Enhanced orbit embedding

Let $\tilde{G}$ be an algebraic group acting on a variety $\tilde{L}$, and $G \subset \tilde{G}$ a subgroup which leaves a subvariety $L \subset \tilde{L}$ stable. For a $G$-orbit $O_G = G u (u \in L)$ in $L$, we can associate an orbit $O_{\tilde{G}} = \tilde{G} u$ of $\tilde{G}$ so that we get a map $L/G \to \tilde{L}/\tilde{G}$ between orbit spaces, though this map is usually not injective. In this note, when $G$ is a symmetric subgroup arising from an involutive anti-automorphism, we give certain sufficient conditions for the map $L/G \to \tilde{L}/\tilde{G}$ to be injective after the method of Ohta (2008). Our main concern here is to produce examples of enhanced Lie algebras (or enhanced $\theta$-representations). We also analyze an obstruction which prevents the orbit space inclusion.Comment: 11 pages, to appear in Commentarii Mathematici Univ. St. Pauli; Minor corrections, change the statement of Corollary 3.5 due to the comment by Anthony Henderso

Theta lifting of two-step nilpotent orbits for the pair O(p, q) x Sp(2n, R)

This is the proceedings of the 2nd Japanese-German Symposium on Infinite Dimensional Harmonic Analysis held from September 20th to September 24th 1999 at the Department of Mathematics of Kyoto University.この論文集は, 1999年9月20日から9月24日の日程で京都大学理学研究科数学教室において開催された第2回日独セミナー「無限次元調和解祈」の成果をもとに編集されたものである.編集 : ハーバート・ハイヤー, 平井 武, 尾畑 信明Editors: Herbert Heyer, Takeshi Hirai, Nobuaki Obata #e

Theta lifting of nilpotent orbits for symmetric pairs

We consider a reductive dual pair (G, G') in the stable range with G' the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent K'_C-orbits, where K' is a maximal compact subgroup of G' and we describe the precise K_C-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair (G, K). As an application, we prove sphericality and normality of the closure of certain nilpotent K_C-orbits obtained in this way. We also give integral formulas for their degrees.Comment: 26 page