| Résume |
I will discuss Kac-Moody Affine Hecke Algebras, which were first constructed as Iwahori-Hecke algebras for p-adic Kac-Moody groups. In the case where the Kac-Moody group is itself affine type, the Kac-Moody Affine Hecke Algebra is a slight variation of Cherednik's DAHA. However, unlike the DAHA, the Kac-Moody Affine Hecke algebra is realized as a convolution algebra.
In particular, it has a "T"-basis corresponding to double cosets. For usual Affine Hecke algebras, this T-basis reflects the Coxeter group structure of the affine Weyl group. However, the Kac-Moody Affine Hecke algebra is not a Coxeter Hecke algebra. Despite this, many Coxeter-like phenomena abound in the Kac-Moody Affine setting. I will present recent results and conjectures with Anna Puskás about pursuing an analogue of Coxeter theory for Kac-Moody Affine Hecke algebras. I will also discuss earlier work with Dan Orr and work in progress with Auguste Hébert.
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