Mickelsson algebras and Zhelobenko operators. (Q2425433)

The classical Mickelsson algebra related to a real reductive group acts on the space of highest weight vectors of its maximal compact subgroup. A similar algebra can be defined for any associative algebra \(A\) containing a universal enveloping algebra (or its \(q\)-analogue) of a contragradient Lie algebra with a fixed triangular decomposition. In the paper under review the authors describe a family of automorphisms for a wide class of Mickelsson algebras satisfying braid group relations. Each generating automorphism is given as a product of the so-called Zhelobenko cocycle and an automorphism of \(A\) which extends the action of the Weyl group on the Cartan subalgebra. The new feature of the approach is the homomorphism property of Zhelobenko maps which was not noticed before. The author's construction of automorphisms of Mickelsson algebras can be applied to the construction of finite-dimensional representations of Yangians and quantum affine algebras.