In this article we prove several reciprocity theorems for some infinite-dimensional dual pairs of representations on Bargmann-Segal-Fock spaces.

Reciprocity theorems for holomorphic representations of some infinite-dimensional groups

In this article we prove several reciprocity theorems for some infinite-dimensional dual pairs of representations on Bargmann-Segal-Fock spaces

Reciprocity Theorems for Holomorphic Representations of Some Infinite-Dimensional Groups

Analogues of the Poincare-Birkhoff-Witt theorem for several pairs of Lie groups acting on complex manifolds are studied. These results lead to algebraic analogues of a theorem of I Segal on the two-sided regular representation of a unimodular locally compact group for dual representations of some infinite-dimensional Lie groups on generalized Bargmann-Segal-Fock spaces. Generalized Casimir invariants for these dual representations are also developed.

Poincaré–Birkhoff–Witt theorems and generalized Casimir invariants for some infinite-dimensional Lie groups: II

Casimir operators for semidirect products of some semisimple groups with Heisenberg groups are computed. The analysis is carried out using dual representations on Fock space, wherein the action of the semidirect products are related to their dual groups, namely certain unitary, orthogonal, and symplectic groups. The compact symplectic group chain is also investigated; by passing to the complexification, groups `between` the symplectic groups are constructed, which are of the form of semidirect products of symplectic groups with Heisenberg groups.

http://iopscience.iop.org/0305-4470/28/23/027/pdf/0305-4470_28_23_027.pdf

E Y Leung

Papers

Casimir operators of semidirect products of semisimple with Heisenberg groups