We introduce and study the Koszul complex for a Hecke R-matrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke R-matrix. Their behaviour with respect to Hecke sum of R-matrices is studied. Given a Hecke R-matrix in n-dimensional vector space, we construct a Hecke R-matrix in 2n-dimensional vector space commuting with a differential. The notion o...

These notes are about applications of the one-dimensional double affine Hecke algebra to q-Gauss integrals and Gaussian sums. The double affine Hecke algebras were designed for a somewhat different purpose: to clarify the quantum Knizhnik-Zamolodchikov equation. Eventually (through the Macdonald polynomials) they led to a unification of the Harish-Chandra transform (the zonal case) and the p-ad...

We construct a family of exact functors from the BernsteinGelfand-Gelfand category O of sln-modules to the category of finite-dimensional representations of the degenerate affine Hecke algebra Hl of GLl. These functors transform Verma modules to standard modules or zero, and simple modules to simple modules or zero. Any simple Hl-module can be thus obtained. Introduction The classical Frobenius...

Lusztig’s presentation of the graded Hecke algebra in terms of generators and relations allows for the definition of algebras associated to noncrystallographic root systems. The representation theory of general graded Hecke algebras is investigated, the expected number of tempered representations for H(H3) are accounted for, and it is shown that one of these representations has the unexpected p...

Let E(p) denote the Eisenstein ideal in the Hecke algebra T(p) of the Drinfeld modular curve X0(p) parameterizing Drinfeld modules of rank two over Fq[T ] of general characteristic with Hecke level p-structure, where p⊳ Fq[T ] is a nonzero prime ideal. We prove that the characteristic p of the field Fq does not divide the order of the quotient T(p)/E(p) and the Eisenstein ideal E(p) is locally ...

This paper provides a survey for the latest developments in the theory of affine q-Schur algebras and Schur–Weyl duality between affine quantum gln and affine type A Hecke algebras. More precisely, we will establish, on the one side, an isomorphism between the double Ringel–Hall algebra D△(n) of a cyclic quiver △(n) and the quantum loop algebra of gln, and establish, on the other side, explicit...

In this paper we prove that the spherical subalgebra eH1,τ e of the double affine Hecke algebra H1,τ is an integral Cohen-Macaulay algebra isomorphic to the center Z of H1,τ , and H1,τ e is a Cohen-Macaulay eH1,τ emodule with the property H1,τ = EndeH1,τ e(H1,τ e) when τ is not a root of unity. In the case of the root system An−1 the variety Spec(Z) is smooth and coincides with the completion o...

This paper is concerned with some important properties of r-polynomials in Hecke algebras with unequal parameters . In chapter one, we briefly introduce some conceptions in Coxeter group W and some notations in Hecke algebra H. In chapter two, we give some important properties of some r-polynomials which satisfy ry,w = ∏l(w)−l(y) i=1 (vsi − v−1 si ) when y ≤ w in W and w = s1s2 · · · sq is any ...

A construction of bases for cell modules of the Birman–Murakami–Wenzl (or B–M–W) algebra Bn(q, r) by lifting bases for cell modules of Bn−1(q, r) is given. By iterating this procedure, we produce cellular bases for B–M–W algebras on which a large Abelian subalgebra, generated by elements which generalise the Jucys–Murphy elements from the representation theory of the Iwahori–Hecke algebra of th...

The theory of congruences for the partition function p(n) depends heavily on the properties of half-integral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P (z) | T (`), where P (z) is the relevant modular generating function. We obtain such formulas using Euler’s Pentagonal Number Theorem and the denominator formula for the Mons...