Schedule for: 16w5039 - Whittaker Functions: Number Theory, Geometry and Physics

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Recall that Jacquet's Whittaker function for a group \(G\), in the non-Archimedean case, is essentially proportional to a character of an irreducible representation of the Langlands dual group - a Schur function in the case of \(\text{GL}_n\). This statement is known as the Shintani-Casselman-Shalika formula. In my opinion, Shintani's proof for \(\text{GL}_n\) is remarkably different from the more general proof by Casselman-Shalika. In this talk, I will present a probabilistic proof that is the natural generalisation of Shintani's. It explains the appearance of the Weyl character formula from a reflection principle for random walks.

I will explain how the theory of crystal bases can be used to understand some of the affine combinatorics that arises in the (co)homology of the affine Grassmannian. In particular, this approach leads to a combinatorial interpretation of the coefficients of a Schur function in the expansion of a Stanley symmetric function. This part is based on joint work with Jennifer Morse. I will also present some new results and conjectures regarding the Schur expansion of the k-Schur functions (which are dual to affine Stanley symmetric functions).

Metaplectic Demazure-Lusztig operators are built on the Chinta-Gunnells action, and (analogously to their nonmetaplectic counterparts) are useful in the study of \(p\)-adic (metaplectic) Whittaker functions. In this talk, I will present joint work with Manish Patnaik that relates metaplectic Iwahori-Whittaker functions to these operators directly. This process gives a metaplectic analogue of earlier work of Brubaker-Bump-Licata in the nonmetaplectic setting. I will also talk about combinatorics that allow the extension of relevant formulae to the affine Kac-Moody groups.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We prove cases of Rietsch mirror conjecture that the Dubrovin quantum connection for projective homogeneous varieties is isomorphic to the pushforward D-module attached to Berenstein-Kazhdan geometric crystals. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. We reveal surprising relations with the works of Frenkel-Gross, Heinloth-Ngo-Yun and Zhu on Kloosterman sheaves. The isomorphism comes from global rigidity results where Hecke eigensheaves are determined by their local ramification. It implies combinatorial identities for the counts of rational curves, the Peterson variety presentation and other consequences. Work with Thomas Lam.

Exponential sums arise various context in algebraic and analytic number theory. In this talk, we explain how they may appear when we study prehomogeneous vector spaces. We then outline a new method for evaluating them explicitly. As an application, we show that there are `many' quartic field discriminants with at most eight prime factors. This is a joint work with Frank Thorne.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In 2012, Brubaker, Bump, Friedberg, Chinta and Gunnells proposed statistical-mechanical models for p-adic Whittaker functions on the degree $n$ metaplectic cover of $\hbox{GL}(r)$. In recent work of Brubaker, Buciumas and Bump, the corresponding Yang-Baxter equations have been found. The corresponding quantum group is identified as a Drinfeld twist of $U_q(\widehat{\mathfrak{gl}}(n))$. The effect of the Drinfeld twisting is to introduce Gauss sums into the R-matrix. The scattering matrix of the intertwining operators on the (nonunique) Whittaker models, previously studied by Kazhdan-Patterson and Chinta-Gunnells, is thus reinterpreted as the R-matrix of this quantum group. Moreover, the internal states of these generalized ice-type models (which are not visible to the intertwining operator) are built up by tensoring from $(n+1)$-dimensional supersymmetric modules for the quantum affine Lie superalgebra $U_q(\widehat{gl}(n|1))$.

Whittaker functions on $p$-adic groups are expressible as partition functions of six-vertex models on a rectangular lattice; at least, this is known for Cartan types $A$ and $B$ and expected more generally. We show that, in type $A$, this may alternately be viewed as the discrete time evolution of a one-dimensional system of free fermions. The Hamiltonian dictating the evolution arises from the Lie superalgebra $\mathfrak{gl}(1 | 1)$ and the Whittaker function may thus be viewed as a kind of generalized ``tau function'' in the terminology of the Kyoto school. All of these notions from statistical mechanics will be explained in the talk and illustrated with pictures. This is joint work with A. Schultz based on {\tt arXiv:1606.00020}.

Let $G$ be a $p$-adic reductive group and $H$ a symmetric subgroup. I will present a criterion for $H$-integrability of matrix coefficients of representations of G. This is joint work with Max Gurevich and a generalization of Casselman's criteria for square integrability. Chong Zhang applied our results to show that for some symmetric subgroups all $H$-invariant linear forms of square integrable representations emerge as $H$-integrals of matrix coefficients. In particular, in a global setting, this provides information on the local components of factorizable period integrals of automorphic forms.

For an arbitrary split group we identify the unramifed Whittaker space with the space of skew-invariant functions on the lattice of coweights and deduce from it the Casselman-Shalika formula.

In a series of papers with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono, we developed two uniform combinatorial models for (tensor products of one-column) Kirillov-Reshetikhin (KR) modules of affine Lie algebras; we also showed that their graded characters coincide with the specialization of symmetric Macdonald polynomials at $t=0$. I will present our latest work, on the extension of the above results corresponding to the non-symmetric Macdonald polynomials. I will then explain the connection of this work to the $q$-Whittaker functions of Braverman-Finkelberg and their results, which extend to quantum $K$-theory Schubert calculus. Other related developments will also be mentioned.

Let W be a finite Weyl group. Associated to W there are two important partial orders - (weak) Bruhat order, and absolute order. These two partial orders are related to a pair of combinatorial lattices, and these lattices are in some (not completely understood) senses "dual" to one another. As a result many other structures related to W come in dual pairs; for example, as we will explain, the braid group of W has a pair of "dual" Garside structures, and a pair of "dual" positive monoids. The goal of this talk will be to explain how these dual structures appear in the higher representation theory of the braid group. (Joint with Hoel Queffelec.)

Let $G$ be a Kac-Moody group associated to a nonsingular, symmetrizable generalized Cartan matrix. First, we consider Eisenstein series on $G$, induced from quasi-characters, and prove the almost-everywhere convergence of Kac-Moody Eisenstein series inside the Tits cone for spectral parameters in the Godement range. For a certain class of Kac-Moody groups satisfying an additional combinatorial property, we show absolute convergence everywhere in the Tits cone for spectral parameters in the Godement range. Next, we consider Eisenstein series on $G$ induced from unramified cusp forms on finite-dimensional Levi subgroups of maximal parabolic subgroups. Under some ``ample" conditions on a maximal parabolic subgroup, we prove that the Eisenstein series are entire on the whole complex plane. This is joint work with L. Carbone, H. Garland, D. Liu and S.D. Miller.

Kazhdan and Patterson constructed generalized theta functions on covers of general linear groups as multi-residues of the Borel Eisenstein series. These representations and their unique models were used by Bump and Ginzburg in the Rankin-Selberg constructions of the symmetric square L-functions for $\text{GL}(r)$. In this talk, we will discuss the two other types of models that the theta representations may support. We first talk about semi-Whittaker models, which generalize the models used in the work of Bump and Ginzburg. Secondly, we determine the unipotent orbits attached to theta functions, in the sense of Ginzburg. We also determine the covers when these models are unique. Time permitting, we will discuss some applications in Rankin-Selberg constructions.

Automorphic forms on exceptional Lie groups appear naturally in string theory compactifications. They manifest themselves as couplings in higher derivative corrections and in terms of generating functions of black hole microstates. I will demonstrate how certain Fourier coefficients attached to the minimal automorphic representations of $E_6$, $E_7$, and $E_8$ are determined by maximally degenerate Whittaker vectors. This fact allows for a simple method for calculating explicit Fourier coefficients which are relevant in string theory. Various recent results, conjectures and open problems are outlined.

Automorphic forms appear in string theory considerations for example in scattering amplitudes. The simplest cases where their form is understood is related to processes that are (partially) controlled by an additional symmetry, called supersymmetry. Using the modern language of exceptional field theory to determine parts of these scattering amplitudes gives alternative expressions for Eisenstein series on exceptional groups (attached to small representations). This approach also gives some insight of non-Eisenstein series that are expected to arise in string theory.

Whittaker's original work involves both exponentially decaying and growing functions, generalizing the K- and I-Bessel functions that occur in the Fourier expansions of cusp forms on rank-1 groups. Decaying Whittaker functions are the bedrock of Fourier expansions of $L^2$ cusp forms for general groups. There is a corresponding beautiful theory of non-decaying Whittaker functions, dating back to work of Kostant and Casselman-Zuckerman in the 1970s. However, there are no known examples of automorphic functions which have non-decaying Whittaker functions in their Fourier expansions outside of rank-1 examples. This is consistent with a tantalizing conjecture of Miatello-Wallach, which asserts that all automorphic eigenfunctions on higher rank groups automatically have moderate growth (and hence decaying Whittaker functions). I'll present some cases of that conjecture for $\text{SL}(3,\mathbb{Z})$, which rule out analogs of the classical modular j-function. [Joint work with Tien Trinh]

In this talk we introduce factorial characters for the classical groups and derive a number of central results. Classically, the factorial Schur function plays a fundamental role in traditional symmetric function theory and also in Schubert polynomial theory. Here we develop a parallel theory for the classical groups, offering combinatorial definitions of the factorial characters for the symplectic and orthogonal groups, and further proving flagged factorial Jacob-Trudi identities and factorial Tokuyama identities. These identities are established by manipulating determinants through the use of certain recurrence relations and by using lattice paths. This work is joint with Ron King.

Recently, Braverman, Kazhdan, and Patnaik have constructed Iwahori-Hecke algebras for p-adic loop groups. Unsurprisingly, the resulting algebra is a variation on Cherednik's DAHA. The p-adic construction also comes with a basis (the double-coset basis) consisting of indicator functions of Iwahori double cosets. Braverman, Kazhdan, and Patnaik also proposed a (double affine) Bruhat preorder on the set of double cosets, which they conjectured to be a poset. I will describe a combinatorial presentation of the double-coset basis and also an alternative way to develop the double affine Bruhat order that is closely related to this combinatorics; from this perspective the order is manifestly a poset. One new feature is a length function that is compatible with the order. I will also discuss joint work with Daniel Orr, where we give a positive answer to a question raised in a previous paper: we prove that the length function can be specialized to take values in the integers. This proves finiteness of chains in the double-affine Bruhat order, and it gives an expected dimension formula for (yet to be defined) transversal slices in the double affine flag variety. If time remains, I will discuss how these results are stepping stones for developing Kazhdan-Lusztig theory in this setting and a number of further open questions.

Casselman's basis is the basis of Iwahori fixed vectors of a spherical representation of a connected reductive $p$-adic group over a non-archimedean local field, which is dual to the intertwining operators at the identity indexed by elements of the Weyl group. The problem of Casselman is to express Casselman's basis in terms of another natural basis, and vice versa. In this talk, using Yang-Baxter basis of Hecke algebra and Kostant-Kumar's twisted group algebra, we will show one solution to Casselman's problem. This is joint work with H. Naruse.

Similar to zeta functions associated to algebraic function fields, Weyl group multiple Dirichlet series associated to algebraic function fields are rational functions in several variables. The denominators of these rational functions are known, but the numerators are not well understood. Like zeta functions, we expect the coefficients of the numerators to encode information about the arithmetic of the defining curve. As a step toward understanding this relationship, in this talk we describe the support of Weyl group multiple Dirichlet series defined over the rational function field ${\mathbb{F}}_q(t)$. In particular, we show that up to a variable change, all such series can be expressed as a finite sum of simpler local series, which act analogue to Euler factors in the construction of the global object.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.