On mod p modular representations which are defined over \F_p

In this paper, we use techniques of Conrey, Farmer and Wallace to find spaces of modular forms $S_k(\Gamma_0(N))$ where all of the eigenspaces have Hecke eigenvalues defined over \F_p, and give a heuristic indicating that these are all such spaces.Comment: This is the final version, accepted and to appear in Glasnik Matematick

Irreducibility of automorphic Galois representations of GL(n), n at most 5

Let pi be a regular, algebraic, essentially self-dual cuspidal automorphic representation of GL_n(A_F), where F is a totally real field and n is at most 5. We show that for all primes l, the l-adic Galois representations associated to pi are irreducible, and for all but finitely many primes l, the mod l Galois representations associated to pi are also irreducible. We also show that the Lie algebras of the Zariski closures of the l-adic representations are independent of l.Comment: Erratum: there is a gap in the proof of the main theorem for n=4,

Diagonal Coinvariants and Double Affine Hecke Algebras

We establish a q-generalization of Gordon's theorem that the space of diagonal coinvariants has a quotient identified with a perfect representation of the rational double affine Hecke algebra. It leads to a simple proof of his theorem and relates it to the Weyl algebras at roots of unity. The universal double affine Hecke algebra and the corresponding universal double Dunkl operators acting in noncommutative polynomials in terms of two sets of variables are introduced.Comment: The final variant to appear in IMR

Macdonald's Evaluation Conjectures, Difference Fourier Transform, and Applications

This paper contains the proof of Macdonald's duality and evaluation conjectures, the definition of the difference Fourier transform, the recurrence theorem generalizing Pieri rules, and the action of GL(2,Z) on the Macdonald polynomials at roots of unity.Comment: AMSTe