Algebras of p-adic distributions and admissible representations

Abstract

Let G be a compact, locally L-analytic group, where L is a finite extension of Qp. Let K be a discretely valued extension field of L. We study the algebra D(G,K) of K-valued locally analytic distributions on G, and apply our results to the locally analytic representation theory of G in vector spaces over K. Our objective is to lay a useful and powerful foundation for the further study of such representations. We show that the noncommutative, nonnoetherian ring D(G,K) "behaves" like the ring of functions on a rigid Stein space, and that (at least when G is Qp-analytic) it is a faithfully flat extension of its subring K\otimes Zp[[G]], where Zp[[G]] is the completed group ring of G. We use this point of view to describe an abelian subcategory of D(G,K) modules that we call coadmissible. We say that a locally analytic representation V of G is admissible if its strong dual is coadmissible as D(G,K)-module. For noncompact G, we say V is admissible if its strong dual is coadmissible as D(H,K) module for some compact open subgroup H. In this way we obtain an abelian category of admissible locally analytic representations. These methods allow us to answer a number of questions raised in our earlier papers on p-adic representations; for example we show the existence of analytic vectors in the admissible Banach space representations of G that we studied in "Banach space representations ...", Israel J. Math. 127, 359-380 (2002). Finally we construct a dimension theory for D(G,K), which behaves for coadmissible modules like a regular ring, and show that smooth admissible representations are zero dimensional.