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 sμ 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 1T of T in sμ. 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 sμ⋅St defined to be 0 at the p-singular elements of G. Let βμ denote the Brauer character of a representation of SLn(q) arising from an irreducible representation of G with highest weight μ. Then we determine the unipotent constituents of the characters βμ⋅St 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 1T in the restriction βμ|T for every maximal torus T of G.