Realization of a holomorphic discrete-series of the Lie group \(SU(1,2)\) as star-representation (Q2704093)

The realization of the method of orbits known as deformation program (1978, Bayen, Flato, Fronsdal, Lichnerowicz, Sternheimer; 1978, Fronsdal) is presented for the case of the group \(G=SU (1,2)\). This method is based on the use of deformation of an algebra of functions on coadjoint orbits, called star products. As a result, the holomorphic discrete series of \(SU(1,2)\) is obtained as a star representation. The star product for an orbit \(O_f= \{gfg^{-1}, g\in G,f\in {\mathfrak g}^*\}\) \(({\mathfrak g}^*\) being the algebraic dual of the Lie algebra \({\mathfrak g}\) of the group \(G)\) associated to the discrete series is defined in the framework of Berezin's symbolic calculus (1995, Yahyai). The star exponential on \({\mathfrak g}^*\) and the adapted Fourier transform are constructed.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00041].