  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 $G$.
