On the asymptotic support of Plancherel measures for homogeneous spaces (Q6634759)

Let \(G\) be a connected, complex reductive group, \(\sigma\) an antiholomorphic involution and let \((G^{\sigma})_e \subseteq G_{\mathbb{R}} \subseteq G^{\sigma}\) be a real form of \(G\). We let \(H \subseteq G\) be a closed, \(\sigma\)-stable, complex algebraic subgroup and \(H_{\mathbb{R}} := H^{\sigma} \cap G_{\mathbb{R}} \subseteq H\). Let \(X_0 := G_{\mathbb{R}} / H_0\) and assume that \(X_0\) admits a nonzero, \(G_{\mathbb{R}}\)-invariant density \(\nu\). The authors study the unitary action of \(G_{\mathbb{R}}\) on the space \(L^2(X_0)\), which admits a decomposition \[L^2(X_0) \simeq \int_{\hat{G}_{\mathbb{R}}}^{\oplus} \pi^{\oplus n(\pi)} dm.\] Here \(\hat{G}_{\mathbb{R}}\) is the unitary dual of \(G_{\mathbb{R}}\). Now \(\mathrm{supp} \ L^2(X_0) \subseteq \hat{G}_{\mathbb{R}}\), the support of \(L^2(X_0)\), is the smallest closed subset of \(\hat{G}_{\mathbb{R}}\) satisfying \(m(\hat{G}_{\mathbb{R}} \ \backslash \ \mathrm{supp} \ L^2(X_0)) = 0\).\N\NThe explicit form of the above decomposition is the Plancherel formula, which has been studied in many contexts. The authors study the behaviour of \(\mathrm{supp} \ L^2(X_0)\) in this paper, and more specifically the asymptotic behaviour. They relate it to the moment map \(\mu: T^*(X_0) \rightarrow \mathfrak{g}^*\) and describe what ``most'' representation look like using coadjoint orbits, and in particular that they arise from a constrution with coadjoint orbits. In a corollary they characterize when \(L^2(X_0)\) has a discrete series representation.\N\NIn particular, there are representations which arise from the quantization of coadjoint orbits \(\mathcal{O}\). Let the asymptotic cone \(\mathrm{AC}(S)\) of \(S\) in \(Z\) be the union of \(\{0\}\) and the elements \(\xi \in Z\) such that \(S \cap C\) is unbounded for any conic neighbourhood \(C\) of \(\xi\). The authors prove that\N\begin{itemize}\N\item[1.] The asymptotic cone \(\mathrm{AC} \left( \bigcup_{\pi \in \mathrm{supp} \ L^2(X_0) \mathrm{ \ from \ a \ coadjoint \ orbit}} \chi_{\pi} \right)\) in \(\mathfrak{j}^*\) a specific Cartan subalgebra of \(\mathfrak{g}\) contains a real semialgebraic set with fixed real dimension \(a\).\N\item[2.] The asymptotic cone \(\mathrm{AC} \left( \bigcup_{\pi \in \mathrm{supp} \ L^2(X_0) \mathrm{ \ not \ from \ a \ coadjoint \ orbit}} \chi_{\pi} \right)\) in \(\mathfrak{j}^*\) is contained in a real algebraic variety with real dimension strictly less than \(a\).\N\end{itemize}\N\NFurthermore, they prove that if the elliptic elements of \(\mu(T^*X_0)\) contain a nonempty open subset of \(\mu(T^*X_0)\) then \(X_0\) has a discrete series. Here we say an element \(\xi \in \mathfrak{g}_{\mathbb{R}}^*\) is elliptic if there exists a Cartan involution \(\theta\) such that \(\theta(\xi) = \xi\).