Venue： Lecture Hall

Speaker：Anne-Marie Aubert（CNRS）

Abstract: In the first part of the talk, I will introduce a notion of cuspidality for enhanced Langlands parameters of $p$-adic reductive groups, which conjecturally puts irreducible supercuspidal representations in bijection with such enhanced $L$-parameters.

Next we will use this notion to construct cuspidal support maps and Bernstein series for enhanced $L$-parameters, in analogy with Bernstein’s theory of representations of $p$-adic groups.

One can associate to every Bernstein series of either irreducible representations of a $p$-adic group or enhanced Langlands parameters, an affine Hecke algebra (possibly extended by the twisted group algebra of a finite group) such that its simple modules are naturally in bijection with the members of the Bernstein series.

In the second part of the talk, I will explain how these results can be applied to the study of the Howe correspondence.