# An Analogue of Bernstein-Zelevinsky Derivatives to Automorphic Forms

Zhuohui Zhang
Tel Aviv University<EMAIL_ADDRESS>
###### Abstract.

In this paper, a construction to imitate the Bernstein-Zelevinsky derivative
for automorphic representations on $GL_{n}(\mathbb{A})$ is introduced. We will
later consider the induced representation

$I(\tau_{1},\tau_{2};\underline{s})=\mathrm{Ind}_{P_{[n_{1},n_{2}]}}^{G_{n}}(\Delta(\tau_{1},n_{1})|\cdot|^{s_{1}}\boxtimes\Delta(\tau_{2},n_{2})|\cdot|^{s_{2}}).$

from the discrete spectrum representations of $GL_{n}(\mathbb{A})$, and apply
our method to study the degenerate Whittaker coefficients of the Eisenstein
series constructed from such a representation as well as of its residues. This
method can be used to reprove the results on Whittaker supports of automorphic
forms of such kind proven by D. Ginzburg, Y. Cai and B. Liu. This method will
also yield new results on the Eulerianity of certain degenerate Whittaker
coefficients.

##  1\. Introduction

The Bernstein-Zelevinsky derivative is a tool introduced by Bernstein and
Zelevinsky in [BZ77] to study the representations of general linear groups
over a $p$-adic field. A classification of irreducible representations of
$GL_{n}$ modulo supercuspidal representations was obtained with this tool in
[Zel80]. Following the notations introduced in [BZ77], denoting by
$G_{n}=GL_{n}$ and $P_{n}=G_{n-1}\rtimes k^{n-1}$ the _mirabolic_ subgroup in
$G_{n}$, defined as the collection of elements in $G_{n}$ with last row
$(0,0,\ldots,0,1)$. The _Bernstein-Zelevinsky derivative_ is based on a
collection of induction-restriction functors:

and is defined as the composition of functors:

$(\cdot)^{(k)}=\Psi^{-}\circ(\Phi^{-})^{k-1}:\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
17.29247pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Rep}G_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
71.29247pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.29247pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 71.29247pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Rep}G_{n-k}.}$}}}}}}}\ignorespaces}}}}\ignorespaces$

These functors satisfy a collection of adjunction properties as described in
[BZ77, 3.2] that are ultimately related to the properties of the Whittaker
models of representations over a $p$-adic group. Calculation with these
functors is based on the _geometric lemma_ , which is a combinatorial rule for
calculating the composition of a restriction functor applied after an
induction functor. For two standard Levi subgroups $M_{\alpha},M_{\beta}$
whose sizes of Levi blocks are prescribed by two partitions $\alpha,\beta$, it
was shown in [BZ77, 2.12] that the functor $F=r_{G,M_{\beta}}\circ
i_{G,M_{\alpha}}$ is _glued_ from the functors
$F_{w}=i_{M_{\beta},M_{\beta^{\prime}}}\circ w\circ
r_{M_{\alpha^{\prime}},M_{\alpha}}$, where

  * •

The partitions
$\alpha=(\alpha_{1},\ldots,\alpha_{r}),\beta=(\beta_{1},\ldots,\beta_{s})$
prescribe the block sizes of the standard Levi subgroups
$M_{\alpha},M_{\beta}$.

  * •

The Weyl group element $w$ are those that sends any positive root $\gamma$ of
$M_{\alpha}$ to a positive root $w\alpha$, and $w^{-1}$ sending a positive
root $\epsilon$ of $M_{\beta}$ to $w^{-1}\epsilon$.

  * •

The refined partition $\gamma^{\prime}$ corresponds to the Levi subgroup
$M_{\gamma}\cap wM_{\beta}w^{-1}$, while $\beta^{\prime}$ corresponds to the
Levi subgroup $M_{\beta}\cap w^{-1}M_{\gamma}w$.

In this paper, we will replace the functorial operations for $p$-adic group
representations by Eisenstein series and their Whittaker coefficients, and
imitate the method in [BZ77] and [Zel80] to prove results about the Whittaker
support of automorphic forms and the Eulerianity of Whittaker coefficients of
automorphic forms for $GL_{n}$.

By the work of Langlands summarized in [MW95, Section V.3], a square-
integrable automorphic form can be described as residues of Eisenstein series
induced from discrete spectrum data on the Levi subgroups of parabolic
subgroups. In particular, it is shown by Mœglin-Waldspurger in [MW89] that the
discrete spectrum of $GL_{n}$ consists of the generalized Speh representations
$\Delta(\tau,n)$, which is the irreducible representation generated by the
residue at
$\underline{s}=(s_{1},\ldots,s_{n})\rightarrow(\frac{n-1}{2},\ldots,-\frac{n-1}{2})$
of an Eisenstein series

$E(\cdot,\underline{s}):\mathrm{Ind}_{P_{[a^{n}]}}^{G_{an}}\left(\tau|\cdot|^{s_{1}}\boxtimes\ldots\boxtimes\tau|\cdot|^{s_{n}}\right)\rightarrow\mathcal{A}(G_{n}(k)\backslash
G_{n}(\mathbb{A})).$

Using the analogues of Bernstein-Zelevinsky derivative developed in this
paper, we can mimic some arguments in [Zel80] for induced representations
attached to discrete spectra data. This method can be applied to prove results
regarding nilpotent orbit associated to the top Fourier coefficient. Some of
them have already been provided in [Gin06], [Liu13], [Cai18] and [LX20].

##  2\. Eisenstein Series and Whittaker Coefficients

###  2.1. Eisenstein Series and Degenerate Whittaker Models

For any standard parabolic subgroup $P=MN\subset G_{n}$ and an automorphic
representation $\pi=\bigotimes_{v}\pi_{v}$ of $M$, as an irreducible
subrepresentation of the space $\mathcal{A}\left(M(k)\backslash
M(\mathbb{A})\right)$ of all automorphic forms on $M$. For the group
$G_{n}=GL_{n}$ and a partition $\underline{\alpha}=(n_{1},n_{2},\ldots,n_{r})$
of the integer $n$, denote by $P_{\underline{\alpha}}$ the standard parabolic
subgroup with a Levi subgroup $M_{\underline{\alpha}}\cong GL_{n_{1}}\times
GL_{n_{2}}\times\ldots\times GL_{n_{r}}$ embedded diagonally in $G_{n}$. The
representation $\pi_{\underline{s}}$ on $M_{n}$ is defined on a pure tensor
$v_{1}\otimes\ldots\otimes v_{r}$ as

$\pi_{\underline{s}}(\mathrm{diag}\left(m_{1},\ldots,m_{r}\right))\left(v_{1}\otimes\ldots\otimes
v_{r}\right)=\pi_{1}^{s_{1}}(m_{1})v_{1}\otimes\ldots\otimes\pi_{r}^{s_{r}}(m_{r})v_{r}$

where $\pi_{i}^{s_{i}}=\pi_{i}\left|\cdot\right|^{s_{i}}$ for
$\mathrm{Re}s_{i}>0$. The induced representation $I(\pi_{\underline{s}})$ is
the space of $K$-finite functions
$\varphi_{\underline{s}}:G_{n}\rightarrow\pi$ satisfying

$\varphi_{\underline{s}}(nmg)=\pi_{\underline{s}}(m)\varphi_{\underline{s}}(g).$

One can define $\tilde{\varphi}_{\underline{s}}$ on $G_{n}$ as $\varphi(g)$
evaluated at the identity element of $M_{\underline{n}}$. The Eisenstein
series $E_{\underline{s}}$ is thus an operator from $I(\pi_{\underline{s}})$
to the space of automorphic forms $\mathcal{A}(G_{n}(k)\backslash
G_{n}(\mathbb{A}))$ defined as

$E(\varphi_{\underline{s}},g)=\sum_{\gamma\in P(k)\backslash
G(k)}\tilde{\varphi}_{\underline{s}}(\gamma g).$

Picking a different standard parabolic subgroup $Q_{\beta}=L_{\beta}U_{\beta}$
corresponding to a different partition $\beta$, and an unramified additive
character $\psi$ on $U$, the _Whittaker coefficient_ of any automorphic form
$f\in\mathcal{A}\left(G(k)\backslash G(\mathbb{A})\right)$ can be defined as

$\mathcal{W}_{\psi}(f)(g)=\int_{U_{\beta}(k)\backslash
U_{\beta}(\mathbb{A})}f(ug)\overline{\psi(u)}du.$

Denoting by $L_{\psi}$ the stabilizer of $\psi$ in $L$, the function
$\mathcal{W}_{\psi}(f)$ lives in the space
$\mathcal{A}(L_{\psi}(k)U(\mathbb{A})\backslash G(\mathbb{A}))_{\psi}$, which
is defined as the $L_{\psi}(k)$-invariant functions $w$ satisfying the
property $w(ug)=\psi(u)w(g)$ for each $u\in U(\mathbb{A})$.

The Bernstein-Zelevinsky derivatives are a collection of operators first
introduced in [BZ77] for representations of $GL_{n}$ over a non-archimedian
local field. Following our notations introduced above, let $P_{[n_{1},n_{2}]}$
be a standard parabolic subgroup with Levi blocks
$M_{[n_{1},n_{2}]}=GL_{n_{1}}\times GL_{n_{2}}$, and $Q_{[n-1,1]}$ a standard
parabolic subgroup with Levi blocks $L_{[n-1,1]}=GL_{n-1}\times GL_{1}$.
Denoting the character

$\psi_{m_{1},\ldots,m_{n-1}}\begin{pmatrix}1&\ldots&0&u_{n-1}\\\
&\ldots&0&\cdots\\\ &&1&u_{1}\\\
&&&1\end{pmatrix}=\psi\left(\sum_{i}m_{i}u_{i}\right).$

on the corresponding unipotent radical $U_{[n-1,1]}\cong\mathbb{A}^{n-1}$ by
$\psi_{m_{1},\ldots,m_{n-1}}$, similar to the definition of the corresponding
operators in [BZ77], the following two operators $\Psi$ and $\Psi$ are defined
as special cases of the Whittaker operator $\mathcal{W}_{\psi}$:

(1) $\displaystyle\Psi:\mathcal{A}\left(G_{n}\left(k\right)\backslash
G_{n}\left(\mathbb{A}\right)\right)$
$\displaystyle\longrightarrow\mathcal{A}\left(L_{[n-1,1]}(k)U_{[n-1,1]}(\mathbb{A})\backslash
G_{n}\left(\mathbb{A}\right)\right)$ $\displaystyle f$
$\displaystyle\longmapsto\mathcal{W}_{\psi_{0\ldots 0}}(f)$

and

(2) $\displaystyle\Phi:\mathcal{A}\left(G_{n}\left(k\right)\backslash
G_{n}\left(\mathbb{A}\right)\right)$
$\displaystyle\longrightarrow\mathcal{A}\left(Q^{\prime}(k)U_{[n-1,1]}(\mathbb{A})\backslash
G_{n}\left(\mathbb{A}\right)\right)_{\psi_{10\ldots 0}}$ $\displaystyle f$
$\displaystyle\longmapsto\mathcal{W}_{\psi_{10\ldots 0}}(f)$

where $Q^{\prime}$ is the stabilizer $GL_{n-2}\ltimes V_{n-2}$, a _mirabolic_
subgroup of $GL_{n-1}$. The $r$-th Bernstein-Zelevinsky derivative operator

$D^{(m)}:\mathcal{A}\left(G_{n}(k)\backslash
G_{n}(\mathbb{A})\right)\longrightarrow\mathcal{A}\left(G_{n-m}(k)\times
Z(G_{m})U^{\prime}_{m}(\mathbb{A})\backslash G(\mathbb{A})\right)_{\psi_{m}}$

($r$ is the length of the partition) is defined as the composition
$\Psi\circ(\Phi)^{m-1}$, where $G_{n-m}$ is the subgroup in $GL_{m}$ embedded
on the top-left corner of $GL_{n}$, $U^{\prime}_{m}$ the nilpotent radical of
a standard parabolic subgroup of corresponding to the partition $[n-m,1^{m}]$,
and $\psi_{m}$ the character

$\psi_{m}\left(\begin{smallmatrix}1&\ldots&0&u_{k,n-k}&&\ldots&u_{1,n-1}\\\
&1&\ldots&&&&\\\ &&1&u_{k,1}&&&\\\ &&&1&&\ldots&\ldots\\\ &&&&&1&u_{1,1}\\\
&&&&&&1\end{smallmatrix}\right)=\psi\left(0\cdot
u_{k,1}+u_{k-1,1}+\ldots+u_{1,1}\right)$

on $U^{\prime}_{m}$. One can also write the derivative operator just as a
Whittaker coefficient $\mathcal{W}_{\psi_{m}}(f)$. For any automorphic form
$f$ of $G$, the Bernstein-Zelevinsky derivative $D^{(m)}(f)$ can be restricted
to an automorphic function on $G_{n-m}$.

###  2.2. Constant Term Formula

The Bernstein-Zelevinsky derivative operator $D^{(m)}$ can also be viewed as
the composition of the constant term operator $c_{P_{[n-m,m]}}$ on
$P_{[n-m,m]}$ and a generic Whittaker coefficient operator $\mathcal{W}_{m}$
on the second factor $GL_{m}$:

$\textstyle{\mathcal{A}(L_{[n-m,m]}(k)U_{[n-m,m]}(\mathbb{A})\backslash
G_{n}(\mathbb{A}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathcal{W}_{m}}$$\textstyle{\mathcal{A}(G_{n}(k)\backslash
G_{n}(\mathbb{A}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{P_{[n-m,m]}}}$$\scriptstyle{D^{(m)}}$$\textstyle{\mathcal{A}\left(G_{n-m}(k)\times
Z(G_{m})U^{\prime}_{m}(\mathbb{A})\backslash G(\mathbb{A})\right)_{\psi_{m}}}$

Following a similar method as in [MW95, II.1.7] the constant term along any
unipotent radical $U^{\prime}$ of the parabolic subgroup
$P^{\prime}=M^{\prime}U^{\prime}$ of the Eisenstein series
$E(\tilde{\varphi},g)$ can be written as the integral:

$c_{P^{\prime}}E(\tilde{\varphi},g)=\sum_{\gamma\in P(k)\backslash
G(k)}\int_{U^{\prime}(k)\backslash U^{\prime}(\mathbb{A})}\varphi(\gamma
u^{\prime}g)du^{\prime}.$

By the Bruhat decomposition $G(k)=\bigcup_{w\in
W_{M,M^{\prime}}}P(k)wP^{\prime}(k)$ with $W_{M,M^{\prime}}=P\backslash
G/P^{\prime}$, any element $\gamma$ belongs to a coset $Pwp^{\prime}$ for some
$p^{\prime}\in P^{\prime}$. This $p^{\prime}$ can be further decomposed into
$p^{\prime}=m^{\prime}u^{\prime}$:

$\displaystyle c_{P^{\prime}}E(\tilde{\varphi},g)$ $\displaystyle=\sum_{w\in
W_{M,M^{\prime}}}\sum_{p^{\prime}\in
P^{\prime}(k)}\int_{U^{\prime}(k)\backslash
U^{\prime}(\mathbb{A})}\varphi(wp^{\prime}u^{\prime}g)du^{\prime}$
$\displaystyle=\sum_{w\in W_{M,M^{\prime}}}\sum_{m^{\prime}\in
M^{\prime}(k)\cap w^{-1}P^{\prime}(k)w\backslash
M^{\prime}(k)}\int_{U^{\prime}(\mathbb{A})}\varphi(wm^{\prime}u^{\prime}g)du^{\prime}$
$\displaystyle=\sum_{w\in W_{M,M^{\prime}}}\sum_{m^{\prime}\in
M^{\prime}(k)\cap w^{-1}P^{\prime}(k)w\backslash
M^{\prime}(k)}\int_{U^{\prime}(\mathbb{A})}\varphi(wu^{\prime}m^{\prime}g)du^{\prime}$
$\displaystyle=\sum_{w\in W_{M,M^{\prime}}}\sum_{m^{\prime}\in
M^{\prime}(k)\cap w^{-1}P^{\prime}(k)w\backslash
M^{\prime}(k)}\int_{U^{\prime}(\mathbb{A})\cap w^{-1}U(\mathbb{A})w\backslash
U^{\prime}(\mathbb{A})}\varphi(wu^{\prime}m^{\prime}g)du^{\prime}.$

We can decompose the domain of integration as the product of the following
spaces:

$U^{\prime}(\mathbb{A})\cap w^{-1}U(\mathbb{A})w\backslash
U^{\prime}(\mathbb{A})=\left(U^{\prime}(\mathbb{A})\cap
w^{-1}M(\mathbb{A})w\right)\left(U^{\prime}(\mathbb{A})\cap
w^{-1}\overline{U}(\mathbb{A})w\right).$

In our setting, we take $P=P_{[n_{1},n_{2}]}$ and $P^{\prime}=Q_{[n-m,m]}$.
Since the unipotent group $U_{[n-m,m]}$ is abelian, the subgroup
$U^{\prime}(\mathbb{A})\cap w^{-1}M(\mathbb{A})w$ is isomorphic to the
unipotent subgroup $U_{n_{1}}\times U_{n_{2}}$. We can thus decompose the
constant term $c_{P^{\prime}}E(\tilde{\varphi},g)$ as the summation over the
Weyl group double coset $W_{M,M^{\prime}}$ and $M^{\prime}\cap
w^{-1}Pw\backslash M^{\prime}$:

$c_{P^{\prime}}E(\tilde{\varphi},g)=\sum_{w\in
W_{M,M^{\prime}}}\sum_{m^{\prime}\in M^{\prime}(k)\cap
w^{-1}P^{\prime}(k)w\backslash
M^{\prime}(k)}\mathcal{A}_{w}\left(c_{P^{\prime}\cap
w^{-1}Mw}\varphi\right)(wm^{\prime}g),$

where the operator $\mathcal{A}_{w}$

(3) $\mathcal{A}_{w}f(g)=\int_{U^{\prime}(\mathbb{A})\cap
w^{-1}\overline{U}(\mathbb{A})w}f(u^{\prime}g)du^{\prime}.$

is a formal intertwining operator between the following two induced
representations:

$\mathcal{A}_{w}:\mathrm{Ind}_{wM^{\prime}w^{-1}\cap
M}^{G}(\pi)\rightarrow\mathrm{Ind}_{M^{\prime}\cap w^{-1}Mw}^{G}(\pi^{w})$

for any representation $\pi$ on the intersection of Levi subgroups
$wM^{\prime}w^{-1}\cap M$. If the induction data is associated to the
representation $\tau^{s}$, the operator $\mathcal{A}_{w}$ can be analytically
continuated to the whole $\mathbb{C}^{n}$ as a meromorphic function of $s$.

We denote the generic Whittaker operator on automorphic functions on the
lower-right corner $GL_{m}$ of the Levi subgroup by $\mathcal{W}_{m}$, which
sends any function
$f\in\mathcal{A}(M_{[n-m,m]}(k)U_{[n-m,m]}(\mathbb{A})\backslash
G(\mathbb{A}))$ to a function in the space
$\mathcal{A}(M_{[n-m,1^{m}]}(k)U_{[n-m,1^{m}]}(\mathbb{A})\backslash
G(\mathbb{A}))_{\psi_{s}}$. The operator can be written as the integral

$\left(\mathcal{W}_{m}f\right)(g)=\int_{U_{m}(k)\backslash
U_{m}(\mathbb{A})}f\left(\left(\begin{smallmatrix}I_{n-m}&0&0\\\ 0&1&u\\\
0&0&1\end{smallmatrix}\right)g\right)\overline{\psi_{[m]}(u)}du$

where $\psi_{[m]}$ is character on $U_{m}$ belonging to the regular orbit of
the space $GL_{m}$. Now we can view the derivative operator as the sum of
compositions of three operators:

(4) $\displaystyle D^{(m)}E(\tilde{\varphi},g)=\sum_{w\in
W_{M,M^{\prime}}}\mathcal{W}_{m}\left(\sum_{m^{\prime}\in M^{\prime}(k)\cap
w^{-1}P^{\prime}(k)w\backslash
M^{\prime}(k)}\mathcal{A}_{w}\left(c_{P^{\prime}\cap
w^{-1}Mw}\varphi\right)(wm^{\prime}g)\right)$

with

  * •

$P=P_{[n_{1},n_{2}]}=M_{[n_{1},n_{2}]}U_{[n_{1},n_{2}]}$, with the
corresponding $M=M_{[n_{1},n_{2}]}$ and $U=U_{[n_{1},n_{2}]}$,

  * •

$P^{\prime}=P_{[n-m,m]}=M_{[n-m,m]}U_{[n-m,m]}$, with the corresponding
$M^{\prime}=M_{[n-m,m]}$ and $U^{\prime}=U_{[n-m,m]}$,

  * •

$W_{M,M^{\prime}}$ is the double coset space $P\backslash G/P^{\prime}$, which
is the set of all $w\in W(G)$ such that any positive root $\alpha$ of $M$ is
sent to a positive root $w\alpha$, and any positive root $\beta$ of
$M^{\prime}$ is sent to a positive root $w^{-1}\beta$.

  * •

The intertwining operator $\mathcal{A}_{w}$ is defined by the analytic
continuation of the integral (3).

###  2.3. Weyl Group Cosets

In this section, we will give a combinatorial description to the
representatives of the double coset $P_{[n_{1},n_{2}]}\backslash
G/Q_{[n-m,1^{m}]}$, as well as the set of Weyl group representatives in
$W_{M,M^{\prime}}$ such that the corresponding term in the sum of (4) is
nonzero. For each pair of integers $l\leq k$, we denote the _cycle_
$(k+1,k,\ldots,l+1,l)$ in the permutation group by $c_{k,l}$. We need two
technical lemma:

######  Lemma 1.

There is a bijection between the set of interlacings between two strings
$12\ldots n_{1}$ and $(n_{1}+1)\ldots(n_{1}+n_{2})$ and the set of Young
subdiagrams $[n_{1}-k_{1},n_{1}-k_{2},\ldots,n_{1}-k_{n_{2}}]$ with $0\leq
k_{1}\leq k_{2}\ldots\leq k_{n_{2}}\leq n_{1}$ of $[n_{1}^{n_{2}}]$.

###### Proof.

In the permutation group $S_{n_{1}+n_{2}}$, the cycle $c_{l,k}$ is the product
of simple reflections:

$c_{l,k}=s_{l}s_{l+1}\ldots s_{k}.$

We can correspond any Young subdiagram
$[n_{1}-k_{1},n_{1}-k_{2},\ldots,n_{1}-k_{n_{2}}]$ to the element

$w_{[n_{1}-k_{1},n_{1}-k_{2},\ldots,n_{1}-k_{n_{2}}]}=c_{n-n_{1}+k_{n_{2}},n-1}c_{n-n_{1}-1+k_{n_{2}-1},n-2}\ldots
c_{1+k_{1},n-n_{2}}$

which sends the word $12\ldots(n_{1}+n_{2})$ to a word $a_{1}a_{2}\ldots
a_{n_{1}+n_{2}}$ with

$\left(a_{k_{1}+1},a_{k_{2}+2},\ldots,a_{k_{n_{2}}+n_{1}}\right)=\left(n_{1}+1,n_{1}+2,\ldots,n_{1}+n_{2}\right)$

and the complement of
$\left(a_{k_{1}+1},a_{k_{2}+2},\ldots,a_{k_{n_{2}}+n_{1}}\right)$ equal to
$12\ldots n_{1}$. Thus, we have constructed a bijection between the set of
interlacings between the two strings and the set of Young subdiagrams of
$[n_{1}^{n_{2}}]$. ∎

######  Lemma 2.

For the rectangular Young diagram $[n_{1}^{n_{2}}]$:

$\ldots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\ldots$$\ldots$$\ldots$$\ldots$

if we fill the diagram with numbers

$n$-1$n$-2$\ldots$$\ldots$$n$-$n_{1}$$n$-2$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\ldots$$\ldots$$\ldots$$\ldots$$n$-$n_{2}$$\ldots$$\ldots$$1$$\vdots$

.

For the Weyl group element
$w_{[n_{1}-k_{1},n_{1}-k_{2},\ldots,n_{1}-k_{n_{2}}]}$, setting the partition
$[l_{n_{1}},\ldots,l_{1}]$ as the transpose of the partition
$[k_{n_{2}},\ldots,k_{1}]$ (adding zeros to make the length of the partition
equal to $n_{1}$), then the inverse of the element
$w_{[n_{1}-k_{1},n_{1}-k_{2},\ldots,n_{1}-k_{n_{2}}]}$ is equal to the element

$w^{\prime}_{[n_{2}-l_{1},\ldots,n_{2}-l_{n_{1}}]}=c_{n-n_{2},n-1-l_{1}}c_{n-n_{2}-1,n-2-l_{2}}\ldots
c_{1,n-n_{1}-l_{n_{1}}}$

which is the word constructed by reading the filled Young subdiagram of the
rectangular Young diagram from top to bottom along each column, starting from
the left-most column to the right-most column.

###### Proof.

By direct enumeration, in the word $a_{1}a_{2}\ldots a_{n_{1}+n_{2}}$, the
number of elements on the left of 1 is equal to the total number of rows of
length $n_{1}$ in the Young diagram $[n_{1}-k_{1},\ldots,n_{1}-k_{n_{2}}]$,
which is equal to $n_{2}-l_{n_{1}}$. The cycle to move 1 back to the first
entry is thus $c_{1,n_{2}-l_{n_{1}}}$. In general, the number of elements from
$(n_{1}+1)\ldots(n_{1}+n_{2})$ on the left of $s\in\\{1,\ldots,n_{1}\\}$ is
equal to the number of rows of length $\geq n_{1}-s+1$, which is equal to
$n_{2}-l_{n_{1}-s+1}$. Thus, the cycle to move $s$ back to the first entry is
$c_{s,n_{2}-l_{n_{1}-s+1}+s-1}$. ∎

######  Lemma 3.

  1. (1)

The elements of the coset $P_{[n_{1},n_{2}]}\backslash G/B$ can be represented
by the collection of Young subdiagrams of the rectangular Young diagram of
height $n_{2}$ and width $n_{1}$.

  2. (2)

If we fill the above rectangular Young diagram with numbers:

$n$-1$n$-2$\ldots$$\ldots$$n$-$n_{1}$$n$-2$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\vdots$$\ldots$$\ldots$$\ldots$$\ldots$$n$-$n_{2}$$\ldots$$\ldots$$1$$\vdots$

.

the double coset $P_{[n_{1},n_{2}]}\backslash G/Q_{[n-m,1^{m}]}$ corresponds
to the collection of Young subdiagrams with no columns filled with numbers
only from $\\{1,\ldots,n-m-1\\}$.

  3. (3)

In a word $a_{1}a_{2}\ldots a_{n_{1}+n_{2}}$ representing a permutation $w$,
we use the light gray intervals $v_{i}$ to represent consecutive intervals of
elements from $(n_{1}+1)\ldots(n_{1}+n_{2})$, and the dark gray intervals
$u_{i}$ as consecutive intervals of elements coming from $12\ldots n_{1}$ (it
is possible for $u_{0}$ to be empty).

$u_{0}$$v_{1}$$u_{1}$$v_{2}$$u_{2}$$\ldots$$u_{i}$$v_{i+1}$$\ldots$

.

The sequence
$\left(|v_{1}|,|v_{1}|+|v_{2}|+|u_{1}|,\ldots,\sum_{j=1}^{i+1}|v_{j}|+\sum_{j=1}^{i}|u_{j}|\right)$
is equal to the sequence of numbers at the pivot positions of the filled
subdiagram. The starting index of each interval $u_{1},u_{2},\ldots,u_{i}$ is
given by the number _above_ each pivot position in the first row.

###### Proof.

There is a bijection between double cosets $P_{[n_{1},n_{2}]}\backslash G/B$
and $B\backslash G/P_{[n_{1},n_{2}]}$ given by $w\mapsto w^{-1}$. Also, by
representing these elements as permutations, there is a bijection between the
double coset $B\backslash G/P_{[n_{1},n_{2}]}$ and the set of interlacings
between two strings $12\ldots n_{1}$ and $(n_{1}+1)\ldots(n_{1}+n_{2})$. The
first part of this lemma follows from Lemma 1. Under the bijection $w\mapsto
w^{-1}$ between $P_{[n_{1},n_{2}]}\backslash G/B$ and $B\backslash
G/P_{[n_{1},n_{2}]}$, the representatives surviving in the double coset
$P_{[n_{1},n_{2}]}\backslash G/Q_{[n-m,1^{m}]}$ are those elements
$w^{\prime}_{[n_{2}-l_{1},\ldots,n_{2}-l_{n_{1}}]}$ from Lemma 2 with the
right-most cycles not in $S_{n-m}$, which correspond to the filled diagrams
with no column filled with numbers entirely from $\\{1,\ldots,n-m\\}$. The
third part of the lemma follows from the proof of Lemma 2. Since in the cycle
representation of $w_{[n_{1}-k_{1},\ldots,n_{1}-k_{n_{2}}]}$, each cycle
$c_{s+k_{s},n_{1}+s-1}$ moves the element $n_{1}+s$ to the $k_{s}+s$-th
position of the word $a_{1}\ldots a_{n_{1}+n_{2}}$, if two adjacent rows $s-1$
and $s$ have the same length, the action by
$w_{[n_{1}-k_{1},\ldots,n_{1}-k_{n_{2}}]}$ will move these two rows to the
$(k_{s-1}+s-1)$ and $(k_{s-1}+s)$-th position, respectively. If the numbers of
columns of the same length are listed in the sequence $(b_{1},\ldots,b_{r})$
which corresponds to the sequence of pivot positions
$(k_{s_{1}},\ldots,k_{s_{r}})$, then the lengths of each $v_{i}$ is equal to
$b_{i}$, and the sequence $(k_{s_{1}},\ldots,k_{s_{r}})$ marks the starting
indices of every $u_{i}$. ∎

The next lemma will be used to describe the Weyl group representatives
corresponding to nonvanishing summands in (4). We will put a partial order
given by the lexicographic order of the coordinates
$(a_{1},a_{2},\ldots,a_{n-1})$ of any root
$a_{1}\alpha_{1}+\ldots+a_{n-1}\alpha_{n-1}$ on the set of positive roots of
$GL_{n}$.

######  Lemma 4.

For any $w\in P_{[n_{1},n_{2}]}\backslash G/Q_{[n-s,1^{s}]}$, and for any
$0\leq s\leq n$, there are no elements from

$\\{w\alpha_{n-s+1},\ldots,w\alpha_{n-1}\\}$

satisfy $\geq{\alpha_{n-s}}$ only if the Young diagram corresponding to $w$ is
a rectangular diagram of height $n_{2}$.

###### Proof.

The set of elements

$\\{w\alpha_{n-s+1},\ldots,w\alpha_{n-1}\\}$

does not intersect the cone $\alpha_{n-s}$ if and only if in the tail of last
$s$ elements of the corresponding word $v_{0}u_{1}v_{1}\ldots$, any
$i\in\\{1,\ldots,n_{1}\\}$ and $j\in\\{n_{1}+1,\ldots,n_{1}+n_{2}\\}$
appearing in this tail satisfy $w(i)>w(j)$. Therefore, for any $s$, the tail
of the string corresponding to $w$ must take the form of one of the following
cases (if every entry is a positive integer):

$\displaystyle t_{0}=\\{n_{1}-s+1,n_{1}-s+2,$
$\displaystyle\ldots,n_{1}-1,n_{1}\\}$ $\displaystyle
t_{1}=\\{n_{1}+n_{2},n_{1}-s+2,$ $\displaystyle\ldots,n_{1}-1,n_{1}\\}$
$\displaystyle t_{2}=\\{n_{1}+n_{2}-1,n,$
$\displaystyle\ldots,n_{1}-1,n_{1}\\}$ $\displaystyle\ldots$ $\displaystyle
t_{s}=\\{n-s+1,n-s+2,$ $\displaystyle\ldots,n\\}.$

Note that if $s>n_{2}$, the tails listed above exist only up to $t_{n_{2}}$,
if $s>n_{1}$, the list of tails start from $t_{s-n_{1}}$. Denoting the total
number of elements in the tail coming from $\\{n_{1}+1,\ldots,n_{1}+n_{2}\\}$
by $s_{1}$, and the total number of elements in the tail coming from
$\\{1,\ldots,n_{1}\\}$ by $s_{2}$, by the third part of Lemma 3, the pivot
positions of the filled Young diagram corresponding to $w$ must satisfy the
following two properties:

  1. (1)

The bottom-most pivot box has content $n-s_{2}$, corresponding to the index
$n_{1}-s_{2}$ on the first row of the Young diagram,

  2. (2)

The content of the next pivot box is an integer $<n-s$.

However, in the double coset $P_{[n_{1},n_{2}]}\backslash G/Q_{[n-s,1^{s}]}$,
columns entirely consisting of boxes with contents $<n-s$ are not permitted.
Therefore, only one pivot box is allowed, and the only situation which also
allows (1) is when the Young diagram is a rectangular Young diagram of height
$n_{2}$. ∎

Summarizing the two lemmas above, denoting by $\mathbb{W}_{n_{1},n_{2}}^{s}$
the subset consisting of the representatives of the double coset
$P_{[n_{1},n_{2}]}\backslash G/Q_{[n-s,1^{s}]}$ corresponding to the
nonvanishing summands of (4) are those which satisfy the condition:

$\mathrm{Ad}(w)\left(\bigoplus_{n_{1}+1\leq i\leq
n-1}\mathfrak{g}_{\alpha_{i}}\right)\cap\bigoplus_{\alpha\geq\alpha_{n_{1}}}\mathfrak{g}_{\alpha}\neq\left\\{0\right\\}.$

The set $\mathbb{W}_{n_{1},n_{2}}^{s}$ can be described by the following
lemma:

######  Lemma 5.

For $n=n_{1}+n_{2}$ and $\nu=n_{1}-n_{2}$, denoting the Weyl group element
corresponding to a rectangular diagram of width $j$ and height $n_{2}$ by
$w(j)$, the subset $\mathbb{W}_{n_{1},n_{2}}^{s}$ of representatives of the
double coset $P_{[n_{1},n_{2}]}\backslash G/Q_{[n-m,1^{s}]}$ are

  1. (1)

For $s\leq\min(n_{1},n_{2})$, the representatives are

$w(0),w(1),\ldots,w(s),$

  2. (2)

For $\min(n_{1},n_{2})<s\leq\max(n_{1},n_{2})$, the representatives are

$w(0),w(1),\ldots,w(\min(n_{1},n_{2})),$

  3. (3)

For $s>\max(n_{1},n_{2})$, the representatives are

$w(s-\max(n_{1},n_{2})),w(m-\max(n_{1},n_{2})+1),\ldots,w(\min(n_{1},n_{2})).$

The parabolic subgroups $M_{[n-s]}\cap w^{-1}P_{[n_{1},n_{2}]}w\subset
GL_{n-s}$ for $w\in\mathbb{W}_{n_{1},n_{2}}^{s}$ have Levi subgroups listed by
the partitions $s_{1}+s_{2}=s$ such that $[n_{1}-s_{1},n_{2}-s_{2}]$ is a
partition of $n-s$.

###### Proof.

The lemma follows from direct enumeration of the rectangular diagrams. If
$s\leq\min(n_{1},n_{2})$, in the proof of Lemma 4, since no columns completely
consisting of elements from $\\{1,2,\ldots,n-s-1\\}$ are allowed, the largest
rectangular diagram with no column completely filled with numbers in
${1,2,\ldots,n-s}$ is the one corresponding to the partition $[s^{n_{2}}]$. If
$n_{1}\leq n_{2}$, in the proof of Lemma 4, we can learn that if $n_{1}<s\leq
n_{2}$, the list of tails range from $t_{s-n_{1}}$ up to $t_{s}$. But if
$s>n_{2}$, the list of tails ranges from $t_{s-n_{1}}$ up to $t_{n_{2}}$. When
$n_{1}>n_{2}$, if $n_{2}<s\leq n_{1}$, the tails exist from $t_{0}$ to
$t_{n_{2}}$, while if $s>n_{1}$, the list of tails ranges from $t_{s-n_{1}}$
up to $t_{n_{2}}$. In either case, the total number of representatives agrees
with the result in part (2) and (3) of this lemma, and the representatives
match with the representatives described in Lemma 4. A rectangular diagram of
width $j$ sends $n_{1}+t$ to the position $n_{1}+t-j$ of the word
corresponding to the permutation. As in the proof of Lemma 4,
$(s_{1},s_{2})=(s-j,j)$, and if $s,j$ satisfies the conditions as described in
the statement of the lemma, $[n_{1}-s+j,s-j]$ and $[n_{2}-j,j]$ are partitions
of the integers $n_{1}$ and $n_{2}$, respectively. The block sizes of the Levi
subgroup $w^{-1}P_{[n_{1},n_{2}]}w$ are given by the partition
$[n_{1}-s+j,s-j,n_{2}-j,j]$, and the sizes of Levi blocks of the parabolic
subgroup $M_{[n-s]}\cap w^{-1}P_{[n_{1},n_{2}]}w\subset GL_{n-s}$ are given by
the partition $[n_{1}-s+j,n_{2}-j]$. ∎

The reasoning in this section can be summarized in the following corollary:

######  Corollary 6.

The coset $W_{M,M^{\prime}}$ in the formula (4) can be replaced by
$\mathbb{W}_{n_{1},n_{2}}^{s}$:

$\displaystyle
D^{(s)}E(\tilde{\varphi},g)=\sum_{w\in\mathbb{W}_{n_{1},n_{2}}^{s}}\mathcal{W}_{s}\left(\sum_{m^{\prime}\in
M_{[n-s]}\cap w^{-1}P_{[n_{1},n_{2}]}w\backslash
M_{[n-s]}(k)}\mathcal{A}_{w}\left(c_{w^{-1}M_{[n-s]}w\cap
P_{[n_{1},n_{2}]}}\tilde{\varphi}\right)(wm^{\prime}g)\right).$

##  3\. Degenerate Whittaker Coefficients and Derivatives

###  3.1. Overview

The set of nilpotent orbits allowing nonzero Fourier coefficients for an
automorphic representation, as well as its local analogue, is a subject
studied by many mathematicians. For the case of $GL(n)$, we summarize timeline
of results and applications as follows:

  * •

In [Sha74, Theorem 3.1 and 4.5], Shalika proved that the local and global
generic Whittaker model, whenever exists, has multiplicity 1. In particular,
Whittaker model always exists for cuspidal automorphic representation. Based
on this, [Sha74] and [PS79] proved the local and global multiplicity-one
theorem for $GL(n)$. The existence of global Whittaker model is also important
in the construction of automorphic $L$-functions with the Langlands-Shahidi
method.

  * •

In [Gin06, Proposition 5.3], Ginzburg proved that the maximal orbit allowing a
nonzero Fourier coefficient for the generalized Speh representation (as a
residue of cuspidal Eisenstein series) $\Delta(\tau,n)$ is $[a^{n}]$ by
showing the vanishing of the induced local Whittaker functional. In [Liu13,
Theorem 2.5.4-2.5.6], the same result was proven following a global argument
based on the _root exchange_ technique. A recipe for the _root exchange_
technique is available in Section 7.1 of [GSR11], and a systematic reduction
method based on the root exchange technique is developed by Gourevitch et al.
in [GGK+22]. A similar global method also appears in [Cai18] for the orbit of
the top Fourier coefficient of degenerate Eisenstein series of $GL(n)$, and in
[LX20] for the orbit corresponding to the top Fourier coefficient of isobaric
sum representations.

In this section, we will provide a global, inductive proof to the vanishing of
the Fourier coefficients associated with larger orbits, as well as an argument
for the Eulerianity of certain Fourier-Whittaker coefficients.

###  3.2. Whittaker Support and Wave-front Set of Automorphic Forms

This section follows the notations and terminologies in [GGK+22]. For any
$\mathbb{Q}$-semisimple element $S\in\mathfrak{g}$, denoting by
$\mathfrak{g}^{S}_{\mu}$ the $\mu$-eigenspace of the adjoint action
$\mathrm{ad}(S)$ on $\mathfrak{g}$,

  * •

A _Whittaker pair_ $(S,\phi)\in\mathfrak{g}\times\mathfrak{g}^{*}$ contains a
$\mathbb{Q}$-semisimple element $S$ and a nilpotent element
$\phi\in(\mathfrak{g}^{*})^{S}_{-2}$. The element $\phi$ can be constructed by
pairing the Killing form of $\mathfrak{g}$ with a nilpotent element
$f_{\phi}\in\mathfrak{g}^{S}_{-2}$.

  * •

A Whittaker pair $(h,\phi)$ is _neutral_ if it can be completed to a standard
$\mathfrak{sl}_{2}$-triple.

  * •

A Whittaker pair $(S,\phi)$ is _standard_ if the nilpotent subalgebra

$\mathfrak{n}_{(S,\phi)}=\mathfrak{g}^{S}_{>1}\oplus\left(\mathfrak{g}^{S}_{1}\cap\mathfrak{g}_{\phi}\right),$

where $\mathfrak{g}_{\phi}$ is the stabilizer of $\phi$, is the nilpotent
radical of the minimal parabolic subalgebra.

  * •

A nilpotent element $\phi\in\mathfrak{g}^{*}$ is $k$-distinguished if the
corresponding nilpotent element $f_{\phi}\in\mathfrak{g}$ does not belong to a
proper $k$-Levi subalgebra of $\mathfrak{g}$. For a neutral Whittaker pair
$(h,\phi)$, the Whittaker pair $(h+Z,\phi)$, with $Z$ a semisimple element
centralizing the pair $(h,\phi)$ and $\phi$ a $k$-distinguished nilpotent
element, is called _Levi-distinguished_ if

$\displaystyle\mathfrak{g}_{>1}^{h+Z}$ $\displaystyle=\mathfrak{g}_{\geq
2}^{h+Z}=\mathfrak{g}_{>0}^{Z}\oplus\mathfrak{l}_{\geq 2}^{h}$
$\displaystyle\mathfrak{g}_{1}^{h+Z}$ $\displaystyle=\mathfrak{l}_{1}^{h}.$

where $\mathfrak{l}$ is the centralizer of $Z$. In particular, for the case of
$\mathfrak{gl}_{n}$ and $f_{\phi}$ a usual Jordan form, the Whittaker pair
$(H,\phi)$ with $H=\mathrm{diag}\left(n-1,n-3,\ldots,3-n,1-n\right)$ is a
Levi-distinguished Whittaker pair.

For any two Whittaker pairs $(H,\phi)$ and $(S,\phi)$ containing the same
nilpotent element $\phi$, $(H,\phi)$ _dominates_ $(S,\phi)$ (denoted by
$(H,\phi)\prec(S,\phi)$) if $H$ and $S$ commute, and

$\mathfrak{g}_{\phi}\cap\mathfrak{g}_{\geq 1}^{H}\subset\mathfrak{g}_{\geq
0}^{S-H}.$

In [GGK+22, Corollary 3.2.2 and Proposition 3.2.3], it is shown that any
Whittaker pair $(H,\phi)$ is dominated by a neutral Whittaker pair with the
same nilpotent element, and there exists a $Z$ in the centralizer of
$(H,\phi)$ such that $(H,\phi)\prec(H+Z,\phi)$ with $(H+Z,\phi)$ standard.

We can define the Whittaker coefficient $\mathcal{W}_{(H,\phi)}(F)$ of any
automorphic function $F$ corresponding to any Whittaker pair $(H,\phi)$ as the
following integral

$\mathcal{W}_{H,\phi}(F)=\int_{N_{S,\phi}}F(ng)\overline{\phi(n)}dn$

where $N_{S,\phi}$ is the analytic subgroup with Lie algebra
$\mathfrak{n}_{S,\phi}$ the nilpotent radical of the symplectic form

$\omega_{\phi}\left(X,Y\right)=\phi([X,Y])$

restricted to the nilpotent subalgebra $\mathfrak{u}=\mathfrak{g}_{\geq
1}^{S}$.

A nilpotent element $\phi$ is in the _Whittaker support_ $\mathrm{WS}(F)$ of
any automorphic function $F$ if there exists a Whittaker support $(S,\phi)$
such that $\mathcal{W}_{H,\phi}(F)\neq 0$, and for any Whittaker pair
$(H^{\prime},\psi)$ with $\psi$ not in the closure of the nilpotent orbit of
$\phi$ we have $\mathcal{W}_{H^{\prime},\psi}(F)=0$.

The following theorem from [GGK+22, Theorem B] provides a method to calculate
the Whittaker support of an automorphic function.

######  Theorem 7.

If $\phi$ is in $\mathrm{WS}(F)$ and two Whittaker pairs satisfy
$(H,\phi)\prec(S,\phi)$, then there exists an integral operator
$\mathcal{M}_{H}^{S}$ such that

$\mathcal{W}_{H,\phi}[F]=\mathcal{M}_{H}^{S}\left(\mathcal{W}_{S,\phi}[F]\right).$

By this theorem, and [GGK+22, Corollary 3.2.2 and Proposition 3.2.3], the
vanishing of Whittaker coefficient corresponding to a standard Whittaker pair
implies the vanishing of any Whittaker coefficient.

###  3.3. Composition of Derivative Operators

We can compose multiple derivative operators.

######  Proposition 8.

For any two derivative operators

$\displaystyle D^{(m_{2})}:\mathcal{A}(G_{n}(k)\backslash G_{n}(\mathbb{A}))$
$\displaystyle\longrightarrow\mathcal{A}(G_{n-m_{2}}(k)U_{[(n-m_{2})1^{m_{2}}]}(\mathbb{A})\backslash
G_{n}(\mathbb{A}))$ $\displaystyle
D^{(m_{1})}:\mathcal{A}(G_{n-m_{2}}(k)\backslash G_{n-m_{2}}(\mathbb{A}))$
$\displaystyle\longrightarrow\mathcal{A}(G_{n-m_{2}-m_{1}}(k)U_{[(n-m_{2}-m_{1})1^{m_{1}}]}(\mathbb{A})\backslash
G_{n-m_{2}}(\mathbb{A})),$

if we define the composition of two operators as

$D^{(m_{1})}\circ
D^{(m_{2})}(f)=D^{(m_{1})}\left(\left(D^{(m_{2})}f\right)|_{G_{n-m_{2}}}\right),$

then this operator $D^{(m_{1})}\circ D^{(m_{2})}$ is equal to the Whittaker
coefficient $\mathcal{W}_{U_{[(n-m_{1}-m_{2})1^{m_{1}+m_{2}}]},\psi}$ along
the unipotent subgroup

$U_{[(n-m_{1}-m_{2})1^{m_{1}+m_{2}}]}=\left\\{\begin{pmatrix}I_{n-m_{1}-m_{2}}&v\\\
0&u_{m_{1}+m_{2}}\end{pmatrix}\mid
v\in\mathrm{Mat}_{n-m_{1}-m_{2},m_{1}+m_{2}}(\mathbb{A}),u_{m_{1}+m_{2}}\in
U_{[m_{1}+m_{2}]}\right\\}$

with $u_{m_{1}+m_{2}}$ a strictly upper-triangular matrix of size
$m_{1}+m_{2}$, and the character $\psi$ given by the partition $(m_{1},m_{2})$
on $GL_{m_{1}+m_{2}}$:

$\displaystyle\psi\begin{pmatrix}I_{n-m_{1}-m_{2}}&v\\\
0&u_{m_{1}+m_{2}}\end{pmatrix}=\psi_{[m_{1},m_{2}]}(u_{m_{1}+m_{2}}).$

###### Proof.

For any automorphic form $f\in\mathcal{A}(G_{n}(k)\backslash
G_{n}(\mathbb{A}))$, the function $D^{(m_{2})}f$ can be restricted to the
subgroup $GL_{n-m_{2}}$ on the top-left corner. After composing with
$D^{(m_{1})}$, by the matrix decomposition

$\begin{pmatrix}I_{n-m_{1}-m_{2}}&v\\\
0&u_{m_{1}+m_{2}}\end{pmatrix}=\begin{pmatrix}I_{n-m_{1}-m_{2}}&0&v_{3}\\\
0&I_{m_{1}}&v_{2}\\\
0&0&u_{m_{2}}\end{pmatrix}\begin{pmatrix}I_{n-m_{1}-m_{2}}&v_{1}&0\\\
0&u_{m_{1}}&0\\\ 0&0&1\end{pmatrix},$

thus we can decompose the integral as

$\displaystyle\int_{[U_{[(n-m_{1}-m_{2})1^{m_{1}+m_{2}}]}]}f\left(\begin{pmatrix}I_{n-m_{1}-m_{2}}&v\\\
0&u_{m_{1}+m_{2}}\end{pmatrix}g\right)\psi_{[m_{1},m_{2}]}\begin{pmatrix}I_{n-m_{1}-m_{2}}&v\\\
0&u_{m_{1}+m_{2}}\end{pmatrix}du_{m_{1}+m_{2}}dv$
$\displaystyle=\int_{[U_{[(n-m_{1}-m_{2})1^{m_{1}}]}]}\int_{[U_{[(n-m_{2})1^{m_{2}}]}]}f\left(\begin{pmatrix}I_{n-m_{1}-m_{2}}&0&v_{3}\\\
0&I_{m_{1}}&v_{2}\\\
0&0&u_{m_{2}}\end{pmatrix}\begin{pmatrix}I_{n-m_{1}-m_{2}}&v_{1}&0\\\
0&u_{m_{1}}&0\\\ 0&0&1\end{pmatrix}g\right)\times$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\psi_{m_{2}}\begin{pmatrix}I_{n-m_{1}-m_{2}}&0&v_{3}\\\
0&I_{m_{1}}&v_{2}\\\
0&0&u_{m_{2}}\end{pmatrix}\psi_{m_{1}}\begin{pmatrix}I_{n-m_{1}-m_{2}}&v_{1}&0\\\
0&u_{m_{1}}&0\\\ 0&0&1\end{pmatrix}du_{m_{2}}du_{m_{1}}dv_{1}dv_{2}dv_{3}.$

∎

Using this proposition, for any Whittaker pair $(H,\phi)$ with
$H=\mathrm{diag}(n-1,n-3,\ldots,3-n,1-n)$ and $\phi$ a nilpotent element with
Jordan type $(\lambda_{1},\ldots,\lambda_{r})$, the corresponding Fourier-
Whittaker coefficient operator can be identified with a composition of
derivative operators:

$\mathcal{W}_{H,\phi}=D^{(\lambda_{1})}\circ\ldots\circ D^{(\lambda_{r})},$

which is the degenerate Whittaker coefficient along the maximal unipotent
radical with respect to the character

$\displaystyle\psi_{\alpha}\left(\begin{smallmatrix}1&u_{1}&\ldots&&&\\\
&1&u_{2}&\ldots&&\\\ &&1&u_{3}&\ldots&&\\\ &&&&\ldots&&\\\ &&&&&1&u_{n-1}\\\
&&&&&&1\end{smallmatrix}\right)=$
$\displaystyle\psi(u_{1}+\ldots+u_{\alpha_{1}-1})\psi(u_{\alpha_{1}+1}+\ldots+u_{\alpha_{1}+\alpha_{2}-1})\ldots$
$\displaystyle\psi(u_{\alpha_{1}+\ldots+\alpha_{s-1}+1}+\ldots+u_{\alpha_{1}+\ldots+\alpha_{s}-1}).$

###  3.4. A First Example: Degenerate Eisenstein Series

For an Eisenstein series attached to the parabolic induction from a trivial
representations on the Levi subgroup $L_{[n_{1},n_{2}]}$:

$E_{\underline{s}}(\cdot):\mathrm{Ind}^{G_{n}}_{P_{[n_{1},n_{2}]}}\left(|\cdot|^{s_{1}}\otimes|\cdot|^{s_{2}}\right)\rightarrow\mathcal{A}_{n},$

we would like to understand the derivative $D^{(m)}E_{\underline{s}}(\phi)$
with $\phi$ the spherical vector of the principal series
$\mathrm{Ind}^{G_{n}}_{P_{[n_{1},n_{2}]}}\left(|\cdot|^{s_{1}}\otimes|\cdot|^{s_{2}}\right)$.
By Corollary 6, the derivative $D^{(m)}E_{\underline{s}}(\phi)$ can be written
as a sum

(5) $\displaystyle
D^{(m)}E_{\underline{s}}(\phi)=\sum_{w\in\mathbb{W}^{m}_{n_{1},n_{2}}}I_{w}(\underline{s},\phi),$

in which each individual integral $I_{w}(\underline{s},\phi)$ is given by the
composition of three operators:

$I_{w}(\underline{s},\phi)=\mathcal{W}_{m}\circ\mathcal{A}_{w}\circ
c_{[n_{1}-m_{1},m_{1}],[n_{2}-m_{2},m_{2}]}\left(\phi\right),$

with the correspondence of $w$ and the pair of integers $(m_{1},m_{2})$ is
given in Lemma 5. It is possible that many terms $I_{w}(\underline{s},\phi)$
in the sum will vanish. The vanishing of such terms is described in the
following proposition:

######  Proposition 9.

All terms $I_{w}(\underline{s},\phi)$ vanish except for the terms
corresponding to the following Weyl group elements:

  1. (1)

When $m=1$:

$\displaystyle w_{[0,1]}$ $\displaystyle=\mathrm{id},$ $\displaystyle
w_{[1,0]}$
$\displaystyle=\left\\{1,\ldots,n_{1}-1,n_{1}+1,\ldots,n_{1}+n_{2},n_{1}\right\\}.$

  2. (2)

When $m=2$:

$\displaystyle
w_{[1,1]}=\left\\{1,\ldots,n_{1}-1,n_{1}+1,\ldots,n_{1}+n_{2}-1,n_{1},n_{1}+n_{2},\right\\}.$

###### Proof.

We prove this proposition by induction on the rank of the group $GL_{n}$.
Setting $n=2$, the case $(n_{1},n_{2})=(2,0)$ or $(0,2)$ corresponds to the
situation when the representation is a character with no induction involved.
In this case, $D^{(m)}E_{\underline{s}}(\phi)$ is simply an integral along the
unipotent radical, and is nonzero only if $m=1$. For $(n_{1},n_{2})=(1,1)$,
the only nonvanishing term in $D^{(m)}E_{\underline{s}}(\phi)$ corresponds to
the cases $(m_{1},m_{2})=(1,0)$, $(0,1)$ and $(1,1)$. By (4), the case
$(m_{1},m_{2})=(1,0)$ or $(0,1)$ corrresponds to the constant term of the
$GL_{2}$ Eisenstein series, and the case $(1,1)$ corresponds to the
calculation of the Whittaker coefficient of the Eisenstein series.
Assuming that the proposition holds for all lower-rank cases, by Lemma 5, it
suffices to study the vanishing of each $I_{w}(\underline{s},\phi)$ for every
Weyl group element of the form

$w_{(m_{1},m_{2})}=\left(1,\ldots,n_{1}-m_{1};n_{1}+1,\ldots,n_{1}+n_{2}-m_{2};n_{1}-m_{1}+1,\ldots,n_{1};n_{1}+n_{2}-m_{2}+1,\ldots,n_{1}+n_{2}\right).$

Moreover, by (4) again, after restricted to the Levi subgroup
$M_{[n_{1}-m_{1},n_{2}-m_{2},m_{1},m_{2}]}$, the term
$I_{w}(\underline{s},\phi)$ is the product of an Eisenstein series
$E_{\underline{s}}^{[n-m_{1},n-m_{2}]}$ on $GL_{n-m}$ and the Whittaker
coefficient
$\mathcal{W}_{m_{1}+m_{2}}\left(E_{\underline{s}}^{[m_{1},m_{2}]}\right)$,
which is the generic Whittaker coefficient along the maximal unipotent
subgroup of the Eisenstein series constructed from the degenerate principal
series
$\mathrm{Ind}_{P_{[m_{1},m_{2}]}}^{G_{m}}\left(|\cdot|^{s_{1}}\otimes|\cdot|^{s_{2}}\right)$.
By the induction hypothesis, this coefficient $I_{w}(\underline{s},\phi)$ can
be nonzero only if $m_{1},m_{2}\leq 1$. ∎

Therefore, after restricted to the subgroup $GL_{n-m}$ on the top-left corner,
the vector $I_{w}(\phi,g)$ finds itself in the space of automorphic functions
generated by an Eisenstein series

$E^{[n_{1}-m_{1},n_{2}-m_{2}]}(\underline{s},\phi;g)=\sum_{\gamma\in
P_{[n_{1}-m_{1},n_{2}-m_{2}]}(k)\backslash G_{n-m}(k)}\phi(\gamma g)$

on the subgroup $GL_{n-m}$. In order to connect this observation to the study
of Whittaker support of Eisenstein series, we need the following lemma:

######  Lemma 10.

For any neutral Whittaker pair $(h,\phi_{\lambda})$ with $\phi_{\lambda}$
given by the Killing pairing $\phi_{\lambda}=K(\cdot,f_{\lambda})$ with a
Jordan form $f_{\lambda}$, and another Whittaker pair
$(h^{\prime},\phi_{\lambda}^{\prime})$ obtained from $(h,\phi_{\lambda})$ by
permuting its Jordan blocks, with $h,h^{\prime}$ lying in the same Cartan
subalgebra, then for any Whittaker pair $(H,\phi^{\prime}_{\lambda})$, we have
$\mathcal{W}_{h,\phi_{\lambda}}=0$ if
$\mathcal{W}_{H,\phi_{\lambda}^{\prime}}=0$.

###### Proof.

The Whittaker coefficients $\mathcal{W}_{h,\phi_{\lambda}}$ and
$\mathcal{W}_{h^{\prime},\phi_{\lambda}^{\prime}}$ are Weyl group conjugates
with each other. The result of this lemma follows from the existence of the
integral operator $\mathcal{M}_{h^{\prime},H}^{\prime}$, according to Theorem
7, such that

$\mathcal{W}_{h^{\prime},\phi_{\lambda}^{\prime}}=\mathcal{M}_{h^{\prime},H}^{\prime}\left(\mathcal{W}_{H,\phi_{\lambda}^{\prime}}\right).$

Thus, $\mathcal{W}_{H,\phi_{\lambda}^{\prime}}=0$ implies
$\mathcal{W}_{h^{\prime},\phi_{\lambda}^{\prime}}=0$. ∎

Using the method developed in this section, we can reprove the following
result on the Whittaker support of degenerate Eisenstein series, first proven
by Cai in [Cai18]:

######  Corollary 11.

For the degenerate Eisenstein series $E_{\underline{s}}$ induced from the
trivial representation of the Levi subgroup $L_{[n_{1},n_{2}]}$, the
degenerate Whittaker coefficient
$\mathcal{W}_{U,\phi_{\lambda}}\left(E_{\underline{s}}\right)$ can be nonzero
only if the corresponding partition $\lambda$ contains only $1$ and $2$.

###### Proof.

We prove this corollary by induction on $n$. According the proof Proposition
9, the base cases when $n=1$ or $2$ are straightforward. By Proposition 9, if
neither of $n_{1},n_{2}$ is zero, and without loss of generality, assuming
$n_{1}\geq n_{2}$, the Eisenstein series $E_{\underline{s}}$ allows both
$D^{(1)}$ and $D^{(2)}$. By Lemma 6 and (5), the derivative
$D^{(1)}E_{\underline{s}}$ is the sum of two vectors

$E^{[n_{1}-1,n_{2}]}\boxtimes\chi_{1}+E^{[n_{1},n_{2}-1]}\boxtimes\chi_{2}$

with $\chi_{1},\chi_{2}$ unramified characters of $GL_{1}$. By the induction
hypothesis, the allowed Whittaker coefficient of each one of these two terms
corresponds to the partition $(2^{n_{2}}1^{n_{1}-n_{2}-1})$ and
$(2^{n_{2}-1}1^{n_{1}-n_{2}+1})$, respectively. Therefore, the top orbit
allowing a degenerate Whittaker coefficient corresponds to the partition
$(2^{n_{2}}1^{n_{1}-n_{2}-1})$, with only the first term surviving. The
partial order of a partition containing only 1 or 2 is determined by the total
number of 2’s in the sequence. However, for any partition $(2^{t}1^{n-2t})$
with $t>n_{2}$, the Whittaker coefficient corresponding to the partition
$(2^{t}1^{n-2t-1})$ kills both $E^{[n_{1}-1,n_{2}]}$ and
$E^{[n_{1},n_{2}-1]}$. Therefore, by Theorem 7, for any Whittaker pair
$(S,\phi_{\lambda})$ with $\lambda=(2^{t}1^{n-2t-1}),t>n_{2}$, the
corresponding $\mathcal{W}_{S,\phi_{\lambda}}$ is zero. ∎

The original result of Cai in [Cai18] says that the _Whittaker support_ of
this degenerate Eisenstein series is $[1^{n_{1}}]+[1^{n_{2}}]$. For the
degenerate Eisenstein series induced from a parabolic subgroup with more than
two Levi blocks, we can perform induction by stages and construct Eisenstein
series on a bigger group from smaller groups.

###  3.5. Generalized Speh Representation

The following lemma for the constant term of the generalized Speh
representation can be found in [Liu13]:

######  Lemma 12.

The constant term along $U_{[n_{1},n_{2}]}$ with $n_{1}+n_{2}=n$ of the
residue $\mathrm{Res}_{n}E_{n}(f,\underline{\nu})$ can be realized as a
section in
$\mathrm{Ind}_{P_{[n_{1},n_{2}]}}^{G_{n}}(\Delta(\tau,n_{1})|\cdot|^{-\frac{n_{2}}{2}}\boxtimes\Delta(\tau,n_{2})|\cdot|^{\frac{n_{1}}{2}})$.

Applying the derivative operator $D^{(ra)}$, which is the composition of a
constant term operator $c_{P_{[(n-r)a,ra]}}$ and a generic Whittaker
coefficient operator $\mathcal{W}_{ra}$. By Lemma 12, for any
$f\in\Delta(\tau,n)$, $c_{P_{[(n-r)a,ra]}}f$ is a section of the induced
representation
$\mathrm{Ind}_{P_{[n-r,r]}}^{G}(\Delta(\tau,n-r)|\cdot|^{-\frac{r}{2}}\boxtimes\Delta(\tau,r)|\cdot|^{\frac{n-r}{2}})$.

######  Proposition 13.

For a generalized Speh representation $\Delta(\tau,n)$ realized as the residue
of an Eisenstein series with $\tau$ a cuspidal automorphic representation of
$GL_{a}$, the only allowed derivative is $D^{(a)}$.

###### Proof.

We can prove this proposition by induction on $n$. When $n=1$, since $\tau$ is
a cuspidal automorphic representation of $GL_{a}$, any constant term operator
$c_{P_{[n_{1},n_{2}]}}$ with $n_{1}$ or $n_{2}$ nonzero will vanish.
Therefore, the only nonzero derivative is $D^{(a)}$, which corresponds to
calculating the generic Whittaker coefficient of any vector in the cuspidal
automorphic representation $\tau$. Assuming the proposition holds for all
$k\leq n$, by Lemma 12, the only possibly nonzero constant terms are of the
form $c_{P_{[(n-r)a,ra]}}$. However, when $r>1$, by Lemma 12, Corollary 6 and
the induction hypothesis, the Whittaker coefficient $\mathcal{W}_{ra}$ as in
Corollary 6 vanishes. Thus the only allowed derivative for any vector
$f\in\Delta(\tau,n)$ is $D^{(a)}$. ∎

This process can be iterated. The derivative operator $D^{(a)}$ of
$\Delta(\tau,n)$ can be realized as a composition of two operators

$\Delta(\tau,n)\xrightarrow{c_{[(n-1)a,a]}}\mathrm{Ind}_{P_{[(n-1)a,a]}}^{G_{na}}(\Delta(\tau,n-1)|\cdot|^{-\frac{1}{2}}\boxtimes\tau|\cdot|^{\frac{n-1}{2}})\xrightarrow{\mathcal{W}_{a}}\mathrm{Ind}_{P_{[(n-1)a,a]}}^{G_{na}}(\Delta(\tau,n-1)|\cdot|^{-\frac{1}{2}}\boxtimes|\cdot|^{\frac{n-1}{2}})$

As a result, the only nonzero degenerate Whittaker coefficients
$\mathcal{W}_{U,\phi_{\lambda}}(f)$ for $f\in\Delta(\tau,n)$ are those with a
partition $\lambda$ containing only $a$, as shown in [Liu13, Theorem 2.3.3].
This result can also be proven by our method, as stated in the following
lemma:

######  Corollary 14.

The only nonzero degenerate Whittaker coefficient of the generalized Speh
representation $\Delta(\tau,n)$ is attached to the orbit $[a^{n}]$.

###### Proof.

For any pure tensor $f\in\Delta(\tau,n)$, by Proposition 13, the only allowed
derivative is $D^{(a)}$. When restricting $D^{(a)}f$ to the subgroup
$GL_{(n-1)a}$ embedded in the top-left corner, according to the proof of
Proposition 13, we can see that $D^{(a)}f$ belongs to the generalized Speh
representation $\Delta(\tau,n-1)$. For the Whittaker pair $(H,\phi)$ with
$H=(n-1,\ldots,1-n)$ and $\phi$ corresponding to the orbit $[a^{n}]$, for any
pure tensor $f$, the corresponding degenerate Fourier coefficient
$\mathcal{W}_{H,\phi}(f)$ is equal to the derivative $D^{(a)}\circ\ldots\circ
D^{(a)}f$. ∎

###  3.6. Induction from Discrete Spectrum Data

We will study the Whittaker support of the Eisenstein series constructed from
the induced representation from generalized Speh representations on a maximal
parabolic subgroup:

$I(\tau_{1},\tau_{2};\underline{s})=\mathrm{Ind}_{P_{[n_{1},n_{2}]}}^{G_{n}}(\Delta(\tau_{1},n_{1})|\cdot|^{s_{1}}\boxtimes\Delta(\tau_{2},n_{2})|\cdot|^{s_{2}}).$

We can construct an Eisenstein series $E_{\underline{s}}(\tilde{\phi})$ for
any vector $\phi\in I(\tau_{1},\tau_{2};\underline{s})$. In the following two
cases, this Eisenstein series will exhibit no poles:

  1. (1)

$\tau_{1}\not\cong\tau_{2}$,

  2. (2)

$\tau_{1}\cong\tau_{2}\cong\tau$, with $s_{1}-s_{2}\gg 0$.

####  3.6.1. Case $\tau_{1}\not\cong\tau_{2}$

Assuming that $\tau_{i}$ is a cuspidal automorphic representation on
$GL_{a_{i}}$, the generalized Speh representation $\Delta(\tau_{i},n_{i})$ is
an automorphic representation for the group $GL_{a_{i}n_{i}}$. By Corollary 6,

$\displaystyle
D^{(s)}E_{\underline{s}}(\tilde{\varphi},g)=\sum_{w\in\mathbb{W}_{n_{1},n_{2}}^{s}}\mathcal{W}_{s}\left(\sum_{m^{\prime}\in
M_{[n-s]}\cap w^{-1}P_{[n_{1},n_{2}]}w\backslash
M_{[n-s]}(k)}\mathcal{A}_{w}\left(c_{w^{-1}M_{[n-s]}w\cap
P_{[n_{1},n_{2}]}}\tilde{\varphi}\right)(wm^{\prime}g)\right),$

the permitted constant terms are those along the parabolic subgroup
$P_{[a_{1}(n_{1}-r_{1}),a_{1}r_{1}]}\times
P_{[a_{2}(n_{2}-r_{2}),a_{2}r_{2}]}$ of the subgroup $GL_{a_{1}n_{1}}\times
GL_{a_{2}n_{2}}$ for permitted $r_{1},r_{2}$. Therefore, for the $r$-th
derivative, the permitted constant terms in the sum correspond to any block
partition

$[n_{1},n_{2}]\rightsquigarrow[n_{1}-p_{1},p_{1};n_{2}-p_{2},p_{2}]$

satisfying $p_{1}a_{1}+p_{2}a_{2}=r$. The operator $\mathcal{A}_{w}$
corresponds to the Weyl group element

$w=(1,2,\ldots,n_{1}-r_{1};n_{1}+1,\ldots,n_{1}+n_{2}-r_{2};n_{1}-r_{1}+1,\ldots,n_{1};n_{1}+n_{2}-r_{2}+1;n_{1}+n_{2})$

that intertwines between the following two principal series:

$\displaystyle
I\left(\Delta(\tau_{1},n_{1}-r_{1})|\cdot|^{-\frac{r_{1}}{2}+s_{1}}\boxtimes\Delta(\tau_{1},r_{1})|\cdot|^{\frac{n_{1}-r_{1}}{2}+s_{1}}\boxtimes\Delta(\tau_{2},n_{2}-r_{2})|\cdot|^{-\frac{r_{2}}{2}+s_{2}}\boxtimes\Delta(\tau_{2},r_{2})|\cdot|^{\frac{n_{2}-r_{2}}{2}+s_{2}}\right)$
$\displaystyle\xrightarrow{\mathcal{A}_{w}}$ $\displaystyle
I\left(\Delta(\tau_{1},n_{1}-r_{1})|\cdot|^{-\frac{r_{1}}{2}+s_{1}}\boxtimes\Delta(\tau_{2},n_{2}-r_{2})|\cdot|^{-\frac{r_{2}}{2}+s_{2}}\boxtimes\Delta(\tau_{1},r_{1})|\cdot|^{\frac{n_{1}-r_{1}}{2}+s_{1}}\boxtimes\Delta(\tau_{2},r_{2})|\cdot|^{\frac{n_{2}-r_{2}}{2}+s_{2}}\right).$

Since $\tau_{1}\not\cong\tau_{2}$, the Eisenstein series constructed from the
induced representation
$I(\Delta(\tau_{1},r_{1})|\cdot|^{\frac{n_{1}-r_{1}}{2}+s_{1}}\boxtimes\Delta(\tau_{2},n_{2}-r_{2})|\cdot|^{-\frac{r_{2}}{2}+s_{2}})$
does not have any pole in the region $\mathrm{Re}(s_{1}-s_{2})\gg 0$, and the
intertwining operator $\mathcal{A}_{w}$ does not introduce any extra
vanishing. In this situation, the nonzero derivatives can be described in the
following proposition:

######  Proposition 15.

If $\tau_{1}\not\cong\tau_{2}$, the only possibly nonzero derivatives of
$I(\tau_{1},\tau_{2};s)$ are $D^{(a_{1})},D^{(a_{2})}$ and
$D^{(a_{1}+a_{2})}$.

###### Proof.

In the base cases when $(n_{1},n_{2})=(1,0)$ or $(n_{1},n_{2})=(0,1)$, the
principal series $I(\tau_{1},\tau_{2};s)$ is a single cuspidal automorphic
representation, and the only nonzero derivative is the highest derivative
$D^{(a_{1})}$ or $D^{(a_{2})}$. If $(n_{1},n_{2})=(1,1)$, the principal series
$I(\tau_{1},\tau_{2};s)$ is generic, and the derivative $D^{(a_{1}+a_{2})}$
calculates the generic Whittaker coefficient of the corresponding Eisenstein
series. For other derivatives, since the constant term along any parabolic
subgroup $Q$ is zero unless $Q$ and $P_{[n_{1},n_{2}]}$ are associate, there
are only two other possibly nonzero derivatives $D^{(a_{1})}$ and
$D^{(a_{2})}$. Assuming the induction hypothesis holds for all lower-rank
cases, the operator $\mathcal{W}_{r}$ calculates the generic Whittaker
coefficient for the induced representation
$I(\Delta(\tau_{1},r_{1})|\cdot|^{\frac{n_{1}-r_{1}}{2}+s_{1}}\boxtimes\Delta(\tau_{2},r_{2})|\cdot|^{\frac{n_{2}-r_{2}}{2}+s_{2}})$.
By the induction hypothesis, this Whittaker coefficient is nonzero only if
$r_{1}\leq 1$ and $r_{2}\leq 1$. When corresponded to the order
$r=r_{1}a_{1}+r_{2}a_{2}$ of the derivative, the only possibilities for $r$
are $r=a_{1},a_{2}$ and $a_{1}+a_{2}$. ∎

####  3.6.2. Case $\tau_{1}\cong\tau_{2}\cong\tau$

In this case, by [Zha22, Theorem A], the residues of the Eisenstein series
$E_{\underline{s}}(\tilde{\varphi})$ at the simple poles
$s_{1}-s_{2}=\frac{n_{1}+n_{2}}{2}-\alpha$ for
$\alpha\in\\{0,1,\ldots,n_{1}-1\\}$ is the irreducible quotient of the
principal series $I(\tau_{1},\tau_{2};\underline{s})$. In this case, the
intertwining operator $\mathcal{A}_{w}$ in (4) may introduce extra zeros. We
will analyze the possible zeros or poles introduced by the intertwining
operator $\mathcal{A}_{w}$.

######  Proposition 16.

The derivative $D^{(m)}$ of the residue of the Eisenstein series
$E_{\underline{s}}$ at $s_{1}-s_{2}=\frac{n_{1}+n_{2}}{2}-\alpha$, with
$\alpha\in\\{0,1,\ldots,\min\\{n_{1},n_{2}\\}-1\\}$ is possibly nonzero only
if

  1. (1)

$m=a$ or $2a$ when $\alpha>0$,

  2. (2)

$m=a$ when $\alpha=0$.

However, if $n_{2}=1$, in which case the only allowed $\alpha$ is 0, then
$D^{(a)}$ also vanishes.

###### Proof.

Since the integral operators $c_{P^{\prime}\cap w^{-1}Mw},\mathcal{A}_{w}$ and
$\mathcal{W}_{s}$ in Corollary 6 are either over a compact domain or can be
analytically continued to the whole complex plane, they all commute with the
residue operator. The result from Proposition 15 still holds, but extra
vanishing may be introduced by the operator $\mathcal{A}_{w}$ and the residue
operator. According to the proof of the Proposition 15, the nonzero
derivatives $D^{(m)}$ correspond to the cases

  1. (1)

If $m=a$, then $(r_{1},r_{2})=(1,0)$ or $(0,1)$.

  2. (2)

If $m=2a$, then $(r_{1},r_{2})=(1,1)$.

When $m=a$, for the term $I_{w}$ corresponding to the case
$(r_{1},r_{2})=(1,0)$, the constant term operator sends the vector of the
induced representation to the induced representation following the
intertwining operator:

$\displaystyle
I\left(\Delta(\tau_{1},n_{1}-1)|\cdot|^{-\frac{1}{2}+s_{1}}\boxtimes\tau|\cdot|^{\frac{n_{1}-1}{2}+s_{1}}\boxtimes\Delta(\tau_{2},n_{2})|\cdot|^{s_{2}}\right)$
$\displaystyle\xrightarrow{\mathcal{A}_{w}}$ $\displaystyle
I\left(\Delta(\tau_{1},n_{1}-1)|\cdot|^{-\frac{1}{2}+s_{1}}\boxtimes\Delta(\tau_{2},n_{2})|\cdot|^{s_{2}}\boxtimes\tau|\cdot|^{\frac{n_{1}-1}{2}+s_{1}}\right).$

If $\alpha<n_{1}-1$, the pole of the original Eisenstein series
$E_{\underline{s}}$ at $s_{1}-s_{2}=\frac{n_{1}+n_{2}}{2}-\alpha$ is still a
pole of the Eisenstein series $E^{[n_{1}-1,n_{2}]}_{\underline{s}}$
constructed from the principal series
$I\left(\Delta(\tau_{1},n_{1}-1)|\cdot|^{-\frac{1}{2}+s_{1}}\boxtimes\Delta(\tau_{2},n_{2})|\cdot|^{s_{2}}\right)$
when restricted to the subgroup $GL_{n_{1}+n_{2}-1}$ on the top-left corner.
The intertwining operator $\mathcal{A}_{w}$ has no pole unless when
$\alpha=n_{1}-1$. When $\alpha=n_{1}-1$,
$s_{1}-s_{2}=\frac{n_{1}+n_{2}}{2}-\alpha$ is no longer a pole of
$E_{\underline{s}}^{[n_{1}-1,n_{2}]}$. The intertwining operator can be
written as the product $\mathcal{A}_{w}=r(w,\underline{s})N(w,\underline{s})$
(cf. [MW89]) of the normalized intertwining operator $N(w,\underline{s})$ with
the normalization factor

$r(w,\underline{s})=\prod_{\begin{subarray}{c}i<j\\\
w(i)>w(j)\end{subarray}}\frac{L(\nu_{i}-\nu_{j},\tau\times\hat{\tau})}{L(1+\nu_{i}-\nu_{j},\tau\times\hat{\tau})\epsilon(\nu_{i}-\nu_{j},\tau\times\hat{\tau},\psi)}.$

In this case, assuming $s_{1}-s_{2}=\frac{n_{1}+n_{2}}{2}-\alpha+t$, and after
cancellation between numerators and denominators, the normalization factor

$r(w,\underline{s})=\frac{L\left(\frac{n_{1}-n_{2}}{2}+s_{1}-s_{2}\right)}{L\left(\frac{n_{1}+n_{2}}{2}+s_{1}-s_{2}\right)}\times\epsilon\text{
factors}=\frac{L(n_{1}-\alpha+t)}{L(n_{1}+n_{2}-\alpha+t)}\times\epsilon\text{
factors}.$

When $\alpha=n_{1}-1$, the intertwining operator $\mathcal{A}_{w}$ has a
simple pole at $s=\frac{n_{1}+n_{2}}{2}-\alpha$.

In the case $(r_{1},r_{2})=(0,1)$, the intertwining operator $\mathcal{A}_{w}$
is a constant operator on any vector in the principal series
$I\left(\Delta(\tau_{1},n_{1})|\cdot|^{s_{1}}\boxtimes\Delta(\tau_{2},n_{2}-1)|\cdot|^{-\frac{1}{2}+s_{2}}\boxtimes\tau|\cdot|^{\frac{n_{2}-1}{2}+s_{2}}\right)$,
and the whole term corresponding to this case in Corollary 6 is killed by the
residue operator if $\alpha=0$.

For $m=2a$, the intertwining operator $\mathcal{A}_{w}$ intertwines between
the following two principal series:

$\displaystyle
I\left(\Delta(\tau_{1},n_{1}-1)|\cdot|^{-\frac{1}{2}+s_{1}}\boxtimes\tau|\cdot|^{\frac{n_{1}-1}{2}+s_{1}}\boxtimes\Delta(\tau_{2},n_{2}-1)|\cdot|^{-\frac{1}{2}+s_{2}}\boxtimes\tau|\cdot|^{\frac{n_{2}-1}{2}+s_{2}}\right)$
$\displaystyle\xrightarrow{\mathcal{A}_{w}}$ $\displaystyle
I\left(\Delta(\tau_{1},n_{1}-1)|\cdot|^{-\frac{1}{2}+s_{1}}\boxtimes\Delta(\tau_{2},n_{2}-1)|\cdot|^{-\frac{1}{2}+s_{2}}\boxtimes\tau|\cdot|^{\frac{n_{1}-1}{2}+s_{1}}\boxtimes\tau|\cdot|^{\frac{n_{2}-1}{2}+s_{2}}\right).$

In this case, assuming $s_{1}-s_{2}=\frac{n_{1}+n_{2}}{2}-\alpha+t$, and after
cancellation between numerators and denominators, the normalization factor

$r(w,\underline{s})=\frac{L\left(\frac{n_{1}-n_{2}}{2}+1+s_{1}-s_{2}\right)}{L\left(\frac{n_{1}+n_{2}}{2}+s_{1}-s_{2}\right)}\times\epsilon\text{
factors}=\frac{L(n_{1}+1-\alpha+t)}{L(n_{1}+n_{2}-\alpha+t)}\times\epsilon\text{
factors}.$

This normalization factor will not introduce any pole to the intertwining
operator. The only complication occurs when $\alpha=0$, in which case
$s_{1}-s_{2}=\frac{n_{1}+n_{2}}{2}$ is no longer a pole of the resulting
Eisenstein series, thus the derivative $D^{(2a)}$ will be killed by the
residue operator. ∎

###  3.7. Whittaker Support of Eisenstein Series with Discrete Spectrum Data

For the principal series $I(\tau_{1},\tau_{2};\underline{s})$ induced from
discrete spectrum data, if $\tau_{1}\not\cong\tau_{2}$, by Proposition 15, the
allowed derivatives are $D^{(a_{1})}$, $D^{(a_{2})}$ and $D^{(a_{1}+a_{2})}$,
sending the Eisenstein series on $GL_{n}$ to an Eisenstein series
$E^{[n_{1}-1,n_{2}]}$, $E^{[n_{1},n_{2}-1]}$ and $E^{[n_{1}-1,n_{2}-1]}$,
respectively. Applying Proposition 15 iteratively, a degenerate Whittaker
coefficient $\mathcal{W}_{H,\phi_{\lambda}}$ is zero if the partition
$\lambda$ contains numbers other than $a_{1},a_{2}$ and $a_{1}+a_{2}$.

######  Proposition 17.

Without loss of generality, assuming $n_{1}\geq n_{2}$, for
$\tau_{1}\not\cong\tau_{2}$ two cuspidal automorphic representations of
$GL_{a_{1}},GL_{a_{2}}$, respectively, the Eisenstein series
$E_{\underline{s}}$ associated to the principal series
$I(\tau_{1},\tau_{2};\underline{s})$ has no poles when
$\mathrm{Re}(s_{1}-s_{2})>0$, and the maximal orbit that allows a degenerate
Whittaker coefficient corresponds to the partition
$((a_{1}+a_{2})^{n_{2}}a_{1}^{n_{1}-n_{2}})$.

###### Proof.

If $(n_{1},n_{2})=(1,0)$ or $(0,1)$, the result reduces to the case of
cuspidal automorphic representations. If $(n_{1},n_{2})=(1,1)$, the maximal
orbit allowing a Whittaker coefficient is the generic orbit of
$GL_{a_{1}+a_{2}}$, and hence corresponds to the partition $(a_{1}+a_{2})$. If
$(n_{1},n_{2})=(2,0)$ or $(0,2)$, the result reduces to the cases of
generalized Speh representations, and the maximal orbit allowing a Whittaker
coefficient corresponds to the partition $(a_{1}^{2})$ and $(a_{2}^{2})$,
respectively. We now apply induction on $n=n_{1}+n_{2}$ to prove the general
case. Among all the partitions containing $a_{1},a_{2}$ and $a_{1}+a_{2}$, the
largest partitions are those with prefix $(a_{1}+a_{2})^{t}$ for some $t>0$.
Assuming $n_{1}>n_{2}$, for any derivative $D^{(a_{1})}$, by Proposition 4,

$D^{(a_{1})}E_{\underline{s}}(g)=E^{[n_{1}-1,n_{2}]}\boxtimes\mathcal{W}_{a_{1}}+\begin{cases}0&\text{if
}a_{1}\not=a_{2}\\\ E^{[n_{1},n_{2}-1]}\boxtimes\mathcal{W}_{a_{2}}&\text{if
}a_{1}=a_{2}\end{cases}$

By the induction hypothesis, each one of the two terms on the right hand side
has Whittaker support corresponding to the partition
$[(a_{1}+a_{2})^{n_{2}}a_{1}^{n_{1}-n_{2}-1}]$ and
$[(a_{1}+a_{2})^{n_{2}-1}a_{1}^{n_{1}-n_{2}+1}]$, respectively. In either
case, the second term is killed by the degenerate Whittaker coefficient
operator corresponding to the partition
$[(a_{1}+a_{2})^{n_{2}}a_{1}^{n_{1}-n_{2}-1}]$. If $n_{1}=n_{2}$, applying the
derivative operator $D^{(a_{1}+a_{2})}$, we have

$D^{(a_{1}+a_{2})}E(\underline{s},g)=E^{[n_{1}-1,n_{2}-1]}\boxtimes\mathcal{W}_{a_{1}+a_{2}}$

with $\mathcal{W}_{a_{1}+a_{2}}$ the Whittaker coefficient of the cuspidal
Eisenstein series $E^{[1,1]}$. By the induction hypothesis, the maximal orbit
allowing a degenerate Whittaker coefficent for $E^{[n_{1}-1,n_{2}-1]}$ is
$[(a_{1}+a_{2})^{n_{1}-1}]$. Since the only allowed derivatives are
$D^{(a_{1})},D^{(a_{2})}$ and $D^{(a_{1}+a_{2})}$, the maximal orbit allowing
a degenerate Whittaker coefficient is thus $[(a_{1}+a_{2})^{n_{1}}]$. ∎

In the situation when $\tau_{1}=\tau_{2}$, we can calculate the Whittaker
support of the residue of the Eisenstein series at
$s=\frac{n_{1}+n_{2}}{2}-\alpha$.

######  Proposition 18.

The Whittaker support of the residue of the Eisenstein series
$\mathrm{Res}_{s=\frac{n_{1}+n_{2}}{2}-\alpha}E_{\underline{s}}(g)$ is
$[a^{n-\alpha}]+[a^{\alpha}]$.

###### Proof.

The first residue appears in the the case when $(n_{1},n_{2})=(1,1)$. By
Proposition 16, the only allowed $\alpha$ with a residue at
$s=\frac{n_{1}+n_{2}}{2}-\alpha$ appears when $\alpha=0$, from which we obtain
a generalized Speh representation $\Delta(\tau,2)$. The only allowed
derivative for the residue in this case is $D^{(a)}$. Through the proof of
Proposition 16, there is a pole of the intertwining operator $\mathcal{A}_{w}$
for the case $(r_{1},r_{2})=(1,0)$, and it is clear that the derivative of the
residue
$D^{(a)}\mathrm{Res}_{s=\frac{n_{1}+n_{2}}{2}-\alpha}E_{\underline{s}}$ is a
vector in $\tau$ after restricted to $GL_{a}$. Assuming the proposition holds
for all smaller $n=n_{1}+n_{2}$, the derivative $D^{(a)}E_{\underline{s}}$ can
be written as a sum of two terms:

$D^{(a)}E_{\underline{s}}=E_{[n_{1}-1,n_{2}]}\boxtimes\mathcal{W}_{a}+E_{[n_{1},n_{2}-1]}\boxtimes\mathcal{W}_{a}^{\prime}.$

If $\alpha<n_{1}-1$, the pole $s=\frac{n_{1}+n_{2}}{2}-\alpha$ is still a pole
of the new Eisenstein series $E_{[n_{1}-1,n_{2}]}$, but if $\alpha=n_{1}-1$,
it is no longer a pole of $E_{[n_{1}-1,n_{2}]}$ but instead a pole of the
intertwininig operator $\mathcal{A}_{w}$. If $\alpha>0$ the pole
$s=\frac{n_{1}+n_{2}}{2}-\alpha$ is still a pole of $E_{[n_{1},n_{2}-1]}$.
Therefore, when $n_{2}>1$,

$D^{(a)}\mathrm{Res}_{s=\frac{n_{1}+n_{2}}{2}-\alpha}E(\underline{s},g)=\begin{cases}\mathrm{Res}_{s=\frac{n_{1}+n_{2}}{2}-\alpha}E_{[n_{1}-1,n_{2}]}\boxtimes\mathcal{W}_{2}&\alpha=0\\\
\mathrm{Res}_{s=\frac{n_{1}+n_{2}}{2}-\alpha}E_{[n_{1}-1,n_{2}]}\boxtimes\mathcal{W}_{1}+\mathrm{Res}_{s=\frac{n_{1}+n_{2}}{2}-\alpha}E_{[n_{1},n_{2}-1]}\boxtimes\mathcal{W}_{2}&0<\alpha<n_{1}-1\\\
c\mathrm{Res}_{s=\frac{n_{1}+n_{2}}{2}-\alpha}E_{[n_{1}-1,n_{2}]}\boxtimes\mathcal{W}_{1}+\mathrm{Res}_{s=\frac{n_{1}+n_{2}}{2}-\alpha}E_{[n_{1},n_{2}-1]}\boxtimes\mathcal{W}_{2}&\alpha=n_{1}-1\end{cases}$

By the induction hypothesis, for $n_{1}>1$, if $\alpha=n_{1}-1$, the Whittaker
supports of each of the two terms in
$D^{(a)}\mathrm{Res}_{s=\frac{n_{1}+n_{2}}{2}-\alpha}E(\underline{s},g)$ are
$[a^{n_{1}-1}]+[a^{n_{2}}]$ and
$[a^{n-\alpha}]+[a^{\alpha-1}]=[a^{n_{1}-2}]+[a^{n_{2}+1}]$, respectively. If
$\alpha=0$, the Whittaker support of
$D^{(a)}\mathrm{Res}_{s=\frac{n_{1}+n_{2}}{2}-\alpha}E(\underline{s},g)$ is
$[a^{n-1}]$. In other cases, each term admitting Whittaker coefficient
corresponds to $[a^{n-\alpha}]+[a^{\alpha-1}]$ and
$[a^{n-\alpha-1}]+[a^{\alpha}]$, respectively. Since
$0\leq\alpha\leq\min\\{n_{1},n_{2}\\}-1$, it turns out that any case would not
allow a Whittaker support corresponding to an orbit larger than
$[a^{n-\alpha}]+[a^{\alpha}]$. ∎

###  3.8. Eulerianity of Whittaker Coefficients

For any Whittaker pair $(H,\phi)$ and any pure tensor $f=\bigotimes_{v}f_{v}$
in an automorphic representation $\pi=\bigotimes_{v}\pi_{v}$, we say its
Whittaker coefficient $\mathcal{W}_{H,\phi}(f)$ is _Eulerian_ if
$\mathcal{W}_{H,\phi}(f)$ factorizes into a tensor product

$\mathcal{W}_{H,\phi}(f)\left(\prod_{v}g_{v}\right)=\bigotimes_{v}\mathcal{W}_{v}(f)\left(g_{v}\right)$

with $\mathcal{W}_{v}$ a function on the group $G(k_{v})$. By multiplicity one
theorems in [Sha74],[PS79], when $\phi$ is a generic character of the maximal
unipotent subgroup $U$, the Whittaker coefficient is Eulerian. A larger
variety of cases are considered in [GGK+21]. In particular, in [GGK+21,
Theorem F], the case when $\pi$ is a discrete spectrum automorphic
representation is considered. We will be applying our method to cover this
case as well as Eisenstein series induced from discrete spectrum data.

######  Theorem 19.

For $\pi$ in the discrete spectrum of $GL_{n}(\mathbb{A})$, any degenerate
Whittaker coefficient of $\pi$ is Eulerian.

###### Proof.

By (4), the only nonzero degenerate Whittaker coefficient of
$f\in\Delta(\tau,n)$ is associated to the orbit $[a^{n}]$. This degenerate
Whittaker coefficient is Eulerian because $D^{(a)}f$ decomposes as the product
of a vector $f^{\prime}\in\Delta(\tau,n-1)$ and a top Fourier coefficient of
cuspidal representation $\tau$. The Eulerianity result follows from applying
$D^{(a)}$ iteratively and the Eulerianity of the top Whittaker coefficient of
a cuspidal automorphic representation. ∎

######  Theorem 20.

For Eisenstein series constructed from the isobaric sum representation
$I(\tau_{1},\tau_{2};s)$ with $\tau_{1}\not\cong\tau_{2}$, if $a_{1}\neq
a_{2}$, then any nonzero Whittaker coefficent is Eulerian.

###### Proof.

The only nonzero Whittaker coefficients of the Eisenstein series are those
which contains only $a_{1},a_{2}$ and $a_{1}+a_{2}$. In the proof of
Proposition 17, if $a_{1}\neq a_{2}$, the derivative $D^{(a_{1})},D^{(a_{2})}$
of the Eisenstein series contains only one term
$E^{[n_{1}-1,n_{2}]}\otimes\mathcal{W}_{a_{1}}$ or
$E^{[n_{1},n_{2}-1]}\otimes\mathcal{W}_{a_{2}}$, with
$\mathcal{W}_{a_{1}},\mathcal{W}_{a_{2}}$ top Whittaker coefficients of
$\tau_{1},\tau_{2}$, respectively, and thus are Eulerian functions on
$GL_{a_{1}},$ and $GL_{a_{2}}$, respectively. Similarly, for the derivative
$D^{(a_{1}+a_{2})}$, the derivative of the Eisenstein series contains only one
term $E^{[n_{1}-1,n_{2}-1]}\boxtimes\mathcal{W}_{a_{1}+a_{2}}$, where
$\mathcal{W}_{a_{1}+a_{2}}$ is a top Whittaker coefficient for a cuspidal
Eisenstein series on $GL_{a_{1}+a_{2}}$. The Euleranity of any nonzero
Whittaker coefficient of the Eisenstein series follows from applying this
process iteratively. ∎

######  Theorem 21.

In all cases discussed in this paper, the top degenerate Whittaker coefficient
is Eulerian.

###### Proof.

The partition associated to the largest orbit allowing a top degenerate
Whittaker coefficient is
$[(a_{1}+a_{2})^{\min\\{n_{1},n_{2}\\}}a_{i}^{|n_{1}-n_{2}|}]$. The derivative
$D^{(a_{1}+a_{2})}E_{\underline{s}}$ is an Eulerian function
$E^{[n_{1}-1,n_{2}-1]}\boxtimes\mathcal{W}_{a_{1}+a_{2}}$. After applying
$D^{(a_{1}+a_{2})}$ multiple times such that the resulting function belongs to
the generalized Speh representation $\Delta(\tau_{i},|n_{1}-n_{2}|)$ , whose
Whittaker support on $GL_{a_{i}|n_{1}-n_{2}|}$ is $[a_{i}^{|n_{1}-n_{2}|}]$.
By [GGK+21, Theorem F], it is an Eulerian function on
$GL_{a_{i}|n_{1}-n_{2}|}$. ∎

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