We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of p-adic groups and R-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on p-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of R-matrices of quantum groups depending on the cover degree and associated root system.
Bibliographical noteFunding Information:
Acknowledgements This work was supported by NSF grants DMS-1406238 (Brubaker), DMS-1601026 (Bump), and DMS-1500977 (Friedberg) and by the Max Planck Institute for Mathematics (Buciumas). We would like to thank Sergey Lysenko and Anna Puskás for useful conversations, and the referee for helpful comments.
- Hecke algebra
- Metaplectic group
- Quantum group
- Whittaker function