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. An essential role in this construction is played by a non-singular character χ of the maximal nilpotent subalgebra n+ ⊂ g. The main result is that the center of U(g) is isomorphic to a commutative subalgebra in U(b_), where b_ ⊂ g is the opposite Borel subalgebra. 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.We show that the Whittaker model introduced by Kostant has a natural homological interpretation in terms of Hecke algebras. Moreover, we introduce a general definition of a Hecke algebra Hk'(A, B, χ) associated to the triple of an associative algebra A, a subalgebra B ⊂ A and a character χ of B. In particular, the Whittaker model of the center of U(g) is identified with Hk0(U(g), U(n+), χ)opp.The goal of this thesis is to generalise the Kostant's construction to quantum groups. An obvious obstruction is the fact that the subalgebra in Uh(g) generated by positive root generators (subject to the quantum Serre relations) does not have non-singular characters. In order to overcome this difficulty we introduce a family of new realisations of quantum groups, one for each Coxeter element of the corresponding Weyl group. The modified quantum Serre relations allow for non-singular characters, and we are able to construct the Whittaker model of the center of Uh(g).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 Uh(g). A remarkable new result is a cross-section theorem for the action of a complex simple Lie group on itself by conjugations. We are able to prove this theorem for all such Lie groups except for the case of E6! Using the cross-section theorem we establish a relation between the Whittaker model and the set of conjugacy classes of regular elements in the corresponding Lie group G.