Etale geometry of Jordan classes closures
Let G be a connected reductive algebraic group over an algebraically closed field k. Lusztig (1984) partitioned G into subvarieties which play a fundamental role in the study of representation theory, the Jordan classes. An analogue partition of the Lie algebra Lie(G) into subvarieties, called decomposition classes, dates back to Borho-Kraft (1979). When k = C the study of geometric properties (e.g., smoothness) of a point g in the closure of a Jordan class J in G can be reduced to the study of the geometry of an element x in the closure of the union of finitely many decomposition classes in Lie( M), where M is a connected reductive subgroup of G depending on g. The talk aims at introducing such objects and at generalizing this reduction procedure to the case char(k) > 0.
Geometric rigidity of simple modules for algebraic groups
Easy examples show that if V is a simple module for an algebraic group G over a field k and E/k is a field extension, then we should not expect the base change VE to be simple (or even semisimple) as a module for 𝐺E. However, for all the simple modules involved in the classification described in David Stewart's talk, the structure of V_{\bar k} is under quite tight control - this is the "geometric rigidity" of the title. It turns out that this rigidity is a feature of simple modules for arbitrary affine algebraic k-groups, with no need for any smoothness or connectedness hypotheses. The argument boils down to some basic ring theory and field theory, and in particular doesn't need any of the structure theory which underpins our classification results in part 1.
Overgroups of distinguished unipotent elements in connected reductive groups
Let G be a simple algebraic group and and let Char(k)=p be good for G. In 2018, M. Korhonen proved that a connected reductive subgroup H of G that contains a distinguished unipotent element of G of order p is G-irreducible. Korhonen's proof depends on checks for the various possible Dynkin types for simple G.
We give a short uniform proof of Korhonen's theorem without resorting to further case-by-case checks, but imposing an extra hypothesis which allows us to use a landmark result by G. Seitz. This is joint work with M. Bate, B. Martin and G. Röhrle.
Nichols algebras over finite simple groups of Lie type
Given a vector space V and a solution c of the braid equation on V ⊗ V, one naturally defines a so-called Nichols (small shuffle) algebra. Notable examples of Nichols algebras are: the symmetric algebra, the exterior algebra, and the positive part of quantized enveloping algebras of a semisimple Lie algebra.
A subclass of Nichols algebras is crucial for the classification program of finite-dimensional Hopf algebras developed by Andruskiewitsch and Schneider. In this case V is a suitable graded representation of a finite group G and c depends on the structure of V. It is in general a very hard problem to establish for which choices of G and V one obtains a finite-dimensional Nichols algebra.
A folklore conjecture states that this is never the case if G is a non-abelian simple group. I will report on progress on this conjecture, based on a collaborations with N. Andruskiewitsch, G. García and M. Costantini.
Local character expansions and asymptotic cones over finite fields
This is joint work with Emile Okada, motivated by the computation of the wavefront set of the positive-depth representations of reductive p-adic groups. We generalise the construction of Gelfand-Graev characters to graded Lie algebras and lift them to produce new test functions to probe the local character expansion in positive depth. We show that these test functions are well adapted to compute the leading terms of the local character expansion and relate their determination to the asymptotic cone of elements in Z/n-graded Lie algebras.
A reduction of the local Langlands correspondence for finite groups of Lie type
The Local Langlands correspondence establishes (conjecturally, in general) a surjective map from the set of smooth admissible representations of a p-adic group to the set of Langlands parameters. The fibers of this map, known as L-packets, are finite and are parametrized by the irreducible representations of a finite group associated with the corresponding Langlands parameter.
In 2020, Vogan proposed a conjecture extending this framework to representations of finite groups of Lie type, aiming for a parameterization compatible with the p-adic case. In this talk, we will provide a overview of Vogan's conjecture. We will focus on two leading examples: the case of GLn, where the conjecture has been established through the work of Macdonald, Silberger, and Zink, and the one of SLn, where further progress has been made in my own research.
Connected components of the moduli space of L-parameters
When studying a category of representations, a natural first problem is to describe its blocks. For the category of smooth complex representations of a p-adic group, this problem was solved by Bernstein in terms of the so-called inertial support. Generalizations of Bernstein's solution to fields (or even rings) of coefficients beyond the complex numbers form a topic of current research. However, the conjectural categorical local Langlands correspondence predicts a close relationship between these blocks and the connected components of the moduli space of L-parameters for the Langlands dual group, and an explicit description of the latter was conjectured by Dat-Helm-Kurinczuk-Moss. I will describe Bernstein's block decomposition, the connection to the moduli space of L-parameters, and a resolution of DHKM's conjecture.
Jordan classes in the enhanced and the exotic setting
I relate on joint work in progress with F. Ambrosio, G. Carnovale, N. Saunders and L. Topley. The enhanced and the exotic modules have been considered by various authors (e.g. Achar-Henderson, Kato, Mautner, Springer) in the study of the respective nilcones. We define a stratification by Jordan classes of these modules. We describe closures of strata in terms of an induction operation, and we study the local geometry of these closures.
Characters and conjugacy class representatives for finite groups of Lie type
Lusztig's geometric theory of character sheaves leads to an explicit program for determining the character table of a finite group of Lie type. This involves the delicate matter of choosing appropriate representatives in the various rational conjugacy classes of the underlying algebraic group. We discuss a number of examples and report on some recent progress.
Generic direct summands of tensor products and multiplicity conjectures
Let G be a reductive algebraic group over an algebraically closed field of positive characteristic. In this talk, I will define a canonical "generic direct summand" in the tensor product of two simple G-modules in the principal block, and I will explain how the generic direct summand appears "generically" as a direct summand of tensor products of simple G-modules in arbitrary blocks. Then I will state a conjecture that describes certain composition multiplicities in generic direct summands geometrically, in terms of Schubert structure constants for the homology ring of the affine Grassmannian, or combinatorially, in terms of structure constants for k-Schur functions. Finally, I will discuss some evidence for the conjecture based on a forthcoming project of Alex Sherman, Geordie Williamson et al., which can be interpreted as constructing a monoidal functor from the category of G-modules to the category of modules over the cohomolgy ring of the affine Grassmannian.
Fixed point spaces for actions of algebraic and finite groups
I will present some old and new results giving bounds for the sizes of fixed point spaces of elements of algebraic and finite groups G in their actions on various G-sets, G-modules and G-varieties.
Towards G-complete reducibility in Kac-Moody groups
G-complete reducibility has been an indispensable tool in studying subgroups of reductive algebraic groups. Completely reducible subgroups are well-behaved and can be used as a stepping stone towards understanding all reductive subgroups, directly generalising the representation-theoretic approach of understanding modules by first classifying irreducible modules, then studying extensions.
It is therefore of interest to ask where else methods of G-complete reducibility can be brought to bear. A particularly natural class is that of Kac-Moody groups, which are defined and behave like infinite-dimensional versions of simple algebraic groups. In this talk, we will look at all the things that go wrong when one attempts to do this, and initial attempts at a remedy.
The Jacobson-Morozov Theorem and G-complete reducibility
Let 𝔤 be the Lie algebra of a reductive algebraic group 𝐺. Over a field of characteristic zero, the Jacobson-Morozov Theorem states that every nilpotent element in 𝔤 is contained in a unique sl2-triple up to conjugacy by 𝐺. In this talk, we investigate how this fails in positive characteristic, and the interaction between non-uniqueness and non-G-complete reducibility.
Tensor triangular geometry for finite group schemes
I’ll describe a (tt) geometric application of a recent theorem of W. van der Kallen on finite generation of cohomology for a finite group scheme defined over an arbitrary commutative Noetherian ring. A short introduction to tensor triangular geometry will be included. This is joint work with T. Barthel, D. Benson, S. Iyengar and H. Krause.
Hesselink strata and nilpotent pieces
Let 𝐺𝕜 be a simple algebraic group over an algebraically closed field 𝕜 of characteristic p at least 0 and 𝔤𝕜 = Lie(𝐺𝕜). We assume that 𝐺𝕜is obtained by base change from a Chevalley group scheme 𝐺ℤ. Write N(𝔤𝕜) for the nullcone of 𝔤𝕜 Being the set of all unstable vectors of the adjoint 𝐺𝕜-module 𝔤𝕜, the variety N(𝔤𝕜) admits a natural stratification introduced by Hesselink. In my talk I will compare the Hesselink strata of N(𝔤𝕜) with the so-called Lusztig-Xue pieces and Mizuno-Spaltenstein pieces of N(𝔤𝕜).
Affine G-varieties of ''small'' dimension
Our base field is the field of complex numbers. Let G be a simple algebraic group. A G-orbit of an affine variety X is called small if it is G-isomorphic to the orbit of the highest weight vector in an irreducible representation V of G. In this talk we will study small affine $G$-varieties, i.e., those G-varieties which admit only small G-orbits and fixed points. In particular, we will show that if G is a simple algebraic group of type A_n, then any affine G-variety of dimension less than 2n is small (there is an analog statement for some other simple algebraic groups). This talk is based on a joint work with Hanspeter Kraft and Susanna Zimmermann.
Generic stabilizers and how to find them
One of the most fundamental ways to understand the action of an algebraic group on a variety, is to determine its generic stabilizer, if it exists. Recent work by Guralnick and Lawther has settled the problem for actions on subspaces of irreducible modules. I will talk about an extension of these results to actions on subspaces with a form, focusing on some of the computational methods involved.
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 of the group. In joint work with Anne-Marie Aubert and Dan Ciubotaru, we propose a potential lift of Lusztig's Fourier transform to the setting of p-adic groups. In my talk, I'll introduce these ideas via examples and talk a bit about motivation from the local Langlands correspondence.
Simple G-Modules for the Sophisticated Nincompoop
I will nutshell the classification of simple modules for smooth connected affine group schemes over a field; the journey immediately disgorges the spectre of the pseudo-reductive group, against which work by Conrad-Gabber-Prasad provides an effective talisman.
Decomposition Classes in Arbitrary Characteristic
Decomposition Classes provide a natural way of partitioning a Lie algebra into finitely many pieces, collecting together adjoint orbits with similar Jordan decompositions. The current literature surrounding these tends to only cover certain cases - such as in characteristic zero, or under the Standard Hypotheses. Building on the prior work of Spaltenstein, Premet-Stewart, Topley-Saunders and Ambrosio, we have adapted many of the useful properties of Decomposition Classes to work in greater generality.
In this talk, we will introduce Decomposition Classes for the Lie algebras of connected reductive algebraic groups - defined over arbitrary algebraically closed fields. We will then present some structure results about Decomposition Classes and their closures, which are called Decomposition Varieties. This connects to Generalised Lusztig-Spaltenstein Induction, and highlights the importance of these concepts to the problem of classifying infinitesimally isolated nilpotent orbits.
The Dynamics of Complete Reducibility in Kac-Moody Groups
A Kac-Moody group G generalises a split reductive group. By introducing a group ind-variety structure, one can define R-parabolic subgroups via a dynamic approach, and therefore a notion of G-complete reducibility. I will explore the behaviour of such R-parabolic subgroups and this theory. As a particular case, I will present a characterisation of G-completely reducible algebraic subgroups, recovering a result of D. Dawson for twin Euclidean buildings. We will also see that the dynamic approach to G-complete reducibility faithfully generalises a separate notion of 'complete reducibility' defined by P.-E. Caprace.
Modular Reduction of Nilpotent Orbits
We consider a split connected reductive algebraic ℤ-group 𝐺 and 𝑉 a 𝐺-module which is either the Lie algebra 𝔤 or its dual 𝔤*. If 𝕜 is an algebraically closed field then, by base change, we get a group 𝐺𝕜 and a corresponding module 𝑉𝕜. Hesselink has defined a partition of the nullcone 𝒩(𝑉𝕜) of 𝑉𝕜 into strata 𝒩(𝑉𝕜 | 𝒪) which can be indexed, thanks to Clarke–Premet, by 𝐺(ℂ)-orbits 𝒪 ⊆ 𝒩(𝔤ℂ), such that 𝒩(𝔤ℂ | 𝒪) = 𝒪 . Each stratum is a union of 𝐺(𝕜)-orbits and contains a unique largest orbit, which we call the Spaltenstein orbit.
In this talk I will describe joint work with Adam Thomas (Warwick) which produces for each orbit 𝒪 ⊆ 𝒩(𝔤ℂ), via a case-by-case analysis, an explicit integral representative 𝑒 ∈ 𝑉∩𝒩(𝑉ℂ | 𝒪) whose reduction 𝑒𝕜 ∈ 𝒩(𝑉𝕜 | 𝒪) is in the Spaltenstein orbit for every algebraically closed field 𝕜. Along the way we give two other variations on this based on the ideas of Dynkin–Richardson and Bala–Carter.
Small dimensional epimorphic subgroups in simple algebraic groups
A closed subgroup H of an algebraic group G is epimorphic if every morphism from G to an algebraic group G' is uniquely determined by its restriction to H. For simple G, of rank at least 2 and defined over a field of characteristic 0, Bien and Borel showed that any proper epimorphic subgroup in G has dimension at least three. They went on to construct 3-dimensional epimorphic subgroups for all such G. We now turn to simple groups G defined over fields of positive characteristic. In joint work with Iulian Simion, we show that for G simple of rank 2, any epimorphic subgroup of G has dimension at least 3 and we exhibit 3-dimensional epimorphic subgroups in rank 2 simple groups. In joint work with Adam Thomas, we consider the problem of constructing "small-dimensional'' epimorphic subgroups in simple groups of rank at least 3. We report both on the work with Simion and on the progress to date for groups of rank at least 3, describing the method we use for constructing small-dimensional epimorphic subgroups, based upon a recipe given by Bien and Borel.