A remark on a trace Paley-Wiener theorem (Q827448)

Let \(G\) denote a reductive \(p\)-adic group and let \(\mathrm{Rep}(G)\) denote the category of admissible complex representations of finite length of \(G\). Let \(\mathrm{Rep}_t(G)\) denote the full subcategory of \(\mathrm{Rep}(G)\) consisting of representations having all irreducible subquotients tempered and let \(R_t(G)\) denote the corresponding Grothendieck group. Let us denote by \(R^i_t(G)\) the subgroup of \(R_t(G)\) generated by all parabolically induced representations of the form \(i_{GM}(\sigma)\), where \(M\) is the standard Levi subgroup of \(G\) and \(\sigma\) is a discrete series representation. Let \(f : R_t(G) \rightarrow \mathbb{C}\) be a \(\mathbb{Z}\)-linear form such that there is an open compact subgroup of \(G\) which dominates \(f\) and for each standard maximal Levi subgroup \(M\) and a discrete series \(\sigma\) of \(M\), the function \(\psi \mapsto f(i_{GM}(\psi \sigma))\) is regular on the unitary group of unramified characters of \(G\), and for any other proper standard Levi subgroup \(N\) and a discrete series \(\tau\) of \(N\) we have \(f(i_{GN}(\tau)) = 0\). The author proves that then there exists \(F \in C^{\infty}_c(G)\) such that \(f(\pi) = \mathrm{tr}(\pi(F))\) for all \(\pi \in R^i_t(G)\). This version of the trace Paley-Wiener theorem is then used to study the Plancherel measure as an invariant of the \(L\)-packet of a discrete series representation.