Induced representations of double affine Hecke algebras and applications
The main results (Cherednik, 1991-1993) about the structure of double affine Hecke algebras are applied to induced representations. The technique used is based on the theory of Weyl groups, the matrix Demazure-Lusztig operators and the difference-matrix counterparts of the operators (Heckman, 1991; Opdam, 1989). The Hecke algebra and its affine extensions are introduced by starting from a given root system. It is proved the difference version of the generalized Matsuo (1991) theorem which gives an isomorphism between the affine Knizhnik-Zamolodchikov (AKZ) equation and the eigenvalue problem (SHOHS) for operators unifying the Sutherland-Heckman-Opdam and Haldany-Shastry ones. As a result, the equivalence of the difference SHOHS with the values in an arbitrary representation of the Hecke algebra and the difference AKZ considered in the corresponding induced representation of the affine Hecke algebra is established (Theorem 4.3). The AKZ equation is connected with Smirnov- Frenkel-Reshetikhin equations for the simplest tensor products. The scalar version of above theorem is directly related to the Macdonald theory (1988).
scientific article; zbMATH DE number 817698