TY - JOUR
T1 - Weyl group multiple dirichlet series of type C
AU - Beineke, Jennifer
AU - Brubaker, Benjamin
AU - Frechette, Sharon
PY - 2011
Y1 - 2011
N2 - We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For a root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. They conjecturally arise as Whittaker coefficients of Eisenstein series on a metaplectic group with cover degree n. For type C and n odd, we construct an infinite family of Dirichlet series and prove they satisfy the above analytic properties in many cases. The coefficients are exponential sums built from Gelfand-Tsetlin bases of certain highest weight representations. Previous attempts to define such series by Brubaker, Bump, and Friedberg required n sufficiently large, so that coefficients were described by Weyl group orbits. We demonstrate that these two radically different descriptions match when both are defined. Moreover, for n = 1, we prove our series are Whittaker coefficients of Eisenstein series on SO(2r+1).
AB - We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For a root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. They conjecturally arise as Whittaker coefficients of Eisenstein series on a metaplectic group with cover degree n. For type C and n odd, we construct an infinite family of Dirichlet series and prove they satisfy the above analytic properties in many cases. The coefficients are exponential sums built from Gelfand-Tsetlin bases of certain highest weight representations. Previous attempts to define such series by Brubaker, Bump, and Friedberg required n sufficiently large, so that coefficients were described by Weyl group orbits. We demonstrate that these two radically different descriptions match when both are defined. Moreover, for n = 1, we prove our series are Whittaker coefficients of Eisenstein series on SO(2r+1).
KW - Eisenstein series
KW - Gelfand-Tsetlin pattern
KW - Metaplectic group
KW - Weyl group multiple Dirichlet series
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U2 - 10.2140/pjm.2011.254.11
DO - 10.2140/pjm.2011.254.11
M3 - Article
AN - SCOPUS:84858422369
SN - 0030-8730
VL - 254
SP - 11
EP - 46
JO - Pacific Journal of Mathematics
JF - Pacific Journal of Mathematics
IS - 1
ER -