Constant term in Harish-Chandra's limit formula
Let \(G_{\mathbb R}\) be a real form of a complex semisimple Lie group \(G\), \(\mathfrak h_{\mathbb R}\) a Cartan subalgebra of its Lie algebra \(\mathfrak g_R\). A classical formula of Harish-Chandra for the delta function at 0 states that \(\lim_{\lambda \to 0} \Pi_{\alpha > 0}\partial(\alpha) m_\lambda = cm_0\), where \(\lambda\in ih_{\mathbb R}^*\) is regular, \(m_\lambda\) is the canonical measure on the coadjoint orbit \(G_{\mathbb R}\cdot\lambda\) and \(\partial(\alpha)\) is the differential operator defined by the root \(\alpha\). See \textit{Harish-Chandra} [Am. J. Math. 79, 733--760 (1957; Zbl 0080.10201)]. The purpose of this paper is to compute explicitly the constant \(c\), which is \(\neq 0\) iff \(\mathfrak h_{\mathbb R}\) is a fundamental Cartan subalgebra. For this end, the action is used of the Weyl group on Lagrangian cycles supported on the conormal bundle of the flag variety of \(G\) defined in [\textit{W. Rossmann}, Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honour of J. Dixmier, Paris/Fr. 1989, Prog. Math. 92, 263--287 (1990; Zbl 0744.22012)]. The approach of the present paper requires more theory, namely the one developed in [\textit{W. Schmid} and \textit{K. Vilonen}, J. Am. Math. Soc. 11, 799--867 (1998; Zbl 0976.22010)] and [\textit{W. Rossmann}, Invent. Math. 121, 531--578 (1995; Zbl 0861.22008)].
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