Closed orbits and tempered orbits. II (Q1915349)

In Part I of this paper [Ann. Éc. Norm. Supér., IV. Sér. 23, 123-149 (1990; Zbl 0760.22010)] the proof of the main theorem: ``A closed coadjoint orbit of any real, finite dimensional Lie algebra is tempered.'' was not complete. The goal of this article is to complete the proof. Let \(G\) be a connected subalgebraic group in \(GL(V)\) where \(V\) is a real, finite dimensional vector space. If the orbit \(G.v\) in \(V\) carries an invariant measure \(\beta\), the product of \(\beta\) by some polynomial function \(p\) on \(V\) is a tempered measure. The annihilator \(L\) of \(\beta\) in the ring \(A(V)\) of differential operators with polynomial coefficients is a left ideal for which the \(A(V)\)-module \(A(V)/L\) is holonomic; hence at the neighbourhood of each point at the infinity of the closure of \(G.v\), the holomorphic function \(s \mapsto p^s \beta\), with values in the space of tempered distributions on this neighbourhood, satisfies a functional equation. So this function has a meromorphic continuation to \(\mathbb{C}\). Let \(\beta_0\) be the constant term of the Laurent series of this function in zero. Then \(\beta_0\) is a tempered measure, the restriction of which to \(G.v\) is equal to \(\beta\). Let \(H\) be a closed, connected subgroup in \(G\) which contains the derived subgroup of \(G\). Then \(H.v\) carries an invariant measure \(\mu\). Let us suppose that the orbit \(H.v\) is closed. So \(\mu\) is a distribution on \(V\). For a good \(H\), \(H.v\) is an intersection of a countable family \(\{D_\nu; \nu \in \mathbb{N}\}\) where \(D_\nu\) is the closure in \(V\) of an open subset in \(G.v\). Then for \(\nu\) big enough and for any rapidly decreasing \(C^\infty\) function on \(V\), the value at \(\varphi\) of \(\beta_0\) is defined and equal to the integral of \(\varphi\) on \(D_\nu\). So the restriction of \(\beta\) to \(D_\nu\) is tempered. Hence by desintegration, \(\mu\) is tempered.