Algebra Seminar

A difficult question in the local Langlands framework is to understand the interplay between the characters of irreducible smooth representations of a reductive group over a local field and the geometry of the dual space of Langlands parameters. An important invariant of the character (viewed as a distribution, i.e, a continuous linear functional on the space of smooth compactly supported functions) is the wavefront set, a measure of its singularities along with their directions. Motivated by the work of Adams, Barbasch, and Vogan for real reductive groups, it is natural to expect that the wavefront set is dual (in a certain sense) to the geometric singular support of the Langlands parameter. Dan Ciubotaru will give an overview of these ideas and describe recent progress in establishing a precise connection for representations of reductive p-adic groups. 

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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