K^F-invariants in irreducible representations of G^F, when G=GL_n

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