Resolutions for Locally Analytic Representations

Abstract

The purpose of this paper is to study resolutions of locally analytic representations of a $p$-adic reductive group $G$. Given a locally analytic representation $V$ of $G$, we modify the Schneider-Stuhler complex (originally defined for smooth representations) so as to give an `analytic' variant ${\mathcal{S}}_{\bullet }^A(V)$. The representations in this complex are built out of spaces of analytic vectors $A_{\sigma }(V)$ for compact open subgroups $U_{\sigma }$, indexed by facets $\sigma$ of the Bruhat-Tits building of $G$. These analytic representations (of compact open subgroups of $G$) are then resolved using the Chevalley-Eilenberg complex from the theory of Lie algebras. This gives rise to a resolution ${\mathcal{S}}_{q,\bullet }^{\mathrm{CE}}(V)\to {\mathcal{S}}_q^A(V)$ for each representation ${\mathcal{S}}_q^A(V)$ in the analytic Schneider-Stuhler complex. In a last step we show that the family of representations ${\mathcal{S}}_{q,j}^{\mathrm{CE}}(V)$ can be given the structure of a Wall complex. The associated total complex ${\mathcal{S}}_{\bullet }^{\mathrm{CE}}(V)$ has then the same homology as that of ${\mathcal{S}}_{\bullet }^A(V)$. If the latter is a resolution of $V$, then one can use ${\mathcal{S}}_{\bullet }^{\mathrm{CE}}(V)$ to find a complex which computes the extension group ${\underline{Ext}}_G^n(V,W)$, provided $V$ and $W$ satisfy certain conditions which are satisfied when both are admissible locally analytic representations.