\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(2.5,0.5)(2.0,-0.2)(2.5,-0.2)\qbezier(2.5,0.5)(3.0,-0.2)(2.5,-0.2)\end{picture}\\\
\chi_{3}\end{array}$
$\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(0.5,0.5)(0.0,-0.2)(0.5,-0.2)\qbezier(0.5,0.5)(1.0,-0.2)(0.5,-0.2)\qbezier(1.5,0.5)(1.0,-0.2)(1.5,-0.2)\qbezier(1.5,0.5)(2.0,-0.2)(1.5,-0.2)\end{picture}\\\
\chi_{1}\chi_{2}\end{array}\quad\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(0.5,0.5)(0.0,-0.2)(0.5,-0.2)\qbezier(0.5,0.5)(1.0,-0.2)(0.5,-0.2)\qbezier(2.5,0.5)(2.0,-0.2)(2.5,-0.2)\qbezier(2.5,0.5)(3.0,-0.2)(2.5,-0.2)\end{picture}\\\
\chi_{1}\chi_{3}\end{array}\quad\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(1.5,0.5)(1.0,-0.2)(1.5,-0.2)\qbezier(1.5,0.5)(2.0,-0.2)(1.5,-0.2)\qbezier(2.5,0.5)(2.0,-0.2)(2.5,-0.2)\qbezier(2.5,0.5)(3.0,-0.2)(2.5,-0.2)\end{picture}\\\
\chi_{2}\chi_{3}\end{array}\quad\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(0.5,0.5)(1.0,-0.2)(1.5,0.5)\end{picture}\\\
\frac{\hbar^{2}q_{1}q_{2}}{(q_{1}-q_{2})^{2}}\end{array}\quad\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(0.5,0.5)(1.5,-0.5)(2.5,0.5)\end{picture}\\\
\frac{\hbar^{2}q_{1}q_{3}}{(q_{1}-q_{3})^{2}}\end{array}\quad\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(1.5,0.5)(2.0,-0.2)(2.5,0.5)\end{picture}\\\
\frac{\hbar^{2}q_{2}q_{3}}{(q_{2}-q_{3})^{2}}\end{array}$
$\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(0.5,0.5)(0.0,-0.2)(0.5,-0.2)\qbezier(0.5,0.5)(1.0,-0.2)(0.5,-0.2)\qbezier(1.5,0.5)(1.0,-0.2)(1.5,-0.2)\qbezier(1.5,0.5)(2.0,-0.2)(1.5,-0.2)\qbezier(2.5,0.5)(2.0,-0.2)(2.5,-0.2)\qbezier(2.5,0.5)(3.0,-0.2)(2.5,-0.2)\end{picture}\\\\[4.30554pt]
\chi_{1}\chi_{2}\chi_{3}\end{array}\quad\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(0.5,0.5)(1.0,-0.2)(1.5,0.5)\qbezier(2.5,0.5)(2.0,-0.2)(2.5,-0.2)\qbezier(2.5,0.5)(3.0,-0.2)(2.5,-0.2)\end{picture}\\\\[4.30554pt]
\chi_{3}\frac{\hbar^{2}q_{1}q_{2}}{(q_{1}-q_{2})^{2}}\end{array}\quad\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(0.5,0.5)(1.5,-1.2)(2.5,0.5)\qbezier(1.5,0.5)(1.1,-0.2)(1.5,-0.2)\qbezier(1.5,0.5)(1.9,-0.2)(1.5,-0.2)\end{picture}\\\\[4.30554pt]
\chi_{2}\frac{\hbar^{2}q_{1}q_{3}}{(q_{1}-q_{3})^{2}}\end{array}\quad\begin{array}[]{c}\begin{picture}(3.0,1.0)\put(0.3,0.5){${}^{1}$}
\put(0.5,0.5){\circle*{0.3}} \put(1.3,0.5){${}^{2}$}
\put(1.5,0.5){\circle*{0.3}} \put(2.3,0.5){${}^{3}$}
\put(2.5,0.5){\circle*{0.3}}
\qbezier(1.5,0.5)(2.0,-0.2)(2.5,0.5)\qbezier(0.5,0.5)(0.0,-0.2)(0.5,-0.2)\qbezier(0.5,0.5)(1.0,-0.2)(0.5,-0.2)\end{picture}\\\\[4.30554pt]
\chi_{1}\frac{\hbar^{2}q_{2}q_{3}}{(q_{2}-q_{3})^{2}}\end{array}$

###  5.2. Type $B_{n}$ and type $C_{n}$

The weight lattice is naturally recognized as

$\mathbb{Z}\mathbf{e}_{1}\oplus\cdots\oplus\mathbb{Z}\mathbf{e}_{n}$

with the set of simple coroots

$\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i}=\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i}-\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i+1}\quad(i=1,\ldots,n-1)\qquad\alpha_{n}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}=p\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{n}$

where $p=2$ for type $B_{n}$ and $p=1$ for type $C_{n}$. We denote

$q_{i}=q^{\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i}},\qquad
x_{i}=D_{-\mathbf{e}_{i}},\qquad\chi_{i}=\Delta_{-\mathbf{e}_{i}},\qquad
t_{i}=-\mathbf{e}_{i}.$

With respect to the ordered set

$W^{-\mathbf{e}_{1}}=\big{\\{}\operatorname{id},s_{1},{\ldots},s_{n}{\cdot}{\cdot}{\cdot}s_{1},s_{n\texttt{-}1}s_{n}{\cdot}{\cdot}{\cdot}s_{1},{\ldots},s_{1}{\cdot}{\cdot}{\cdot}s_{n\texttt{-}1}s_{n}{\cdot}{\cdot}{\cdot}s_{1}\big{\\}},$

we have

$M(\chi):=\Theta(-\mathbf{e}_{1})=\left[\begin{array}[]{@{}c@{\,}c@{\,}c|c@{\,}c<EMAIL_ADDRESS>\vdots&\ddots&\vdots&\vdots&\reflectbox{$\ddots$}&\vdots\\\\[4.30554pt]
\frac{\hbar}{1-q_{n}/q_{1}}&\cdots&\chi_{n}&\frac{p\hbar}{1-q_{n}^{p}}&\cdots&\frac{\hbar}{1-q_{1}q_{n}}\\\\[4.30554pt]
\hline\cr\rule{0.0pt}{12.91663pt}\frac{\hbar}{1-q_{n}^{-1}q_{1}^{-1}}&\cdots&\frac{p\hbar}{1-q_{n}^{-p}}&-\chi_{n}&\cdots&\frac{\hbar}{1-q_{1}/q_{n}}\\\\[4.30554pt]
\vdots&\reflectbox{$\ddots$}&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
\frac{p\hbar}{1-q_{1}^{-p}}&\cdots&\frac{\hbar}{1-q_{1}^{-1}q_{n}^{-1}}&\frac{\hbar}{1-q_{n}/q_{1}}&\cdots&-\chi_{1}\end{array}\right].$
(5.4)

Here in the upper right (resp. lower left) block, the $(i,j)$-entry is given
by $\frac{\hbar}{1-q_{i}q_{n+1-j}}$ (resp.
$\frac{\hbar}{1-q_{n+1-i}^{-1}q_{j}^{-1}}$) when $i+j\neq n+1$ and
${p\hbar\over 1-q_{i}^{p}}$ (resp. ${p\hbar\over 1-q_{j}^{-p}}$) otherwise.
Similar to the case of type $A$, we have

$\det\bigg{(}y1_{2n}+M(\chi)\bigg{)}=\prod_{i=1}^{n}(y^{2}-t_{i}^{2}).$

Expand the characteristic polynomial

$\det\bigg{(}y1_{2n}+M(\chi)\bigg{)}=y^{2n}+\mathcal{E}_{1}(\chi)y^{2n-1}+\cdots+\mathcal{E}_{2n-1}(\chi)y+\mathcal{E}_{2n}(\chi).$

Thus for each $k\in\\{1,\ldots,n\\}$, $\mathcal{E}_{2k-1}(\chi)=0$, and the
following relation holds in $QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$,

$\mathcal{E}_{2k}(\chi)=(-1)^{k}e_{k}(t^{2}).$

Since the algebra of $W$-invariant polynomial is generated by $k$-th
elementary symmetric polynomials in squares. By Deformation Principle in
Proposition 4.12, we can conclude the following presentation of quantum
cohomology.

######  Theorem 5.4.

For $G$ of type $B_{n}$ or $C_{n}$, we have

$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})=\frac{\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})[\hbar,x_{1},\ldots,x_{n},t_{1},\ldots,t_{n}]}{\left<\mathcal{E}_{2k}(\chi)-(-1)^{k}e_{k}(t^{2}),k=1,\ldots,n\right>}.$
(5.5)

###  5.3. Type $D_{n}$

The weight lattice is naturally recognized as

$\bigg{\\{}\lambda_{1}\mathbf{e}_{1}+\cdots+\lambda_{n}\mathbf{e}_{n}\,\bigg{|}\,\lambda_{i}\in\frac{1}{2}\mathbb{Z},\lambda_{i}-\lambda_{j}\in\mathbb{Z}.\bigg{\\}}$

with the set of simple coroots

$\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i}=\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i}-\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i+1}\quad(i=1,\ldots,n-1)\qquad\alpha^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{n}=\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{n-1}+\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{n}.$

We denote

$q_{i}=q^{\mathbf{e}^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{i}},\qquad
x_{i}=D_{-\mathbf{e}_{i}},\qquad\chi_{i}=\Delta_{-\mathbf{e}_{i}},\qquad
t_{i}=-\mathbf{e}_{i}.$

With respect to the ordered set

$W^{-\mathbf{e}_{1}}=\big{\\{}\operatorname{id},s_{1},{\ldots},s_{n\texttt{-}1}s_{n\texttt{-}2}{\cdot}{\cdot}{\cdot}s_{1},s_{n}s_{n\texttt{-}2}{\cdot}{\cdot}{\cdot}s_{1},s_{n}{\cdot}{\cdot}{\cdot}s_{1},s_{n\texttt{-}2}s_{n}{\cdot}{\cdot}{\cdot}s_{1},{\ldots},s_{1}{\cdot}{\cdot}{\cdot}s_{n\texttt{-}2}s_{n}{\cdot}{\cdot}{\cdot}s_{1}\big{\\}},$

we have

$M(\chi):=\Theta(-\mathbf{e}_{1})=\left[\begin{array}[]{@{}c@{\,}c@{\,}c|c@{\,}c<EMAIL_ADDRESS>\vdots&\ddots&\vdots&\vdots&\reflectbox{$\ddots$}&\vdots\\\\[4.30554pt]
\frac{\hbar}{1-q_{n}/q_{1}}&\cdots&\chi_{n}&0&\cdots&\frac{\hbar}{1-q_{n}q_{1}}\\\\[4.30554pt]
\hline\cr\rule{0.0pt}{12.91663pt}\frac{\hbar}{1-q_{n}^{-1}q_{1}^{-1}}&\cdots&0&-\chi_{n}&\cdots&\frac{\hbar}{1-q_{1}/q_{n}}\\\\[4.30554pt]
\vdots&\reflectbox{$\ddots$}&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
0&\cdots&\frac{\hbar}{1-q_{1}^{-1}q_{n}^{-1}}&\frac{\hbar}{1-q_{n}/q_{1}}&\cdots&-\chi_{1}\end{array}\right].$
(5.6)

Here in the upper right (resp. lower left) block, the $(i,j)$-entry is given
by $\frac{\hbar}{1-q_{i}q_{n+1-j}}$ (resp.
$\frac{\hbar}{1-q_{n+1-i}^{-1}q_{j}^{-1}}$) when $i+j\neq n+1$ and $0$
otherwise. Similar to the case of type $A$, we have

$\det\bigg{(}y1_{2n}+M(\chi)\bigg{)}=\prod_{i=1}^{n}(y^{2}-t_{i}^{2}).$

Expand the characteristic polynomial

$\det\bigg{(}y1_{2n}+M(\chi)\bigg{)}=y^{2n}+\mathcal{E}_{1}(\chi)y^{2n-1}+\cdots+\mathcal{E}_{2n-1}(\chi)y+\mathcal{E}_{2n}(\chi).$

Thus for each $k\in\\{1,\ldots,n\\}$, $\mathcal{E}_{2k-1}(\chi)=0$ and the
following relation holds in $QH_{\mathbb{T}}^{*}(T^{*}\mathcal{B})$,

$\mathcal{E}_{2k}(\chi)=(-1)^{k}e_{k}(t^{2}).$

For type $D_{n}$, the polynomial $e_{n}(t)$ is also $W$-invariant. In
particular,

$(-1)^{n}\mathcal{E}_{2n}(\chi)=(-1)^{n}\det
M(\chi)=e_{n}(t^{2})=e_{n}(t)^{2}.$

Consider

$A(\chi)=\left[\begin{array}[]{@{}c@{\,}c@{\,}c@{\,}c@{}}\chi_{1}&\frac{\hbar(1-q_{1})(1+q_{2})}{(1-q_{1}/q_{2})(1-q_{1}q_{2})}&\cdots&\frac{\hbar(1-q_{1})(1+q_{n})}{(1-q_{1}/q_{n})(1-q_{1}q_{n})}\\\\[4.30554pt]
\frac{\hbar(1-q_{2})(1+q_{1})}{(1-q_{2}/q_{1})(1-q_{1}q_{2})}&\chi_{2}&\cdots&\frac{\hbar(1-q_{2})(1+q_{n})}{(1-q_{2}/q_{n})(1-q_{2}q_{n})}\\\\[4.30554pt]
\vdots&\vdots&\ddots&\vdots\\\
\frac{\hbar(1-q_{n})(1+q_{1})}{(1-q_{n}/q_{1})(1-q_{1}q_{n})}&\frac{\hbar(1-q_{n})(1+q_{2})}{(1-q_{n}/q_{2})(1-q_{2}q_{n})}&\cdots&\chi_{n}\end{array}\right].$
(5.7)

######  Lemma 5.5.

We have

$\det M(\chi)=\det A(\chi)\cdot\det A(-\chi).$

###### Proof.

Denote

$L=\left[\begin{array}[]{@{}c@{\,}c@{\,}c@{\,\,}|@{\,}c@{\\!\\!}c<EMAIL_ADDRESS>&\ddots&&&\reflectbox{$\ddots$}\\\ &&1&-q_{n}^{-1}&\\\ \hline\cr-
q_{1}&&&&&-1\\\\[-4.30554pt] &\ddots&&&\reflectbox{$\ddots$}\\\
&&-q_{n}&-1&&\end{array}\right],\qquad
R=\left[\begin{array}[]{c@{\,\,\,}c@{\,\,\,}c@{\,\,\,}|c@{}c<EMAIL_ADDRESS>&\ddots&&&\ddots\\\ &&1&&&q_{n}^{-1}\\\ \hline\cr&&q_{n}&&&-1\\\\[-4.30554pt]
&\reflectbox{$\ddots$}&&&\reflectbox{$\ddots$}\\\\[-4.30554pt]
q_{1}&&&-1&&\end{array}\right].$

By direct computation, we have

$\frac{1}{2}LM(\chi)R=\left[\begin{matrix}A(\chi)\\\ &A(-\chi)^{\tt
t}\end{matrix}\right].$

Note that

$\frac{1}{2}RL=\left[\begin{array}[]{@{}c@{\,\,\,}c@{\,\,\,}c@{\,\,\,}|@{\,}c@{\\!\\!\\!}c<EMAIL_ADDRESS>&&&&\reflectbox{$\ddots$}\\\ &&&-q_{n}^{-1}&\\\
\hline\cr&&q_{n}&&&\\\\[-4.30554pt] &\reflectbox{$\ddots$}&&&\\\
q_{1}&&\end{array}\right]$

whose determinant is $1$. As a result, we get the assertion. ∎

Consider the expansion of

$\det
A(\chi)=\sum_{\sigma\in\mathfrak{S}_{n}}(-1)^{\ell(\sigma)}m_{\sigma},\qquad
m_{\sigma}=A(\chi)_{1\sigma(1)}\cdots A(\chi)_{n\sigma(n)}.$

######  Lemma 5.6.

We have

$\det A(\chi)=\sum_{\sigma}(-1)^{\ell(\sigma)}m_{\sigma}$

with $\sigma$ going through permutations whose nontrivial cycles all have even
lengths.

###### Proof.

If $\sigma$ is a permutation having a non-trivial odd cycle, we define
$\sigma^{\prime}$ by inverting the unique odd cycle of $\sigma$ which contains
the smallest index. Since the correspondence $\sigma\mapsto\sigma^{\prime}$ is
an involution, it suffices to show

$m_{\sigma}+m_{\sigma^{\prime}}=0.$

We can assume without loss of generality that the odd cycle is $(1\cdots k)$
for odd $k$. Note that

$\displaystyle{A(\chi)_{12}A(\chi)_{23}\cdots A(\chi)_{k1}}$
$\displaystyle=\prod_{i}\frac{\hbar(1-q_{i})(1+q_{i+1})}{(1-q_{i}/q_{i+1})(1-q_{i}q_{i+1})}$
$\displaystyle=\prod_{i}\frac{\hbar(1-q_{i}^{2})q_{i}}{(q_{i+1}-q_{i})(1-q_{i}q_{i+1})}$
$\displaystyle=(-1)^{k}{A(\chi)_{1k}A(\chi)_{21}\cdots A(\chi)_{k,k-1}}.$

where the indices are understood as elements of $\mathbb{Z}/k\mathbb{Z}$. So
we can deduce $m_{\sigma}+m_{\sigma^{\prime}}=0$. ∎

In particular, we have

$\det A(\chi)=\det\big{(}{-}A(-\chi)\big{)}=(-1)^{n}\det A(-\chi).$

Combining Lemma 5.5, we have

$\big{(}\det
A(\chi)\big{)}^{2}=(-1)^{n}\det\big{(}M(\chi)\big{)}=e_{n}(t)^{2}.$

Note that $e_{n}(t)\in H_{G}^{*}(\operatorname{pt})$ and we can view $\det
A(\chi)$ an element of $QH^{*}_{\mathbb{G}}(T^{*}\mathcal{B})$, see Remark
4.3. Since $QH^{*}_{\mathbb{G}}(T^{*}\mathcal{B})$ is a domain by Proposition
2.2 (3), we have $\det A(\chi)=\pm e_{n}(t)$. As $\det
A(\chi)=e_{n}(x)\bmod\hbar$, the only possibility is $\det A(\chi)=e_{n}(t)$.

######  Theorem 5.7.

For $G$ of type $D_{n}$, we have

$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})=\frac{\mathcal{O}(T^{\mathchoice{\vee}{\vee}{\vee}{\mathsf{v}}}_{\operatorname{reg}})[\hbar,x_{1},\ldots,x_{n},t_{1},\ldots,t_{n}]}{\left<\begin{array}[]{l}\mathcal{E}_{2k}(\chi)-(-1)^{k}e_{k}(t^{2}),k=1,\ldots,n-1\\\
\det A(\chi)-e_{n}(t)\end{array}\right>}.$ (5.8)

##  Appendix A Toda limit

The equivariant quantum cohomology of $\mathcal{B}$ can be obtained from that
of $T^{*}\mathcal{B}$, by taking a Toda limit as follows.

######  Definition A.1.

The _Toda limit_ of a class $\gamma\in QH^{*}_{\mathbb{G}}(T^{*}\mathcal{B})$
is defined to be

$\lim\nolimits^{\mathrm{td}}\gamma=\lim_{\hbar\to\infty}\gamma(\hbar,\hbar^{-2\rho}q)\in
QH_{G}^{*}(\mathcal{B}),$

where $\gamma(\hbar,\hbar^{-2\rho}q)$ denotes the class obtained from $\gamma$
by the substitution

$q^{\beta}\mapsto\hbar^{-\left<2\rho,\beta\right>}q^{\beta}.$

When $\gamma\in QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$, $\gamma\in
QH^{*}_{T}(\mathcal{B})$ is also called the Toda limit.

######  Proposition A.2 ([BMO11, Proposition 8.1]).

For any classes $\gamma_{1},\gamma_{2}\in
QH^{*}_{\mathbb{G}}(T^{*}\mathcal{B})$ whose Toda limits both converge, we
have

$\big{(}\lim\nolimits^{\mathrm{td}}\gamma_{1}\big{)}*\big{(}\lim\nolimits^{\mathrm{td}}\gamma_{2}\big{)}=\lim\nolimits^{\mathrm{td}}(\gamma_{1}*\gamma_{2})$

These also hold if we replace $\mathbb{G}$ by $\mathbb{T}$.

######  Remark A.3.

Proposition 8.1 of [BMO11] only stated for the case when $\gamma_{1}$ is a
divisor. Nevertheless, its proof therein works in general. This also follows
from the fact that $QH^{*}_{\mathbb{G}}(T^{*}\mathcal{B})$ is generated by
divisors.

By taking the Toda limit, we can obtain a ring presentation of the equivariant
cohomology $QH^{*}_{T}(\mathcal{B})$ from
$QH^{*}_{\mathbb{T}}(T^{*}\mathcal{B})$. Below we discuss such presentation
for $\mathcal{B}$ of classical type, and compare them with the nice
description by using integrals of motions of the Toda lattice for the
Langlands-dual Lie group by Kim [Kim99].

### Type $A_{n-1}$

Note $\rho=(n-1)\mathbf{e}_{1}+(n-2)\mathbf{e}_{2}+\cdots+\mathbf{e}_{n-1}.$
We conjugate (5.1) by

$\operatorname{diag}\big{(}1,-\hbar,\dots,(-\hbar)^{n-1}\big{)}.$

Note that

$\lim\nolimits^{\mathrm{td}}\frac{(-1)^{i-j}\hbar^{1+i-j}}{1-q_{i}/q_{j}}=\lim_{\hbar\to\infty}\frac{(-1)^{j-i}\hbar^{1+i-j}}{1-\hbar^{2(i-j)}q_{i}/q_{j}}=\begin{cases}q_{j}/q_{j+1},&i=j+1,\\\
-1,&j=i+1,\\\ 0&\text{otherwise}.\end{cases}$

After taking the Toda limit, we obtain the following matrix

$\left[\begin{matrix}x_{1}&-1&0&\cdots&0\\\\[4.30554pt]
q_{1}/q_{2}&x_{2}&-1&\cdots&0\\\\[4.30554pt]
0&q_{2}/q_{3}&x_{3}&\cdots&0\\\\[4.30554pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
0&0&0&\cdots&x_{n}\end{matrix}\right].$ (A.1)

By Proposition A.2, this covers the representation of
$QH^{*}_{T}(\mathcal{B})$ of type $A_{n-1}$ in [Kim99].

### Type $B_{n}$

Note
$\rho=(n-\tfrac{1}{2})\mathbf{e}_{1}+(n-\tfrac{3}{2})\mathbf{e}_{2}+\cdots+\tfrac{1}{2}\mathbf{e}_{n}.$
We conjugate (5.4) by

$\operatorname{diag}\big{(}\hbar^{-n+1/2},\dots,\hbar^{-1/2},\hbar^{1/2},\dots,\hbar^{n-1/2}\big{)}.$

By taking the Toda limit entry-wise, we obtain the following matrix

$\left[\begin{array}[]{cccc|cccc}x_{1}&1&&&&\\\\[4.30554pt]
-q_{1}/q_{2}&x_{2}&\ddots&&&&\\\\[4.30554pt] &\ddots&\ddots&1\\\\[4.30554pt]
&&-q_{n-1}/q_{n}&x_{n}&2\\\\[4.30554pt]
\hline\cr\rule{0.0pt}{12.91663pt}&&&-2q_{n}^{2}&-x_{n}&1\\\\[4.30554pt]
&&&&-q_{n}/q_{n-1}&\ddots&\ddots\\\\[4.30554pt]
&&&&&\ddots&-x_{2}&1\\\\[4.30554pt] &&&&&&-q_{1}/q_{2}&-x_{1}\\\\[4.30554pt]
\end{array}\right].$ (A.2)

We need the following lemma.

######  Lemma A.4 ([Mui60]).

The determinant of a tridiagonal matrix is given by

$\det\left[\begin{matrix}x_{1}&a_{1}&0&\cdots&0\\\\[4.30554pt]
b_{1}&x_{2}&a_{2}&\cdots&0\\\\[4.30554pt]
0&b_{2}&x_{3}&\cdots&0\\\\[4.30554pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
0&0&0&\cdots&x_{n}\end{matrix}\right]=\sum_{\sigma}\bigg{(}\prod_{i=\sigma(i)}x_{i}\prod_{\sigma(i)=i+1}(-a_{i}b_{i})\bigg{)},$

where $\sigma$ goes over involutions over $\\{1,\ldots,n\\}$ such that
$|\sigma(i)-i|\leq 1$ for any $1\leq i\leq n$.

As a corollary, replacing $(a_{i},b_{i})$ by $(-a_{i},-b_{i})$ for any $1\leq
i\leq n$ does not affect the characteristic polynomial of a tridiagonal
matrix. Using this Lemma A.4, by suitable re-assigning sign for (A.2) and
rearranging columns and rows if necessary, we recover the matrix in
$\mathfrak{sp}_{n}$ obtained in [Kim99].

### Type $C_{n}$

Note $\rho=n\mathbf{e}_{1}+(n-1)\mathbf{e}_{2}+\cdots+\mathbf{e}_{n}.$ We can
conjugate via the same diagonal matrix

$\operatorname{diag}\big{(}\hbar^{-n+1/2},\dots,\hbar^{-1/2},\hbar^{1/2},\dots,\hbar^{n-1/2}\big{)}.$

Since the $p$ is different from type $B$, we obtain the following matrix

$M=\left[\begin{array}[]{<EMAIL_ADDRESS>-q_{1}/q_{2}&x_{2}&\ddots&&&&\\\\[4.30554pt] &\ddots&\ddots&1\\\\[4.30554pt]
&&-q_{n-1}/q_{n}&x_{n}&1\\\\[4.30554pt]
\hline\cr\rule{0.0pt}{12.91663pt}&&&-q_{n}&-x_{n}&1\\\\[4.30554pt]
&&-q_{n-1}&&-q_{n}/q_{n-1}&\ddots&\ddots\\\\[4.30554pt]
&\reflectbox{$\ddots$}&&&&\ddots&-x_{2}&1\\\\[4.30554pt]
-q_{1}&&&&&&-q_{1}/q_{2}&-x_{1}\\\\[4.30554pt] \end{array}\right].$ (A.3)

######  Lemma A.5.

The polynomial $y\det(y1_{2n}+M)$ is the characteristic polynomial of the
following matrix

$\left[\begin{array}[]{@{}c@{\,\,}c@{\,\,}c@{\,\,}c|c|c@{\,}c@{\,}c<EMAIL_ADDRESS>q_{1}/q_{2}&x_{2}&\ddots&&&\\\\[4.30554pt] &\ddots&\ddots&-1&\\\\[4.30554pt]
&&q_{n-1}/q_{n}&x_{n}&-1/2\\\\[4.30554pt]
\hline\cr\rule{0.0pt}{12.91663pt}&&&q_{n}&0&1/2\\\\[4.30554pt]
\hline\cr\rule{0.0pt}{12.91663pt}&&&&-q_{n}&-x_{n}&1\\\\[4.30554pt]
&&&&&-q_{n}/q_{n-1}&\ddots&\ddots\\\\[4.30554pt]
&&&&&&\ddots&-x_{2}&1\\\\[4.30554pt] &&&&&&&-q_{1}/q_{2}&-x_{1}\\\\[4.30554pt]
\end{array}\right].$ (A.4)

###### Proof.

We just give a sketch here. By expanding the determinant, the characteristic
polynomial $\det(y1_{2n}+M)$ equals

$q_{1}+q_{2}\det\big{[}\begin{subarray}{c}y+x_{1}\\\\[2.15277pt]
\phantom{y-x_{0}}\end{subarray}\begin{subarray}{c}\phantom{y-x_{0}}\\\\[2.15277pt]
y-x_{1}\end{subarray}\big{]}+q_{3}\det\left[\begin{subarray}{c}y+x_{1}\\\\[2.15277pt]
-q_{1}/q_{2}\\\\[2.15277pt] \phantom{y-x_{0}}\\\\[2.15277pt]
\phantom{y-x_{0}}\\\\[2.15277pt]
\end{subarray}\,\,\begin{subarray}{c}1\\\\[2.15277pt] y+x_{2}\\\\[2.15277pt]
\phantom{y-x_{0}}\\\\[2.15277pt] \phantom{y-x_{0}}\\\\[2.15277pt]
\end{subarray}\,\,\begin{subarray}{c}\phantom{y-x_{0}}\\\\[2.15277pt]
\phantom{y-x_{0}}\\\\[2.15277pt] y-x_{2}\\\\[2.15277pt]
-q_{1}/q_{2}\\\\[2.15277pt]
\end{subarray}\,\,\begin{subarray}{c}\phantom{y-x_{0}}\\\\[2.15277pt]
\phantom{y-x_{0}}\\\\[2.15277pt] 1\\\\[2.15277pt] y-x_{1}\\\\[2.15277pt]
\end{subarray}\right]+\cdots$

Let us write $yq_{i}$ as

$\frac{1}{2}(q_{i}/q_{i+1})\cdots(q_{n-1}/q_{n})q_{n}(y+x_{i})+\frac{1}{2}(q_{i}/q_{i+1})\cdots(q_{n-1}/q_{n})q_{n}(y-x_{i}).$

By Lemma A.4 and tedious computation, we can conclude that $y\det(y1_{2n}+M)$
coincides with the characteristic polynomial of the matrix (A.4). ∎

By suitable re-assigning sign for (A.4) and rearranging columns and rows if
necessary, we recover the matrix in $\mathfrak{so}_{2n+1}$ obtained in
[Kim99].

### Type $D_{n}$

Note $\rho=(n-1)\mathbf{e}_{1}+(n-2)\mathbf{e}_{2}+\cdots+\mathbf{e}_{n-1}.$
We can conjugate the diagonal matrix

$\operatorname{diag}\big{(}\hbar^{-n+1},\dots,-\hbar^{-1},1,1,\hbar,\dots,\hbar^{n-1}\big{)}.$

We obtain the following matrix

$\left[\begin{array}[]{@{}cc@{\,}c<EMAIL_ADDRESS>-q_{1}/q_{2}&x_{2}&\ddots&&&&\\\\[4.30554pt] &\ddots&\ddots&1&1\\\\[4.30554pt]
&&-q_{n-1}/q_{n}&x_{n}&&1\\\\[4.30554pt]
\hline\cr\rule{0.0pt}{12.91663pt}&&-q_{n-1}q_{n}&&-x_{n}&1\\\\[4.30554pt]
&&&-q_{n-1}q_{n}&-q_{n}/q_{n-1}&\ddots&\ddots\\\\[4.30554pt]
&&&&&\ddots&-x_{2}&1\\\\[4.30554pt] &&&&&&-q_{1}/q_{2}&-x_{1}\\\\[4.30554pt]
\end{array}\right].$ (A.5)

By expanding determinant directly, it is not hard to show that it has the same
characteristic polynomial as that of

$\left[\begin{array}[]{@{}cc@{\,}c<EMAIL_ADDRESS>q_{1}/q_{2}&x_{2}&\ddots&&&&\\\\[4.30554pt]
&\ddots&\ddots&-1&-1\\\\[4.30554pt] &&q_{n-1}/q_{n}&x_{n}&&1\\\\[4.30554pt]
\hline\cr\rule{0.0pt}{12.91663pt}&&q_{n-1}q_{n}&&-x_{n}&1\\\\[4.30554pt]
&&&-q_{n-1}q_{n}&-q_{n}/q_{n-1}&\ddots&\ddots\\\\[4.30554pt]
&&&&&\ddots&-x_{2}&1\\\\[4.30554pt] &&&&&&-q_{1}/q_{2}&-x_{1}\\\\[4.30554pt]
\end{array}\right].$ (A.6)

which is the matrix for $\mathfrak{so}_{2n}$ in [Kim99].

##  Appendix B Proof of Theorem 5.2

For type $A_{n-1}$, the Weyl group $W$ is given by the symmetric group
$\mathfrak{S}_{n}$.

######  Lemma B.1.

For $n\geq 2$, we have

$\sum_{\begin{subarray}{c}\sigma\in\mathfrak{S}_{n}\\\
\sigma(1)=1\end{subarray}}\frac{1}{t_{\sigma(1)}{-}t_{\sigma(2)}}\cdots\frac{1}{t_{\sigma(n{-}1)}{-}t_{\sigma(n)}}\frac{1}{t_{\sigma(n)}{-}t_{\sigma(1)}}=\begin{cases}\frac{-1}{(t_{1}-t_{2})^{2}},\\!\\!\\!&n=2,\\\
0,&n>2.\end{cases}$ (B.1)

###### Proof.

This assertion is obvious for $n=2$. Now we assume $n>2$. It suffices to show

$\sum_{\sigma\in\mathfrak{S}_{n}}\frac{1}{t_{\sigma(1)}-t_{\sigma(2)}}\cdots\frac{1}{t_{\sigma(n-1)}-t_{\sigma(n)}}\frac{1}{t_{\sigma(n)}-t_{\sigma(1)}}=0$
(B.2)

for $n>2$. In fact, each term of the sum (B.2) is invariant under the action
of the cycle $(12\cdots n)$, so it implies (B.1). Following the idea of
[FN16], we consider the following class in $H_{\mathbb{T}}^{*}(\mathcal{B})$,

$\gamma=-\prod_{\begin{subarray}{c}i<j\\\ j-i\neq
1,n-1\end{subarray}}(x_{i}-x_{j}),$

where we recall $x_{i}=D_{-\mathbf{e}_{i}}$ and $t_{i}=-\mathbf{e}_{i}$. By
Atyiah–Bott localization theorem [AB84], we have

$\int_{\mathcal{B}}\gamma=\sum_{\sigma\in\mathfrak{S}_{n}}\frac{\gamma|_{\sigma}}{\prod_{i<j}(t_{\sigma(i)}-t_{\sigma(j)})}\in
H_{\mathbb{T}}^{*}(\operatorname{pt})$

which is exactly the left-hand side of (B.2). But
$\deg_{\mathbb{C}}\gamma=\frac{n(n-1)}{2}-n<\dim\mathcal{B}$, thus
$\int_{\mathcal{B}}\gamma=0$. ∎

######  Lemma B.2.

We have

$\det\left[\begin{matrix}0&\frac{1}{t_{1}-t_{2}}&\frac{1}{t_{1}-t_{3}}&\cdots&\frac{1}{t_{1}-t_{n}}\\\\[4.30554pt]
\frac{1}{t_{2}-t_{1}}&0&\frac{1}{t_{2}-t_{3}}&\cdots&\frac{1}{t_{2}-t_{n}}\\\\[4.30554pt]
\frac{1}{t_{3}-t_{1}}&\frac{1}{t_{3}-t_{2}}&0&\cdots&\frac{1}{t_{3}-t_{n}}\\\\[4.30554pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
\frac{1}{t_{n}-t_{1}}&\frac{1}{t_{n}-t_{2}}&\frac{1}{t_{n}-t_{3}}&\cdots&0\end{matrix}\right]=\sum_{\pi}\prod_{\pi(i)=j>i}\frac{1}{(t_{i}-t_{j})^{2}}$
(B.3)

where the sum is over all perfect matchings $\pi$ of $\\{1,\ldots,n\\}$.

###### Proof.

Let us consider the expansion of the above determinant

$\sum_{\sigma\in\mathfrak{S}_{n}}(-1)^{\ell(\sigma)}m_{\sigma}.$

Firstly since the diagonal entries vanish, we have $m_{\sigma}=0$ if $\sigma$
has a fixed point. Notice that the right-hand side of (B.3) equals the sum
over permutations $\sigma$ such that $\sigma^{2}=\operatorname{id}$.

For any permutation $\sigma$ such that $\sigma^{2}\neq\operatorname{id}$, the
cycle decomposition of $\sigma$ must have a cycle of length $>2$. Let
$\eta_{\sigma}$ be the cycle of $\sigma$ of length $>2$ containing the
smallest index, and denote by $A_{\sigma}$ the set of indices of
$\eta_{\sigma}$. Let $[\sigma]$ be the set of permutations $\sigma^{\prime}$
such that $A_{\sigma}=A_{\sigma^{\prime}}$ and $\sigma(i)=\sigma^{\prime}(i)$
for all $i\notin A_{\sigma}$. In other words, $\sigma^{\prime}\in[\sigma]$ is
obtained by a permutation of indices of $\eta_{\sigma}$ of $\sigma$. By Lemma
B.1 above,

$\sum_{\sigma^{\prime}\in[\sigma]}m_{\sigma^{\prime}}=0$

Note that all permutations $\sigma^{\prime}\in[\sigma]$ have the same sign.
The proof is complete. ∎

######  Lemma B.3.

Theorem 5.2 is true for $k=n$, i.e.

$\displaystyle\det\left[\begin{matrix}\chi_{1}&\frac{\hbar}{1-q_{1}/q_{2}}&\cdots&\frac{\hbar}{1-q_{1}/q_{n}}\\\\[4.30554pt]
\frac{\hbar}{1-q_{2}/q_{1}}&\chi_{2}&\cdots&\frac{\hbar}{1-q_{2}/q_{n}}\\\\[4.30554pt]
\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
\frac{\hbar}{1-q_{n}/q_{1}}&\frac{\hbar}{1-q_{n}/q_{2}}&\cdots&\chi_{n}\end{matrix}\right]$
$\displaystyle=\sum_{\pi}J(\pi)V(\pi),$

with the sum over all matchings $\pi$ over $\\{1,\ldots,n\\}$.

###### Proof.

By the expansion in polynomials in $\chi_{i}$’s, the determinant is equal to

$\displaystyle\sum_{A}{\chi}_{a_{1}}\cdots{\chi}_{a_{s}}\det\left[\begin{matrix}0&\frac{\hbar}{1-q_{b_{1}}/q_{b_{2}}}&\cdots&\frac{\hbar}{1-q_{b_{1}}/q_{b_{t}}}\\\\[4.30554pt]
\frac{\hbar}{1-q_{b_{2}}/q_{b_{1}}}&0&\cdots&\frac{\hbar}{1-q_{b_{2}}/q_{b_{t}}}\\\\[4.30554pt]
\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
\frac{\hbar}{1-q_{b_{d}}/q_{b_{1}}}&\frac{\hbar}{1-q_{b_{d}}/q_{b_{2}}}&\cdots&0\end{matrix}\right]$

where the sum goes over all subsets $A=\\{a_{1},\ldots,a_{s}\\}$ of
$\\{1,\ldots,n\\}$ with complement $\\{b_{1}<\cdots<b_{t}\\}$. Note that by
Lemma B.2

$\displaystyle\quad\det\left[\begin{matrix}0&\frac{\hbar}{1-q_{b_{1}}/q_{b_{2}}}&\cdots&\frac{\hbar}{1-q_{b_{1}}/q_{b_{t}}}\\\\[4.30554pt]
\frac{\hbar}{1-q_{b_{2}}/q_{b_{1}}}&0&\cdots&\frac{\hbar}{1-q_{b_{2}}/q_{b_{t}}}\\\\[4.30554pt]
\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
\frac{\hbar}{1-q_{b_{t}}/q_{b_{1}}}&\frac{\hbar}{1-q_{b_{t}}/q_{b_{2}}}&\cdots&0\end{matrix}\right]$
$\displaystyle=\hbar^{t}q_{b_{1}}\cdots
q_{b_{t}}\det\left[\begin{matrix}0&\frac{1}{q_{b_{2}}-q_{b_{1}}}&\cdots&\frac{1}{q_{b_{t}}-q_{b_{1}}}\\\\[4.30554pt]
\frac{1}{q_{b_{1}}-q_{b_{2}}}&0&\cdots&\frac{1}{q_{b_{t}}-q_{b_{2}}}\\\\[4.30554pt]
\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
\frac{1}{q_{b_{1}}-q_{b_{t}}}&\frac{1}{q_{b_{2}}-q_{b_{t}}}&\cdots&0\end{matrix}\right]=\sum_{\pi}V(\pi),$

where the sum is over perfect matchings $\pi$ over $\\{b_{1}<\ldots<b_{t}\\}$.
We can extend $\pi$ to a (not necessarily perfect) matching over
$\\{1,\ldots,n\\}$ by setting $\pi(a_{i})=a_{i}$ for $i=1,\ldots,s$. Thus the
determinant equals

$\sum J(\pi)V(\pi)$

with sum over all matchings of $\pi$ over $\\{1,\ldots,n\\}$. ∎

###### Proof of Theorem 5.2.

Similarly to the proof of Lemma B.3, the coefficient of $y^{n-k}$ in
$\det\big{(}y1_{n}+M(\chi)\big{)}$ is

$\displaystyle\sum_{K}\det\left[\begin{matrix}\chi_{c_{1}}&\frac{\hbar}{1-q_{c_{1}}/q_{c_{2}}}&\cdots&\frac{\hbar}{1-q_{c_{1}}/q_{c_{k}}}\\\\[4.30554pt]
\frac{\hbar}{1-q_{c_{2}}/q_{c_{1}}}&\chi_{c_{2}}&\cdots&\frac{\hbar}{1-q_{c_{2}}/q_{c_{k}}}\\\\[4.30554pt]
\vdots&\vdots&\ddots&\vdots\\\\[4.30554pt]
\frac{\hbar}{1-q_{c_{k}}/q_{c_{1}}}&\frac{\hbar}{1-q_{c_{k}}/q_{c_{2}}}&\cdots&\chi_{c_{k}}\end{matrix}\right]$

where the sum goes over all subsets
$K=\\{c_{1}<\cdots<c_{k}\\}\subset\\{1,\ldots,n\\}$ of order $k$. By Lemma
B.3, it is equal to

$\sum_{K}\sum_{\pi}J(\pi)V(\pi)$

with $K$ a $k$-subset of $\\{1,\ldots,n\\}$ and $\pi$ going through the
matchings of $K$. This finishes the proof. ∎

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