Symmetric subgroup invariants in irreducible representations of GF, when G=GLn

Using a general result of Lusztig, we give explicit formulas for the dimensions of KF-invariants in irreducible representations of GF, when G=GLn, F :G→G is a Frobenius map, and K is an F-stable subgroup of finite index in Gθ for some involution θ :G→G commuting with F. The proofs use some combinatorial facts about characters of symmetric groups.