Let G = GL(N), K = GL(p)× GL(q), where p + q = N , and let n be a positive integer. We construct a functor from the category of HarishChandra modules for the pair (G,K) to the category of representations of the degenerate affine Hecke algebra of type Bn, and a functor from the category of K-monodromic twisted D-modules on G/K to the category of representations of the degenerate double affine He...

It is a well known fact from the group theory that irreducible tensor representations of classical groups are suitably characterized by irreducible representations of the symmetric groups. However, due to their different nature, vector and spinor representations are only connected and not united in such description. Clifford algebras are an ideal tool with which to describe symmetries of multip...

We determine the unitary dual of the geometric graded Hecke algebras with unequal parameters which appear in Lusztig’s classification of unipotent representations for exceptional p-adic groups. The largest such algebra is of type F4. Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent represent...

This paper reviews a result of Jensen on characters of some tilting modules for SL3(k), where k has characteristic at least five and fills in some gaps in the proof of this result. We then apply the result to finding some decomposition numbers for three part partitions for the symmetric group and the Hecke algebra. We review what is known for characteristic two and three. The quantum case is al...

The main purpose of this paper is to produce a geometric realization for the tempered modules of the affine Hecke algebra of type C (1) n with arbitrary, non-root of unity, unequal parameters, using the exotic DeligneLanglands correspondence ([Ka08a]). Our classification has several applications to the Weyl group module structure of the tempered Hecke algebra modules. In particular, we provide ...

This paper concerns the combinatorics of the orbit Hecke algebra associated with the orbit of a two sided Weyl group action on the Renner monoid of a finite monoid of Lie type, M . It is shown by Putcha in [12] that the Kazhdan-Lusztig involution ([6]) can be extended to the orbit Hecke algebra which enables one to define the R-polynomials of the intervals contained in a given orbit. Using 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 ...

For certain algebraic Hecke characters χ of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F . By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-...

We introduce two new packages, Nemo and Hecke, written in the Julia programming language for computer algebra and number theory. We demonstrate that high performance generic algorithms can be implemented in Julia, without the need to resort to a lowlevel C implementation. For specialised algorithms, we use Julia’s efficient native C interface to wrap existing C/C++ libraries such as Flint, Arb,...

In this paper we use the Hecke algebra of type B to define a new algebra S which is an analogue of the q–Schur algebra. We construct Weyl modules for S and obtain, as factor modules, a family of irreducible S–modules over any field.