Related papers: Inequalities characterizing distinguished unipoten…
We use the Euler characteristic of the orbit category of a finite group to establish equivalences between theorems of Frobenius and K.S. Brown and between theorems of Steinberg and L. Solomon.
We establish a relation between the known parametrization of a family of irreducible representations of a Weyl group and Springer's correspondence. We outline a parametrization of unipotent character sheaves on a connected reductive group…
We show that various properties of unipotent elements in a reductive group over the complex numbers can be recovered purely in terms of the affine Weyl group of the dual group.
Together with David Schlang we computed the discriminants of the invariant Hermitian forms for all indicator $o$ even degree absolutely irreducible characters of the ATLAS groups supplementing the tables of orthogonal determinants computed…
Let G be any connected reductive group over a non-archimedean local field. We analyse the unipotent representations of G, in particular in the cases where G is ramified. We establish a local Langlands correspondence for this class of…
A computably presented algebraic field $F$ has a \emph{splitting algorithm} if it is decidable which polynomials in $F[X]$ are irreducible there. We prove that such a field is computably categorical iff it is decidable which pairs of…
For a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced…
We study groups definable in existentially closed geometric fields with commuting derivations. Our main result is that such a group can be definably embedded in a group interpretable in the underlying geometric field. Compared to earlier…
Let $G$ be a complex reductive algebraic group. In arXiv:2108.03453, we have defined a finite set of irreducible admissible representations of $G$ called `unipotent representations', generalizing the special unipotent representations of…
This paper is an introduction, in a simplified setting, to Lusztig's theory of character sheaves. It develops a notion of character sheaves on reductive Lie algebras which is more general then such notion of Lusztig, and closer to Lusztig's…
We determine approximations to the decomposition matrices for unipotent $\ell$-blocks of several series of finite reductive groups of classical and exceptional type over $\FF_q$ of low rank in non-defining good characteristic~$\ell$.
We begin the study of character sheaves on a not necessarily connected reductive group, extending the known theory for connected groups.
We obtain two related characterizations of discrete quantum groups and discrete quantum groups of Kac type as allegorical group objects in the symmetric monoidal dagger category of quantum sets and relations, of interest to quantum…
For simple algebraic groups defined over algebraically closed fields of good characteristic, we give upper bounds on the covering numbers of unipotent conjugacy classes in terms of their (co)ranks and in terms of their dimensions.
The irreducible characters of a finite reductive group are partitioned into Harish-Chandra series that are labelled by cuspidal pairs. In this note, we describe how one can algorithmically calculate those cuspidal pairs using results of…
In this paper we give a complete classification of cyclically graded semisimple Lie algebras that afford cuspidal character sheaves and determine the support of the cuspidal character sheaves. This constitutes a major step towards the…
The local Euler obstructions and the Euler characteristics of linear sections with all hyperplanes on a stratified projective variety are key geometric invariants in the study of singularity theory. Despite their importance, in general it…
Let $F$ be a non-Archimedean local field and let $\mathcal{O}_{F}$ be its ring of integers. The orbit of an irreducible representation $\rho$ of $\mathrm{GL}_n(\mathcal{O}_F)$ is a conjugacy class in $\mathfrak{gl}_n(\mathcal{O}_F)$…
We give a uniform description of the decomposition of the unipotent variety of a classical group in arbitrary characteristic into pieces (considered in a non-uniform way in the earlier parts of this paper).
The local topological dynamics of subgroups of ${\rm Diff} ({\mathbb C^n}, 0)$, with special emphasis on ${\rm Diff} ({\mathbb C^2}, 0)$, is discussed with a view towards integrability questions. It is proved in particular that a subgroup…