The principal investigator intends to study the irreducible unitary representations of the universal covering of the symplectic group. The classification of irreducible unitary representations of higher rank symplectic groups is an important open problem in mathematics and it has a wide range of applications in number theory, geometric quantization and harmonic analysis. In this project, the P.I. intends to construct and study irreducible unitary representations using invariant tensor product. The motivation comes from the theta correspondence. Due to the work of R. Howe and J-S Li, a large class of small unitary representations of the symplectic groups can be realized as invariant tensor products of the oscillator representation with unitary representations of the orthogonal groups. The proposed research deals with the invariant tensor products associated with elementary representations---unitarizable subquotients of representations on the Shilov boundary. Due to the work of Sahi and many others, elementary representations fall into four classes: unitary principal series, complementary series, small constituents and large constituents. The principal investigator will study the behavior of the four types of invariant tensor functors associated to four classes of elementary representations and relate these invariant tensor functors to the unitary parabolic induction, complementary induction, Howe's correspondence and cohomological induction, respectively. This unified approach should shed lights on the structure of the unitary dual of the universal covering of the symplectic group. Other classical groups will be treated as time allows.
A Lie group is a fundamental object used in mathematics to describe continuous symmetries. The theory of Lie group representations provides a mathematical foundation for quantum mechanics. The sympletic group, in particular, plays a central role in mechanics, optics and other area of physics. Many important problems in analysis, mathematical physics and number theory are closely related to irreducible unitary representations of the symplectic groups. The proposed research will improve the general understanding of the irreducible unitary representations of the symplectic groups and other classical groups.
