Quadratic unipotent blocks in general linear, unitary and symplectic groups

TL;DR

This paper establishes a correspondence between quadratic unipotent characters of general linear, unitary, and symplectic groups, extending to certain blocks, revealing structural similarities across these groups.

Contribution

It proves a bijection between quadratic unipotent characters of GL, U, and Sp groups, and extends this to specific bbl-blocks, advancing understanding of their representation theory.

Findings

Bijection between quadratic unipotent characters of GL, U, and Sp groups.

Extension of correspondence to bbl-blocks for certain bbl.

Structural insights into unipotent characters across classical groups.

Abstract

An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element $s$ in a dual group such that $s^2=1$. We prove that there is a bijection between, on the one hand the set of quadratic unipotent characters of $GL(n,q)$ or $U(n,q)$ for all $n \geq 0$ and on the other hand, the set of quadratic unipotent characters of $Sp(2n,q)$ for all $n \geq 0$. We then extend this correspondence to $\ell$-blocks for certain $\ell$ not dividing $q$.