# Hypergeometric integrals, hook formulas and Whittaker vectors

G. Felder⋄, A. Smirnov†, V. Tarasov∘, A. Varchenko⋆

###### Abstract.

We determine the coefficient of proportionality between two multidimensional
hypergeometric integrals. One of them is a solution of the dynamical
difference equations associated with a Young diagram and the other is the
vertex integral associated with the Young diagram. The coefficient of
proportionality is the inverse of the product of weighted hooks of the Young
diagram. It turns out that this problem is closely related to the question of
describing the action of the center of the universal enveloping algebra of
$\mathfrak{gl}_{n}$ on the space of Whittaker vectors in the tensor product of
dual Verma modules with fundamental modules, for which we give an explicit
basis of simultaneous eigenvectors.

⋄ Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland

${}^{\circ}\mskip-0.99998mu$Department of Mathematical Sciences, Indiana
University –Purdue University Indianapolis
402 North Blackford St, Indianapolis, IN 46202-3216, USA

†,⋆ Department of Mathematics, University of North Carolina at Chapel Hill
Chapel Hill, NC 27599-3250, USA

Key words: Singular vectors, Young diagrams, excited diagrams, hooks, master
function, hypergeometric integrals, Whittaker vectors

2020 Mathematics Subject Classification:

††footnotetext: ${}^{\diamond}\mskip-0.99998mu$E-mail:
<EMAIL_ADDRESS>supported in part by the SNSF under grants
196892, 205607
${}^{\dagger}\mskip-0.99998mu$E-mail<EMAIL_ADDRESS>supported in
part by the NSF under grant DMS - 2054527 and by the RSF under grant
19-11-00062
${}^{\circ}\mskip-0.99998mu$E-mail<EMAIL_ADDRESS>supported in part by
the Simons Foundation under grants 430235, 852996
${}^{\star}\mskip-0.99998mu$E-mail<EMAIL_ADDRESS>supported in part by
the NSF under grant DMS-1954266

In memory of Igor Krichever (1950 –2022)

###### Contents

  1. 1 Introduction
  2. 2 Singular vectors
    1. 2.1 Linear function $\psi$
    2. 2.2 Fundamental representations
    3. 2.3 Young diagrams
    4. 2.4 Singular vectors
    5. 2.5 Hooks
    6. 2.6 Recurrence relations
    7. 2.7 Excited diagrams
    8. 2.8 Proof of Theorem 2.5
    9. 2.9 Change of variable and weight shift
  3. 3 Applications
    1. 3.1 Master function
    2. 3.2 Weight function of $u_{\lambda}$
    3. 3.3 Two integrals
    4. 3.4 Whittaker vectors

##  1\. Introduction

We determine the coefficient of proportionality between two multidimensional
hypergeometric integrals. One of them is a solution of the dynamical
difference equations associated with a Young diagram and the other is the
vertex integral associated with the Young diagram. The coefficient of
proportionality is the inverse of the product of weighted hooks of the Young
diagram. The same coefficient appears in the problem of diagonalizing the
action of the center of the universal enveloping algebra of
$\mathfrak{gl}_{n}$ on the space of Whittaker vectors in the tensor product of
dual Verma modules with fundamental modules..

The standard basis $(u_{\lambda})$ of a fundamental $\mathfrak{gl}_{n}$-module
$U_{r}$ is labeled by the Young diagrams $\lambda$ inscribed in an
$(n-r)\times r$-rectangle. One assigns to a basis vector $u_{\lambda}$ a
system of dynamical difference equations

(1.1) $\displaystyle
I(z_{1},\dots,z_{i}+\kappa,\dots,z_{n-1},\kappa)=a_{i}(z_{1},\dots,z_{n-1},\kappa)I(z_{1},\dots,z_{n-1},\kappa),\quad
i=1,\dots,n-1,$

where $I(z_{1},\dots,z_{n-1},\kappa)$ is an unknown scalar function and
$a_{i}$ are suitable coefficients defined in terms of the
$\mathfrak{gl}_{n}$-action on $U_{r}$. The equations were introduced in [TV],
and solutions were constructed in [MV]. A solution $I_{\lambda}(z,\kappa)$ to
(1.1) is given by a hypergeometric integral of dimension equal to the number
of boxes in $\lambda$.

We also introduce another hypergeometric integral $V_{\lambda}(z,\kappa)$ of
the same dimension associated with $\lambda$. We show that it is proportional
to $I_{\lambda}(z,\kappa)$ and determine the coefficient of proportionality
between the two integrals in Theorem 3.1

(1.2) $\displaystyle
V_{\lambda}(z,\kappa)\,=\,\frac{1}{\prod_{\square\in\lambda}h(\square)(z)}\,I_{\lambda}(z,\kappa)\,,$

where $h(\square)(z)$ is the hook-weight of a box $\square$ of $\lambda$, see
the definition in (2.5).

Our motivation for considering (1.2) is the following. On the one hand, the
enumerative geometry of quiver varieties is controlled by two important
objects: the vertex function and the capping operator of a quiver variety [O].
These are the generating functions counting quasimaps to the quiver variety
with nonsingular or relative boundary conditions.

On the other hand, with any quiver variety $X$ one can associate a
hypergeometric integral. The 3D-mirror symmetry predicts that this integral
computes the vertex function of the mirror variety $X^{!}$. The capping
operator of $X^{!}$ is obtained in the same way with additional insertion of
the stable envelope functions to the integral [AO] (also known as weight
functions in the theory of qKZ equations). For example, see [SmV1, SmV2] where
the integral formulas of this type are discussed in the case of cotangent
bundle over Grassmannian $X^{!}=T^{*}Gr(k,n)$.

Let $X$ be the zero-dimensional Nakajima quiver variety of type $A$ associated
to a Young diagram $\lambda$ [DS1, DS2]. The hypergeometric integral assigned
to $X$ is given by the function $V_{\lambda}(z,\kappa)$ and the integral for
the capping operator is given by $I_{\lambda}(z,\kappa)$. Since the cohomology
of $X_{\lambda}$ are one-dimensional, it is expected from [O] that both
integrals are proportional. Our formula (1.2) establishes the coefficient of
proportionality explicitly.

Note, however, that in this case, the mirror $X^{!}_{\lambda}$ is a
2$|\lambda|$-dimensional variety which can not be realized as a quiver
variety. Thus, the vertex function for $X^{!}_{\lambda}$ is not defined by the
methods of [O]. To clarify this point, we refer to the function
$V_{\lambda}(z,\kappa)$ as the vertex integral, instead of the “vertex
function of $X^{!}_{\lambda}$”.

We note also that the coefficient in (1.2) has a geometric meaning: the mirror
variety $X^{!}_{\lambda}$ is equipped with a torus action with unique fixed
point. The denominator of the coefficient in (1.2) is the product over the
half of the tangent weights at this point with parameters $z_{1},\dots,z_{n}$
understood as the equivariant parameters of the torus [DS2].

To determine the coefficient of proportionality we consider the tensor product
$M\otimes U_{r}$ , where $M$ is a Verma module and analyze singular weight
vectors in $M\otimes U_{r}$ of the form

$\displaystyle v(\lambda)=\sum_{\mu\leqslant\lambda}v_{\mu}\otimes
u_{\mu}\,,\qquad\operatorname{with}\ v_{\mu}\in M.$

The collection of vectors $(v_{\mu})$ is quite a nontrivial object. We
simplify it by choosing a suitable linear function $\psi:M\to{\mathbb{C}}$ and
considering instead the collection of numbers $(\psi(v_{\mu}))$. We develop
simple recurrence relations and formulas for these numbers. We also show that

$\displaystyle
V_{\lambda}(z,\kappa)\big{/}I_{\lambda}(z,\kappa)=\psi(v_{\emptyset})/\psi(v_{\lambda}).$

Together with formulas for $\psi(v_{\mu})$ this equation proves formula (1.2).

The numbers $\psi(v_{\mu})$ are functions of the highest weight of $M$
associated to the skew Young diagram $\lambda/\mu$. We show that these numbers
arise in the problem of diagonalizing the action of the center $Z$ of the
universal enveloping algebra on the space of Whittaker vectors of
$M^{\prime}\otimes U_{r}$ where $M^{\prime}$ is the dual of the Verma module
$M$.

A Whittaker vector in a $\mathfrak{g}l_{n}$-module is a vector on which the
nilpotent subalgebra of lower triangular matrices acts via a fixed regular
character, see Section 3.4. The center $Z$ acts on the space of Whittaker
vectors $\operatorname{Wh}(M^{\prime}\otimes U_{r})$. A Whittaker vector
$\beta\in M^{\prime}\otimes U_{r}$ is uniquely determined by its contraction
$\beta(v)\in U_{r}$ with the highest weight vector $v$ of $M$. For generic
highest weight of $M$, we show that a basis of eigenvectors is given by

$\beta_{\lambda}(v)=\sum_{\mu\leqslant\lambda}\sum_{\nu\in
E(\lambda/\mu)}\frac{1}{\prod_{\square\in\lambda\smallsetminus\nu}h(\square)(z)}\,u_{\mu},$

where $\lambda$ runs over the set of Young diagrams fitting in an $(n-r)\times
r$ rectangle and $z$ is an affine function of the highest weight of $M$. The
set $E(\lambda/\mu)$ is the set of Ikeda–Naruse excited diagrams, which are
subsets of $\lambda$ obtained from $\mu$ by moving boxes according to certain
rules. The coefficient of $u_{\mu}$ is $\psi(v_{\mu})$. In particular the
coefficient of $u_{\emptyset}$ is the coefficient of proportionality in (1.2)
(the coefficient of $u_{\lambda}$ is normalized to be 1).

### Aknowledgements

The fourth author thanks FIM at ETH Zurich and IHES in Bures-sur-Yvette for
hospitality in June-July 2023. The fourth author also thanks E. Mukhin for
useful discussions.

##  2\. Singular vectors

###  2.1. Linear function $\psi$

Consider the complex Lie algebra $\mathfrak{gl}_{n}$ with standard generators
$e_{ij}$, $i,j=1,\dots,n$, simple roots $\alpha_{i}$, $i=1,\dots,n-1$, half-
sum $\rho$ of positive roots. Denote $e_{i}=e_{i,i+1}$, $f_{i}=e_{i+1,i}$,
$h_{i}=e_{i,i}-e_{i+1,i+1}$ for $i=1,\dots,n-1$.

Let $M$ be a $\mathfrak{gl}_{n}$ Verma module with highest weight vector $v$.
Define a linear function $\psi:M\to{\mathbb{C}}$ as follows. Any vector
$v^{\prime}\in M$ can be written (in general non-uniquely) as a finite linear
combination of the products of elements $f_{1},\dots,f_{n-1}$ applied to $v$,

$\displaystyle v^{\prime}=\sum
c_{i_{m},i_{m-1},\dots,i_{1}}f_{i_{m}}f_{i_{m-1}}\dots f_{i_{1}}v\,,$

where $1\leqslant i_{j}\leqslant n-1$ and
$c_{i_{m},i_{m-1},\dots,i_{1}}\in{\mathbb{C}}$. Set

(2.1) $\displaystyle\psi(v^{\prime})=\sum c_{i_{m},i_{m-1},\dots,i_{1}}\,.$

The function $\psi$ is well-defined since it is zero on Serre’s relations
$f_{i}^{2}f_{i\pm 1}-2f_{i}f_{i\pm 1}f_{i}+f_{i\pm 1}f_{i}^{2}=0$. It is in
fact a Whittaker vector in the dual of $M$, as will be discussed in Section
3.4.

###  2.2. Fundamental representations

Let $U_{r}$, $r=1,\dots,n-1$, be the $r$-th fundamental representation of
$\mathfrak{gl}_{n}$. Its highest weight is $(1,\dots,1,0,\dots,0)$ with $r$
ones.

The $U_{1}$ is the vector representation ${\mathbb{C}}^{n}$ with standard
basis $u_{i}$, $i=1,\dots,n$. The $U_{r}$ is the $r$-th exterior power
$\wedge^{r}{\mathbb{C}}^{n}$ of the vector representation with standard basis

(2.2) $\displaystyle u_{I}:=u_{i_{1}}\wedge u_{i_{2}}\wedge\dots\wedge
u_{i_{r}}\,,$

where $I=\\{i_{1}<i_{2}<\dots<i_{r}\\}$ is any $r$-element subset of
$\\{1,\dots,n\\}$. Denote by $\mathcal{I}_{r}$ the set of such subsets.

The decomposition

$\displaystyle U_{r}=\oplus_{I\in\mathcal{I}_{k}}{\mathbb{C}}u_{I}\,$

is the weight decomposition. We have $e_{ii}u_{I}=u_{I}$ if $i\in I$ and
$e_{ii}u_{I}=0$ otherwise. Thus, the weight $w(u_{I})$ of $u_{I}$ is the
$n$-vector whose $i$-th coordinate is 1 if $i\in I$ and is 0 otherwise. The
vector $u_{I^{min}}$ with $I^{min}=\\{1<2<\dots<r\\}$ is a highest weight
vector.

###  2.3. Young diagrams

The set $\mathcal{I}_{r}$ is identified with the set of sequences of
nonnegative integers

$\displaystyle\\{0\leqslant\lambda_{1}\leqslant\dots\leqslant\lambda_{r}\leqslant
n-r\\}$

by the formula $\\{i_{1}<i_{2}<\dots<i_{r}\\}\,\mapsto\,\\{i_{1}-1\leqslant
i_{2}-2\leqslant\dots\leqslant i_{r}-r\\}$. The set of such sequences is
identified with the set of Young diagrams inscribed in the $(n-r)\times
r$-rectangle $R$. Thus the set $\mathcal{I}_{r}$ is identified with the set of
Young diagrams inscribed in the rectangle $R$.

For example, $I^{min}$ corresponds to the empty Young diagram $\emptyset$, and
the rectangle $R$ corresponds to the subset $\\{n-r+1<n-r+2<\dots<n\\}$.

We conclude that the basis $(u_{I})$ of $U_{r}$ is labeled by the Young
diagrams. A vector $u_{I}$ will also be denoted $u_{\lambda}$ if $I$
corresponds to a Young diagram $\lambda$. The weight $w(u_{I})$ of $u_{I}$
will also be denoted by $w(\lambda)$.

The set of Young diagrams is partially ordered with respect to inclusion of
the diagrams. We write $\mu\leqslant\lambda$ if the Young diagram $\lambda$
contains the Young diagram $\mu$.

###  2.4. Singular vectors

Let $M$ be the $\mathfrak{gl}_{n}$ Verma module with highest weight $t-\rho$
and highest weight vector $v$ , where $t=(t_{1},\dots,t_{n})$.

Fix a Young diagram $\lambda\in\mathcal{I}_{r}$. Consider the vector subspace
$\cap_{i=1}^{n-1}\operatorname{Ker}e_{i}$ of singular vectors in $M\otimes
U_{r}$ of weight $w(\lambda)+t-\rho$. For generic $t$, this space is one-
dimensional with a generator of the form

(2.3) $\displaystyle v(\lambda):=v\otimes
u_{\lambda}+\sum_{\mu<\lambda}v_{\mu}\otimes u_{\mu}\,$

for suitable vectors $v_{\mu}\in M$. Recall the linear function
$\psi:M\to{\mathbb{C}}$. Define the following scalar functions
$g_{\lambda/\mu}$ of $t$:

(2.4) $\displaystyle
g_{\lambda/\mu}=\psi(v_{\mu})\quad\operatorname{for}\quad\mu<\lambda\quad\operatorname{and}\quad
g_{\lambda/\lambda}=1.$

More precisely, each $g_{\lambda/\mu}$ is a function of
$z=(z_{1},\dots,z_{n-1})$ where $z_{i}=t_{i+1}-t_{i}$.

The main result of this paper is recurrence relations and a formula for
functions $g_{\lambda/\mu}$.

###  2.5. Hooks

The $(n-r)\times r$-rectangle $R$ lies in the positive quadrant in
${\mathbb{R}}^{2}$ and consists of unit boxes $\square_{i,j}$,
$i=1,\dots,n-r$, $j=1,\dots,r$. The center of a box $\square_{i,j}$ has
coordinates $\big{(}i-\frac{1}{2},j-\frac{1}{2}\big{)}$. Every nonempty Young
diagram $\lambda\in\mathcal{I}_{r}$ contains the corner box $\square_{1,1}$.

To every box $\square_{i,j}$ we assign one of $z_{1},\dots,z_{n-1}$ by the
rule:

$\displaystyle z(\square_{i,j})\,:=\,z_{i-j+r}\,.$

For example, $z(\square_{1,r})=z_{1}$, $z(\square_{1,1})=z_{r}$,
$z(\square_{n-r,1})=z_{n-1}$, $z(\square_{n-r,r})=z_{n-r}$. We say that
$z_{i-j+r}$ is the $z$-label of a box $\square_{i,j}$.

Recall that a hook $H_{\lambda}(\square_{i,j})$ of a box $\square_{i,j}$ in a
Young diagram $\lambda$ is the set of all boxes $\square_{a,b}$ in $\lambda$
such that $a=i$, $b\geqslant j$, or $a\geqslant i$, $b=j$. We define the hook-
weight of a box $\square$ of $\lambda$ by the formula

(2.5) $\displaystyle h(\square)=1+\sum_{\square^{\prime}\in
H_{\lambda}(\square)}z(\square^{\prime})\,.$

######  Theorem 2.1.

We have

(2.6) $\displaystyle
g_{\lambda/\emptyset}=\frac{1}{\prod_{\square\in\lambda}h(\square)}\,.$

For example, if $(r,n)=(2,4)$ and $\lambda=(2,1)$. Then $\lambda$ consists of
the three boxes $\square_{1,1}$, $\square_{1,2}$, $\square_{2,1}$, and

$\displaystyle
g_{\lambda/\emptyset}=\frac{1}{(z_{1}+z_{2}+z_{3}+1)(z_{1}+1)(z_{3}+1)}\,.$

In Section 2.7 we give a formula for all coefficients $g_{\lambda/\mu}$.
Theorem 2.1 follows from Theorem 2.5 below.

###  2.6. Recurrence relations

Let $\lambda/\mu$ be a skew-diagram. Let $k_{i}$ be the number of boxes in
$\lambda/\mu$ with $z$-label $z_{i}$, where $i=1,\dots,n-1$. We put $k_{n}=0$.
Define the $z$-content of $\lambda/\mu$ by the formula

$\displaystyle s_{\lambda/\mu}=\sum_{i=1}^{n-1}k_{i}(k_{i}-k_{i+1}+z_{i})\,.$

For example, if $(r,n)=(2,4)$, $\lambda=(2,2)$, $\mu_{1}=\emptyset$,
$\mu_{2}=(1)$, $\mu_{3}=(1,1)$, $\mu_{4}=(2)$, $\mu_{5}=(2,1)$, then

$\displaystyle\phantom{aaa}s_{\lambda/\mu_{1}}=z_{1}+2z_{2}+z_{3}+2,\qquad
s_{\lambda/\mu_{2}}=z_{1}+z_{2}+z_{3}+1,$ $\displaystyle
s_{\lambda/\mu_{3}}=z_{1}+z_{2}+1,\qquad
s_{\lambda/\mu_{4}}=z_{2}+z_{3}+1,\qquad s_{\lambda/\mu_{5}}=z_{2}+1.$

######  Theorem 2.2.

The following recurrence relations hold:

(2.7) $\displaystyle
g_{\lambda/\mu}=\frac{1}{s_{\lambda/\mu}}\sum_{\mu^{\prime}}g_{\lambda/\mu^{\prime}}\,,$

where the sum is over all the Young diagrams $\mu^{\prime}$ such that
$\mu<\mu^{\prime}\leqslant\lambda$ and the skew-diagram $\mu^{\prime}/\mu$
consists of one box.

For example, if $(r,n)=(2,4)$, $\lambda=(2,2)$, and $\mu_{i}$ are as before,
then we have

$\displaystyle g_{\lambda/\emptyset}$ $\displaystyle=$
$\displaystyle\frac{1}{z_{1}+2z_{2}+z_{3}+2}\,g_{\lambda/\mu_{2}}=\frac{1}{(z_{1}+2z_{2}+z_{3}+2)(z_{1}+z_{2}+z_{3}+1)}\,[g_{\lambda/\mu_{3}}+g_{\lambda/\mu_{4}}]$
$\displaystyle=\frac{1}{(z_{1}+2z_{2}+z_{3}+2)(z_{1}+z_{2}+z_{3}+1)}\Big{(}\frac{1}{z_{1}+z_{2}+1}+\frac{1}{z_{2}+z_{3}+1}\Big{)}g_{\lambda/\mu_{5}}$
$\displaystyle=\frac{1}{(z_{1}+z_{2}+z_{3}+1)(z_{1}+z_{2}+1)(z_{2}+z_{3}+1)(z_{2}+1)}\,$

where in the last step we used $g_{\lambda/\lambda}=1$.

###### Proof.

Denote $\beta=t-\rho-\sum_{i=1}^{n-1}k_{i}\alpha_{i}$ where $k_{i}$ are some
nonnegative integers. Let $M[\beta]$ be the weight subspace of $M$ of weight
$\beta$.

######  Lemma 2.3.

For any $v^{\prime}\in M[\beta]$, we have

(2.9)
$\displaystyle\psi((e_{1}+\dots+e_{n-1})v^{\prime})\,=\,-\sum_{i=1}^{n-1}k_{i}(k_{i}-k_{i+1}+z_{i})\,\psi(v^{\prime}).$

###### Proof.

The proof is straightforward. It is enough to check formula (2.9) for
$v^{\prime}=f_{m_{1}}\dots f_{m_{k}}v$ where $1\leqslant m_{j}\leqslant n-1$
and for any $i$ the sequence $m_{1},\dots,m_{k}$ has exactly $k_{i}$ elements
equal to $i$.

For example, for $\beta=t-\rho-2\alpha_{1}-\alpha_{2}$ and
$v^{\prime}=f_{1}f_{1}f_{2}v$ we have

$\displaystyle\psi((e_{1}+e_{2}+e_{3})v^{\prime})$ $\displaystyle=$
$\displaystyle\psi(h_{1}f_{1}f_{2}v+f_{1}h_{1}f_{2}v+f_{1}f_{1}h_{2}v)$
$\displaystyle=$
$\displaystyle-\psi((z_{1}+2)f_{1}f_{2}v+z_{1}f_{1}f_{2}v+(z_{2}+1)f_{1}f_{1}v)=-(2z_{1}+z_{2}+3),$

while $\psi(v^{\prime})=1$. ∎

To prove the theorem notice that the vector $v(\lambda)$ is singular and hence
$\psi((e_{1}+\dots+e_{n-1})v(\lambda))=0$. By Lemma 2.3, we also have

(2.10)
$\displaystyle\psi((e_{1}+\dots+e_{n-1})v(\lambda))=\sum_{\mu<\lambda}\Big{(}-s_{\lambda/\mu}\,g_{\lambda/\mu}+\sum_{\mu^{\prime}}g_{\lambda/\mu^{\prime}}\Big{)}u_{\mu}\,,$

where the second sum is over all the Young diagrams $\mu^{\prime}$ such that
$\mu<\mu^{\prime}\leqslant\lambda$ and the skew-diagram $\mu^{\prime}/\mu$
consists of one box. Since $(u_{\mu})_{\mu\in\mathcal{I}_{r}}$ is a basis of
$U_{r}$, the coefficient of each $u_{\mu}$ in (2.10) must be equal to zero.
This proves the theorem. ∎

######  Corollary 2.4.

Let $d$ be the number of boxes in $\lambda/\mu$. Then

(2.11) $\displaystyle
g_{\lambda/\mu}=\sum_{\mu=\mu_{1}<\mu_{2}<\dots<\mu_{d}<\lambda}\frac{1}{\prod_{i=1}^{d}s_{\lambda/\mu_{i}}}\,.$

See an example in formula (LABEL:ex).

###  2.7. Excited diagrams

Let $\lambda/\mu$ be a skew-diagram and $D$ a subset of the Young diagram
$\lambda$. A box $\square_{i,j}$ of $D$ is called active if the boxes
$\square_{i+1,j},\,\square_{i+1,j+1},\,\square_{i,j+1}$ are all in
$\lambda-D$. Let $b=\square_{i,j}$ be an active box of $D$, define $D_{b}$ to
be the set obtained by replacing $\square_{i,j}$ in $D$ by
$\square_{i+1,j+1}$. We call this replacement an elementary excitation. An
excited diagram of $\lambda/\mu$ is a subset of boxes of $\lambda$ obtained
from the Young diagram $\mu$ after a sequence of elementary excitations on
active boxes. Let $E(\lambda/\mu)$ be the set of excited diagrams of
$\lambda/\mu$, see this definition in [IN, Na, MPP].

######  Theorem 2.5.

We have

(2.12) $\displaystyle
g_{\lambda/\mu}=\frac{1}{\prod_{\square\in\lambda}h(\square)}\,\sum_{\nu\in
E(\lambda/\mu)}\prod_{\square\in\nu}h(\square)\,.$

For example, in the notation of formula (LABEL:ex), the set
$E(\lambda/\mu_{2})$ consists of two elements $\\{\square_{1,1}\\}$ and
$\\{\square_{2,2}\\}$. Then

$\displaystyle
g_{\lambda/\mu_{2}}=\frac{z_{1}+2z_{2}+x_{3}+2}{(z_{1}+z_{2}+z_{3}+1)(z_{1}+z_{2}+1)(z_{2}+z_{3}+1)(z_{2}+1)}\,,$

where

$\displaystyle h(\square_{1,1})+h(\square_{2,2})=z_{1}+2z_{2}+x_{3}+2.$

######  Remark.

The equality between (2.11) and (2.12) in the case $\mu=\emptyset$ is a
generalization of the classical hook-length formula relating the number of
standard Young tableaux of shape $\lambda$ to the inverse product of hook-
lengths. It converges to it in the limit where all $z_{i}$ are equal and tend
to infinity. In the same limit for general $\mu\subset\lambda$ we obtain
Naruse’s generalization for skew diagrams [Na].

###  2.8. Proof of Theorem 2.5

Theorem 2.5 follows from Corollary 2.4 and Naruse’s formula in [Na] by a
change of parameters $z_{1},\dots,z_{n-1}$.

More precisely, let $\lambda\in\mathcal{I}_{r}$ be a nonempty Young diagram.
We say that a box $\square_{i,j}\in\lambda$ is a boundary box if
$\square_{i+1,j+1}\notin\lambda$. Let $\square_{i,j}\in\lambda$ be a boundary
box. If $\square_{i,j+1}\notin\lambda$ and $\square_{i+1,j}\notin\lambda$,
then $\square_{i,j}$ is an active boundary box according to the definition in
Section 2.7 (with $D=\lambda$). If $\square_{i,j+1}\in\lambda$ and
$\square_{i+1,j}\in\lambda$, then we say that $\square_{i,j}$ is a corner
boundary box. If $\square_{i,j+1}\in\lambda$ and
$\square_{i+1,j}\notin\lambda$, or if $\square_{i,j+1}\notin\lambda$ and
$\square_{i+1,j}\in\lambda$, then we say that $\square_{i,j}$ is a flat
boundary box.

If $\lambda$ has a box with $z$-label $z_{i}$, then $\lambda$ has an exactly
one boundary box with $z$-label $z_{i}$. We define new parameters $y_{i}$ by
the following formulas. We define

$\displaystyle z_{i}$ $\displaystyle=$ $\displaystyle
y_{i}-1,\quad\operatorname{if\,the\,boundary\,box\,with\,label}\,z_{i}\,\operatorname{is\,an\,active\,boundary\,box}\,,$
$\displaystyle z_{i}$ $\displaystyle=$ $\displaystyle
y_{i}+1,\quad\operatorname{if\,the\,boundary\,box\,with\,label}\,z_{i}\,\operatorname{is\,a\,corner\,boundary\,box}\,,$
$\displaystyle z_{i}$ $\displaystyle=$ $\displaystyle y_{i}\,,\quad\quad\
\,\operatorname{if\,the\,boundary\,box\,with\,label}\,z_{i}\,\operatorname{is\,a\,flat\,boundary\,box}\,.$

To every box $\square_{i,j}\in\lambda$ we assign one of $y_{1},\dots,y_{n-1}$
by the rule:

$\displaystyle y(\square_{i,j})\,:=\,y_{i-j+r}\,.$

We say that $y_{i-j+r}$ is the $y$-label of a box $\square_{i,j}$.

######  Lemma 2.6.

  * (i)

Let $\square$ be a box in $\lambda$ with hook-weight
$h(\square)(z)=1+z_{a}+z_{a+1}+\dots+z_{b}$ for some $a,b$. Then

(2.13) $\displaystyle h(\square)(z(y))=y_{a}+y_{a+1}+\dots+y_{b}\,.$

  * (ii)

Let $\mu<\lambda$ and let

$\displaystyle s_{\lambda/\mu}(z)=\sum_{i=1}^{n-1}k_{i}(k_{i}-k_{i+1}+z_{i})$

be the $z$-content of the skew-diagram $\lambda/\mu$. Then

(2.14) $\displaystyle s_{\lambda/\mu}(z(y))=\sum_{i=1}^{n-1}k_{i}y_{i}\,.$

For example, in the notation of formula (LABEL:ex), the change of variables
for $\lambda$ is

$\displaystyle z_{1}=y_{1},\qquad z_{2}=y_{2}-1,\qquad z_{3}=y_{3}\,.$

Then $s_{\lambda/\emptyset}(z)=z_{1}+2z_{2}+z_{3}+2$ and
$s_{\lambda/\emptyset}(z(y))=y_{1}+2y_{2}+y_{3}$ . Similarly,
$s_{\lambda/\mu_{2}}(z)=z_{1}+z_{2}+z_{3}+1$ and
$s_{\lambda/\mu_{2}}(z(y))=y_{1}+y_{2}+y_{3}$ .

###### Proof.

The proof of the lemma is straightforward. For example, we prove part (i). Let
$\square_{i,j}$ be a box in $\lambda$ with hook-weight
$h(\square_{i,j})(z)=1+z_{a}+z_{a+1}+\dots+z_{b}$ for some $a,b$. The boxes
$\square_{i,a}$ and $\square_{b,j}$ are boundary boxes of $\lambda$. Let us
walk from the box $\square_{i,a}$ to the box $\square_{b,j}$ through the
boundary boxes of $\lambda$. This walk consists of $b-a+1$ boundary boxes with
$z$-labels $z_{a},z_{a+1},\dots,z_{b}$. Let $\ell$ be the number of active
boundary boxes in this walk. Then the walk has exactly $\ell-1$ corner
boundary boxes. Hence our change of variables transforms
$h(\square)(z)=1+z_{a}+z_{a+1}+\dots+z_{b}$ to
$1+y_{a}+y_{a+1}+\dots+y_{b}-\ell+(\ell-1)=y_{a}+y_{a+1}+\dots+y_{b}$. Part
(i) is proved. ∎

Having Lemma 2.6 we rewrite Corollary 2.4 in terms of the variables $y_{i}$.
Namely, define the $y$-hook-weight of a box $\square\in\lambda$ by the formula

$\displaystyle\tilde{h}(\square)=\sum_{\square^{\prime}\in
H_{\lambda}(\square)}y(\square^{\prime})\,,$

and the $y$-content of a skew-diagram $\lambda/\mu$ by the formula

$\displaystyle\tilde{s}_{\lambda/\mu}=\sum_{i=1}^{n-1}k_{i}y_{i}\,,$

if $k_{i}$ is the number of boxes in $\lambda/\mu$ with $y$-label $y_{i}$.
Then

$\displaystyle\tilde{h}(\square)(y)=h(\square)(z(y)),\qquad\tilde{s}_{\lambda/\mu}(y)=s_{\lambda/\mu}(z(y))$

by Lemma 2.6. Formula (2.11) takes the form:

(2.15) $\displaystyle g_{\lambda/\mu}(z(y))$ $\displaystyle=$
$\displaystyle\sum_{\mu=\mu_{1}<\mu_{2}<\dots<\mu_{d}<\lambda}\frac{1}{\prod_{i=1}^{d}\tilde{s}_{\lambda/\mu_{i}}(y)}\,.$

On the other hand, H. Naruse’s formula [Na, page 13] states that

(2.16)
$\displaystyle\sum_{\mu=\mu_{1}<\mu_{2}<\dots<\mu_{d}<\lambda}\frac{1}{\prod_{i=1}^{d}\tilde{s}_{\lambda/\mu_{i}}(y)}$
$\displaystyle=$
$\displaystyle\frac{1}{\prod_{\square\in\lambda}\tilde{h}(\square)(y)}\,\sum_{\nu\in
E(\lambda/\mu)}\prod_{\square\in\nu}\tilde{h}(\square)(y)\,,$

see also [IN, MPP]. Hence,

(2.17) $\displaystyle g_{\lambda/\mu}(z(y))$ $\displaystyle=$
$\displaystyle\frac{1}{\prod_{\square\in\lambda}h(\square)(z(y))}\,\sum_{\nu\in
E(\lambda/\mu)}\prod_{\square\in\nu}h(\square)(z(y))\,,$

and Theorem 2.2 is proved.

###  2.9. Change of variable and weight shift

The change of variables $y\mapsto z=z(y)$ can be understood in terms of
weights as follows:

######  Lemma 2.7.

Let $\zeta\colon\mathbb{C}^{n}\to\mathbb{C}^{n-1}$ be the linear map
$t\mapsto(t_{2}-t_{1},\dots,t_{n}-t_{n-1})$. If $z=\zeta(t)$ then
$y=\zeta(t-w(\lambda))$.

###### Proof.

The weight corresponding to the Young diagram $\lambda$ is
$w(\lambda)=(\epsilon_{1},\dots,\epsilon_{n})$ where
$\epsilon_{i}\in\\{0,1\\}$ with $\epsilon_{i}=1$ iff
$i\in\\{i_{1}<\dots<i_{r}\\}$ where $i_{k}=\lambda_{k}+k$ ($k=1,\dots,r$), see
Section 2.3. Then for each $i=1,\dots,n-1$, we have

  * •

$\epsilon_{i+1}-\epsilon_{i}=-1$ if $i=i_{k}$ for some $k\in\\{1,\dots,r\\}$
and $i_{k}<i_{k+1}-1$, where we set $i_{r+1}=n+1$,

  * •

$\epsilon_{i+1}-\epsilon_{i}=1$ if $i+1=i_{k}$ for some $k\in\\{1,\dots,r\\}$
and $i_{k-1}<i_{k}-1$, where we set $i_{0}=0$,

  * •

$\epsilon_{i+1}-\epsilon_{i}=0$, otherwise.

The first alternative occurs iff $i=i_{k}$ and $\lambda_{k}<\lambda_{k+1}$
where we set $\lambda_{r+1}=n-r$. This is exactly the condition for the box
with coordinates $(\lambda_{k},r-k)$, which has $z$-label $z_{i}$, to be an
active boundary box.

The second alternative occurs iff $i+1=i_{k}$ and $\lambda_{k-1}<\lambda_{k}$
where we set $\lambda_{0}=0$. This is exactly the condition for the boundary
box with coordinates $(\lambda_{k},r-k+1)$, which has $z$-label $z_{i}$, to be
a corner boundary box. ∎

##  3\. Applications

###  3.1. Master function

Let $\lambda$ be a Young diagram inscribed in the $(n-r)\times r$ rectangle
$R$. Let $\lambda$ have $k_{i}$ boxes with $z$-label $z_{i}$ for
$i=1,\dots,n-1$. Denote $k=k_{1}+\dots+k_{n-1}$.

Consider ${\mathbb{C}}^{k}$ with coordinates $x=(x_{i,j})$, $i=1,\dots,n-1$,
$j=1,\dots,k_{i}$. Define the master function 111The superpotential in the
terminology of enumerative geometry.

(3.1)
$\displaystyle\Phi_{\lambda}(x,z)=\prod_{i=1}^{n-1}\prod_{j=1}^{k_{i}}x_{i,j}^{z_{i}+1}\prod_{j=1}^{k_{r}}(x_{k,j}-1)^{-1}\prod_{i=1}^{n-1}\prod_{j<j^{\prime}}(x_{i,j}-x_{i,j^{\prime}})^{2}\prod_{i=1}^{n-2}\prod_{j=1}^{k_{i}}\prod_{j^{\prime}=1}^{k_{i+1}}(x_{i,j}-x_{i+1,j^{\prime}})^{-1}\,.$

The linear functions $x_{i,j}$, $x_{k,j}-1$, $x_{i,j}-x_{i,j^{\prime}}$,
$x_{i,j}-x_{i+1,j^{\prime}}$ appearing in the master function define an
arrangement $\mathcal{C}$ of hyperplanes in ${\mathbb{C}}^{k}$.

The group $G=S_{k_{1}}\times\dots\times S_{k_{n-1}}$ acts on
${\mathbb{C}}^{k}$ by permuting the coordinates $(x_{i,j})$ with the same
first index $i$. The arrangement $\mathcal{C}$ and master function
$\Phi_{\lambda}(x,z)$ are $G$-invariant.

For $\kappa\in{\mathbb{C}}^{\times}$, the multivalued function
$\Phi_{\lambda}(x,z)^{1/\kappa}$ defines a rank one local system
$\mathcal{L}_{\kappa}$ on the complement
$X={\mathbb{C}}^{k}\setminus\mathcal{C}$ to the arrangement. The group $G$
acts on the homology $H_{*}(X;\mathcal{L}_{\kappa})$ and cohomology
$H^{*}(X;\mathcal{L}_{\kappa})$. Let $H_{k}(X;\mathcal{L}_{\kappa})^{-}\subset
H_{k}(X;\mathcal{L}_{\kappa})$ and $H^{k}(X;\mathcal{L}_{\kappa})^{-}\subset
H^{k}(X;\mathcal{L}_{\kappa})$ be the isotypical components corresponding to
the sign representation. It is known that for generic $\kappa$, we have $\dim
H^{k}(X;\mathcal{L}_{\kappa})^{-}=\dim H_{k}(X;\mathcal{L}_{\kappa})^{-}=1$
since the space $H^{k}(X;\mathcal{L}_{\kappa})^{-}$ can be identified with the
space of singular vectors in $M\times U_{r}$ of weight $w(\lambda)+t-\rho$,
which is of dimension 1, see [SV].

###  3.2. Weight function of $u_{\lambda}$

Let $u_{\lambda}$ be the basis vector of $U_{r}$ corresponding to the diagram
$\lambda$. The vector $u_{\lambda}$ is related to the highest weight vector
$u_{\emptyset}$ by the formula

(3.2) $\displaystyle u_{\lambda}=f_{\ell_{k}}\dots
f_{\ell_{2}}f_{\ell_{1}}u_{\emptyset}\,,$

where $f_{\ell_{k}},\dots,f_{\ell_{2}},f_{\ell_{1}}$ is a certain (admissible)
sequence of Cartan generators $f_{1},\dots,f_{n-1}$ in which there are exactly
$k_{i}$ elements $f_{i}$ for every $i=1,\dots,n-1$. Let
$\mathcal{F}_{\lambda}$ be the set of all such admissible sequences.

Let $f=\\{f_{\ell_{k}},\dots,f_{\ell_{1}}\\}$ be an admissible sequence.
Define the function $W^{\circ}_{f}(x)$,

(3.3) $\displaystyle
W^{\circ}_{f}(x)=\frac{1}{(x_{a_{k},b_{k}}-x_{a_{k-1},b_{k-1}})\dots(x_{a_{3},b_{3}}-x_{a_{2},b_{2}})(x_{a_{2},b_{2}}-x_{a_{1},b_{1}})(x_{a_{1},b_{1}}-1)}$

such that

  1. (i)

each variable $x_{i,j}$ is present in (3.3),

  2. (ii)

if $(x_{a_{c},b_{c}}-x_{a_{c-1},b_{c-1}})$ is any of the factors, then
$a_{c}=\ell_{c}$,

  3. (iii)

for any $i$ and $1\leqslant j<j^{\prime}\leqslant k_{i}$, the variable
$x_{i,j}$ appears in (3.3) on the right from the variable $x_{i,j^{\prime}}$ .

These properties determine the function $W^{\circ}_{f}(x)$ uniquely.

Define

$\displaystyle
W_{\lambda}(x)=\operatorname{Sym}_{x_{1,1},\dots,x_{1,k_{1}}}\dots\operatorname{Sym}_{x_{n-1,1},\dots,x_{n-1,k_{n-1}}}\Big{[}\sum_{f\in\mathcal{F}_{\lambda}}W^{\circ}_{f}(x)\Big{]}\,,$

where we use the notation
$\operatorname{Sym}_{t_{1},\dots,t_{j}}P({t_{1},\dots,t_{j}}):=\sum_{\sigma\in
S_{j}}P(t_{\sigma(1)},\dots,t_{\sigma(j)})$. The function $W_{\lambda}(x)$ is
called the weight function of the vector $v\otimes u_{\lambda}$ in $M\otimes
U_{r}$.

For example, if $(r,n)=(2,4)$, $\lambda=(2,2)$, then
$x=(x_{1,1},x_{2,1},x_{2,2},x_{3,1})$. There are two admissible sequences

$\displaystyle
u_{\lambda}=f_{2}f_{3}f_{1}f_{2}u_{\emptyset}=f_{2}f_{1}f_{3}f_{2}u_{\emptyset}\,,$

and

$\displaystyle W_{\lambda}(x)$ $\displaystyle=$
$\displaystyle\frac{1}{(x_{2,2}-x_{3,1})(x_{3,1}-x_{1,1})(x_{1,1}-x_{2,1})(x_{2,1}-1)}$
$\displaystyle+$
$\displaystyle\frac{1}{(x_{2,1}-x_{3,1})(x_{3,1}-x_{1,1})(x_{1,1}-x_{2,2})(x_{2,2}-1)}$
$\displaystyle+$
$\displaystyle\frac{1}{(x_{2,2}-x_{1,1})(x_{1,1}-x_{3,1})(x_{3,1}-x_{2,1})(x_{2,1}-1)}$
$\displaystyle+$
$\displaystyle\frac{1}{(x_{2,1}-x_{1,1})(x_{1,1}-x_{3,1})(x_{3,1}-x_{2,2})(x_{2,2}-1)}\,.$

###  3.3. Two integrals

Let $\gamma\in H_{k}(X;\mathcal{L}_{\kappa})^{-}$ be a generator. Let

$\displaystyle\wedge_{i,j}dx_{i,j}\,$

denote the wedge product in the lexicographic order of all the differentials
$dx_{i,j}$ . Define two functions

$\displaystyle
I_{\lambda}(z,\kappa)=\int_{\gamma}\Phi_{\lambda}(x,z)^{1/\kappa}W_{\lambda}(x)\big{(}\wedge_{i,j}dx_{i,j}\big{)}\,,\qquad
V_{\lambda}(z,\kappa)=\int_{\gamma}\Phi_{\lambda}(x,z)^{1/\kappa}\frac{1}{\prod_{i,j}x_{i,j}}\big{(}\wedge_{i,j}dx_{i,j}\big{)}\,.$

Both function are multiplied by the same nonzero constant if we choose a
different generator.

As shown in [MV], the first function is a hypergeometric solution of the
dynamical difference equations associated with the weight subspace
$U_{r}[w(\lambda)]$ of the $\mathfrak{gl}_{n}$-module $U_{r}$. The dynamical
equations were introduced in [TV]. The (hypergeometric) solutions of the
dynamical equations were constructed in [MV]. The dynamical equations is a
system of difference equations of the form

$\displaystyle
I(z_{1},\dots,z_{i}+\kappa,\dots,z_{n-1},\kappa)=a_{i}(z_{1},\dots,z_{n-1},\kappa)I(z_{1},\dots,z_{n-1},\kappa),\qquad
i=1,\dots,n-1,$

for suitable coefficients $a_{i}$ defined in terms of the
$\mathfrak{gl}_{n}$-action on $U_{r}$.

We call the second function $V_{\lambda}(z,\kappa)$ – the vertex integral
associated with the weight subspace $U_{r}[w(\lambda)]$ of the
$\mathfrak{gl}_{n}$-module $U_{r}$.

######  Theorem 3.1.

We have

(3.4) $\displaystyle
V_{\lambda}(z,\kappa)\,=\,\frac{1}{\prod_{\square\in\lambda}h(\square)(z)}\,I_{\lambda}(z,\kappa)\,.$

The starting goal of this project was to find the coefficient of
proportionality between the vertex integral $V_{\lambda}(z,\kappa)$ and the
hypergeometric solution $I_{\lambda}(z,\kappa)$ which turned out to be the
inverse of the product of the hook-weights of the boxes of the Young diagram
$\lambda$.

###### Proof.

In [SV], given $M\otimes U_{r}$ and $\lambda\in\mathcal{I}_{r}$, a vector
$\bar{v}(\lambda)$ is constructed,

$\displaystyle\bar{v}(\lambda):=\bar{v}_{\lambda}\otimes
u_{\lambda}+\sum_{\mu<\lambda}\bar{v}_{\mu}\otimes
u_{\mu}\,,\qquad\bar{v}_{\lambda},\bar{v}_{\mu}\in
H^{k}(X;\mathcal{L})^{-}\otimes M.$

Thus $\bar{v}(\lambda)\in H^{k}(X;\mathcal{L}_{\kappa})^{-}\otimes M\otimes
U_{r}$. The vector $\bar{v}(\lambda)$ has $\mathfrak{gl}_{n}$-weight
$w(\lambda)+t-\rho$ and is singular with respect to the factors $M\otimes
U_{r}$. The vector $\bar{v}(\lambda)$ is a cohomological version of the vector
$v(\lambda)$ defined in (2.3) and studied in the previous sections.

The vector $\bar{v}_{\lambda}$ is represented by the differential form

$\displaystyle\big{(}\Phi(x,z)^{1/\kappa}W_{\lambda}(x)\big{(}\wedge_{i,j}dx_{i,j}\big{)}\big{)}\otimes
v,$

see [SV].

The vector $\bar{v}_{\emptyset}$ is represented by a differential form
constructed as follows. A sequence
$f_{\ell_{k}},\dots,f_{\ell_{2}},f_{\ell_{1}}$ is called weakly admissible if
for $i=1,\dots,n-1$, the sequence contains exactly $k_{i}$ elements $f_{i}$.
Let $\mathcal{F}^{\star}_{\lambda}$ be the set of all weakly admissible
sequences.

For example, if $(r,n)=(2,4)$ and $\lambda=(2,2)$, then
$\mathcal{F}^{\star}_{\lambda}$ consists of 12 sequences:
$\\{f_{2},f_{2},f_{1},f_{3}\\}$, …, $\\{f_{3},f_{1},f_{2},f_{2}\\}$.

Let $f=\\{f_{\ell_{k}},\dots,f_{\ell_{1}}\\}$ be a weakly admissible sequence.
Define the function $W^{\star}_{f}(x)$ by the formula

(3.5) $\displaystyle
W^{\star}_{f}(x)=\frac{1}{(x_{a_{k},b_{k}}-x_{a_{k-1},b_{k-1}})\dots(x_{a_{3},b_{3}}-x_{a_{2},b_{2}})(x_{a_{2},b_{2}}-x_{a_{1},b_{1}})x_{a_{1},b_{1}}}$

such that

  1. (i)

each variable $x_{i,j}$ is present in (3.5),

  2. (ii)

if $(x_{a_{c},b_{c}}-x_{a_{c-1},b_{c-1}})$ is any of the factors, then
$a_{c}=\ell_{c}$,

  3. (ii’)

$(a_{1},b_{1})=(\ell_{1},1)$,

  4. (iii)

for any $i$ and $1\leqslant j<j^{\prime}\leqslant k_{i}$, the variable
$x_{i,j}$ appears in (3.3) on the right from the variable $x_{i,j;}$.

These properties determine the function $W^{\star}_{f}(x)$ uniquely.

Notice that the last factors in (3.3) and (3.5) are different.

Define the function $W_{f}(x)$ by the formula

(3.6) $\displaystyle
W_{f}(x)=\operatorname{Sym}_{x_{1,1},\dots,x_{1,k_{1}}}\dots\operatorname{Sym}_{x_{n-1,1},\dots,x_{n-1,k_{n-1}}}\big{[}W^{\star}_{f}(x)\big{]}\,.$

Then the vector $v_{\emptyset}$ is represented by the differential form

$\displaystyle\sum_{f=\\{f_{\ell_{k}},\dots,f_{\ell_{1}}\\}\in\mathcal{F}^{\star}_{\lambda}}\big{(}\Phi_{\lambda}(x,z)^{1/\kappa}W_{f}(x)\big{(}\wedge_{i,j}dx_{i,j}\big{)}\big{)}\otimes
f_{\ell_{k}}\dots f_{\ell_{1}}v,$

see [SV].

Let $\gamma\in H_{k}(X;\mathcal{L}_{\kappa})^{-}$ be a generator. The integral
of $\bar{v}(\lambda)$ over $\gamma$ is a scalar multiple the vector
$v(\lambda)$,

$\displaystyle\int_{\gamma}\bar{v}(\lambda)=c(z,\kappa)\,v(\lambda).$

We apply the linear function $\psi:M\to{\mathbb{C}}$ to both sides of this
equation and equate the coefficients of $u_{\lambda}$ and $u_{\emptyset}$.
Then

$\displaystyle c(z,\kappa)$ $\displaystyle=$
$\displaystyle\int_{\gamma}\Phi(x,z)^{1/\kappa}W_{\lambda}(x)\big{(}\wedge_{i,j}dx_{i,j}\big{)},$
$\displaystyle c(z,\kappa)\,g_{\lambda/\emptyset}(z)$ $\displaystyle=$
$\displaystyle\int_{\gamma}\Phi(x,z)^{1/\kappa}\Big{(}\sum_{f\in\mathcal{F}^{\star}_{\lambda}}W_{f}(x)\Big{)}\big{(}\wedge_{i,j}dx_{i,j}\big{)}\,.$

Using the formula

$\displaystyle\sum_{\sigma\in
S_{k}}\frac{1}{(s_{\sigma(k)}-s_{\sigma(k-1)})(s_{\sigma(k-1)}-s_{\sigma(k-2)})\dots(s_{\sigma(2)}-s_{\sigma(1)})s_{\sigma(a)}}=\frac{1}{\prod_{j=1}^{k}s_{j}}$

and the definition of $W_{f}(x)$ we conclude that

$\displaystyle\sum_{f\in\mathcal{F}^{\star}_{\lambda}}W_{f}(x)=\frac{1}{\prod_{i,j}x_{i,j}}\,.$

Hence

$\displaystyle c(z,\kappa)=I_{\lambda}(z,\kappa),\qquad
c(z,\kappa)\,g_{\lambda/\emptyset}(z)=V_{\lambda}(z,\kappa).$

Now formula (2.6) implies Theorem 3.1. ∎

###  3.4. Whittaker vectors

Let $\mathfrak{n}^{-}$ be the maximal nilpotent subalgebra of
$\mathfrak{gl}_{n}$ of lower triangular matrices. It is generated by
$f_{1},\dots,f_{n-1}$. Let $\eta\colon\mathfrak{n}^{-}\to\mathbb{C}$ be the
character of the Lie algebra $\mathfrak{n}^{-}$ such that $\eta(f_{i})=-1$ for
all $i$.

A Whittaker vector in a $\mathfrak{gl}_{n}$-module $V$ is a vector $u\in V$ so
that $xu=\eta(x)u$ for all $x\in\mathfrak{n}^{-}$. This notion was introduced
and studied by B. Kostant, [Ko]. The space of Whittaker vectors in $V$ is
denoted $\operatorname{Wh}(V)$. It is a module over the center $Z$ of the
universal enveloping algebra of $\mathfrak{gl}_{n}$. A Whittaker vector $u\neq
0$ such that $zu=\chi(z)u$ for all $z\in Z$ and some character $\chi\colon
Z\to\mathbb{C}$ is said to have infinitesimal character $\chi$.

For example let $M^{\prime}=\mathrm{Hom}_{\mathbb{C}}(M,\mathbb{C})$ be the
dual of a Verma module $M$. It is a $\mathfrak{gl}_{n}$-module for the action
$(x\alpha)(m)=-\alpha(xm)$, $\alpha\in M$, $m\in M$, $x\in\mathfrak{gl}_{n}$.
Central elements $z\in Z$ act on $M$ as multiples $\chi_{M^{\prime}}(z)$ of
the identity, for some character $\chi_{M^{\prime}}\colon Z\to\mathbb{C}$. The
linear function $\psi\in M^{\prime}$ of Section 2.1 is defined by the
conditions $f_{i}\psi=-\psi$ and $\psi(v)=1$ and is in particular a Whittaker
vector. On the other hand, any Whittaker vector in $M^{\prime}$ is uniquely
determined by its value on $v$ since $v$ generates $M$ as a module over
$U(\mathfrak{n}^{-})$. Thus:

######  Lemma 3.2.

The space of Whittaker vectors $\operatorname{Wh}(M^{\prime})$ is one-
dimensional, spanned by the Whittaker vector $\psi$ of infinitesimal weight
$\chi_{M^{\prime}}$.

More generally let us consider the problem of describing the $Z$-module of
Whittaker vectors in the $\mathfrak{gl}_{n}$-module $M^{\prime}\otimes
U\cong\operatorname{Hom}_{\mathbb{C}}(M,U)$ for a Verma module $M$ and a
fundamental module $U$. By definition $\alpha\colon M\to U$ is a Whittaker
vector if and only if

$x\alpha(m)=\alpha(xm)+\eta(x)\alpha(m),\quad\forall m\in
M,x\in\mathfrak{n}^{-}.$

It follows that a Whittaker vector $\alpha$ is again uniquely determined by
its value on the highest weight vector $v\in M$.

Let $M_{t-\rho}$ denote the Verma module of highest weight
$t-\rho\in\mathbb{C}^{n}$. Let $\chi(t)=\chi_{M^{\prime}_{t-\rho}}$ be the
infinitesimal character of its dual.

######  Proposition 3.3.

Let $r\in\\{1,\dots,n-1\\}$ and $t\in\mathbb{C}^{n}$ be generic and set
$z_{i}=t_{i+1}-t_{i}$, $(i=1,\dots,n-1)$. Then for each Young diagram
$\lambda\in\mathcal{I}_{r}$ there is a unique Whittaker vector
$\alpha_{\lambda,t}\in\operatorname{Hom}_{\mathbb{C}}(M_{w(\lambda)+t-\rho},U_{r})$
of infinitesimal character $\chi(t)$ such that

$\alpha_{\lambda,t}(v)=\sum_{\mu\leqslant\lambda}g_{\lambda/\mu}(z)u_{\mu},$

where

$g_{\lambda/\mu}(z)=\frac{1}{\prod_{\square\in\lambda}h(\square)}\sum_{\nu\in
E(\lambda/\mu)}\prod_{\square\in\nu}h(\square),$

and $h(\square)=1+\sum_{\square^{\prime}\in
H_{\lambda}(\square)}z(\square^{\prime})$.

###### Proof.

The morphism of $\mathfrak{gl}_{n}$-modules $M_{w(\lambda)+t-\rho}\to
M_{t-\rho}\otimes U_{r}$ sending the highest weight vector to the singular
vector $v(\lambda)=\sum_{\mu\leqslant\lambda}v_{\mu}\otimes u_{\mu}$, see
Section 2.4, induces a morphism

$M^{\prime}_{t-\rho}\to\operatorname{Hom}_{\mathbb{C}}(M_{w(\lambda)+t-\rho},U_{r}).$

The morphism property implies that it sends the Whittaker vector $\psi$ of
infinitesimal character $\chi(t)$ to a Whittaker vector $\alpha$ with the same
infinitesimal character. By definition
$\alpha(v)=\sum_{\mu\leqslant\lambda}\psi(v_{\mu})u_{\mu}$ and
$g_{\lambda/\mu}=\psi(v_{\mu})$ is given in Theorem 2.5. ∎

We thus obtain an explicit diagonalization of the action of the center $Z$ on
the space of Whittaker vectors in $M^{\prime}\otimes U$ for a generic Verma
module $M$ and a fundamental module $U$:

######  Theorem 3.4.

Let $t\in\mathbb{C}^{n}$, $r\in\\{1,\dots,n-1\\}$ and $y_{i}=t_{i+1}-t_{i}$
$(i=1,\dots,n-1)$. Then

$W=\operatorname{Wh}(\operatorname{Hom}_{\mathbb{C}}(M_{t-\rho},U_{r}))$

has dimension $\operatorname{dim}(U_{r})$. For generic $t$, $W$ decomposes
into a direct sum $W=\oplus_{\lambda\in\mathcal{I}_{r}}W_{\lambda}$ of
$Z$-invariant one-dimensional subspaces on which $Z$ acts by the character
$\chi(t-w(\lambda))$. The subspace $W_{\lambda}$ is spanned by the Whittaker
vector $\beta_{\lambda,t}$, such that
$\beta_{\lambda,t}(v)=\sum_{\mu\leqslant\lambda}\tilde{g}_{\lambda/\mu}(y)u_{\mu}$
with

$\tilde{g}_{\lambda/\mu}(y)=\frac{1}{\prod_{\square\in\lambda}\tilde{h}(\square)}\sum_{\nu\in
E(\lambda/\mu)}\prod_{\square\in\nu}\tilde{h}(\square),$

and $\tilde{h}(\square)=\sum_{\square^{\prime}\in
H_{\lambda}(\square)}y(\square^{\prime})$.

###### Proof.

By Proposition 3.3 the Whittaker vectors
$\beta_{\lambda,t}=\alpha_{\lambda,t-w(\lambda)}$ belong to $W$ and have
infinitesimal character $\chi(t-w(\lambda))$. Since
$\beta_{\lambda}(v)=u_{\lambda}$ plus a linear combination of $u_{\mu}$ with
$\mu<\lambda$, these vectors are linearly independent. We need to show that
they span $W$. Let $\beta$ is a Whittaker vector. Since the vectors
$\beta_{\lambda}(v)$ form a basis of $U_{r}$ there exist coefficients
$c_{i}\in\mathbb{C}$ so that
$\gamma=\beta-\sum_{\lambda\in\mathcal{I}_{r}}c_{\lambda}\beta_{\lambda}$ is a
Whittaker vector which vanishes on $v$. Since a Whittaker vector is uniquely
determined by its value on $v$, $\gamma$ must be zero.

Let $t^{\prime}=t-w(\lambda)$ and
$z_{i}^{\prime}=t^{\prime}_{i+1}-t^{\prime}_{i}$, $(i=1,\dots,n-1)$. We need
to compute the coeffcients $g_{\lambda/\mu}(z^{\prime})$. By Lemma 2.7,
$z^{\prime}=z(y)$ defined in Section 2.8 and therefore
$g_{\lambda/\mu}(z^{\prime})=\tilde{g}_{\lambda/\mu}(y)$. ∎

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