The geometric meaning of Zhelobenko operators.

4 June 2013
Alexey Sevastyanov
<p>Let g be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, b a Borel subalgebra of g, h the Cartan sublagebra contained in b and N the unipotent subgroup corresponding to the nilradical n of b. Extremal projection operators are projection operators onto the subspaces of n-invariants in certain g-modules the action of n on which is locally nilpotent. Zhelobenko also introduced a family of operators which are analogues to extremal projection operators. These operators are called now Zhelobenko operators.<br />I shall show that the explicit formula for the extremal projection operator for g obtained by Asherova, Smirnov and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence (N, h) -&gt; b given by the restriction of the adjoint action. Simple geometric proofs of &nbsp;formulas for the ``classical'' counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.</p>