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Abstract: We develop the theory of Weyl group multiple Dirichlet series for rootsystems of type C. For an arbitrary root system of rank r and a positiveinteger n, these are Dirichlet series in r complex variables with analyticcontinuation and functional equations isomorphic to the associated Weyl group.In type C, they conjecturally arise from the Fourier-Whittaker coefficients ofminimal parabolic Eisenstein series on an n-fold metaplectic cover of SO2r+1.For any odd n, we construct an infinite family of Dirichlet seriesconjecturally satisfying the above analytic properties. The coefficients ofthese series are exponential sums built from Gelfand-Tsetlin bases of certainhighest weight representations. Previous attempts to define such series byBrubaker, Bump, and Friedberg in 6 and 7 required n to be sufficientlylarge, so that coefficients could be described by Weyl group orbits. Wedemonstrate that our construction agrees with that of 6 and 7 in the casewhere both series are defined, and hence inherits the desired analyticproperties for n sufficiently large. Moreover our construction is valid evenfor n=1, where we prove our series is a Whittaker coefficient of an Eisensteinseries. This requires the Casselman-Shalika formula for unramified principalseries and a remarkable deformation of the Weyl character formula of Hamel andKing 20.

Author: Jennifer Beineke, Ben Brubaker, Sharon Frechette

Source: https://arxiv.org/

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