# Whittaker Functions: Number Theory, Geometry and Physics (16w5039)

Arriving in Banff, Alberta Sunday, July 24 and departing Friday July 29, 2016

## Organizers

Benjamin Brubaker (University of Minnesota)

Gautam Chinta (City University of New York)

Solomon Friedberg (Boston College)

Paul Gunnells (University of Massachusetts Amherst)

Daniel Bump (Stanford University)

## Objectives

Whittaker functions appear in a wide variety of different contexts. The goals of the proposed workshop are to report on the latest results in these interrelated fields, to identify the main obstructions to progress and to chart a course for future developments. Due to the interdisciplinary nature of the proposed topics, we will supplement the research talks with more introductory lectures explaining the motivations coming from automorphic forms, combinatorial representation theory and mathematical physics. By doing so, we hope to emulate the successes of the previous workshops in helping junior participants become productive researchers in these areas.

As an indication of the richness of the topics to be explored, we briefly explain how Whittaker functions arise in number theory, geometry, combinatorial representation theory and physics.

In number theory Whittaker functions appear as Fourier coefficients of automorphic forms, and are used in constructing the associated $L$-functions. In particular, they arise in both the Langlands-Shahidi method and in Rankin-Selberg constructions. Two important directions of generalization must be mentioned. Adapting this to the metaplectic group leads to both local and global questions which are at the heart of our topic. Metaplectic Whittaker functions appear both in the theory of period integrals of automorphic forms and in the theory of zeta functions of prehomogeneous vector spaces. The arithmetic implications of the relationship between metaplectic Whittaker functions and prehomogeneous vector spaces remain little explored and deserve further study.

In another direction there has been a push to carry out the program initiated by Garland to develop Eisenstein series on infinite-dimensional groups, mainly in work of Garland-Miller-Patnaik and of Braverman-Garland-Kazhdan-Patnaik. Though their first appearance was in number theory, Eisenstein series also arise in string theory, where they are used compute scattering amplitudes -- see e.g. the work of Green-Miller-Russo-Vanhove.

For nonarchimedean Whittaker functions, the connection with combinatorics and representation theory is provided by the Casselman-Shalika formula for spherical Whittaker functions which gives a connection with the representation theory of the Langlands dual group. In the course of generalizing this result to the case of Eisenstein series on certain metaplectic covers of reductive groups, the organizers, in collaborations with J. Hoffstein, P. J. McNamara, L. Zhang among others, have uncovered a rich structure to formulas for Whittaker functions. This has lead to an influx of ideas from combinatorial representation theory and statistical physics into the study of Whittaker functions such as Kashiwara's theory of crystal bases, Mirkovic-Vilonen polytopes, and the Yang-Baxter equation. Notably the interplay runs in both directions: work of Brubaker-Schultz, Friedberg-Zhang, Friedlander-Gaudet-Gunnells, Hamel-King, Ivanov and others give combinatorial formulas (e.g. deformations of the Weyl character formula) motivated by the theory of Whittaker functions. These formulas may be expressed in terms of either crystal graphs or partition functions of statistical physics.

Like the nonarchimedean theory, archimedean Whittaker functions also have connections with geometry, crystal bases and mathematical physics. It was realized by Kostant and by Kazhdan that the eigenfunctions of the Toda lattice Hamiltonian are archimedean Whittaker functions. Through work of Givental, this connects with the quantum cohomology of flag varieties and mirror symmetry. Recently various researchers (Gerasimov-Lebedev-Oblezin, Ishii-Oda, Feigin-Jimbo-Miwa, Kedem-DiFrancesco) have obtained results understanding archimedean Whittaker functions in terms of crystals. These results provide a reinterpretation of integrals studied by Stade which were originally motivated by number theory. Moreover, beginning with work of Biane, Bougerol and O'Connell on Brownian motion, a deep connection was found with the Littelmann Path Method and the theory of crystals. Archimedean Whittaker functions occur naturally in this theory. After further work of O'Connell and his collaborators this led to Chhaibi's work relating Whittaker functions to "geometric" crystals, which are schemes with structures whose tropicalizations are Kashiwara crystals. Recent work of Rietsch and of Lam also connects the quantum Toda lattice to geometric crystals and the quantum cohomology of the flag variety.

The geometry of flag, Schubert and Bott-Samelson varieties connects with this subject in different ways. We mention only those related to the flag variety of the L-group. This geometry connects with Demazure operators and Demazure characters which appear in recent work on Whittaker functions. Work of Cherednik-Orr, Ion, Brubaker-Bump-Licata, Brubaker-Bump-Friedberg and others relate Demazure-Lusztig operators to the theory of Whittaker and other special functions. Through representations of the affine Hecke algebra this topic connects with Kazhdan-Lusztig theory and potentially the Springer correspondence. Meanwhile Chinta, Gunnells and Puskas gave metaplectic analogs of Demazure-Lusztig operators, but the geometric significance remains mysterious in this case.

The earlier workshops 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.

As an indication of the richness of the topics to be explored, we briefly explain how Whittaker functions arise in number theory, geometry, combinatorial representation theory and physics.

In number theory Whittaker functions appear as Fourier coefficients of automorphic forms, and are used in constructing the associated $L$-functions. In particular, they arise in both the Langlands-Shahidi method and in Rankin-Selberg constructions. Two important directions of generalization must be mentioned. Adapting this to the metaplectic group leads to both local and global questions which are at the heart of our topic. Metaplectic Whittaker functions appear both in the theory of period integrals of automorphic forms and in the theory of zeta functions of prehomogeneous vector spaces. The arithmetic implications of the relationship between metaplectic Whittaker functions and prehomogeneous vector spaces remain little explored and deserve further study.

In another direction there has been a push to carry out the program initiated by Garland to develop Eisenstein series on infinite-dimensional groups, mainly in work of Garland-Miller-Patnaik and of Braverman-Garland-Kazhdan-Patnaik. Though their first appearance was in number theory, Eisenstein series also arise in string theory, where they are used compute scattering amplitudes -- see e.g. the work of Green-Miller-Russo-Vanhove.

For nonarchimedean Whittaker functions, the connection with combinatorics and representation theory is provided by the Casselman-Shalika formula for spherical Whittaker functions which gives a connection with the representation theory of the Langlands dual group. In the course of generalizing this result to the case of Eisenstein series on certain metaplectic covers of reductive groups, the organizers, in collaborations with J. Hoffstein, P. J. McNamara, L. Zhang among others, have uncovered a rich structure to formulas for Whittaker functions. This has lead to an influx of ideas from combinatorial representation theory and statistical physics into the study of Whittaker functions such as Kashiwara's theory of crystal bases, Mirkovic-Vilonen polytopes, and the Yang-Baxter equation. Notably the interplay runs in both directions: work of Brubaker-Schultz, Friedberg-Zhang, Friedlander-Gaudet-Gunnells, Hamel-King, Ivanov and others give combinatorial formulas (e.g. deformations of the Weyl character formula) motivated by the theory of Whittaker functions. These formulas may be expressed in terms of either crystal graphs or partition functions of statistical physics.

Like the nonarchimedean theory, archimedean Whittaker functions also have connections with geometry, crystal bases and mathematical physics. It was realized by Kostant and by Kazhdan that the eigenfunctions of the Toda lattice Hamiltonian are archimedean Whittaker functions. Through work of Givental, this connects with the quantum cohomology of flag varieties and mirror symmetry. Recently various researchers (Gerasimov-Lebedev-Oblezin, Ishii-Oda, Feigin-Jimbo-Miwa, Kedem-DiFrancesco) have obtained results understanding archimedean Whittaker functions in terms of crystals. These results provide a reinterpretation of integrals studied by Stade which were originally motivated by number theory. Moreover, beginning with work of Biane, Bougerol and O'Connell on Brownian motion, a deep connection was found with the Littelmann Path Method and the theory of crystals. Archimedean Whittaker functions occur naturally in this theory. After further work of O'Connell and his collaborators this led to Chhaibi's work relating Whittaker functions to "geometric" crystals, which are schemes with structures whose tropicalizations are Kashiwara crystals. Recent work of Rietsch and of Lam also connects the quantum Toda lattice to geometric crystals and the quantum cohomology of the flag variety.

The geometry of flag, Schubert and Bott-Samelson varieties connects with this subject in different ways. We mention only those related to the flag variety of the L-group. This geometry connects with Demazure operators and Demazure characters which appear in recent work on Whittaker functions. Work of Cherednik-Orr, Ion, Brubaker-Bump-Licata, Brubaker-Bump-Friedberg and others relate Demazure-Lusztig operators to the theory of Whittaker and other special functions. Through representations of the affine Hecke algebra this topic connects with Kazhdan-Lusztig theory and potentially the Springer correspondence. Meanwhile Chinta, Gunnells and Puskas gave metaplectic analogs of Demazure-Lusztig operators, but the geometric significance remains mysterious in this case.

The earlier workshops 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.