Harald Upmeier: Generalized Fock Spaces and Unipotent Representations
From CRCG-Wiki
For any semisimple Lie group G of hermitian type, realized as the holomorphic automorphism group of a hermitian symmetric domain B=G/K of rank r, the scalar holomorphic discrete series (Bergman spaces of holomorphic functions on B) has an analytic continuation for discrete parameters, which leads to unipotent representations of G. For the real symplectic group, these spaces are the Fock spaces of entire functions, endowed with the metaplectic representation. For the general case, we propose a geometric realization of these unipotent representations on certain algebraic varieties, defined in terms of the complex Jordan algebra (or Jordan triple) which is associated with B. These varieties are not natural G-orbits but are the Matsuki-dual KC-orbits for the boundary G-orbits ("limits of discrete series") of B. On these Jordan varieties we construct a measure with a Bessel function density, and define a non-holomorphic G-action with the following properties:
- There exists a G-equivariant field (bundle) of Hilbert spaces over the base manifold B, such that the fiber at 0 is the closure of the K-finite vectors, corresponding to partitions of length k<r;
- This Hilbert space bundle carries a projectively flat connection over B, which induces the G-representation in the projective sense;
- The parallel transport "Bogolyubov" transformations can be described in terms of the Jordan theoretic Bergman operators.
