The type C Weyl group multiple Dirichlet series conjecture

Let $\Phi=C_r$ for any positive integer $r$, let $n$ be odd, and let $Z_\Psi(\mathbf{s};\mathbf{m})$ be the Dirichlet series described in the source, with coefficients $H^{(n)}(p^{\mathbf{k}};p^{\mathbf{l}})$ defined from the prime-power construction. Here $\mathbf{s}$ has $r$ complex variables, and $W(Sp(2r))$ denotes the Weyl group of $Sp(2r)$.

Type C multiple-Dirichlet-series conjecture. The series $Z_\Psi(\mathbf{s};\mathbf{m})$ should have meromorphic continuation to $\mathbb{C}^r$ and satisfy a group of functional equations isomorphic to $W(Sp(2r))$, with the specified action on $\mathbb{C}^r$; moreover, it should be the Whittaker coefficient of a minimal parabolic Eisenstein series on an $n$-fold metaplectic cover of $SO_{2r+1}(F_S)$.

The conjecture gives the desired analytic and automorphic properties for the Gelfand–Tsetlin-based construction in type $C$. The paper constructs the series for odd $n$ and proves agreement with earlier definitions, as well as the required properties in the cases established in the paper; the general assertion stated here is not presented as fully proved.