# Invariants of maximal tori and unipotent constituents of some quasi-projective characters for finite classical groups

## TL;DR

This paper analyzes the decomposition of certain characters of finite classical groups, focusing on orbit characters, their multiplicities on maximal tori, and the unipotent constituents of their products with the Steinberg character.

## Contribution

It provides explicit computations of multiplicities and unipotent constituents for orbit and Brauer characters associated with specific weights in finite classical groups.

## Key findings

- Computed multiplicities of trivial characters in orbit characters for all maximal tori.
- Determined unipotent constituents of characters formed by orbit characters times the Steinberg character.
- Analyzed the structure of characters arising from strongly q-restricted weights.

## Abstract

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let ${\mathbf G}$ be an algebraic group of classical type with defining characteristic $p>0$, $\mu$ a dominant weight and $W$ the Weyl group of ${\mathbf G}$. Let $G=G(q)$ be a finite classical group, where $q$ is a $p$-power. For a weight $\mu$ of ${\mathbf G}$ the sum $s_\mu$ of distinct weights $w(\mu)$ with $w\in W$ viewed as a function on the semisimple elements of $G$ is known to be a generalized Brauer character of $G$ called an orbit character of $G$. We compute, for certain orbit characters and every maximal torus $T$ of $G$, the multiplicity of the trivial character $1_T$ of $T$ in $s_\mu$. The main case is where $\mu=(q-1)\omega$ and $\omega$ is a fundamental weight of ${\mathbf G}$. Let $St$ denote the Steinberg character of $G$. Then we determine the unipotent characters occurring as constituents of $s_\mu\cdot St$ defined to be 0 at the $p$-singular elements of $G$. Let $\beta_\mu$ denote the Brauer character of a representation of $SL_{n}(q)$ arising from an irreducible representation of ${\mathbf G}$ with highest weight $\mu$. Then we determine the unipotent constituents of the characters $\beta_\mu\cdot St$ for $\mu=(q-1)\omega$, and also for some other $\mu$ (called strongly $q$-restricted). In addition, for strongly restricted weights $\mu$, we compute the \mult of $1_T$ in the restriction $\beta_\mu|_T$ for every maximal torus $T$ of $G$.