Tempered orbital measures and asymptotic expansion (Q1902309)

Let \(G\) be an irreducible algebraic group defined over \(\mathbb{R}\), \(X\) an algebraic variety defined over \(\mathbb{R}\). Assume there is a regular \(G\)-action on \(X\). \(X(\mathbb{R})\) is dense in \(X\) for the Zariski topology. Consider Hausdorff topologies on \(G(\mathbb{R})\) and \(X(\mathbb{R})\). Let \(G\) be a connected, closed subgroup of \(G(\mathbb{R})\), and \(T_a\) the anisotropic component of a maximal torus defined over \(\mathbb{R}\) of \(G\). Let \(F\) be a regular function on \(X\) whose restriction to \(X(\mathbb{R})\) assumes real values \(\geq 1\), and is proper (for the Hausdorff topologies on \(X(\mathbb{R})\) and \(\mathbb{R}\)). Let \(x_0\) be a point of \(X(\mathbb{R})\). Assume: (1) There is a rational character \(\gamma\) of \(G\) for which the character of \(G(\mathbb{R}): g\mapsto |\gamma(g)|\) is the multiplier of a relatively invariant positive Radon measure on the orbit \(G(\mathbb{R}).x_0\); (2) the orbit \(G.x_0\) is dense in \(X(\mathbb{R})\) for the Hausdorff topology; (3) the Lie algebra \(\mathfrak g\) is split or semisplit relative to its natural imbedding in the Lie algebra of \(G(\mathbb{R})\). Denote by \(\widetilde G_1\) the subgroup of \(G(\mathbb{R})\) generated by \(H(\mathbb{R})\), \(T_a(\mathbb{R})\), \(G\), \(G(\mathbb{R})(x_0)\) (the stabilizer of \(x_0\)); herein \(H\) is an algebraic subgroup of \(G\) which contains its derived subgroup. Theorem: Let \(\mu\) be a relatively invariant measure on the orbit \(\widetilde G_1.x_0\) whose multiplier is the homomorphism \(g\mapsto |\gamma(g)|\). Then there is a real positive \(A\) for which the integral \(\int_{\widetilde G_1.x_0} F(x)^{- z} d\mu(x)\) is absolutely convergent for \(\text{Re } z> A\), and admits a meromorphic continuation to the whole of \(\mathbb{C}\). The asymptotic behaviour of the function on \(\mathbb{R}_+\), \(t\mapsto \{x\in \widetilde G_1.x_0; F(x)\leq t\}\) is studied. The existence of \(\mu\) results from earlier results of the author [Ann. Sci. Éc. Norm. Supér., IV. Sér. 23, 123-149 (1990; Zbl 0760.22010), Lemma 1.2, and Méthodes des orbites. Applications exponentielles et cônes polyédraux (unpublished preprint)]. The burden of proof rests on analytical results established in Propositions 1.7 and 1.8. See pp. 340-360, in particular p. 356. The theorem is of interest in the study of unitary representations of a Lie group \(G\) by the Kirillov method of orbits.