R – Conformal Symmetries . Ii . Geometric Quantization and Hidden Symmetries of Verma Modules over Virasoro Algebra

This article being a continuation of the first part [1] is addressed to specialists in representation theory, infinite dimensional geometry, quantum algebra, mathematical physics and informatics of interactive systems. The interrelations of the infinite dimensional geometry of the homogeneous Kähler manifold M = Diff+(S )/S (see e.g. [2] and numerous refs wherein), its quotient M1 = Diff+(S )/PSL(2,R) and the qR–conformal symmetries are discussed. Considering qR–conformal symmetries in the Verma modules as Nomizu operators [1] one receives a connection in a bundle over M1 with fibers isomorphic to Verma modules over the Lie algebra sl(2,C). An interpretation of this connection in terms of geometric quantization is proposed. Simultaneously, another interpretation of qR–conformal symmetries based on a geometric construction of the Verma modules over Virasoro algebra by the orbit method [3] as hidden symmetries is formulated. Such hidden symmetries are local over M1, i.e. commute with the natural action of the commutative ring O(M1) of all polynomial germs of holomorphic functions on M1 in the Verma modules over Virasoro algebra.