Prequantizations and the Berezin star product on \(GL(n,\mathbb{C})\) (Q2771245)

A prequantization of a symplectic variety \((M,\omega)\) is a triple \((L, \nabla, h)\) where \(L\) is a complex line bundle over \(M\), \(\nabla\) is a connection on \(L\) with curvature \(\omega\) and \(h\) is a \(\nabla\)-invariant Hermitian structure on \(L\). Such a prequantization exists if and only if \(\omega\) satisfies an integrality condition. The present paper is devoted to the case where \(M=GL(n, \mathbb C)\) and \(\omega= \sum_{r,s}dx_{rs}\wedge dy_{rs}\) (here \(g=(x_{rs}+iy_{rs})_{r,s}\) is the generic element of \(GL(n, \mathbb C)\)). NEWLINENEWLINENEWLINEThe authors classify up to equivalence the prequantizations of \((GL(n, \mathbb C), \omega)\). More precisely, they give an explicit representative system of equivalence classes \((L_{\lambda}=L, \nabla_{\lambda}, h_{\lambda})\) \(({\lambda}\in {\mathbb R}/{\mathbb Z} \) where \(L\) is the trivial bundle over \(GL(n, \mathbb C)\). The action by left translations of \(SU(n)\) on \(GL(n, \mathbb C)\) is Hamiltonian. For each value of \(\lambda\), this action is quantized to a representation of the Lie algebra of \(SU(n)\) in a Hilbert space of polarized sections on \(L_{\lambda}\). These representations can be integrated to the universal covering group of \(SU(n)\), one of them to \(SU(n)\) itself. NEWLINENEWLINENEWLINETo conclude: this paper does not contain very deep results but it is interesting to see how geometric quantization techniques work in the given example. Moreover, the paper is well written and contains some general definitions on geometric quantization and lemmas which are rather difficult to find elsewhere in the literature.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00047].