Geometric quantization, cohomology groups and interwining operators. (English) Zbl 1018.53042

Holod, Petro I.; Skrypnyk, T. V.

Mladenov, I. M. (ed.) et al., Proceedings of the international conference on geometry, integrability and quantization, Varna, Bulgaria, September 1-10, 1999. Sofia: Coral Press Scientific Publishing. 95-104 (2000).

The Borel-Weil theorem and its generalizations in geometric quantization give geometric realizations of discrete series representations of Lie groups. In some cases, to obtain a space of quantum states one has to pass to higher Dolbeault cohomology groups. (E.g. it is necessary if the phase space coincides with the orbit of the semisimple Lie group and the character does not “agree” with the chosen polarization.) In this paper the authors give a brief sketch of general facts about representations in Dolbeault cohomology groups, give explicit description of the phase space of globally holomorphic sections of induced linear bundle (zeroth Dolbeault cohomology), then they give an analogous description for the highest Dolbeault cohomology group and corresponding highest harmonic forms, and construct integral representatives of cohomological classes and corresponding higher harmonic forms of other cohomology groups. The main tool used in this paper are the so-called interwining operators.
For the entire collection see [Zbl 0940.00039].

Reviewer: Oganes M.Khudaverdian (Manchester)

MSC:

53D50	Geometric quantization
81T70	Quantization in field theory; cohomological methods

Keywords:

interwining operators; higher Dolbeault cohomology; harmonic forms; flag manifolds