Alvis-Curtis duality, central characters, and real-valued characters

TL;DR

This paper proves that Alvis-Curtis duality preserves Frobenius-Schur indicators in certain groups, extending previous results and enabling computation of these indicators for specific characters of finite groups of Lie type.

Contribution

It establishes the invariance of Frobenius-Schur indicators under Alvis-Curtis duality for connected reductive groups with connected center, extending prior work and applying it to compute indicators for various characters.

Findings

Alvis-Curtis duality preserves Frobenius-Schur indicators.

Extended Prasad's result relating indicators to central characters.

Computed indicators for specific real-valued characters of finite groups.

Abstract

We prove that Alvis-Curtis duality preserves the Frobenius-Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius-Schur indicator of a regular real-valued character to its central character. We apply these results to compute the Frobenius-Schur indicators of certain real-valued, irreducible, Frobenius-invariant Deligne-Lusztig characters, and the Frobenius-Schur indicators of real-valued regular and semisimple characters of finite unitary groups.