Geometry, Number Theory, and Representation Theory Seminar

It is a classical result of Gelfand-Kazhdan, Rodier and Shalika that for a local linear algebraic group, every irreducible representation admits at most one Whittaker model. Such a unique model, if it exists and thus entitles the representation generic, plays a crucial role in the Langlands-Shahidi method of L-functions. It is also important for some Rankin-Selberg type L-functions. However, for a finite-degree central cover of a linear algebraic group, its irreducible genuine representations rarely admit unique Whittaker models. In this talk, we concentrate on Iwahori-spherical genuine representations of a cover, and discuss about some known results regarding their Whittaker dimensions. We also mention several problems and some speculations on such Whittaker dimensions.

Suppose that one has a supercuspidal representation of a Levi subgroup of some reductive \(p\)-adic group \(G\). Bernstein associated to this a block Rep\((G)^s\) in the category of smooth \(G\)-representations. We address the question: what does Rep\((G)^s\) look like? Usually this is investigated with Bushnell--Kutzko types, but these are not always available. Instead, we approach it via a progenerator of Rep\((G)^s\). We will discuss the structure of the \(G\)-endomorphism algebra of such a progenerator in detail. We will show that Rep\((G)^s\) is "almost" equivalent with the module category of an affine Hecke algebra -- a statement that will be made precise in several ways. In the end, this leads to a classification of the irreducible representations in Rep\((G)^s\) in terms of the complex torus and the finite group that are canonically associated to this Bernstein component.

The Bernstein decomposition of a reductive \(p\)-adic group \(G\) gives a parametrization, in terms of cuspidal representations of Levi groups, on the indecomposable subcategories of the category smooth \(G\)-representations. Those subcategories are so-called Bernstein blocks, and are usually equivalent to the module category of some affine Hecke algebras.
In this talk, I shall first describe the Bernstein components of the Gelfand-Graev representation, which are usually close to the sign projector of an affine Hecke algebra. Then I will explain how this gives insights on the structure of representations restricted from \(GL_{n+1}\) to \(GL_n\), resolving some problems related to Gan-Gross-Prasad conjectures. In particular, each Bernstein component of an irreducible \(GL_{n+1}\)-representation restricted to \(GL_n\) is indecomposable. Part of the work is joint with Gordan Savin.

We will present a refinement of Riemann-Roch-Grothendieck theorem on the level of differential forms for families of curves with hyperbolic cusps. The study of spectral properties of the Kodaira Laplacian on those surfaces, and more precisely of its determinant, lies in the heart of our approach.
When our result is applied directly to the moduli space of punctured stable curves, it expresses the extension of the Weil-Petersson form (as a current) to the boundary of the moduli space in terms of the first Chern form of a Hermitian line bundle. This provides a generalisation of a result of Takhtajan-Zograf.
We will also explain how our results imply some bounds on the growth of Weil-Petersson form near the compactifying divisor of the moduli space of punctured stable curves. This would permit us to give a new approach to some well-known results of Wolpert on the Weil-Petersson geometry.

Mirror symmetry, in a crude formulation, is usually presented as a correspondence between curve counting on a Calabi-Yau variety X, and some invariants extracted from a mirror family of Calabi-Yau varieties. After the physicists Bershadsky-Cecotti-Ooguri-Vafa, this is organised according to the genus of the curves in X we wish to enumerate, and gives rise to an infinite recurrence of differential equations. In this talk, I will give a general introduction to these problems based on joint work with Gerard Freixas and Christophe Mourougane. I will explain the main ideas of the proof of the conjecture for Calabi-Yau hypersurfaces in projective space, relying on the Riemann-Roch theorem in Arakelov geometry. Our results generalise from dimension 3 to arbitrary dimensions previous work of Fang-Lu-Yoshikawa.

The motivic Chern classes are K-theoretic generalization of the MacPherson classes in homology. The motivic Chern classes of Schubert cells have a Langlands dual description in the Iwahori invariants of principal series representation of the p-adic Langlands dual group. In joint works with Aluffi, Mihalcea, and Schurmann, we use this relation to solve conjectures of Bump, Nakasuji and Naruse about Casselman's basis, and also relate the Euler characteristics of the motivic Chern classes to the Iwahori Whittaker functions.

We construct a new family of irreducible representations of U_q(g_R) and its modular double by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the generators of U_q(g_R) act by positive self-adjoint operators on a Hilbert space. This generalizes the well-established positive representations introduced by [Frenkel-Ip] which correspond to induction by the minimal parabolic (i.e. Borel) subgroup. We also study in detail the special case of type A_n acting on L^2(R^n) with minimal functional dimension, and establish the properties of its central characters and universal R operator. We construct R a positive version of the evaluation module of the affine quantum group U_q(\widehat{sl}_{n+1}) modeled over this minimal positive representation of type A_n.

Let G be a split reductive group over a non-Archimedean local field and H be its Iwahori-Hecke algebra. Principal series representations of H, introduced by Matsumoto at the end of 1970's, are important in the representation theory of H. Every irreducible representation of H is the quotient of and can be embedded in some principal series representation of H and thus studying these representations enables to get information on the irreducible representations of H. S.Kato provided an irreducibility criterion for these representations in the beginning of the 1980's.
Kac-Moody groups are interesting infinite dimensional generalizations of reductive groups. Their study over non-Archimedean local field began in 1995 with the works of Garland. Let G be a split Kac-Moody group (à la Tits) over a non-Archimedean local field. Braverman, Kazhdan and Patnaik and Bardy-Panse, Gaussent and Rousseau associated an Iwahori-Hecke algebra to G in 2014. I recently defined principal series representations of these algebras. In this talk, I will talk of these representations, of a generalization of Kato's irreducibility criterion for these representations and of how they decompose when they are reducible.

In 1976, Victor Guillemin published a paper discussing geometric scattering theory, in which he related the Lax-Phillips Scattering matrices (associated to a noncompact hyperbolic surface with cusps) and the sojourn times associated to a set of geodesics which run to infinity in either direction. Later, the work of Guillemin was generalized to locally symmetric spaces by Lizhen Ji and Maciej Zworski. In the case of a Q-rank one locally symmetric space G/X they constructed a class of scattering geodesics which move to infinity in both directions and are distance minimizing near both infinities. An associated sojourn time was defined for such a scattering geodesic, which is the time it spends in a fixed compact region. One of their main results was that the frequencies of oscillation coming from the singularities of the Fourier transforms of scattering matrices on G\X occur at sojourn times of scattering geodesics on the locally symmetric space.
In this talk I will review the work of Guillemin, Ji and Zworski as well as discuss the work from my doctoral dissertation on analogous results for higher rank locally symmetric spaces. In particular, I will describe higher dimensional analogues of scattering geodesics called Scatter- ing Flat and study these flats in the case of the locally symmetric space given by the quotient SL(3, Z)\SL(3, R)/SO(3). A parametrization space is discussed for such scattering flats as well as an associated vector valued parameter (bearing similarities to sojourn times) called sojourn vector and these are related to the frequency of oscillations of the associated scattering matrices coming from the minimal parabolic subgroups of SL(3, R). The key technique is the factorization of higher rank scattering matrices.

In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.
Recently, a family of global integrals that represent the tensor product L-functions for classical groups (joint with Friedberg, Ginzburg, and Kaplan) and the tensor product L-functions for covers of symplectic groups (Kaplan) was discovered. These can be viewed as generalizations of the doubling method. In this talk, we explain how to develop the doubling integrals for Brylinski-Deligne extensions of connected classical groups. This gives a family of Eulerian global integrals for this class of non-linear extensions.

Shimura varieties attached to orthogonal groups (of which modular curves are examples) are interesting objects of study for many reasons, not least of which is the fact that they possess an abundance of “special” cycles. These cycles are at the centre of a conjectural program proposed by Kudla; roughly speaking, Kudla’s conjectures suggest that upon passing to an (arithmetic) Chow group, the special cycles behave like the Fourier coefficients of automorphic forms. These conjectures also include more precise identities; for example, the arithmetic Siegel-Weil formula relates arithmetic heights of special cycles to derivatives of Eisenstein series. In this talk, I’ll describe a construction (in joint work with Luis Garcia) of Green currents for these cycles, which are an essential ingredient in the “Archimedean” part of the story; I’ll also sketch a few applications of this construction to Kudla’s conjectures.

There are many formulas that express interesting properties of a finite group \( G \) in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio \( \mathrm{Trace}(ρ(g)) / \mathrm{dim}(ρ) \) for an irreducible representation \( ρ \) of \( G \) and an element \( g \in G\). For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G. Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.

Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms”, which is (since the 60s) the main organization principle, and is based on the (huge collection) of “Large” representations.

This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.

This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried with Steve Goldstein (Madison) and John Cannon (Sydney).