In this paper, we study certain Fuchsian groups H (p1, . . . , pn) , called generalized Hecke groups. These groups are isomorphic to ∏∗ n j=1Zpj . Let Γ be a subgroup of finite index in H (p1, . . . , pn) . By Kurosh’s theorem, Γ is isomorphic to Fr ∗ ∏∗ k i=1Zmi , where Fr is a free group of rank r , and each mi divides some pj . Moreover, H/Γ is Riemann surface. The numbers m1, . . . , mk are...

Starting with a numerically non-critical (at p) Hecke eigenclass f in the homology of a congruence subgroup Γ of SL3(Z) (where p divides the level of Γ) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of f to a Hecke eigenclass F with coefficients in a module of p-adic distributions. Then we show how to find to any desired degree of accuracy th...

In the previous paper [EK1], we formulated a conjecture on the relations between certain classes of irreducible representations of affine Hecke algebras of type B and symmetric crystals for gl ∞ . In the first half of this paper (sections 2 and 3), we give a survey of the LLTA type theorem of the affine Hecke algebra of type A. In the latter half (sections 4, 5 and 6), we review the constructio...

Let E be an elliptic curve over Q with complex multiplication by the ring of integers of an imaginary quadratic eld. Let be the Hecke character associated to E by the theory of complex multiplication. Let be the complex conjugate character of. For a pair of integers k and j, deene the Hecke character ' = '(k; j) = k j. Let p be a prime where E has good, ordinary reduction. Let p be a xed prime ...

A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for Kronecker coefficients, which are multiplicities of the decomposition of the tensor product of two Sr-irreducibles into irreducibles. Mulmuley and Sohoni attempt to solve this problem using canonical basis theory, by first constructing a nonstandard Hecke algebra Br, which, though not a Hopf al...

Let $(\lambda_f(n))_{n\geq 1}$ be the Hecke eigenvalues of either a holomorphic eigencuspform or Hecke-Maass cusp form $f$. We prove that, for any fixed $\eta>0$, under Ramanujan-Petersson conjecture $\rm GL_2$ Maass forms, Rankin-Selberg coefficients $(\lambda_f(n)^2)_{n\geq admit level distribution $\theta=2/5+1/260-\eta$ in arithmetic progressions.

We construct the q-analogue of a certain class of group cohomology and introduce the action of Hecke operators on such cohomology. We also show that such an action determines a representation of a Hecke ring in each of the associated group cohomology spaces. MSC 2000: 20G10, 11F60 (primary); 11F75, 18G99 (secondary)

We examine super symmetric representations of the B-type Hecke algebra. We exploit such representations to obtain new non-diagonal solutions of the reflection equation associated to the super algebra Uq(gl(m|n)). The boundary super algebra is briefly discussed and it is shown to be central to the super symmetric realization of the B-type Hecke algebra