Algebra Seminar

The well-known "Dunkl operators" associated to a finite real reflection group constitute a commutative parameter family of deformations of the directional derivatives in Euclidean space. These operators are "differential-reflection" operators. Heckman and Cherednik have defined trigonometric versions of Dunkl's operators. The interest for these operators lies in their deep ties to Macdonald polynomials and hypergeometric functions, to the Calogero-Moser quantum integrable system, and to the representation theory of Hecke algebras. 

"Hypergeometric shift operators" are powerful tools to study Weyl group symmetric structures and functions in these contexts. In this talk, Eric Opdam presents a theorem of existence and uniqueness of ''nonsymmetric shift operators'' for the Dunkl operators. These are themselves differential reflection operators which "shift" the parameters of the Dunkl operators by integers by means of a "transmutation relation".

(Joint work with Valerio Toledano Laredo) 

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