Cuspidal representations of reductive \(p\)-adic groups are relatively injective and projective (Q2792390)

Let \(G\) be a reductive \(p\)-adic group and \(R\) a discrete commutative ring containing \(p^{-1}\). The author shows that cuspidal \(R\)-linear representations of \(G\) are relatively injective and projective in \(\mathfrak{Rep}_R(G)\), in the sense that the functors \(\mathrm{Hom}_{R,G}(-,V)\) and \(\mathrm{Hom}_{R,G}(V,-)\) are exact for cuspidal \(V\) on the class of exact sequences in \(\mathfrak{Rep}_R(G)\) which become split upon restriction to \(\mathfrak{Rep}_R(U)\) for all compact-modulo-center subgroups \(U\) of \(G\). The proof proceeds by realizing \(V\) as a direct summand of relatively injective and projective representations of \(V\)-valued functions on the affine Bruhat--Tits building of \(G\), the construction uses the support projections defined in [the author and \textit{M. Solleveld}, J. Reine Angew. Math. 647, 115--150 (2010; Zbl 1210.22012)]. Specializing to the case when \(R\) is a field with characteristic \(\ell\) not dividing the order of any finite quotient of \(U\), one recovers classical results of Casselman for \(R=\mathbb{C}\) and Vignéras for \(\ell\neq0\), which leads to an orthogonal decomposition of \(\mathfrak{Rep}_R(G)\) à la Bernstein.