Resolutions for Locally Analytic Representations

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 \(A_\sigma(V)\) 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 \({\mathcal S}^{\rm CE}_{q,\bullet}(V) \rightarrow {\mathcal S}^A_q(V)\) 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.