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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Character sheaves on neutrally solvable groups
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by Tanmay Deshpande
Represent. Theory 21 (2017), 534-589
Published electronically: December 8, 2017


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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Bibliographic Information
  • Tanmay Deshpande
  • Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India
  • MR Author ID: 900930
  • Email:
  • Received by editor(s): July 20, 2016
  • Received by editor(s) in revised form: September 20, 2017
  • Published electronically: December 8, 2017
  • © Copyright 2017 American Mathematical Society
  • Journal: Represent. Theory 21 (2017), 534-589
  • MSC (2010): Primary 20C33
  • DOI:
  • MathSciNet review: 3733826