Schedule

All scheduled events will take place in the Alan Turing Building at the University of Manchester. 

On the 28th and 29th August: all events will take place in Frank Adams 1.

On the 30th August: all events will take place in Frank Adams 2. 

There will be a subsidised conference dinner at 7pm on Thursday at La Vina, and the menu can be found here. 

Titles and Abstracts

An introduction to unipotent classes and nilpotent orbits for reductive algebraic groups

David Stewart

3:00 - 4:00 P.M. 28/08/24 

Abstract: 

Let G be a reductive algebraic group over an algebraically closed field k. It is a miraculous and extremely useful fact that there are only finitely many conjugacy classes of unipotent elements in G(k) and orbits of G(k) on nilpotent elements of g=Lie(G), no matter the characteristic of k. This fact underpins many major results in Lie theory. Some examples:

• the Liebeck–Seitz–Testerman classification of the conjugacy classes of maximal subgroups of G (which builds on those of Dynkin);

• our understanding of the dimensions of simple modules for g in positive characteristic (such as Premet's proof of conjectures due to Kac–Weisfeiler and Humphreys);

• the Springer correspondence, determining the representations of Weyl groups, thence Lusztig's classification of the complex irreducible representations of finite groups of Lie type.

This talk will nutshell the classification of unipotent classes and nilpotent orbits and give an overview of some of the most important parts of the literature. 

G-cr over a field and representations of pseudo-reductive groups

Michael Bate

10:00 - 11:00 A.M. 29/08/24 

Abstract: 

For a reductive group G over a possibly non-algebraically closed field k, we define "G-complete reducibility over k" in terms of the k-parabolic and k-Levi subgroups of G. There are some interesting problems which do not arise in the case that k is algebraically closed, and many of these problems can be understood through an analysis of the first obvious case, where G = GL_n. Here, finding completely reducible subgroups is equivalent to finding completely reducible representations of algebraic k-groups, and many naturally occurring examples illustrate some of the generic features of the theory over non-algebraically closed fields.

I'll describe some aspects of joint work with David Stewart studying simple modules for algebraic groups (which quickly reduces to the case of pseudo-reductive groups). Some of what we have found in this work suggests interesting problems in the study of complete reducibility over a field, which I will try to draw out. 

Overgroups of distinguished unipotent elements in reductive groups

Ben Martin

2:00 - 3:00 P.M. 29/08/24 

Abstract:

Suppose G is a simple algebraic group defined over an algebraically closed field of good characteristic p. In 2018 Korhonen showed that if H is a connected reductive subgroup of G which contains a distinguished unipotent element u of G of order p, then H is G-irreducible in the sense of Serre. We present a short uniform proof of this result under an additional natural hypothesis. 

Moreover, we generalize Korhonen's results by removing the restriction on the order of u under a mild condition on p depending on the rank of G, and we present an analogue of Korhonen's theorem for Lie algebras. This is joint work with Mike Bate, Sören Böhm and Ben Martin.

Saturation and G-complete reducibility

Gerhard Röhrle

10:00 - 11:00 A.M. 30/08/24  

Abstract:

Let G be a connected reductive group over an algebraically closed field k of characteristic p and let u be a unipotent element of G.  If p= 0 or p is sufficiently large then we can define unipotent elements u^t of G for t in k, via a logarithm map and an exponential map.  We say that a subgroup H of G is saturated if u^t belongs to H for all unipotent u in H and all t in k.  We define  H^sat to be the smallest saturated subgroup of G containing H, and we call H^ sat the saturation of H.  I will discuss interactions between G-complete reducibility and saturation, including some joint work with Michael Bate, Sören Böhm, Sebastian Herpel and Gerhard Röhrle.

Varying flavours of complete reducibility

Alastair Litterick

2:00 - 3:00 P.M. 30/08/24 

Abstract:

Originating with the familiar representation-theoretic concept, we have nowadays come to understand G-complete reducibility as a geometric property, characterised using cocharacters and orbits of G on its Cartesian powers G^n, and as a combinatorial property, characterised via the action of subgroups on the spherical building of G. This allows a number of generalisations. When a subgroup K of G acts on G^n, we are led to 'relative complete reducibility.' In the presence of a Frobenius endomorphism \sigma acting on G and its building, we arrive at so-called '\sigma-complete reducibility'. And finally, we can replace G by groups such as Kac-Moody groups: Infinite-dimensional analogues with a twin building, in which the notion of opposition (and hence complete reducibility) can still be defined and studied.

This talk consists of joint work with Bate, Martin, Roehrle and their former students Attenborough, Bannuscher, Gruchot and Uchiyama, and incipient joint work with my PhD student Harvey Sykes.