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Hecke modules from metaplectic ice. (English) Zbl 1466.20002

Summary: 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.

MSC:

20C08 Hecke algebras and their representations
11F68 Dirichlet series in several complex variables associated to automorphic forms; Weyl group multiple Dirichlet series
16T20 Ring-theoretic aspects of quantum groups
16T25 Yang-Baxter equations
22E50 Representations of Lie and linear algebraic groups over local fields
20G42 Quantum groups (quantized function algebras) and their representations

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