Orateur : Stefan Dawydiak
Établissement : Université de Bonn ()
Dates : 2025-01-30 – 2025-01-30
Heures : 14:00 – 14:00
Lieu : Salle 0-6
Résumé :
Affine Hecke algebras play a prominent role the representation theory of p-adic groups, where a large subcategory of representations are equivalent to modules over the affine Hecke algebra. Moreover, as we will recall, this ring can then be studied via algebraic geometry, in keeping with the Langlands philosophy. The important subcategory of tempered representations is equivalent to modules over a larger ring, the Harish-Chandra Schwartz algebra. However, this ring is a Fréchet algebra not definable in purely algebraic terms. Braverman-Kazhdan proposed an algebraic analogue of the Schwartz algebra in the form of Lusztig’s asymptotic Hecke algebra $J$. We will explain this proposal and our results relating to it, and then give some applications of the connection between the asymptotic Hecke algebra and p-adic groups. In particular, the algebra again $J$ admits an algebro-geometric presentation, which we will recall and apply to coherent sheaves on Springer fibres, giving a conceptual explanation of previous joint work with Bezrukavnikov and Dobrovolska. Time permitting, we will outline future work building on the enlargement of the role of affine Hecke algebras in the theory of p-adic groups due to Aubert-Moussaoui-Solleveld, Solleveld, and Adler-Fintzen-Mishra-Ohara.