Multiplicities and Plancherel formula for the space of nondegenerate Hermitian matrices (Q2235722)

Let \(E/F\) be a quadratic extension of local fields. The space \(X=X_n\) of nondegenerate Hermitian matrices carries a natural action of \(G=\mathrm{GL}_n(E)\). Let \(G'=\mathrm{GL}_n(F)\) and let \(BC: \operatorname{Irr}(G') \to \operatorname{Irr}(G')\) be Arthur-Clozel's base-change map between the smooth duals of \(G'\) and \(G\). This paper contains two results concerning the spectral decomposition of \(X\). The first result is an an explicit Plancherel decomposition of \(L^2(X)\) as the direct integral of unitary \(G\)-representations \[ L^2(X) \simeq \int^\oplus_{\operatorname{Temp}(G')} BC(\sigma) d\mu_{G'}(\sigma), \] where \(\operatorname{Temp}(G')\subset \operatorname{Irr}(G')\) is the tempered dual of \(G'\) and \(d\mu_{G'}\) is the Plancherel measure for \(G'\). The second result is a formula for the multiplicities of generic representations in the \(p\)-adic case that extends previous work of Feigon-Lapid-Offen [\textit{B. Feigon} et al., Publ. Math., Inst. Hautes Étud. Sci. 115, 185--323 (2012; Zbl 1329.11053)]. For the proof, the author develops the local analogue of the Kuznetsov trace formula for an arbitrary quasi-split group and also the relative Kuznetsov trace formula for \(X\).