Lie Groups and Representation Theory Archives for Fall 2023 to Spring 2024
Compatibility of semisimple local Langlands parameters with parahoric Satake parameters
When: Mon, September 12, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Qihang Li (UMD) -
Abstract: In this talk, I will explain how formal properties of semisimple local Langlands parameters imply that there is at most one correspondence between parahoric-spherical representations and semisimple local Langlands parameters. After sketching the proof, I will explain how one can use a similar result to prove the compatibility of semisimple local Langlands parameters with parahoric Satake parameters.
Gerbes and rigid inner forms
When: Mon, September 26, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Peter Dillery (UMD) -
Abstract: We discuss gerbes over a local or global field F, focusing on how studying torsors on them can give refined conjectures concerning the local and global Langlands correspondence. We will focus primarily on the local case, discussing how one can use such torsors to parametrize compound L-packets and the role they play in local endoscopy. We will also discuss the relation between rigid inner forms and extended pure inner forms, which provides some additional geometric perspective.
Convex polytopes, toric varieties and combinatorics of Arthur's trace formula
When: Wed, October 19, 2022 - 2:00pmWhere: Kirwan 1308
Speaker: Kiumars Kaveh (University of Pittsburgh) -
Abstract: I start by discussing two beautiful well-known theorems about decomposing a convex polytope into a signed sum of cones, namely the classical Brianchon-Gram theorem and the Lawrence-Varchenko theorem. I will explain the connection with toric varieties and how Brianchon-Gram theorem can be thought of as a computation of Euler characteristic using Cech cohomology. I will then explain a generalization of the Brianchon-Gram which can be summarized as "truncating a function on the Euclidean space with respect to a polytope". This is an extraction of the combinatorial ingredients of Arthur's ''convergence'' and ''polynomiality'' results in his famous trace formula. Arthur's trace formula concerns the trace of the left action of a reductive group G on the space L^2(G/Γ) where Γ is a discrete (arithmetic) subgroup. The combinatorics involved is closely related to compactifications of ''locally symmetric spaces''. Our ''combinatorial truncation'' can be thought of as an analogue of Arthur's truncation over a toric variety (in place of a compactification of a locally symmetric space). More generally, in a work in progress, we define a "truncation" operation for "nice" classes of stratified spaces together with actions of discrete groups. This contains toric varieties and compactifications of locally symmetric spaces as examples. We prove convergence/polynomiality results in this context.This is a joint work (in progress) with Mahdi Asgari (OkSU).
Applications of the Endoscopic Classification to Statistics of Cohomological Automorphic Representations on Unitary Groups
When: Mon, October 24, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Rahul Dalal (Johns Hopkins University) - https://math.jhu.edu/~dalal/
Abstract: Consider the family of automorphic representations on some unitary group with fixed (possibly non-tempered) cohomological representation $\pi_0$ at infinity and level dividing some finite upper bound. We compute statistics of this family as the level restriction goes to infinity. For unramified unitary groups and many different $\pi_0$, we are able to compute the exact leading term for both counts of representations and averages of Satake parameters. We get bounds on our error term similar to the previous work of Shin-Templier for the case of discrete series at infinity.
The main technical tool is an extension of an inductive argument using Mok and Kaletha-Minguez-Shin-White's endoscopic classification that was originally developed by Taïbi to count unramified representations on Sp and SO.
This is joint work with Mathilde Gerbelli-Gauthier
Nonemptiness of affine Deligne-Lusztig varieties
When: Fri, November 11, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: DongGyu Lim (Berkeley) -
Abstract: The mod p points of a Shimura variety have a conjectural description called the Langlands-Rapoport conjecture. In relation to the conjecture, Rapoport defined (a suitable union of) affine Deligne-Lusztig varieties as the (conjectural) p-part of the description, in one of his Astérisque papers. One of the most basic questions is 'when they are nonempty'. As Rapoport pointed out, 'the individual affine Deligne-Lusztig varieties are very difficult to understand, whereas a suitable finite union of them are more accessible'. For the certain union, the nonemptiness criterion is completely known (by the so-called Mazur's inequality or B(G,μ)). However, the question about the "individual" ones is moderately open (with no general conjecture). I will discuss old and new nonemptiness results and suggest a new conjecture, for the individual ones, in the basic case. As an application, I will briefly mention a new explicit dimension formula in the rank 2 case (for which no conjectural formula was stated before).
Connected components of affine Deligne-Lusztig varieties
When: Mon, November 14, 2022 - 2:00pmWhere: Kirwan Hall B0421
Speaker: DongGyu Lim (Berkeley) -
Abstract: Affine Deligne-Lusztig varieties show up naturally in the study of Shimura varieties, Rapoport-Zink spaces, and moduli spaces of local shtukas. Among various questions on its geometric properties, the question on the connected components turns out to be a fairly important problem. For example, Kisin, in his proof of the Langlands-Rapoport conjecture (in a weak sense) for abelian type Shimura variety with the hyperspecial level structure, crucially used the description of the set of connected components. Since then, many authors have answered this question in various restricted cases. I will discuss these previous works and my new result (joint work with Ian Gleason and Yujie Xu) which finishes the question in the mixed characteristic case.
Semi-modules and crystal bases via affine Deligne-Lusztig varieties
When: Wed, November 16, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Ryosuke Shimada (University of Tokyo) -
Abstract: There are two combinatorial ways of parameterizing the$J_b$-orbits of the irreducible components of affine Deligne-Lusztig varieties for $GL_n$ and superbasic $b$.
One way is to use the extended semi-modules introduced by Viehmann.
The other way is to use the crystal bases introduced by Kashiwara and Lusztig.
In this paper, we give an explicit correspondence between them using
the crystal structure.
Comparison of local Langlands correspondences for odd unitary groups
When: Mon, December 5, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Alexander Bertoloni-Meli (University of Michigan) -
Abstract: I will speak on joint work with Linus Hamann and Kieu Hieu Nguyen. For odd unramified unitary groups over a p-adic field, there is a Langlands correspondence using trace formula techniques due to Mok and Kaletha--Minguez--Shin--White. Another correspondence was constructed recently by Fargues--Scholze using p-adic geometry. We show these correspondences are compatible by first proving an analogous result for unitary similitude groups by studying the cohomology of local Shimura varieties. If time permits, we will discuss applications to the Kottwitz conjecture and eigensheaf conjecture of Fargues.
Hecke orbits on Shimura varieties of Hodge type
When: Mon, December 12, 2022 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Pol van Hoften (Stanford) -
Abstract: Oort conjectured in 1995 that isogeny classes in the moduli space A_g of principally polarised abelian varieties in characteristic p are Zariski dense in the Newton strata containing them. There is a straightforward generalisation of this conjecture to the special fibres of Shimura varieties of Hodge type, and in this talk, I will present a proof of this conjecture. I will mostly focus on the case of A_g since most of the new ideas can already be explained in this special case. This is joint work with Marco D'Addezio.
Geometry of local Arthur packets and Vogan's conjecture for GL_n
When: Wed, February 22, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Mishty Ray (University of Calgary) -
Abstract: The local Langlands correspondence for a connected reductive p-adic group G partitions the set of equivalence classes of smooth irreducible representations of G(F) into L-packets using Langlands parameters. Arthur’s work introduces the notion of local A-packets, which are sets of smooth irreducible representations of G(F) attached to Arthur parameters. On the other hand, Vogan’s geometric perspective on the Langlands correspondence establishes a bijection between equivalence classes of smooth irreducible representations of G(F) (along with its pure inner forms) and simple equivariant perverse sheaves on a moduli space of Langlands parameters. This gives us the notion of an ABV-packet, which is also a set of smooth irreducible representations of G(F), but now attached to any Langlands parameter. Conjecturally, ABV-packets are generalized A-packets; we call this Vogan’s conjecture. In recent work joint with Clifton Cunningham, we prove this conjecture for p-adic GL_n. In this talk, we will explore the geometry of the moduli space of Langlands parameters for GL_n via examples. We will provide comments on the proof of Vogan’s conjecture in this setting and discuss the scope of further generalizations.
Geometric Eisenstein Series and the Cohomology of Shimura Varieties, I
When: Mon, March 13, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Linus Hamann (Princeton) -
Abstract: Given a connected reductive group G and a Levi subgroup M,
Braverman-Gaistgory and Laumon constructed geometric Eisenstein
functors which take Hecke eigensheaves on the moduli stack Bun_{M} of
M-bundles on a curve to eigensheaves on the moduli stack Bun_{G} of
G-bundles. Recently, Fargues and Scholze constructed a general
candidate for the local Langlands correspondence by doing geometric
Langlands on the Fargues-Fontaine curve. In this talk, we explain recent work
on carrying the theory of geometric Eisenstein series over to the
Fargues-Scholze setting. In particular, we explain how, given the
eigensheaf S_{\chi} on Bun_{T} attached to a smooth character \chi of
the maximal torus T, one can construct an eigensheaf on Bun_{G} under
a certain genericity hypothesis on \chi, by applying a geometric
Eisenstein functor to S_{\chi}. Assuming the Fargues-Scholze
correspondence satisfies certain expected properties, we fully
describe the stalks of this eigensheaf in terms of normalized
parabolic inductions of the generic \chi.
Geometric Eisenstein Series and the Cohomology of Shimura Varieties, II
When: Wed, March 15, 2023 - 2:00pmWhere: Kirwan Hall 1311
Speaker: Linus Hamann (Princeton) -
Abstract: We will explore some of the consequences of the theory
described in the first talk to the cohomology of local and global
Shimura varieties. In particular, the Hecke eigensheaf property for
the S_{\chi} described in the previous talk recovers certain cases of
an averaging formula of Shin for the cohomology of local Shimura
varieties. Moreover, under the generic condition introduced in the
first talk, it actually refines the averaging formula describing the
exact degrees of cohomology that the parabolic inductions attached to
\chi contribute to cohomology. By combing this with the geometry of
the Hodge-Tate period morphism and local-global compatibility of the
Fargues-Scholze correspondence, one can obtain very strong control
over the generic part of the cohomology of global Shimura varieties,
and this works even with torsion coefficients. We will conclude by
describing joint work with Si-Ying Lee where we apply this perspective
to generalize the known torsion vanishing results of
Caraiani-Scholze/Koshikawa, and obtain a formula for the generic part
that bears a remarkable similarity to work of Xiao-Zhu, but on the
generic fiber of the Shimura variety.
Finite conductor models for zeros near the central point of elliptic curve $L$-functions
When: Wed, March 29, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Steven J. Miller (Williams College) -
Abstract: Random Matrix Theory has successfully modeled the behavior of
zeros of elliptic curve L-functions in the limit of large conductors. We
explore the behavior of zeros near the central point for one-parameter
families of elliptic curves with rank over Q(T) and small conductors.
Zeros of L-functions are conjectured to be simple except possibly at the
central point for deep arithmetic reasons; these families provide a
fascinating laboratory to explore the effect of multiple zeros on nearby
zeros. Though theory suggests the family zeros (those we believe exist due
to the Birch and Swinnerton-Dyer Conjecture) should not interact with the
remaining zeros, numerical calculations show this is not the case;
however, the discrepancy is likely due to small conductors, and unlike
excess rank is observed to noticeably decrease as we increase the
conductors. We shall mix theory and experiment and see some surprising
results, which leads us to conjecture that a discretized Jacobi ensemble
correctly models the small conductor behavior. If time permits we will
discuss generalizations to other cuspidal newforms.
Prismatic realization functors on Shimura varieties of abelian type (joint with Naoki Imai and Hiroki Kato)
When: Wed, April 19, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Alex Youcis (University of Tokyo) -
Abstract: Shimura varieties are certain spaces associated to a reductive group G which intuitively parameterize 'G-motives' of a certain type, and have played a pivotal role in the Langlands program. Central to the theory, especially its applications to the Langlands program, are certain p-adic local systems on these Shimura varieties. In this talk I discuss ongoing joint work with Imai and Kato studying how the prismatic theory of Bhatt, Scholze, and others provides a robust framework to define 'integral models' of these local systems, and discuss the relationship to previous work of other authors.
Local-global compatibility over function fields
When: Mon, April 24, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Siyan Daniel Li-Huerta (Harvard University) -
Abstract: The Langlands program predicts a relationship between automorphic forms and Galois representations that vastly generalizes class field theory. We present a proof that V. Lafforgue's global Langlands correspondence is compatible with Fargues–Scholze's semisimplified local Langlands correspondence, generalizing local-global compatibility from class field theory. As a consequence, we canonically lift Fargues–Scholze's construction to a non-semisimplified local Langlands correspondence for fields of characteristic p ≥ 5. We also deduce that Fargues–Scholze's construction agrees with that of Genestier–Lafforgue, answering a question of Fargues–Scholze, Hansen, Harris, and Kaletha.
Newton stratification on the $B_{dR}^+$-Grassmannian
When: Wed, April 26, 2023 - 2:00pmWhere: Kirwan Hall 3206
Speaker: Serin Hong (University of Michigan) -
Abstract: The $B_{dR}^+$-Grassmannian is a p-adic (perfectoid) analogue of the classical affine Grassmannian. It plays an important role in the geometrization of the local Langlands program and the study of local Shimura varieties. In this talk, we discuss its geometry in terms of a natural stratification called the Newton stratification, with a particular focus on the case where the underlying group is GLn.