SOME CONSEQUENCES OF HARISH-CHANDRA SUBMERSION PRINCIPLE

Let G be a reductive p-adic group, K a good maximal compact subgroup, K1 subset-of K any open subgroup, and pi an admissible representation of G of finite type. In A submersion principle and its applications, H arish-Chandra proves the theorem that f(K)(pi)(kgk-1) dk is a finite-r ank operator for g in the regular set G' in order to show that the cha racter THETA(pi)(g) is a locally constant class function on G'. From t his, the authors derive the formula theta(1)THETA(g) = d(pi) integral( G/Z) integral(k1) theta(xkgk-1 x-1) dk dk (G is-an-element-of G') for any K-finite matrix coefficient theta of a discrete series representat ion pi with formal degree d(pi). They use another technical result of the paper to prove that invariant integrals of Schwartz space function s converge absolutely. None of these results depends upon a characteri stic zero assumption.