For a Hecke pair (G, H) and a character σ of H we consider a generalised Hecke algebra Hσ(G, H), which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ on Hσ. The image of σ in Cc(Gσ) is a non-full projection pσ, and Hσ(G, H) is isomorphic to pσCc(Gσ)pσ. We study the structure and properties of C *-completions of a...

This paper is a report on a computer check of some important positivity properties of the Hecke algebra in type H4, including the nonnegativity of the structure constants in the Kazhdan-Lusztig basis. This answers a long-standing question of Lusztig’s. The same algorithm, carried out by hand, also allows us to deal with the case of dihedral Coxeter groups. 1 Statement of the problem 1.1. Let W ...

A geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g. a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundle over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally ...

We study a class of representations called “calibrated representations” of the degenerate double affine Hecke algebra and those of the rational Cherednik algebra of type GLn. We give a realization of calibrated irreducible modules as spaces of coinvariants constructed from integrable modules over the affine Lie algebra ĝl m . Moreover, we give a character formula of these irreducible modules in...

The Lascoux-Leclerc-Thibon conjecture, reformulated and solved by S. Ariki, asserts that the K-group of the representations of the affine Hecke algebras of type A is isomorphic to the algebra of functions on the maximal unipotent subgroup of the group associated with a Lie algebra g where g is gl ∞ or the affine Lie algebra A (1) l , and the irreducible representations correspond to the upper g...

In this paper, we study the Kazhdan–Lusztig cells of a Coxeter group W in a “relative” setting, with respect to a parabolic subgroup WI ⊆ W . This relies on a factorization of the Kazhdan–Lusztig basis {Cw} of the corresponding (multi-parameter) Iwahori–Hecke algebra with respect to WI . We obtain two applications to the “asymptotic case” in type Bn, as introduced by Bonnafé and Iancu: we show ...

In this paper we are interested in Hochschild cohomology of finite-dimensional algebras; the main motivation is to generalize group cohomology to larger classes of algebras. If suitable finite generation holds, one can define support varieties of modules as introduced by [SS]. Furthermore, when the algebra is self-injective, many of the properties of group representations generalize to this set...

We determine the finite-dimensional simple modules for two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl n and sln, and give a complete reducibility result. These quantum groups have a natural n-dimensional module V . We prove an analogue of Schur-Weyl duality in this setting: the centralizer algebra of the quantum group action on the k-fold ten...

Two actions of the Hecke algebra of type A on the corresponding polynomial ring are introduced. Both are deformations of the natural action of the symmetric group on polynomials, and keep symmetric functions invariant. An explicit combinatorial formula is given for the resulting graded characters on the coinvariant algebra.