Resolutions of tempered representations of reductive \(p\)-adic groups (Q393072)

Let \(G\) be the group of rational points of a connected reductive group over a non-archimedean local field \(F\), and suppose that \(V\) is a smooth, complex representation of \(G\). Then one can view \(V\) as a non-degenerate module over the Hecke algebra \(\mathcal{H}(G)\). If \(V\) is admissible, then it is tempered if and only if it extends (necessarily uniquely) to a module over the Schwartz algebra \(\mathcal{S}(G)\). Following \textit{P. Schneider} and \textit{U. Stuhler} [Publ. Math., Inst. Hautes Étud. Sci. 85, 97--191 (1997; Zbl 0892.22012)], one can thus view the category \(\text{Mod-}\mathcal{S}(G)\) of \(\mathcal{S}(G)\)-modules as a category of tempered representations. Moreover, regarding \(\mathcal{S}(G)\) as an LF-algebra, \(V\) has a canonical structure as an LF-space, making it a complete topological \(\mathcal{S}(G)\)-module. The authors introduce an exact category \(\text{Mod}_{\text{LF}}(\mathcal{S}(G))\) that respects this structure, and that contains all admissible \(\mathcal{S}(G)\)-modules.NEWLINENEWLINEAn abbreviated form of the main result is that if \(V\) and \(W\) are admissible representations of \(G\), then \(\text{Ext}^n(V,W)\) doesn't depend on whether one works in the category \(\text{Mod-}\mathcal{H}(G)\), \(\text{Mod-}\mathcal{S}(G)\), or \(\text{Mod}_{\text{LF}}(\mathcal{S}(G))\).NEWLINENEWLINESome of the results already appear in works of \textit{R. Meyer} [in: Noncommutative geometry and number theory. Where arithmetic meets geometry and physics. Based on two workshops, Bonn, Germany, August 2003 and June 2004. Wiesbaden: Vieweg. 263--300 (2006; Zbl 1111.22016)] and of \textit{P. Schneider} and \textit{E.-W. Zink} [J. Inst. Math. Jussieu 6, No. 4, 639--688 (2007; Zbl 1126.22008); Geom. Funct. Anal. 17, No. 6, 2018--2065 (2008; Zbl 1144.22012)]. However, the proofs here are new, relying on an explicit construction of a continuous contraction of the Schneider-Stuhler resolutions.NEWLINENEWLINEAlthough the main results concern complex representations of \(G\), much of the machinery is valid for representations over any unital ring in which the residual characteristic of \(F\) is invertible.NEWLINENEWLINEFinally, the authors show that all of their results extend to nonconnected groups that satisfy a mild technical condition (Condition 6.3), imposed to overcome the fact that the enlarged Bruhat-Tits building of \(G\) has no natural polysimplicial structure. There is a minor misprint in the paragraphs before and after this condition: Every instance of \( X_* ( G^\circ / Z(G^\circ) ) \) should actually be \( X_* ( Z(G^\circ) ) \).