A. Kirillov (Research Institute of Mathematical Sciences, Kyoto U)

Saga of Dunkl operators

Abstract:

The Dunkl operators (DO for short) play crucial role in different fields 
of Integrable Systems, Representation Theory, Algebraic Geometry and 
many others. The basic property of Dunkl operators is that they form a 
commutative family. There are three types of DO : rational, 
trigonometric and elliptic. We construct a certain noncommutative 
quadratic algebra and together with a family of mutually commuting 
elements such that images of these elements under the corresponding 
representation of the quadratic algebra in question, coincides 
respectively with rational, trigonometric and elliptic DO. These 
universal Dunkl elements have many interesting properties, e.g. they
generate a commutative subalgebra which can be considered as "universal 
cohomology theory" of flag varieties (of type A in my talk). Namely, the 
images of the commutative subalgebra generated by Dunkl elements under 
the corresponding representation, are isomorphic respectively to 
classical, quantum, elliptic (?) cohomology and/or K-theory, as well as 
their equivariant versions. Another application of universal Dunkl 
elements related with a "theory of universal hypergeometric series."