Abstract: Iwahori-Hecke algebras are generalizations of some of the most
important group algebras -- those of Coxeter groups like the symmetric group
or dihedral groups. So they appear all over mathematics where actions of
these groups are involved. They appeared in Jones' work on polynomial invariants for knots, on Kazhdan and Lusztig's work in computing cohomology of nice classes of varieties, and in the study of representations of Lie groups over finite and local
fields. In the latter case, they can turn problems about infinite dimensional representations into beautiful algebraic combinatorics. We'll define them and discuss some of their remarkable appearances, and at the end describe some recent joint work (with Bump and Friedberg) on computing matrix coefficients of certain group representations using Hecke algebras. No prior knowledge about Hecke algebras will be assumed.
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