  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.
