The determination of the Iwahori-spherical unitary representations for split p-adic groups can be reduced to the classification of unitary representations with real infinitesimal character for associated graded Hecke algebras. We determine the unitary modules with real infinitesimal character for the graded Hecke algebra of type E6.

Let G denote a connected reductive group over a nonarchimedean local field F . Let K denote a special maximal parahoric subgroup of G(F ). We establish a Satake isomorphism for the Hecke algebra HK of K-bi-invariant compactly supported functions on G(F ). The key ingredient is a Cartan decomposition describing the double coset space K\G(F )/K. We also describe how our results relate to the trea...

There is a very beautiful correspondence between branched covers of the Riemann sphere P1 and subgroups of the fundamental group π1(P1 − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F ). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois g...

We further develop the theory of inducing W -graphs worked out by Howlett and Yin (Math. Z. 244(2):415–431, 2003 and Manuscr. Math. 115(4):495– 511, 2004), focusing on the case W = Sn. Our main application is to give two W -graph versions of tensoring with the Sn defining representation V , one being H ⊗HJ − for H ,HJ the Hecke algebras of Sn,Sn−1 and the other ( ̂ H +⊗H −)1, where ̂ H + is a sub...

The purpose of this paper is to give a reciprocity between Uq(h) and Hn,r, the Hecke algebra of (Z/rZ) ≀ Sn introduced by Ariki and Koike [1]. The Schur-Weyl reciprocity was originally discovered for GL(m) and Sn [17, p.130]. This is the first example of dual pairs and has been generalized to various pairs of groups and algebras. Jimbo [10] proved a q-analogue of the original reciprocity, namel...

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of the book N. Chriss, V. Ginzburg ‘Representation Theory and Complex Geometry’ , Birkhäuser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a Quantum group or the Iwaho...

We introduce the notion of automorphic symbol generalizing the classical modular symbol and use it to attach very general p-adic L-functions to nearly ordinary Hilbert automorphic forms. Then we establish an exact control theorem for the p-adically completed cohomology of a Hilbert modular variety localized at a suitable nearly ordinary maximal ideal of the Hecke algebra. We also show its freen...

In this paper a q{analogue of the Coxeter complex of a nite Coxeter group W is constructed for the (generic) Hecke algebra H associated to W. It is shown that the homology of this chain complex, together with that of its truncations, vanishes away from top dimension. The remainder of the paper investigates the representations of H aaorded by the top homology modules of these complexes. In parti...

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of the book by N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry , Birkhäuser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal enveloping algebra of a complex semisimple Lie algebra, a quantum group or the ...

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Yd,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Yd,n(u). Surprisingly, the trace does ...