Character sheaves on neutrally solvable groups

Author:
Tanmay Deshpande

Journal:
Represent. Theory **21** (2017), 534-589

MSC (2010):
Primary 20C33

DOI:
https://doi.org/10.1090/ert/510

Published electronically:
December 8, 2017

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Abstract: Let be an algebraic group over an algebraically closed field of characteristic . In this paper we develop the theory of character sheaves on groups such that their neutral connected components are solvable algebraic groups. For such algebraic groups (which we call neutrally solvable) we will define the set of character sheaves on as certain special (isomorphism classes of) objects in the category of -equivariant -complexes (where we fix a prime ) on . We will describe a partition of the set into finite sets known as -packets and we will associate a modular category with each -packet of character sheaves using a truncated version of convolution of character sheaves. In the case where and is equipped with an -Frobenius we will study the relationship between -stable character sheaves on and the irreducible characters of (all pure inner forms of) . 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 -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 -stable -packets. Moreover, we will prove that the block in this transition matrix corresponding to any -stable -packet can be described as the crossed S-matrix associated with the auto-equivalence of the modular category induced by .

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Additional Information

**Tanmay Deshpande**

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

Email:
tanmaynd2001@gmail.com

DOI:
https://doi.org/10.1090/ert/510

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