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

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 $S{L}_n(q)$ 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 ${\beta }_{\mu }{\mathrm{\mid }}_T$ for every maximal torus $T$ of $G$.