On the irreducibility of locally analytic principal series representations

Abstract:

Let $\mathbf {G}$ be a $p$-adic connected reductive group with Lie algebra $\mathfrak {g}$. For a parabolic subgroup $\mathbf {P} \subset \mathbf {G}$ and a finite-dimensional locally analytic representation $V$ of a Levi subgroup of $\mathbf {P}$, we study the induced locally analytic $\mathbf {G}$-representation $W = \operatorname {Ind}_{\mathbf {P}}^{\mathbf {G}}(V)$. Our result is the following criterion concerning the topological irreducibility of $W$: If the Verma module $U(\mathfrak {g}) \otimes _{U(\mathfrak {p})} V’$ associated to the dual representation $V’$ is irreducible, then $W$ is topologically irreducible as well.