Affine Hecke Algebras of Type a and Their Cyclotomic Quotients

These are notes for a talk in the MIT-NEU Graduate seminar on Hecke Algebras and Affine Hecke Algebras (AHAHA) held in Fall 2014. This talk is divided into three parts. In the first, we introduce the affine Hecke algebras and describe a useful basis for the algebra over the ground ring. We then give a complete description of the center of the affine Hecke algebra and prove Kato’s Theorem regarding unique irreducibility of the Kato module in its central character block. The main reference for this part of the talk is [Kle05, 3.1-3.4, 4.1-4.3]. The second part of the talk is related to cyclotomic Hecke algebras, also called Ariki-Koike algebras in the references provided. We prove a basis theorem for these algebras and use the basis theorem to show that the cyclotomic Hecke algebras are symmetric algebras. The main references for this part of the talk is [GJ11, 5.1-5.2]. Auxilliary useful references include [AK94, BM97, MM98] and [Mac95, 1, Appendix B]. In the final part of the talk, we construct an equivalence of categories between affine Hecke algebra modules in the Kato block and modules over the center of particular character. The main references for this part of the talk is [CR04, 3.1-3.2].