Invariant differential equations on reductive Lie groups and algebras. II
[Part I by the authors in Ann. Inst. Fourier 50, 1799--1857 (2000; Zbl 0970.22009).] Let \(D\) be a biinvariant differential operator on a reductive Lie group \(G\) which satisfies the Benabdallah-Rouvière condition. The authors prove that it is possible to solve the equation \(Du=v\) in the space \({\mathcal D}'(G)^G\) of invariant distributions on \(G\). In a previous paper they established the same statement for the space \({\mathcal D}'(G)^G_F\) of invariant distributions with finite order. By using properties of orbital integrals, it amounts to proving the same statement for a Cartan subgroup \(H\). Hence the larger part of the paper is a proof of the following statement: Let \(D\) be an invariant differential operator on \(H={\mathbb T}^p\times {\mathbb R}^n\), and \(\Omega \) a connected open set in \(H\). If \(D\) has a fundamental solution, then \(D{\mathcal D}'(\Omega )={\mathcal D}'(\Omega )\). For \(H={\mathbb R}^n\) it is a classical result by Malgrange and Ehrenpreis. For \(H={\mathbb T}^p\), it has been proved by Cérézo and Rouvière. The proof follows the lines of the one by Malgrange.
scientific article; zbMATH DE number 2107271
scientific article