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Schrödinger-type equations and unitary highest weight representations of the metaplectic group. (English) Zbl 1401.22014

Christensen, Jens Gerlach (ed.) et al., Representation theory and harmonic analysis on symmetric spaces. AMS special session on harmonic analysis, in honor of Gestur Ólafsson’s 65th birthday, Atlanta, GA, USA, January 4, 2017. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4070-1/pbk; 978-1-4704-4884-4/ebook). Contemporary Mathematics 714, 157-174 (2018).
Authors’ abstract: By the work of Kashiwara-Vergne and Enright-Parthasarathy, every unitary highest weight representation of the metaplectic group \(\mathrm{Mp}(n,\mathbb{R})\) can be embedded in \({\mathcal L}^{2}(\mathrm{M}_{n,k})\) for some \(k\geq 1,\) where \(\mathrm{M}_{n,k}\) denotes the space of real \(n\times k\) matrices. Furthermore, every unitary highest weight representation can be embedded in a space of sections of a holomorphic vector bundle on the Siegel upper half-space or, via boundary values, in a degenerate principal series representation. In this paper, we give a new realization of unitary highest weight representations in the kernel of a system of Schrödinger-type equations on the space \(\mathrm{M}_{n,k}\times \text{Sym}_{n},\) where \(\text{Sym}_{n}\) denotes the space of symmetric real \(n\times n\) matrices. Our realization has simple intertwining maps to the previously known realizations mentioned above.
For the entire collection see [Zbl 1400.22001].

MSC:

22E46 Semisimple Lie groups and their representations
22E30 Analysis on real and complex Lie groups
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