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-...

In this paper we investigate non-central elements of the Iwahori-Hecke algebra of the symmetric group whose squares are central. In particular, we describe a commutative subalgebra generated by certain non-central square roots of central elements, and the generic existence of a rank-three submodule of the Hecke algebra contained in the square root of the centre, but not in the centre. The gener...

Hecke’s theory is concerned with a family of finite-dimensional vector spaces Sk(N,χ), indexed by weights, levels, and characters. The Hecke operators on such spaces already provide a very rich theory. It will be very advantageous to pass to the adelic setting, however, for the same reasons that Hecke characters on number fields should be studied in the adelic setting (rather than as homomorphi...

Iwahori-Hecke algebras for the innnite series of complex reeection groups G(r; p; n) were constructed recently in the work of Ariki and Koike, Brou e and Malle, and Ariki. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper in which we derived Murnaghan-Nakayama rules for t...

For a prime p we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space V of modular forms of weight 2 and level p2. For p ≡ 3 mod 4 we define a special Hecke stable subspace V0 of V which contains the space of modular forms with CM by the ring of integers of Q( √−p) and we describe the calculation of the corresponding Brandt matrices.

By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index. Résumé. En étudiant l’action de l’algèbre de 0-Hecke sur l’algèbre coinvariante et la variété de drapeaux complète, nous interprétons les fonctions généra...

The explicit formulas for the transformation of theta-functions of integral positive definite quadratic forms under the action of regular Hecke operators, obtained in the author’s earlier paper (1996), are converted to transformation formulas for the theta-functions with rational characteristics (the theta-series) viewed as Siegel modular forms. As applications, sequences of invariant subspaces...

Fix ρ : GQ,S → GL2(k) absolutely irreducible, and let ρ : GQ,S → GL2(R) be the universal deformation. We’re interested in the map R → Tm for some Hecke algebra defined in terms of ρ. Note that the Hecke algebra is 1-dimensional, and even finite free over Zp. The universal deformation ring R, however, often has dimension > 1 and nonzero p-torsion. In other words, the surjection R ։ Tm is not eve...

In this paper, we compute the center of the infinitesimal Hecke algebras Hz associated to sl2; then using nontriviality of the center, we study representations of these algebras in the framework of the BGG category O. We also discuss central elements in infinitesimal Hecke algebras over gl n and sp(2n) for all n. We end by proving an analogue of Duflo’s theorem for Hz.

We would like to recall that in the case of Hecke eigenforms on Γ1 non-vanishing results for their Hecke L-functions at an arbitrary point s0 in the critical strip (not on the critical line) have been proved in [4] (cf. also [7]), using holomorphic kernel functions. This method was carried over to the case of half-integral weight in [8], for arbitrary level. However, in this approach for given ...