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Representation Theory

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Character sheaves on neutrally solvable groups

Author: Tanmay Deshpande
Journal: Represent. Theory 21 (2017), 534-589
MSC (2010): Primary 20C33
Published electronically: December 8, 2017
MathSciNet review: 3733826
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Abstract: Let $ G$ be an algebraic group over an algebraically closed field $ \mathtt {k}$ of characteristic $ p>0$. In this paper we develop the theory of character sheaves on groups $ G$ such that their neutral connected components $ G^\circ $ are solvable algebraic groups. For such algebraic groups $ G$ (which we call neutrally solvable) we will define the set $ \operatorname {CS}(G)$ of character sheaves on $ G$ as certain special (isomorphism classes of) objects in the category $ \mathscr {D}_G(G)$ of $ G$-equivariant $ \overline {\mathbb{Q}}_{\ell }$-complexes (where we fix a prime $ \ell \neq p$) on $ G$. We will describe a partition of the set $ \operatorname {CS}(G)$ into finite sets known as $ \mathbb{L}$-packets and we will associate a modular category $ \mathscr {M}_L$ with each $ \mathbb{L}$-packet $ L$ of character sheaves using a truncated version of convolution of character sheaves. In the case where $ \mathtt {k}=\overline {\mathbb{F}}_q$ and $ G$ is equipped with an $ \mathbb{F}_q$-Frobenius $ F$ we will study the relationship between $ F$-stable character sheaves on $ G$ and the irreducible characters of (all pure inner forms of) $ G^F$. In particular, we will prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for neutrally solvable groups and that these almost characters coincide with the ``trace of Frobenius'' functions associated with $ F$-stable character sheaves. We will also prove that the matrix relating the irreducible characters and almost characters is block diagonal where the blocks on the diagonal are parametrized by $ F$-stable $ \mathbb{L}$-packets. Moreover, we will prove that the block in this transition matrix corresponding to any $ F$-stable $ \mathbb{L}$-packet $ L$ can be described as the crossed S-matrix associated with the auto-equivalence of the modular category $ \mathscr {M}_L$ induced by $ F$.

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Tanmay Deshpande
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India

Received by editor(s): July 20, 2016
Received by editor(s) in revised form: September 20, 2017
Published electronically: December 8, 2017
Article copyright: © Copyright 2017 American Mathematical Society