Algebra Seminar

Given a module V over a complex reductive Lie algebra a, the extremal projection p=p(a) takes V into the subspace of its extreme (highest) vectors. The projection p can be given by a universal explicit formula independent of the module V. A theorem of Zhelobenko states that p is a unique element of an extension of the enveloping algebra U(a) satisfying certain natural conditions. The method of extremal projections and its applications will be discussed in the talk. In particular, a simple construction of a basis in the harmonic polynomials will be given with the use of the projection p=p(sl(2)).