Dipendra Prasad (IIT Bombay)
Degenerate Whittaker models
The notion of a Whittaker model and the more general degenerate Whittaker model (which were introduced for p-adic groups by Moeglin-Waldspurger) is of fundamental importance to representation theory and automorphic forms. They also appear prominently in the GGP conjectures as Bessel and FJ models. The talk is meant to introduce the subject, discuss some of the less well-known examples, and talk about some open questions.
Beth Romano (King's College London)
A Fourier transform for unipotent representations of p-adic groups
In the representation theory of finite reductive groups, an essential role is played by Lusztig's nonabelian Fourier transform, an involution on the space of unipotent characters. In joint work with Anne-Marie Aubert and Dan Ciubotaru, we propose a potential lift of Lusztig's Fourier transform to the setting of split p-adic groups and their pure inner twists. Our work generalizes a construction of Moeglin–Waldspurger for orthogonal groups. In my talk, I'll introduce these ideas via examples and talk about motivation from the local Langlands correspondence. I'll focus on our recent work, which extends the proposed lift from the setting of elliptic characters to that of compact characters.
Ian Petrow (University College London)
Some applications of Petersson/Bruggeman–Kuznetsov formulas for finite places
The Petersson and Bruggeman–Kuznetsov (PBK) formulas are classical tools in the analytic theory of automorphic forms that relate an average of GL(2) automorphic forms to certain exponential sums called Kloosterman sums. Recently Y. Hu, M.P. Young and I have given versions of these formulas in which one may specify the local representation types of the automorphic forms at non-archimedean places, cf. the classical formulas specifying the archimedean representation type.
In this talk I would like to present some applications of the new PBK formulas to moments of L-functions and subconvexity for L-functions. The first of these applications are estimates the cubic moment of central values of L-functions for specified local component families, leading to Weyl subconvexity for PGL(2) L-functions with square conductor, and in depth aspect. The second of these applications are to the large sieve inequality for PGL(2)/Q, i.e. with an average over automorphic representations of all conductors at most X and bounded archimedean parameters, say.
Subhajit Jana (ISI Bangalore)
Fine spectral expansion of a Rankin–Selberg period and application
We will talk about the fine spectral expansion of the absolute square of a maximal degenerate Eisenstein series as a Schwartz distribution, and also, how this expansion gives rise to certain partial reciprocity formula. Joint work with Ramon Nunes and Jakub Dobrowolski.
Ana Caraiani (Imperial College London)
Igusa stacks and intersection cohomology
I will discuss joint work with Linus Hamann and Mingjia Zhang, where we use the Igusa stack diagram and the work of Fargues-Scholze to study the intersection cohomology of Shimura varieties with both torsion and characteristic 0 coefficients.
Si Ying Lee (National University of Singapore)
p-isogenies with G structure
I will talk about a general framework for defining p-isogenies with G-structure, as well as applications to counting mod p points for Shimura varieties for which integral models are available, including those of exceptional type. This is joint work in progress with Keerthi Madapusi.
Alexander Stasinski (Durham)
Explicit and geometric constructions of Deligne–Lusztig representations of reductive groups over finite local rings
I will outline the main ideas of the recently completed proof, joint with Zhe Chen, that the representations of a (sufficiently nice) reductive group scheme G over the ring of integers O of a non-archimedean local field, constructed purely algebraically and explicitly by Gérardin, coincide (up to sign) with the 'higher-level' regular/generic Deligne–Lusztig representations defined via the ℓ-adic cohomology of Deligne–Lusztig varieties over truncations of O modulo powers of the maximal ideal. This resolves a problem raised by Lusztig and makes it possible to determine the dimensions and orbits of these representations, which cannot be obtained from the cohomological construction alone.
Xinwen Zhu (Stanford)
Deligne-Lusztig induction of tilting perverse sheaves
I will first review tilting perverse sheaves in the Hecke category and discuss a recent theorem, proved independently by A. Eteve and myself, showing that Deligne-Lusztig induction of tilting sheaves in the finite Hecke category yields a family of projective objects in the category of representations of finite groups of Lie type. Then I will discuss an affine analogue of this result, in the context of the categorical local Langlands correspondence, and indicate its arithmetic applications to then cohomology of Shimura varieties. The affine aspect of the work is joint with Ian Gleason and Xiangqian Yang.
Jessica Fintzen (Bonn)
Reduction to depth-zero for \bar{Z}[1/p]-representations of p-adic groups
The category of smooth complex representations of p-adic groups decomposes into Bernstein blocks and by a joint result with Adler, Mishra and Ohara from August 2024 we know that under some minor tameness assumptions each Bernstein block is equivalent to a depth-zero Bernstein block, which are the representations that correspond roughly to representations of finite groups of Lie type. This result allows to reduce a lot of problems about representations of p-adic groups and the Langlands correspondence to their depth-zero counterpart that is often easier to solve or already known.
In this talk we present analogous results for R-representations of p-adic groups where R is any ring that contains all p-power roots of unity, a fourth root of unity and the inverse of a square-root of p, for example, R could be an algebraically closed field of characteristic different from p or the ring \bar{Z}[1/p]. This is a joint work with Jean-François Dat. While the result is analogous to the result with complex coefficients (except for the “blocks” being “larger”), the proof is of a very different nature. In the complex setting the proof is achieved via type theory and an isomorphism of Hecke algebras, which are techniques not available for general R-representations. We sketch in the talk how we deal with the category of R-representations instead.
Tasho Kaletha (Bonn)
Type Deligne–Lusztig varieties
The construction of representations of p-adic groups via Kim-Yu types contains a structural dychotomy: Depth-zero types arise geometrically via Deligne-Lusztig theory, while positive-depth types arise algebraically via the Weil-Heisenberg representation. Some recent work has extended the geometric method to positive depth, but under the limiting assumption that all occurring groups are unramified. In this talk we will present a geometric construction of Kim-Yu types that covers the ramified case. The main input is Deligne-Lusztig theory in the setting of non-reductive groups of the form J=H\rimes G, where H is the Heisenberg group associated to a symplectic vector space (V,η) and G is a reductive group acting on (V,η). We will show that the cohomology of the resulting Deligne-Lusztig varieties realizes the Kim–Yu procedure twisted by a certain explicit sign character. This is joint work with Charlotte Chan and Xinwen Zhu.
Tsao-Hsien Chen (Minnesota)
Loop Spaces, Real Groups, and Langlands Duality
I will begin by reviewing the well-established relationship between loop groups and Langlands duality. I will then discuss recent progress toward extending this relationship to the setting of loop spaces of symmetric spaces (or, more generally, spherical varieties) and Langlands duality for real groups (or Relative Langlands duality). This is joint work with David Nadler and Lingfei Yi.
Raphaël Beuzart-Plessis (Aix-Marseille)
On the surjectivity of the local Genestier–Lafforgue parameterization
Genestier–Lafforgue and Fargues–Scholze have constructed a semi-simple local Langlands correspondence for reductive groups over local fields of positive characteristic.
In this talk, I will explain how, assuming a version of the stable (twisted) trace formula for base change over a function field, one can show the surjectivity of this parametrization for unramified groups and when the characteristic does not divide the order of the Weyl group. This is based on joint work with Michael Harris and Jack Thorne.
Shuichiro Takeda (Osaka)
Iwahori–Hecke algebras for double covers over 2-adic fields
Let G be a simply connected Chevalley group over a p-adic field. It is known that the Iwahori–Hecke algebra of G admits an Iwahori-Matsumoto (IM) presentation. Savin has shown in 2005 that a double cover of G also admits an IM presentation, assuming that G is simply laced and p is odd. In this talk, I will discuss how to obtain the analogous IM presentation when p is even, still assuming G is simply laced. This is a joint work with E. Karasiewicz.
Dan Ciubotaru (Oxford)
Endoscopic transfer and the wavefront upper bound conjecture
This talk is motivated by the elusive relation between the irreducible admissible Harish-Chandra characters of a reductive p-adic group and the Langlands–Arthur parameters. Arguably the best-known instance of this relation is the Hiraga–Ichino–Ikeda formal degree formula for square-integrable characters in terms of adjoint gamma-factors. The formal degree concerns the coefficient of the distribution attached to the zero nilpotent orbit in the Harish-Chandra–Howe local character expansion. At the opposite end, the largest orbits that contribute to the character expansion determine the wavefront set of the character distribution.
I will present a joint result with Hiraku Atobe, proving that the orbits of maximal dimension in the union of the geometric wavefront sets of the representations in any Arthur packet for a split classical p-adic group (p large) equal the Spaltenstein dual of the nilpotent orbit given by the A-parameter, which verifies a local analogue of a conjecture by Jiang.
Anne-Marie Aubert (Paris-Sorbonne)
The local Langlands correspondence through twisted graded Hecke algebras
Graded Hecke algebras, which were introduced by Lusztig in order to study the category of unipotent representations of p-adic groups, are graded analogues of affine Hecke algebras. Twisted graded Hecke algebras are slightly more general versions in which the group algebra of the spherical Weyl group is replaced by a twisted group algebra of an extension by a finite abelian group. The construction of twisted graded Hecke algebras will be recalled in the talk.
I will next explain how these algebras naturally occur in both the general representation theory of p-adic groups and the spectral side of the local Langlands correspondence, and how they can be used to construct the later in a vaste range of situations.
Farrell Brumley (Paris-Jussieu)
Joint Linnik Problems
A well-known class of arithmetic equidistribution problems, attributed to Linnik, is concerned with periodic toric orbits on quaternionic varieties. Classical examples include the equidistribution of CM points of large discriminant on the modular surface and projections to the sphere of integer solutions to the sum of three squares. These problems were essentially solved by Duke using techniques in automorphic forms and analytic number theory. One can combine any two Linnik problems using a diagonal action of the torus, which encodes their simultaneous equidistribution (or disjointness). This creates a new set of problems, first put forward by Michel and Venkatesh, of considerably greater difficulty. We present new work with Blomer and Radziwiłł which uses an array of automorphic and analytic number theoretic techniques to prove the simultaneous equidistribution of two distinct Linnik problems, under a no-Siegel-zero type hypothesis. The latter assumption encodes the abundance of small split primes in quadratic field extensions, a property which interacts directly with competing approaches emanating from ergodic theory.
Paul Nelson (Aarhus)
Equidistribution and moments of L-functions
I will survey how integral representations and period formulas recast equidistribution problems (for instance, involving unipotent shears on (GL(2n), GL(n))) as moment problems for automorphic L-functions, a mechanism originating in earlier work of many authors. I will recall rank-one prototypes in which the varying vectors are pure translates of fixed ones, as well as the microlocal framework from my joint work with Venkatesh on GGP periods such as (U(n+1), U(n)), where recent progress in homogeneous dynamics now yields effective consequences. I will then indicate a higher-rank variant, joint with Subhajit Jana, in which (GL(m), GL(n)) moments appear as L^2-norms of shears of fixed automorphic forms; here, implications for moments and subconvexity remain conditional on dynamical input not yet available.