RELATIVE KLOOSTERMAN INTEGRALS FOR GL(3) .2.

Let G' be a quasi-split reductive group over a local field F, f' the characteristic function of a maximal compact subgroup K' of G', N' a maximal unipotent subgroup of G'. We consider the orbits of maximal dimension for the action of N' x N' on G' and the weighted orbital integral of f' on such an orbit, the weight being a generic character. The resulting integral, we call a Kloosterman integral. A relative version of this construction is to consider a symmetric space S associated to a quasi-split group G, a maximal unipotent subgroup N of G, a maximal compact K of G and the orbits of maximal dimension for the action of N on S. The weighted orbital integral of the characteristic function f of K and S on such an orbit is what we call a relative Kloosterman integral; the weight is an appropriate character of N. We conjecture that a relative Kloosterman integral is actually a Kloosterman integral for an appropriate group G'. We prove the conjecture in a simple case: E is an unramified quadratic extension of F, G is GL(3, E), S is the set of 3 x 3 matrices s such that ssBAR = 1; the group G' is then the quasi-split unitary group in three variables.