Symmetric spaces over finite fields, Frobenius-Schur indices, and symmetric function identities

We give a method to produce Hall-Littlewood symmetric function identities from symmetric spaces over finite fields. A special case of such identities motivates to introduce a notion of q-Frobenius-Schur index of an irreducible character of a finite reflection group G. In the case G is an imprimitive complex reflection group, we calculate the indices explicitly using a new q-Cauchy identity fur Schur functions.