On a symplectic quantum Howe duality

Abstract

We prove a nonsemisimple quantum version of Howe's duality with the rank 2n symplectic and the rank 2 special linear group acting on the exterior algebra of type C. We also discuss the first steps towards the symplectic analog of harmonic analysis on quantum spheres, give character formulas for various fundamental modules, and construct canonical bases of the exterior algebra. Contents1. Introduction 1 2. The module and the two commuting actions 4 3. Howe duality -part 1 17 4. Integral versions and Weyl filtrations 21 5. Howe duality -part II 34 6. Further results I: The dual action via differential operators 38 7. Further results II: On fundamental tilting modules 41 8. Further results III: Canonical bases 44 References 48