Strict quantization of coadjoint orbits

  • Philipp Schmitt

    Leibniz University Hannover, Germany; University of Copenhagen, Denmark
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For every semisimple coadjoint orbit O^\hat{\mathcal{O}} of a complex connected semisimple Lie group G^\hat{G}, we obtain a family of G^\hat{G}-invariant products ^\hat{*}_\hbar on the space of holomorphic functions on O^\hat{\mathcal{O}}. For every semisimple coadjoint orbit O\mathcal{O} of a real connected semisimple Lie group GG, we obtain a family of GG-invariant products *_\hbar on a space A(O)\mathcal{A}(\mathcal{O}) of certain analytic functions on O\mathcal{O} by restriction. A(O)\mathcal{A}(\mathcal{O}), endowed with one of the products *_\hbar, is a GG-Fréchet algebra, and the formal expansion of the products around =0\hbar=0 determines a formal deformation quantization of O\mathcal{O}, which is of Wick type if GG is compact. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules and complex analytic results on the extension of holomorphic functions.

Cite this article

Philipp Schmitt, Strict quantization of coadjoint orbits. J. Noncommut. Geom. 15 (2021), no. 4, pp. 1181–1249

DOI 10.4171/JNCG/429