Character sheaves on neutrally solvable groups
HTML articles powered by AMS MathViewer
- by Tanmay Deshpande PDF
- Represent. Theory 21 (2017), 534-589 Request permission
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 $F$.References
- Mitya Boyarchenko, Characters of unipotent groups over finite fields, Selecta Math. (N.S.) 16 (2010), no. 4, 857–933. MR 2734333, DOI 10.1007/s00029-010-0036-9
- Mitya Boyarchenko, Character sheaves and characters of unipotent groups over finite fields, Amer. J. Math. 135 (2013), no. 3, 663–719. MR 3068399, DOI 10.1353/ajm.2013.0023
- Mitya Boyarchenko and Vladimir Drinfeld, Character sheaves on unipotent groups in positive characteristic: foundations, Selecta Math. (N.S.) 20 (2014), no. 1, 125–235. MR 3147415, DOI 10.1007/s00029-013-0133-7
- Swarnendu Datta, Metric groups attached to skew-symmetric biextensions, Transform. Groups 15 (2010), no. 1, 72–91. MR 2600696, DOI 10.1007/s00031-010-9078-5
- Tanmay Deshpande, Heisenberg idempotents on unipotent groups, Math. Res. Lett. 17 (2010), no. 3, 415–434. MR 2653679, DOI 10.4310/MRL.2010.v17.n3.a4
- Tanmay Deshpande, Minimal idempotents on solvable groups, Selecta Math. (N.S.) 22 (2016), no. 3, 1613–1661. MR 3518560, DOI 10.1007/s00029-016-0229-y
- Tanmay Deshpande, Crossed $S$-matrices and character sheaves on unipotent groups, Adv. Math. 312 (2017), 64–106. MR 3635806, DOI 10.1016/j.aim.2017.03.013
- Tanmay Deshpande, Shintani descent for algebraic groups and almost characters of unipotent groups, Compos. Math. 152 (2016), no. 8, 1697–1724. MR 3542490, DOI 10.1112/S0010437X16007429
- Tanmay Deshpande, Modular categories, crossed S-matrices, and Shintani descent, Int. Math. Res. Not. IMRN 4 (2017), 967–999. MR 3658157, DOI 10.1093/imrn/rnw051
- Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, On braided fusion categories. I, Selecta Math. (N.S.) 16 (2010), no. 1, 1–119. MR 2609644, DOI 10.1007/s00029-010-0017-z
- Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik, On fusion categories, Ann. of Math. (2) 162 (2005), no. 2, 581–642. MR 2183279, DOI 10.4007/annals.2005.162.581
- Pavel Etingof, Dmitri Nikshych, and Victor Ostrik, Fusion categories and homotopy theory, Quantum Topol. 1 (2010), no. 3, 209–273. With an appendix by Ehud Meir. MR 2677836, DOI 10.4171/QT/6
- Ofer Gabber and François Loeser, Faisceaux pervers $l$-adiques sur un tore, Duke Math. J. 83 (1996), no. 3, 501–606 (French). MR 1390656, DOI 10.1215/S0012-7094-96-08317-9
- George Lusztig, Character sheaves. II, III, Adv. in Math. 57 (1985), no. 3, 226–265, 266–315. MR 806210, DOI 10.1016/0001-8708(85)90064-7
- Y.H. Liu. $t$-structures on tensor triangulated categories.
- Toshiaki Shoji, Shintani descent for algebraic groups over a finite field. I, J. Algebra 145 (1992), no. 2, 468–524. MR 1144943, DOI 10.1016/0021-8693(92)90113-Z
Additional Information
- Tanmay Deshpande
- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India
- MR Author ID: 900930
- Email: tanmaynd2001@gmail.com
- 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: https://doi.org/10.1090/ert/510
- MathSciNet review: 3733826