Please use this identifier to cite or link to this item: http://dr.iiserpune.ac.in:8080/xmlui/handle/123456789/8402
Title: Prasad’s Conjecture About Dualizing Involutions
Authors: AROTE, PRASHANT
MISHRA, MANISH
Dept. of Mathematics
Keywords: Reductive groups
Characters
Representations
2024-JAN-WEEK1
TOC-JAN-2024
2024
Issue Date: May-2024
Publisher: Oxford University Press
Citation: International Mathematics Research Notices, 2024(09), 7700–7720.
Abstract: Let $G$ be a connected reductive group defined over a finite field ${\mathbb{F}}_{q}$ with corresponding Frobenius $F$. Let $\iota _{G}$ denote the duality involution defined by D. Prasad under the hypothesis $2\textrm{H}<^>{1}(F,Z(G))=0$, where $Z(G)$ denotes the center of $G$. We show that for each irreducible character $\rho $ of $G<^>{F}$, the involution $\iota _{G}$ takes $\rho $ to its dual $\rho <^>{\vee }$ if and only if for a suitable Jordan decomposition of characters, an associated unipotent character $u_{\rho }$ has Frobenius eigenvalues $\pm $ 1. As a corollary, we obtain that if $G$ has no exceptional factors and satisfies $2\textrm{H}<^>{1}(F,Z(G))=0$, then the duality involution $\iota _{G}$ takes $\rho $ to its dual $\rho <^>{\vee }$ for each irreducible character $\rho $ of $G<^>{F}$. Our results resolve a finite group counterpart of a conjecture of D. Prasad.
URI: https://doi.org/10.1093/imrn/rnad296
http://dr.iiserpune.ac.in:8080/xmlui/handle/123456789/8402
ISSN: 1073-7928
1687-0247
Appears in Collections:JOURNAL ARTICLES

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