# Whittaker Functions: Number Theory, Geometry and Physics (13w5154)

Arriving in Banff, Alberta Sunday, October 13 and departing Friday October 18, 2013

## Organizers

Ben Brubaker (University of Minnesota)

Daniel Bump (Stanford University)

Gautam Chinta (City University of New York)

Solomon Friedberg (Boston College)

Paul Gunnells (University of Massachusetts Amherst)

## Objectives

We intend to focus on the following topics in the new workshop:

* Connections between spherical Whittaker functions and quantum

groups.

* Iwahori fixed vectors in the Whittaker model over a p-adic field

and their relations to Kazhdan-Lusztig theory.

* Automorphic forms and their Whittaker models for Kac-Moody groups.

* Interpolations and further analogies between the archimedean and

nonarchimedean theories of Whittaker functions.

* Relations between Whittaker functions and the theory of

prehomogenous vector spaces, including connections to the theory of

multiple Dirichlet series.

As we now briefly explain, each of these questions is closely

connected to others on the list and should be pursued in parallel.

Spherical Whittaker functions on reductive algebraic groups over

local fields have been the subject of intense study for many

decades. If the ground field is p-adic, then the Casselman-Shalika

formula is, in some sense, the whole story. The formula is a

cornerstone of the theory of automorphic forms and has also appeared

in mathematical physics and geometric Langlands theory.

According to the Casselman-Shalika formula, the values of the

Whittaker function are expressible in terms of the finite-dimensional

irreducible characters of a Lie group, the Langlands dual or

L-group. These may be described by the Weyl character formula.

The proposers and others including Kazhdan, Patterson, and Hoffstein

have worked on metaplectic Whittaker functions for some years. The

Casselman-Shalika formula generalizes to the metaplectic case in two

different ways. Thus two different formulas describe spherical

metaplectic Whittaker functions, and each of these may be regarded as

a generalization of the Weyl character formula. In the Weyl character

formula, the character (of the L-group) is a ratio of a sum over the

Weyl group to a denominator, which is a product over the simple

roots. In one metaplectic generalization, the numerator is still a

sum over the Weyl group, but the terms of the sum are related by an

interesting group action having origin in the work of Kazhdan and

Patterson. In the other generalization, the numerator is deformed to

a sum over a Kashiwara crystal.

Metaplectic Whittaker functions may also be expressed as partition

functions of statistical-mechanical systems, at least for the

classical Cartan types. This description connects with both of the

two descriptions that we have already mentioned. It also leads to a

new tool in their study -- the Yang-Baxter equation. Both the crystal

base description and the appearance of the Yang-Baxter equation in

their theory suggest a deeper connection with quantum groups, but

this has not been fully explained.

On the other hand, archimedean Whittaker functions were used by

Kostant in order to prove the integrability of the Toda

lattice. Givental, in connection with mirror symmetry, considered a

particular maximal commuting family of Hamiltonians for the quantum

Toda lattice and related them to the quantum cohomology of the flag

variety. The connection between Givental's theory and Kostant's

theory of Whittaker functions was pointed out by Kostant and greatly

developed by Gerasimov, Lebedev and Oblezin in a series of papers.

One of the the developments in Gerasimov, Lebedev and Oblezin is an

interpolation between Whittaker functions in the real and

nonarchimedean cases. This is not entirely unprecedented; for

example, Macdonald polynomials interpolate between zonal spherical

functions in the archimedean and nonarchimedean case. The Givental

representation of the archimedean Whittaker function bears strong

resemblance to the latter of the two metaplectic p-adic Whittaker

function representations. This should serve as further evidence from

which connections to quantum groups may be studied.

Iwahori Whittaker functions are p-adic Whittaker functions that are

Iwahori fixed vectors in the Whittaker model. They form a vector

space whose dimension is the order of the Weyl group. There are

various natural bases indexed by the Weyl group. The spherical

Whittaker function is one special vector in this space. Casselman and

Shalika used the Iwahori Whittaker functions to compute the spherical

Whittaker functions, but their computation doesn't clarify the

general Iwahori Whittaker functions. Reeder has shown that the same

geometry used by Kazhdan and Lusztig to classify Iwahori-spherical

representations appears when examining the Whittaker model. In

particular, their geometric realizations involve Chern classes of

vector bundles on certain prehomogeneous vector spaces.

A further connection between the theory of Whittaker functions and

that of prehomogenous vector spaces appears in the global setting,

via multiple Dirichlet series. Indeed, Whittaker functions on a

metaplectic group over a global field are Dirichlet series in several

complex variables, known as Weyl group multiple Dirichlet

series. They possess meromorphic continuation and groups of

functional equations isomorphic to the Weyl group, but are not Euler

products. In some examples, especially in the case of metaplectic

double covers, the coefficients are themselves L-functions or

products of L- functions and as such, have applications in number

theory including moments of L-functions.

Similarly, zeta functions of prehomogenous vector spaces are

Dirichlet series with interesting coefficients. These were introduced

in the 1970's by M. Sato and Shintani, and since have been applied

strikingly by many workers to mean values of class numbers and to

counting number fields of fixed degree ordered by

discriminant. Unlike Weyl group multiple Dirichlet series, they are

usually defined as Dirichlet series in only one variable. Yet many

examples look like specializations of multivariate Dirichlet series,

and multivariate generalizations are possible. The implications of

this observation have been relatively little explored.

There is one known example of a multivariate prehomogenous zeta

function that coincides with a Weyl group multiple Dirichlet series,

which we now describe. This Dirichlet series in two complex

variables, which is now recognized to be a Whittaker coefficient of

the minimal parabolic Eisenstein series on the metaplectic double

cover of GL(3), was first obtained by Siegel as the Mellin transform

of a half-integral weight Eisenstein series on a congruence subgroup

of SL(2,Z). This series was subsequently rediscovered by

Goldfeld-Hoffstein who used it to study mean values of quadratic

Dirichlet L-functions.

A completely different approach to obtaining the analytic properties

of this series was given by Shintani, who realized it as the zeta

function associated to a prehomogeneous vector space of

GL(2)xGL(1). Each approach has its respective advantages and

disadvantages. The prehomogenous vector space gives a clear

definition of a Dirichlet series counting arithmetically interesting

orbits. However, in this approach integrating over the singular set

leads to difficulties in identifying the polar structure. By

contrast, in the multiple Dirichlet series approach passing from a

heuristic to precise definition of the series takes a great deal of

work, but once a precise definition is given, the required analytic

properties of the series follow straightforwardly. It is also worth

noting that the double Dirichlet series described above satisfies a

group of 12 functional equations, but only 4 of these are apparent

from the prehomogenous vector space description of Shintani. Thus the

multiple Dirichlet series method reveals new functional equations of

this series.

Finally, the 10w5096 workshop discussed generalizations of Whittaker

functions to the Kac-Moody case - particularly the Whittaker

coefficients for constant terms of Eisenstein series as a first step.

These may be pursued purely formally by giving a combinatorial recipe

for an analog of the Whittaker function and exploring its properties,

or by defining Whittaker functions on loop groups, using the theory

developed by Garland. New insights into additional factors appearing

in the constant term by Braverman, Kazhdan, and Patnaik, as well as

new alternatives for constructing the constant term, may potentially

allow for long awaited generalizations of the Langlands- Shahidi

method in this context.

The goal of this interdisciplinary workshop is to bring together a

group of researchers in number theory, representation theory,

prehomogeneous vector spaces, and mathematical physics whose work

bears upon open questions in the study of Whittaker functions and who

are active in these areas. More than a third of the proposed

participants attended the 10w5096 with productive results, and the

remaining invitees help to further develop this rapidly evolving

field and broaden its connections to other parts of mathematics. In

particular, we hope that the addition of experts on prehomogeneous

vector spaces will focus attention on the explanations for observed

coincidences between this theory and metaplectic Whittaker

functions. We expect that the level of talks at the conference will

be varied, ranging from expository lectures presenting foundational

theory for the varied interests of the invitees to research level

talks on current work. This mix will serve to allow conference

participants to get quickly up to speed on essential background while

being introduced to open questions and common points of

investigation.

The earlier workshop set in motion a set of projects at the

intersection of several rather different fields of mathematics. The

newly proposed workshop will allow some of these attendees to further

develop and report on the progress since the first workshop. It will

also introduce a completely new set of both accomplished researchers

and junior faculty to the field, in an effort to encourage new

interactions and unusual collaborations. The intimate setting of

BIRS is ideal for fostering this kind of interaction, since it offers

ample opportunity for discussion outside the hours of the formal

program activities.

* Connections between spherical Whittaker functions and quantum

groups.

* Iwahori fixed vectors in the Whittaker model over a p-adic field

and their relations to Kazhdan-Lusztig theory.

* Automorphic forms and their Whittaker models for Kac-Moody groups.

* Interpolations and further analogies between the archimedean and

nonarchimedean theories of Whittaker functions.

* Relations between Whittaker functions and the theory of

prehomogenous vector spaces, including connections to the theory of

multiple Dirichlet series.

As we now briefly explain, each of these questions is closely

connected to others on the list and should be pursued in parallel.

Spherical Whittaker functions on reductive algebraic groups over

local fields have been the subject of intense study for many

decades. If the ground field is p-adic, then the Casselman-Shalika

formula is, in some sense, the whole story. The formula is a

cornerstone of the theory of automorphic forms and has also appeared

in mathematical physics and geometric Langlands theory.

According to the Casselman-Shalika formula, the values of the

Whittaker function are expressible in terms of the finite-dimensional

irreducible characters of a Lie group, the Langlands dual or

L-group. These may be described by the Weyl character formula.

The proposers and others including Kazhdan, Patterson, and Hoffstein

have worked on metaplectic Whittaker functions for some years. The

Casselman-Shalika formula generalizes to the metaplectic case in two

different ways. Thus two different formulas describe spherical

metaplectic Whittaker functions, and each of these may be regarded as

a generalization of the Weyl character formula. In the Weyl character

formula, the character (of the L-group) is a ratio of a sum over the

Weyl group to a denominator, which is a product over the simple

roots. In one metaplectic generalization, the numerator is still a

sum over the Weyl group, but the terms of the sum are related by an

interesting group action having origin in the work of Kazhdan and

Patterson. In the other generalization, the numerator is deformed to

a sum over a Kashiwara crystal.

Metaplectic Whittaker functions may also be expressed as partition

functions of statistical-mechanical systems, at least for the

classical Cartan types. This description connects with both of the

two descriptions that we have already mentioned. It also leads to a

new tool in their study -- the Yang-Baxter equation. Both the crystal

base description and the appearance of the Yang-Baxter equation in

their theory suggest a deeper connection with quantum groups, but

this has not been fully explained.

On the other hand, archimedean Whittaker functions were used by

Kostant in order to prove the integrability of the Toda

lattice. Givental, in connection with mirror symmetry, considered a

particular maximal commuting family of Hamiltonians for the quantum

Toda lattice and related them to the quantum cohomology of the flag

variety. The connection between Givental's theory and Kostant's

theory of Whittaker functions was pointed out by Kostant and greatly

developed by Gerasimov, Lebedev and Oblezin in a series of papers.

One of the the developments in Gerasimov, Lebedev and Oblezin is an

interpolation between Whittaker functions in the real and

nonarchimedean cases. This is not entirely unprecedented; for

example, Macdonald polynomials interpolate between zonal spherical

functions in the archimedean and nonarchimedean case. The Givental

representation of the archimedean Whittaker function bears strong

resemblance to the latter of the two metaplectic p-adic Whittaker

function representations. This should serve as further evidence from

which connections to quantum groups may be studied.

Iwahori Whittaker functions are p-adic Whittaker functions that are

Iwahori fixed vectors in the Whittaker model. They form a vector

space whose dimension is the order of the Weyl group. There are

various natural bases indexed by the Weyl group. The spherical

Whittaker function is one special vector in this space. Casselman and

Shalika used the Iwahori Whittaker functions to compute the spherical

Whittaker functions, but their computation doesn't clarify the

general Iwahori Whittaker functions. Reeder has shown that the same

geometry used by Kazhdan and Lusztig to classify Iwahori-spherical

representations appears when examining the Whittaker model. In

particular, their geometric realizations involve Chern classes of

vector bundles on certain prehomogeneous vector spaces.

A further connection between the theory of Whittaker functions and

that of prehomogenous vector spaces appears in the global setting,

via multiple Dirichlet series. Indeed, Whittaker functions on a

metaplectic group over a global field are Dirichlet series in several

complex variables, known as Weyl group multiple Dirichlet

series. They possess meromorphic continuation and groups of

functional equations isomorphic to the Weyl group, but are not Euler

products. In some examples, especially in the case of metaplectic

double covers, the coefficients are themselves L-functions or

products of L- functions and as such, have applications in number

theory including moments of L-functions.

Similarly, zeta functions of prehomogenous vector spaces are

Dirichlet series with interesting coefficients. These were introduced

in the 1970's by M. Sato and Shintani, and since have been applied

strikingly by many workers to mean values of class numbers and to

counting number fields of fixed degree ordered by

discriminant. Unlike Weyl group multiple Dirichlet series, they are

usually defined as Dirichlet series in only one variable. Yet many

examples look like specializations of multivariate Dirichlet series,

and multivariate generalizations are possible. The implications of

this observation have been relatively little explored.

There is one known example of a multivariate prehomogenous zeta

function that coincides with a Weyl group multiple Dirichlet series,

which we now describe. This Dirichlet series in two complex

variables, which is now recognized to be a Whittaker coefficient of

the minimal parabolic Eisenstein series on the metaplectic double

cover of GL(3), was first obtained by Siegel as the Mellin transform

of a half-integral weight Eisenstein series on a congruence subgroup

of SL(2,Z). This series was subsequently rediscovered by

Goldfeld-Hoffstein who used it to study mean values of quadratic

Dirichlet L-functions.

A completely different approach to obtaining the analytic properties

of this series was given by Shintani, who realized it as the zeta

function associated to a prehomogeneous vector space of

GL(2)xGL(1). Each approach has its respective advantages and

disadvantages. The prehomogenous vector space gives a clear

definition of a Dirichlet series counting arithmetically interesting

orbits. However, in this approach integrating over the singular set

leads to difficulties in identifying the polar structure. By

contrast, in the multiple Dirichlet series approach passing from a

heuristic to precise definition of the series takes a great deal of

work, but once a precise definition is given, the required analytic

properties of the series follow straightforwardly. It is also worth

noting that the double Dirichlet series described above satisfies a

group of 12 functional equations, but only 4 of these are apparent

from the prehomogenous vector space description of Shintani. Thus the

multiple Dirichlet series method reveals new functional equations of

this series.

Finally, the 10w5096 workshop discussed generalizations of Whittaker

functions to the Kac-Moody case - particularly the Whittaker

coefficients for constant terms of Eisenstein series as a first step.

These may be pursued purely formally by giving a combinatorial recipe

for an analog of the Whittaker function and exploring its properties,

or by defining Whittaker functions on loop groups, using the theory

developed by Garland. New insights into additional factors appearing

in the constant term by Braverman, Kazhdan, and Patnaik, as well as

new alternatives for constructing the constant term, may potentially

allow for long awaited generalizations of the Langlands- Shahidi

method in this context.

The goal of this interdisciplinary workshop is to bring together a

group of researchers in number theory, representation theory,

prehomogeneous vector spaces, and mathematical physics whose work

bears upon open questions in the study of Whittaker functions and who

are active in these areas. More than a third of the proposed

participants attended the 10w5096 with productive results, and the

remaining invitees help to further develop this rapidly evolving

field and broaden its connections to other parts of mathematics. In

particular, we hope that the addition of experts on prehomogeneous

vector spaces will focus attention on the explanations for observed

coincidences between this theory and metaplectic Whittaker

functions. We expect that the level of talks at the conference will

be varied, ranging from expository lectures presenting foundational

theory for the varied interests of the invitees to research level

talks on current work. This mix will serve to allow conference

participants to get quickly up to speed on essential background while

being introduced to open questions and common points of

investigation.

The earlier workshop set in motion a set of projects at the

intersection of several rather different fields of mathematics. The

newly proposed workshop will allow some of these attendees to further

develop and report on the progress since the first workshop. It will

also introduce a completely new set of both accomplished researchers

and junior faculty to the field, in an effort to encourage new

interactions and unusual collaborations. The intimate setting of

BIRS is ideal for fostering this kind of interaction, since it offers

ample opportunity for discussion outside the hours of the formal

program activities.