Twisted Dickson-Mui invariants and the Steinberg module multiplicity

Abstract

We determine the invariants, with arbitrary determinant twists, of the parabolic subgroups of the finite general linear group GL_n(q) acting on the tensor product of the symmetric algebra and the exterior algebra of the natural GL_n(q)-module V. In addition, we obtain the graded multiplicity of the Steinberg module of GL_n(q) in the tensor product of the symmetric algebra and the exterior algebra of V, twisted by an arbitrary determinant power.

v2, 15 pages, minor changes, some typos fixed and references added, to appear in Math. Proc. Camb. Phil. Soc