We introduce Hecke operators on de Rham cohomology of compact oriented manifolds. When the manifold is a quotient of a Hermitian symmetric domain, we prove that certain types of such operators are compatible with the usual Hecke operators on automorphic forms.

In this paper we study the branching problems for Hecke algebra H(Dn) of type Dn. We explicitly describe the decompositions of the socle of the restriction of each irreducible H(Dn)-representation to H(Dn−1) into irreducible modules by using the corresponding results for type B Hecke algebras. In particular, we show that any such restrictions are always multiplicity free.

We use the action of the Hecke operators T̃j(p 2) (1 ≤ j ≤ n) on the Fourier coefficients of Siegel modular forms to bound the eigenvalues of these Hecke operators. This extends work of Duke-Howe-Li and of Kohnen, who provided bounds on the eigenvalues of the operator T (p).

The main aim of the paper is to formulate and prove a result about the structure of double affine Hecke algebras which allows its two commutative subalgebras to play a symmetric role. This result is essential for the theory of intertwiners of double affine Hecke algebras.

The determination of the Iwahori-spherical unitary representations for split p-adic groups can be reduced to the classification of unitary representations with real infinitesimal character for associated graded Hecke algebras. We determine the unitary modules with real infinitesimal character for the graded Hecke algebra of type E6.

Invariant subspaces and eigenfunctions for regular Hecke operators actinng on spaces spanned by products of even number of Igusa theta constants with rational characteristics are constructed. For some of the eigenfunctions of genuses g = 1 and 2, the corresponding zeta functions of Hecke and Andrianov are explicitely calculated.

This article gives conceptual statements and proofs relating parabolic induction and Jacquet functors on split reductive groups over a nonArchimedean local field to the associated Iwahori-Hecke algebra as tensoring from and restricting to parabolic subalgebras. The main tool is Bernstein’s presentation of the Iwahori-Hecke algebra.

Given E/F a quadratic extension of number fields and a cuspidal representation π of GL2(AE), we give a full description of the fibers of the Asai transfer of π. We then determine the extent to which the Hecke eigenvalues of all the Hecke operators indexed by integral ideals in F determine the representation π.

We generalize a classical result of Andrianov on the decomposition Hecke polynomials. If G is connected reductive group defined over non-archimedean local field