Stable spectrum for pseudo-Riemannian locally symmetric spaces (Q627264)

Let \(X=G/H\) be a reductive symmetric space, where \(G\) is a connected non-compact reductive linear Lie group and \(H=(G^\sigma)_0\) is the identity component of the set of fixed points of \(G\) under some involutive automorphism \(\sigma\). The space \(X\) carries a \(G\)-invariant pseudo-Riemannian metric which is induced by the Killing form of the Lie algebra \(\mathfrak g\) of \(G\) if \(G\) is semi-simple. A Clifford-Klein form \(X_\Gamma\) of \(X\) is the quotient space \(X_\Gamma=\Gamma\setminus X\), where \(\Gamma\) is a discrete subgroup of \(G\) acting properly discontinuously and freely on \(X\). Any \(G\)-invariant differential operator \(D\) on \(X\) induces a differential operator \(D_\Gamma\) on \(X_\Gamma\), and the map \(D\mapsto D_\Gamma\) is a monomorphism from the ring \(\mathbb D(X)\) of \(G\)-invariant differential operators on \(X\) into the ring \(\mathbb D(X_\Gamma)\) of differential operators on \(X_\Gamma\). The discrete spectrum \(\text{Spec}_d(X_\Gamma)\) on \(X_\Gamma\) is the set of algebra homomorphisms \(\lambda:\mathbb D(X)\to \mathbb C\) such that the set \(L^2(X_\Gamma,{\mathcal M}_\lambda)\) of weak solutions \(f\in L^2(X_\Gamma)\) to the system \(D_\Gamma f=\lambda(D)f\) for all \(D\in\mathbb D(X)\) is non-trivial. In this paper, the authors show how to construct elements of \(L^2(X_\Gamma,{\mathcal M}_\lambda)\), that is, joint eigenfunctions on \(X_\Gamma\) corresponding to \(\text{Spec}_d(X_\Gamma)\), and try to understand the behavior of \(\text{Spec}_d(X_\Gamma)\) under small deformations of \(\Gamma\) in \(G\). Let \(\theta\) be a Cartan involution of \(G\) commuting with \(\sigma\) and let \(K=G^\sigma\) be the corresponding maximal compact subgroup of \(G\) with the Lie algebra \(\mathfrak k\). The authors prove that if \(\operatorname{rank}G/H = \operatorname{rank}K/K \cap H\), then the discrete spectrum \(\text{Spec}_d(X_\Gamma)\) on \(X_\Gamma\) is infinite for any standard compact Clifford-Klein form \(X_\Gamma\) of \(X\). Also, they show that there is an infinite subset of \(\text{Spec}_d(X_\Gamma)\) that is stable under small deformations of \(\Gamma\) in \(G\).