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Register a new userCharacter sheaves on unipotent groups in positive characteristic: foundations
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In this article we formulate and prove the main theorems of the theory of character sheaves on unipotent groups over an algebraically closed field of characteristic p>0. In particular, we show that every admissible pair for such a group G gives rise to an L-packet of character sheaves on G, and that, conversely, every L-packet of character sheaves on G arises from a (non-unique) admissible pair. In the appendices we discuss two abstract category theory patterns related to the study of character sheaves. The first appendix sketches a theory of duality for monoidal categories, which generalizes the notion of a rigid monoidal category and is close in spirit to the Grothendieck-Verdier duality theory. In the second one we use a topological field theory approach to define the canonical braided monoidal structure and twist on the equivariant derived category of constructible sheaves on an algebraic group; moreover, we show that this category carries an action of the surface operad. The third appendix proves that the naive definition of the equivariant constructible derived category with respect to a unipotent algebraic group is equivalent to the correct one.
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Let $k$ be an algebraically closed field of characteristic $p > 2$, let $n in mathbb Z_{>0} $, and take $G$ to be one of the classical algebraic groups $mathrm{GL}_n(k)$, $mathrm{SL}_n(k)$, $mathrm{Sp}_n(k)$, $mathrm O_n(k)$ or $mathrm{SO}_n(k)$, with $mathfrak g = operatorname{Lie} G$. We determine the maximal $G$-stable closed subvariety $mathcal V$ of the nilpotent cone $mathcal N$ of $mathfrak g$ such that the $G$-orbits in $mathcal V$ are in bijection with the $G$-orbits of $mathfrak{sl}_2$-triples $(e,h,f)$ with $e,f in mathcal V$. This result determines to what extent the theorems of Jacobson--Morozov and Kostant on $mathfrak{sl}_2$-triples hold for classical algebraic groups over an algebraically closed field of ``small odd characteristic.
This paper concerns character sheaves of connected reductive algebraic groups defined over non-Archimedean local fields and their relation with characters of smooth representations. Although character sheaves were devised with characters of representations of finite groups of Lie type in mind, character sheaves are perfectly well defined for reductive algebraic groups over any algebraically closed field. Nevertheless, the relation between character sheaves of an algebraic group $G$ over an algebraic closure of a field $K$ and characters of representations of $G(K)$ is well understood only when $K$ is a finite field and when $K$ is the field of complex numbers. In this paper we consider the case when $K$ is a non-Archimedean local field and explain how to match certain character sheaves of a connected reductive algebraic group $G$ with virtual representations of $G(K)$. In the final section of the paper we produce examples of character sheaves of general linear groups and matching admissible virtual representations.
We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $mathbb k$ under the assumption that the characteristic $ell$ of $mathbb k$ is rather good for $G$, i.e., $ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautners characteristic-$ell$ `cleanness conjecture; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.
We establish a Springer correspondence for classical symmetric pairs making use of Fourier transform, a nearby cycle sheaf construction and parabolic induction. In particular, we give an explicit description of character sheaves for classical symmetric pairs.
We consider an analog of the problem Veblen formulated in 1928 at the IMC: classify invariant differential operators between natural objects (spaces of either tensor fields, or jets, in modern terms) over a real manifold of any dimension. For unary operators, the problem was solved by Rudakov (no nonscalar operators except the exterior differential); for binary ones, by Grozman (there are no operators of orders higher than 3, operators of order 2 and 3 are, bar an exception in dimension 1, compositions of order 1 operators which, up to dualization and permutation of arguments, form 8 families). In dimension one, Grozman discovered an indecomposable selfdual operator of order 3 that does not exist in higher dimensions. We solve Veblens problem in the 1-dimensional case over the ground field of positive characteristic. In addition to analogs of the Berezin integral (strangely overlooked so far) and binary operators constructed from them, we discovered two more (up to dualization) types of indecomposable operators of however high order: analogs of the Grozman operator and a completely new type of operators.
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