Locally analytic distributions and 𝑝-adic representation theory, with applications to 𝐺𝐿₂

Source abstract

In this paper we study continuous representations of locally L L -analytic groups G G in locally convex K K -vector spaces, where L L is a finite extension of Q p \mathbb {Q}_p and K K is a spherically complete nonarchimedean extension field of L L . The class of such representations includes both the smooth representations of Langlands theory and the finite dimensional algebraic representations of G G , along with interesting new objects such as the action of G G on global sections of equivariant vector bundles on p p -adic symmetric spaces. We introduce a restricted category of such representations that we call “strongly admissible” and we show that, when G G is compact, our category is anti-equivalent to a subcategory of the category of modules over the locally analytic distribution algebra of G G . As an application we prove the topological irreducibility of generic members of the p p -adic principal series for G L 2 ( Q p ) GL_2(\mathbb {Q}_p) . Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous K K -valued representations of locally L L -analytic groups.