Program and Abstracts

Program

Thursday 17 October (Lecture Room 5)

13.00 - 14.00 Matilde Maccan

14.00 - 14.30 Coffee

14.30 - 15.30 Simon Cooper

15.50 - 16.50 Michel Brion

Friday 18 October (Lecture Room 4 in the morning, Lecture Room 5 in the afternoon)

9.00 - 10.00 Gerhard Röhrle

10.00 - 10.30 Coffee

10.30 - 11.30 Hankyung Ko

11.30 - 13.00 Lunch

13.00 - 14.00 Philippe Gille

Abstracts

Michel Brion: Forms of vector groups

The structure of unipotent algebraic groups over a field is quite intricate if the field is imperfect. In particular, there exist nontrivial forms of vector groups, and these are the building blocks for all smooth connected unipotent groups. The talk will present an approach to forms of vector groups and their subgroups via noncommutative algebra. It gives a new proof and a generalization of a theorem of Tits: every form of G_a^n is the kernel of an epimorphism from G_a^{n+1} to G_a. This is joint work in progress with Stefan Schröer.

Simon Cooper: Tautological rings of Shimura varieties in characteristic p 

Tautological rings of Shimura varieties are subrings of the Chow ring which are generated by classes of representations of the 'Shimura' parabolic. They contain many interesting classes and are open to computation due to group-theoretic extra structure. I'll present a conjecture describing these tautological rings in characteristic p and provide some basic evidence for it. Part of this evidence is a theorem I proved describing the tautological ring of a Hilbert modular variety at an unramified prime, whose proof I'll explain. The techniques rely on characteristic p methods via the theory of G-Zips, which provides a connection between group theory and geometry of Shimura varieties.

Philippe Gille: Loop torsors in Galois cohomology and more

In terms of non-abelian Galois cohomology, quadratic forms of rank n are classified by torsors under the orthogonal group O_n, so that Gauss diagonalization of quadratic forms implies that those torsors actually have a very nice shape called loop torsors. We will review what is known about loop torsors in non-abelian Galois cohomology of algebraic groups and will pursue with other nice rings, for example the localization of a regular complete ring with respect to a system of parameters.

Hankyung Ko: Hecke categories and representation theory of reductive groups

Many classical problems in Lie theory have combinatorial answers in terms of the Weyl group. Such an answer often comes from and is best understood by looking at higher structures. This talk concerns RepG, the category of rational representations of a reductive algebraic group in positive characteristic, and the Hecke category, a (2-)category overlying the associated Weyl group. I will discuss how the Hecke category acts on RepG and how it solves problems in representation theory. This is an overview of major developments in the area but aims to include a recent joint work with Ben Elias, Nicolas Libedinsky, Leonardo Patimo, where we construct a combinatorial basis of Hecke categories.

Matilde Maccan: Isotrivial elliptic surfaces in characteristic p

Studying contracted products of two curves by a finite group is a classical topic in complex algebraic surfaces. As a variant of this approach, we consider minimal surfaces, equipped with a faithful action of an elliptic curve. Such a surface S is isomorphic to a contracted product of an elliptic curve and of an “equivariantly normal” curve, a notion developed recently by Brion. Here, the quotient is by a finite group scheme, possibly infinitesimal. Computing the Betti numbers of S, we see where these surfaces sit in the classification; and obtain information on their automorphism groups. When the group is diagonalizable, we extract additional invariants using a Hurwitz-type formula. This is joint work with Pascal Fong.

Gerhard Röhrle: G-complete reducibility and semisimplification

Let G be a reductive algebraic group---possibly non-connected---over a field k and let H be a subgroup of G.  If G= GL(n) then there is a degeneration process for obtaining from H a completely reducible subgroup H' of G; one takes a limit of H along a cocharacter of G in an appropriate sense.  We generalise this idea to arbitrary reductive G using Serre's notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields. Our construction produces a G-completely reducible subgroup H' of G, unique up to G(k)-conjugacy, which we call a k-semisimplification of H.  This gives a single unifying construction which extends various special cases in the literature (in particular, it agrees with the usual notion for G= GL(n) and with Serre's ``G-analogue'' of semisimplification for subgroups of G(k)).

There is an analogue of the notion of G-complete reducibility over k for a Lie subalgebra h of the Lie algebra Lie(G) of G and an analogous concept of a k-semisimplification h' of h; h' is a Lie subalgebra of Lie(G) associated to h which is G-completely reducible over k. This is the Lie algebra counterpart of the analogous notion for subgroups from above.  As in the subgroup case, we show that h' is unique up to Ad(G(k))-conjugacy in Lie(G). Moreover, we prove that the two concepts are naturally compatible: for H a closed subgroup of G and H' a k-semisimplification of H, the Lie algebra Lie(H') is a k-semisimplification of Lie(H).

This is a report on joint work with Sören Böhm, Michael Bate, Benjamin Martin and Laura Voggesberger.