Locally analytic distributions and $p$-adic representation theory, with applications to $GL_{2}$

Abstract:

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