Recent zbMATH articles in MSC 46F99https://zbmath.org/atom/cc/46F992024-03-13T18:33:02.981707ZWerkzeugVanishing of certain equivariant distributions on spherical spaces for quasi-split groupshttps://zbmath.org/1528.220082024-03-13T18:33:02.981707Z"Lu, Hengfei"https://zbmath.org/authors/?q=ai:lu.hengfeiThis paper contains an important result concerning the vanishing of some eigendistributions on a quasi-split real reductive group. Let \(G\) be a quasi-split real reductive group, \(B\) be a standard Borel subgroup of \(G\), \(U\) be the corresponding unipotent radical of \(B\), and let \(\Psi\) be a non-degenerate character of \(U\). Let \(H\) be a spherical subgroup of \(G\), and let \(\chi\) be a character of \(H\). Let \(Z\) be the complement to the union of open \(B\times H\)-double cosets in \(G\), and let \(\mathfrak{z}\) be the center of the enveloping algebra \(\mathscr{U}(\mathfrak{g})\) of the Lie algebra \(\mathfrak{g}\) of \(G\). The author proves (Theorem 1.1) that there are no non-zero \(\mathfrak{z}\)-eigen \((U\times H\), \(\psi \times \chi\))-equivariant distributions supported on \(Z\).
Theorem 1.1 is a generalization of a theorem from [\textit{A. Aizenbud} and \textit{D. Gourevitch}, Math. Z. 279, No. 3--4, 745--751 (2015; Zbl 1315.22014)], where the authors obtained a similar result for a split real reductive group \(G\).
Reviewer: Allan Merino (Boston)