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Algebras of $$p$$-adic distributions and admissible representations. (English) Zbl 1028.11070
The authors pursue an extensive research project on locally analytic representations of a locally $$L$$-analytic group $$G$$, where $$L$$ is a $$p$$-adic field [Represent. Theory 5, 111-128 (2001; Zbl 1028.17007); J. Am. Math. Soc. 15, 443-468 (2002; Zbl 1028.11071); Isr. J. Math. 127, 359-380 (2002; Zbl 1006.46053); Doc. Math., J. DMV 6, 447-481 (2001; Zbl 1028.11068)]. To obtain an operative theory of such representations, they introduce an additional “finiteness” condition called admissibility. The admissible locally analytic representations form an Abelian category. When $$G$$ is the group of $$L$$-points of an algebraic group, this category includes many important examples, such as the principal series representations, the finite dimensional algebraic representations and the smooth representations of Langlands theory. The approach is algebraic, via the algebra $$D(G,K)$$ of locally analytic distributions on $$G$$. Most of the paper is devoted to studying the structure and properties of the algebra $$D(G,K)$$. The results are applied to representation theory. It is shown, among other results, that all smooth admissible representations in the sense of Langlands theory are admissible locally analytic representations.

##### MSC:
 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 22E50 Representations of Lie and linear algebraic groups over local fields 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis