Suzuki's conjecture on automorphic representations for metaplectic covers

Let ${\bf A}$ be the ring of adeles, let $\tau$ be an irreducible cuspidal representation of $GL_m({\bf A})$, and let $GL_{nm}^{(n)}({\bf A})$ denote the $n$-fold metaplectic covering group. For an unramified place $\nu$, write the unramified parameters of $\tau_\nu$ as $\eta_{1,\nu},\ldots,\eta_{m,\nu}$ and choose $\mu_{i,\nu}$ with $\mu_{i,\nu}^n=\eta_{i,\nu}$. Let $P_{m,n}$ be the parabolic subgroup of $GL_{nm}$ whose Levi part is $GL_n^m$, and let $\Theta_{\mu_{i,\nu}}^{(n)}$ be the local Theta representation of $GL_n^{(n)}(F_\nu)$ twisted by $\mu_{i,\nu}$. Suzuki's conjecture. To $\tau$ one can associate an irreducible cuspidal automorphic representation $\epsilon^{(n)}(\tau)$ of $GL_{nm}^{(n)}({\bf A})$ such that, for almost all places $\nu$, its unramified constituent is the unramified constituent of

$$\operatorname{Ind}_{P_{m,n}^{(n)}(F_\nu)}^{GL_{nm}^{(n)}(F_\nu)}(\Theta_{\mu_{1,\nu}}^{(n)}\otimes\Theta_{\mu_{2,\nu}}^{(n)}\otimes\ldots\otimes\Theta_{\mu_{m,\nu}}^{(n)})\delta_{P_{m,n}}^{\frac{1}{2}}.$$

Moreover, $\epsilon^{(n)}(\tau)$ is generic and its unramified constituent has a unique Whittaker function. The conjecture supplies the residue representation needed in the paper's global integral constructions; its existence is not known in general.