Stability in the high-dimensional cohomology of congruence subgroups

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

Twisted Dickson–Mui invariants and the Steinberg module multiplicity

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