The Whittaker model of the center of the quantum group and Hecke algebras

Abstract

In 1978 Kostant suggested the Whittaker model of the center of the universal enveloping algebra U(g) of a complex simple Lie algebra g. The main result is that the center of U(g) is isomorphic to a commutative subalgebra in U(b), where b is a Borel subalgebra in g. This observation is used in the theory of principal series representations of the corresponding Lie group G and in the proof of complete integrability of the quantum Toda lattice. In this thesis we generalize the Kostant's construction to quantum groups. In our construction we use quantum analogues of regular nilpotent elements defined in math.QA/9812107. We also show that the Whittaker model has a natural homological interpretation in terms of Hecke algebras introduced by the author in math.QA/9805134. The new Whittaker model is applied to the deformed quantum Toda lattice recently studied by Etingof. We give new proofs of his results which resemble the original Kostant's proofs for the quantum Toda lattice. Finally, we study the ``quasi-classical'' limit of the Whittaker model for U_h(g). Using the cross-section theorem proved in q-alg/9702016 we establish a relation between the Whittaker model and the set of conjugacy classes of regular elements in the corresponding Lie group $G$.