Coherent transformations and irreducible representations corresponding to complex structures on a cylinder and a torus. (Q1810004)

The prototype of a coherent transform is the well-known Bargmann transformation of quantum mechanics, which intertwines the Schrödinger representation and the Fock representation of the 3-dimensional Heisenberg algebra, the latter consisting of pseudo-differential operators defined on a Hilbert space of (anti-)analytic functions on the phase space \(\mathbb{R}^{2}\) endowed with a complex structure. This construction has the potential of considerable generalization, as is detailed in the present paper, which may be considered as a continuation of two earlier papers of the authors [in: Coherent Transform, Quantization and Poisson Geometry, Transl., Am. Math. Soc. 187(40), 1--202 (1998; Zbl 1036.81019); \textit{M. V. Karasev}, Lett. Math. Phys. 56, 229--269 (2001; Zbl 0996.53058)]. Specifically, the Heisenberg algebra is generalized to a class of algebras with non-Lie commutation relations whose symplectic leaves are surfaces of revolution (a cylinder and a torus). Bargmann space is generalized to Hilbert spaces of antiholomorphic sections over these surfaces with respect to a family of complex structures on each of them. The Hilbert spaces carry irreducible Hermitian representations of the original algebras, and coherent transforms intertwining these with abstract representations are constructed and expressed in terms of the Riemann theta function.