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 ${\mathcal S}^A_\bullet(V)$. The representations in this complex are built out of spaces of analytic vectors $A_σ(V)$ for compact open subgroups $U_σ$, indexed by facets $σ$ 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 ${\mathcal S}^{\rm CE}_{q,\bullet}(V) \rightarrow {\mathcal S}^A_q(V)$ for each representation ${\mathcal S}^A_q(V)$ in the analytic Schneider-Stuhler complex. In a last step we show that the family of representations ${\mathcal S}^{\rm CE}_{q,j}(V)$ can be given the structure of a Wall complex. The associated total complex ${\mathcal S}^{\rm CE}_\bullet(V)$ has then the same homology as that of ${\mathcal S}^A_\bullet(V)$. If the latter is a resolution of $V$, then one can use ${\mathcal S}^{\rm CE}_\bullet(V)$ to find a complex which computes the extension group $\underline{Ext}^n_G(V,W)$, provided $V$ and $W$ satisfy certain conditions which are satisfied when both are admissible locally analytic representations.