ADVANCES IN MATHEMATICS 94, 24CL255 (1992) Extensions of Representations of p-adic Nilpotent Groups S. GELFAND* AND D. KAZHDAN+ Department of Mathematics, Harvard University, Science Center, One Oxford Street, Cambridge, Massachusetts 02138 1. INTRODUCTION 1. Let E be a local field of characteristic 0 and G be the group of E-rational points of a connected algebraic nilpotent group over E. We study in this paper extensions of complex algebraic representations of G. For two irreducible algebraic (see [BZ], or Definition 2.5 below) representations rrr and rcn, of G (we allow rrr to be equivalent to rr2) one can rather easily prove that Ext’(rrr , rr2) = 0 (a similar result for real nilpotent Lie groups was proved by F. du Cloux [Cl). This implies that Ext”(rc,, 7~~) = 0 for all n > 0 and for all representations rtr and 7~~ of finite length. However, if rcr and/or rr2 are not of finite length, non-trivial exten- sions can exist. As one of the simplest examples, take G to be the additive group of E and K to be the representation of G by translations in the space V= CF( G) of locally constant functions on G with compact support: rr(a)f(x) =f(x + a), UE G, fe V. Let V, be the invariant subspace of V formed by functions with zero integral over E: V0 = {f e V, JEf(x) dx = O}. Then V,, is an invariant subspace of V and rt is a non-trivial extension of the representation rcO in V, by the identity (one-dimensional) representa- tion. Our main result gives, for two representations 7c1and rc2 of G, sufficient conditions for the vanishing of all groups Ext”(n,, rc2), n 2 0. The best way to formulate these conditions is through an analog for G of Kirillov’s orbit theory (see [M], [GK], [HI). 2. Let g be the Lie algebra of G. Since the group G is nilpotent, the exponential mapping exp: g + G is a homeomorphism. Let Ad be the adjoint action of G on g. Considering g as a commutative topological group under addition, denote by g* the Pontryagin dual to g. For a totally * Supported during his stay at MIT by A. P. Sloan Foundation Grant 88-l-l. + Partially supported by an NSF grant. 240 OOOl-8708/92 $9.00 Copyright 0 1992 by Academic Press. Inc. All rights of reproduction in any form reserved.