A proof of De Concini-Kac-Procesi conjecture and Lusztig's partition
If you have a question about this talk, please contact David Craven.
In 1992 De Concini, Kac and Procesi observed that isomorphism classes of irreducible representations of a quantum group at odd primitive root of unity m are parameterized by conjugacy classes in the corresponding algebraic group G. They also conjectured that the dimensions of irreducible representations corresponding to a given conjugacy class O are divisible by m1/2 dim O. In this talk I shall outline a proof of an improved version of this conjecture and derive some important consequences of it related to q-W algebras.
A key ingredient of the proof are transversal slices S to the set of conjugacy classes in G. Namely, for every conjugacy class O in G one can find a special transversal slice S such that O intersects S and dim O = codim S. The construction of the slice utilizes some new combinatorics related to invariant planes for the action of Weyl group elements in the real reflection representation. The condition dim O = codim S is checked using some new mysterious results by Lusztig on intersection of conjugacy classes in algebraic groups with Bruhat cells.
This talk is part of the Algebra Seminar series.
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