Harmonic analysis: automorphic forms and deformed Howe dualities

This research proposal is divided in two areas. The first is dedicated to the representation theory of rational Cherednik algebras. Using the successful theory of Dirac operators in Lie theory, adapted to rational Cherednik algebras, the intentions are twofold: on the one hand, we propose to study deformations of Howe dual pairs from a classical Weyl algebra context to a rational Cherednik context and describe the joint-decomposition of this deformed pair in the natural polynomial-spinor representations. On the other hand, we shall study the problem of computing multiplicities of a simple module in a composition series for a standard module in category O. The second area concerns automorphic forms and the Langlands program. We propose to study the residual automorphic spectrum using the theory of residue distributions introduced by Heckman and Opdam in the late 90's. These methods are well-taylored for residue-calculus with root systems and this approach is expected to yield important simplifications in the description of residual automorphic representations.