Dimensions of some locally analytic representations

TL;DR

This paper calculates the dimensions of specific locally analytic representations of p-adic groups, including principal series and those from line bundles, with applications to Colmez's unitary principal series of GL2(Qp).

Contribution

It provides explicit dimension formulas for classes of locally analytic representations, including new results for representations associated with p-adic symmetric spaces.

Findings

Dimensions of principal series representations are computed.

Dimensions of certain homogeneous line bundle representations are determined.

Application to Colmez's unitary principal series of GL2(Qp) is demonstrated.

Abstract

Let $G$ be the group of points of a split reductive group over a finite extension of ${\mathbb Q}_p$. In this paper, we compute the dimensions of certain classes of locally analytic $G$-representations. This includes principal series representations and certain representations coming from homogeneous line bundles on $p$-adic symmetric spaces. As an application, we compute the dimensions in Colmez' unitary principal series of ${\rm GL}_2({\mathbb Q}_p)$.