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 G be an algebraic group of classical type with defining characteristic p > 0 , μ a dominant weight and W the Weyl group of G. Let G = G ( q ) be a finite classical group, where q is a p-power. For a weight μ of G the sum s μ of distinct weights w ( μ ) with w ∈ 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 μ . The main case is where μ = ( q − 1 ) ω and ω is a fundamental weight of G. Let St denote the Steinberg character of G. Then we determine the unipotent characters occurring as constituents of s μ ⋅ S t defined to be 0 at the p-singular elements of G. Let β μ denote the Brauer character of a representation of S L n ( q ) arising from an irreducible representation of G with highest weight μ. Then we determine the unipotent constituents of the characters β μ ⋅ S t for μ = ( q − 1 ) ω , and also for some other μ (called strongly q-restricted). In addition, for strongly restricted weights μ, we compute the multiplicity of 1 T in the restriction β μ | T for every maximal torus T of G.