We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, Iwahori-Hecke algebras of types A, B, and D, the complex reflection groups G(r, p, n) and the corresponding cyclotomic Hecke algebras Hr,p,n, can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to affine Hec...

We present an explicit set of matrices giving the action of the Hecke operators T (p), Tj(p ) on Siegel modular forms. Introduction It is well-known that the space of elliptic modular forms of weight k has a basis of simultaneous eigenforms for the Hecke operators, and the Fourier coefficients of an eigenform (and hence the eigenform) are completely determined by its eigenvalues and first Fouri...

We study the algebraic properties of the five-parameter family H(t1, t2, t3, t4; q) of double affine Hecke algebras of type C ∨ C1. This family generalizes Cherednik's double affine Hecke algebras of rank 1. It was introduced by Sahi and studied by Noumi and Stokman as an algebraic structure which controls Askey-Wilson polynomials. We show that if q = 1, then the spectrum of the center of H is ...

Let K be a local non-archimedian field, F = K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group G = G(F) and its central extension Ĝ . For instance our spherical Hecke algebra corresponds to the subgroup G(A) ⊂ G(F) where A ⊂ F is the subring OK((t)) where OK ⊂ K is the ring...

We apply the crystal basis theory for Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type G(p, p, n). This yields a classification of simple modules over these cyclotomic Hecke algebras in the non-separated case, generalizing our previous work [J. Hu, J. Algebra 267 (2003), 7–20]. The separated case was completed in [J. Hu, J. Algebra...

The introductory part contains definitions and basic properties of harmonic theta series, Siegel modular forms, and Hecke operators. Then the transformation formulas are recalled, related to the action of modular substitutions and regular Hecke operators on general harmonic theta series, including specialization to the case of Maass theta series. The following new results are obtained: construc...

This paper is concerned with the modular representation theory of the affine Hecke-Clifford superalgebra, the cyclotomic Hecke-Clifford superalgebras, and projective representations of the symmetric group. Our approach exploits crystal graphs of affine Kac-Moody algebras.

We introduce and study the Koszul complex for a Hecke R-matrix. Its cohomologies, called the Berezinian, are used to de ne 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 di erential. The notion of ...

Let O be an arbitrary order in an indeenite quaternion division algebra over Q. If O 1 is the group of elements in O with norm equal to 1, and H the complex upper half-plane, then X O := O 1 nH is a compact Riemann surface. Furthermore, let ? 0 (d) SL 2 (Z) be the Hecke congruence group of level d. Then X d := ? 0 (d)nH is a non-compact Riemann surface with nite volume. Let be the hyperbolic La...