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

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


Ben Brubaker (University of Minnesota)

(Stanford University)

(City University of New York)

(Boston College)

(University of Massachusetts Amherst)


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

* Connections between spherical Whittaker functions and quantum

* 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

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.