Mittagsseminar

A categorical approach to the (p-adic) Langlands program searches for a functor from categories of representations of groups such as G = GL_n(F), F a finite extension of Q_p, to coherent sheaves on stacks of L-parameters. These functors are also supposed to feature in a description of the étale cohomology of Shimura varieties in terms of coherent cohomology of spaces of Galois representations. In this talk we will focus on locally analytic representations of G and we are looking for the description (in terms of coherent sheaves and Galois representations) of the cohomology of Shimura varieties with coefficients in certain overconvergent p-adic coefficient systems (closely related to overconvergent p-adic automorphic forms). I will explain what this description looks like in the case of the modular curve and discuss geometric properties of the spaces of Galois representations involved.

Equivariant motivic characteristic classes of Schubert cells

We explain in the context of complete flag varieties X = G/B the inductive calculation of equivariant motivic characteristic classes of Schubert cells via suitable Demazure-Lusztig operators, fitting with convolution actions of corresponding Hecke-algebras and Weyl groups. Applications include solutions of:
(1) a conjecture of Bump, Nakasuji and Naruse. Here a specialization of the equivariant K-theory of G/B was identified, as a Hecke module, to the Iwahori-fixed part of the the principal series representation of a p-adic Langlands dual group. Under this identification, certain classes related to motivic Chern classes of Schubert cells, were sent to the standard basis elements of the principal series representation, and the fixed point classes were sent to the Casselman basis elements;
(2) a positivity conjecture of Aluffi-Mihalcea for the non-equivariant MacPherson Chern classes of Schubert cells;
(3) a positivity conjecture about the Euler characteristic of generic triple intersections of Schubert cells.
This is joint work with P. Aluffi, L. Mihalcea and C. Su, and for (3) with C. Simpson and B. Wang.

Taming perfectoid fields

Tilting perfectoid fields allows to transfer results between certain henselian fields of mixed characteristic and their positive characteristic counterparts and vice versa. We present a model-theoretic approach to tilting via ultraproducts, which allows to transfer many first-order properties between a perfectoid field and its tilt (and conversely). In particular, our method yields a simple proof of the Fontaine-Wintenberger Theorem which states that the absolute Galois group of a perfectoid field and its tilt are canonically isomorphic. A key ingredient in our approach is an Ax-Kochen/Ershov principle for perfectoid fields (and generalizations thereof).

Square roots of symplectic L-functions and Reidemeister torsion

We give a purely topological formula for the square class of the central value of the L-function of a symplectic representation on a curve. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds. This is related to the theory of epsilon factors in number theory and Meyer’s signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh.

Motivic homotopy theory of the classifying stack of finite groups of Lie type

Let G be a split reductive group over F_p with associated finite group of Lie type G^F. Let T be a split maximal torus contained inside a Borel B of G. We relate the (rational) Tate motives of BG^F with the T-equivariant Tate motives of the flag variety G/B. We explain how this could lead to representation theoretic applications, using recent advances in motivic geometric representation theory of the flag variety. On the way, we show that for a split reductive group G over a field k acting on a smooth k-scheme X, we get an isomorphism Aⁿ_G(X, m)_Q ≅ Aⁿ_T(X, m)_Q^W extending the classical result of Edidin-Graham to higher equivariant Chow groups.

Pentecost

Pentecost is a Christian holiday which takes place on the 50th day (the seventh Sunday) after Easter Sunday. It commemorates the descent of the Holy Spirit upon the Apostles and other followers of Jesus Christ while they were in Jerusalem celebrating the Feast of Weeks, as described in the Acts of the Apostles (Acts 2:1-31).

The prismatic realization for Shimura varieties of abelian type

On a Shimura variety, which is by definition associated to a reductive group G together with some other data, there is a canonical pro-étale torsor parametrizing level structures. When the Shimura variety is of abelian type, it can be thought of as the étale realization of the ‘universal motive with G-structure’ under the guiding principle that those Shimura varieties should be moduli spaces of certain motives with G-structure. When the level is hyperspecial at a prime p, a crystalline realization has been constructed, by Lovering, on the p-adic completion of the integral canonical model, which corresponds to the étale realization via the p-adic Hodge theory. Given recent progress of p-adic Hodge theory, especially that of prismatic cohomology theory, it is natural to expect that Shimura varieties of abelian type admit a prismatic realization that specializes to the étale and crystalline realizations. I will talk about construction of such a realization and some consequences (joint work with Naoki Imai and Alex Youcis).

The affine Grassmannian as a presheaf quotient

The affine Grassmannian of a reductive group G is usually defined as the étale sheafification of the quotient of the loop group LG by the positive loop subgroup. I will discuss various triviality results for G-torsors which imply that this sheafification is often not necessary.

Positive definite invariant forms for generalized Weyl and q-Weyl algebras

Let A be an algebra over complex numbers with an antilinear automorphism ρ, M be a bimodule over A. A positive definite Hermitian form (·, ·) on M is said to be invariant if (am, n) = (m, nρ(a)) for all a ∈ A, m,n ∈ M.
I will discuss classification of positive definite invariant forms in the following cases:
(1) M = A and A is a non-commutative deformation of a Kleinian singularity of type A, sometimes called generalized Weyl algebra.
(2) M = A and A is a q-deformation of a Kleinian singularity of type A (generalized q-Weyl algebra).
(3) A is a Weyl or a q-Weyl algebra with generic parameter.
The first case is joint work with Etingof, Rains, Stryker.

The weight part of Serre’s modularity conjecture for totally real fields

The strong form of Serre’s modularity conjecture states that every two-dimensional continuous, odd, irreducible mod p representation of the absolute Galois group of Q arises from a modular form of a specific minimal weight, level and character. We show this minimal weight is equal to two other notions of minimal weight, one inspired by work of Buzzard, Diamond and Jarvis and one coming from p-adic Hodge theory. We discuss the interplay between these three notions for Galois representations over totally real fields and we investigate the consequences of this for generalised Serre conjectures. We focus on the modularity of partial weight one Hilbert modular forms, extending recent work of Diamond and Sasaki.

Representation theory via 6-functor formalisms

We present recent advances on the abstract theory of 6-functor formalisms and apply them to the representation theory of locally profinite groups. This sheds new light on classical results like preservation of admissibility under various operations, Bernstein-Zelevinsky duality and Second Adjointness. As an application we obtain new results on the p-adic representation theory of p-adic Lie groups.

Decomposability of the de Rham complex in positive characteristic

Deligne and Illusie proved that if a smooth variety over F_p admits a lift over Z/p² then the truncation of its de Rham complex in degrees < p is quasi-isomorphic to the direct sum of its cohomology sheaves. As a consequence, the Hodge-to-de Rham spectral sequence of a smooth proper liftable variety degenerates, provided that the dimension of the variety is ≤ p. However, further truncations of the de Rham complex of a liftable variety need not be decomposable. I will describe the obstruction to decomposing the truncation of the de Rham complex in degrees ≤ p in terms of other invariants of the variety, and will give an example of a smooth projective variety over F_p that lifts to Z_p but whose Hodge-to-de Rham spectral sequence does not degenerate at the first page. The proof relies on the existence of prismatic cohomology, but the key argument is a computation in homotopical algebra, motivated by a construction of Steenrod operations on cohomology of topological spaces.

Gaitsgory’s central functor and Arkhipov-Bezrukavnikov’s equivalence for p-adic groups

In 2002, Arkhipov and Bezrukavnikov established an equivalence between the Iwahori-equivariant derived category of constructible ℓ-adic sheaves on the affine flag variety, the so-called Iwahori-Whittaker category on the affine flag variety, and the Langlands dual group equivariant derived category of coherent sheaves on the Langlands dual Springer resolution for a connected reductive group over an algebraic closure of F_p. In this talk, we discuss this equivalence for p-adic groups by studying a mixed-characteristic Gaitsgory’s central functor. This is a joint work with J. Anschütz, J. Lourenço, and Z. Wu.