Zhelobenko operators, Schubert cells and \(q-W\) algebras

Alexey Sevastyanov (Aberdeen)

Frank Adams 1,

In the beginning of the 80th Zhelobenko suggested a formula for a projection operator onto the subspace of singular vectors for modules from the BGG category O for a complex semisimple Lie algebra. This projection operator and some its modifications called Zhelobenko operators are related to the problem of finding an explicit description for the space of invariant regular functions with respect to a free regular action of an algebraic group on an algebraic variety. In this talk I shall discuss a similar construction in case of the so-called \(q-W\) algebras which are related to the category of generalized Gelfand-Graev representations for quantum groups. The underlying geometry in this case is the geometry of the conjugation action of certain unipotent groups on Schubert cells. Using Zhelobenko operator technique I shall suggest an explicit description for generators of a \(q-W\) algebra in terms of generators of the corresponding quantum group. As an application of the technical tools used in the general construction I shall obtain new explicit formulas for natural coordinates on Schubert cells in terms of matrix elements of finite-dimensional irreducible representations. The simplest case of the Drinfeld-Jimbo quantum group associated to the Lie algebra \(\mathfrak{sl}_2\) will be considered in detail. No quantum group background is assumed in this talk. Surprisingly, the results that will be presented in this talk have no direct analogues for complex semisimple Lie algebras and for ordinary \(W\)-algebras associated to them.

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