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

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 $G$.