CHARACTER DEGREES OF GROUPS ASSOCIATED WITH FINITE SPLIT BASIC ALGEBRAS WITH INVOLUTION

Abstract

Let $𝒜$ be a finite-dimensional split basic algebra over a finite field $\mathbb{k}$ with odd characteristic, and assume that $𝒜$ is endowed with an involution $\sigma :𝒜\to 𝒜$. We determine the degrees of the irreducible characters of the group $C_G\left(\sigma \right)=\left\{x\in G:\sigma \left(x^{-1}\right)=x\right\}$ where $G={𝒜}^{\times }$ is the unit group of $𝒜$, and prove that every irreducible character of $C_G\left(\sigma \right)$ is induced by a linear character of some subgroup. As a particular case, our results hold for the Sylow $p$-subgroups of the finite classical groups of Lie type, and extend (in a uniform way) the results obtained by B. Szegedy in [11].