Degenerate Double Affine Hecke Algebra And Conformal Field Theory

We introduce a class of induced representations of the degenerate double affine Hecke algebra H of glN (C) and analyze their structure mainly by means of intertwiners of H . We also construct them from ŝlm(C)-modules using Knizhnik-Zamolodchikov connections in the conformal field theory. This construction provides a natural quotient of induced modules, which turns out to be the unique irreducible one under a certain condition. Some cunjectual formulas are presented for the symmetric part of these quotients. In this paper, the representations of the degenerate double affine Hecke algebra H are discussed from view points of the conformal field theory associated to the affine Lie algebra of type ŝlm(C). The relations between the KZ-connections of the conformal field theory and the representations of the degenerate affine Hecke algebra was first discussed by Cherednik [1]. And Matsuo [2] succeeded to clarify the relations between the differential equations satisfied by spherical functions and KZ-connections. At first part of this paper we discuss the properties of the parabolic induced modules of H, which are induced from a certain one dimensional representations of parabolic subalgebras of H. Secondly we will give an explicit construction of H-modules from ŝlm(C) modules through KZ-connections. It will be shown that these Hmodules arising from Verma modules of ŝlm(C) correspond to parabolic induced modules of H. In the final part of this paper, we will discuss the structure of the representations of affine Hecke algebra arising from level l integral representations of ŝlm(C). We describe the contents of the paper more precisely: In §1 we introduce basic notions about the degenerate double affine Hecke algebra H. In particular, intertwiners of weight spaces of H play essential roles in the analysis of H-modules.