Let \(F\) denote a non-Archimedean local field of characteristic zero and let \(\operatorname{Irr}(G)\) stand for the set of isomorphism classes of irreducible (genuine) representations of \(G\). Let \(W\) denote a symplectic space of dimension \(2n\) over \(F\) and let \(\mathrm{Mp}(W)\) denote the associated metaplectic group. Also, let \(\psi\) stand for a nontrivial additive character of \(F\).
In the paper under review, the authors prove that there is a bijection \[ \Theta_{\psi} : \operatorname{Irr}(\mathrm{Mp}(W)) \leftrightarrow \operatorname{Irr}(\mathrm{SO}(V^{+})) \cup \operatorname{Irr}(\mathrm{SO}(V^{-})), \] where \(V^{+}\) (respectively, \(V^{-}\)) is the split (respectively, non-split) quadratic space of discriminant \(1\) and dimension \(2n+1\), while \(\mathrm{SO}(V^{+}\) (respectively, \(\mathrm{SO}(V^{-})\)) is the associated special orthogonal group. The bijection \(\Theta_{\psi}\) is given by the theta correspondence with respect to \(\psi\) for the group \(\mathrm{Mp}(W) \times \mathrm{SO}(V^{\pm})\).
This extends the results of J. L. Waldspurger [Prog. Math. 12, 357–369 (1981; Zbl 0454.10015); Forum Math. 3, No. 3, 219–307 (1991; Zbl 0724.11026)] for \(n=1\) and provides a classification of the irreducible representations of \(\mathrm{Mp}(W)\) in terms of those of the special orthogonal groups \(\mathrm{SO}(V^{+})\) and \(\mathrm{SO}(V^{-})\). Also, the authors determine a series of particularly interesting properties of this classification. We list some of them in what follows.
Let \(\pi \in \operatorname{Irr}(\mathrm{SO}(V))\) and \(\sigma \in \operatorname{Irr}(\mathrm{Mp}(W))\) correspond under \(\Theta_{\psi}\). Then the following hold:
\(\pi\) is a discrete series if and only if \(\sigma\) is a discrete series.
\(\pi\) is tempered if and only if \(\sigma\) is tempered.
If \(\pi\) and \(\sigma\) are discrete series, then the formal degrees of \(\pi\) and \(\sigma\), with respect to the appropriate Haar measures, coincide.
If \(\pi\) is a generic representation of \(\mathrm{SO}(V^{+})\) then \(\sigma\) is a \(\psi\)-generic representation of \(\mathrm{Mp}(W)\); if \(\sigma\) is \(\psi\)-generic and tempered, then \(\pi\) is generic.
If \(\rho\) is an irreducible representation of \(\mathrm{GL}(k,F)\) and \(\chi\) a \(1\)-dimensional character of \(\mathrm{GL}(1,F)\), then we have the following equalities of Plancherel measures, \(L\)-factors and \(\varepsilon\)-factors: \(\mu (s, \pi \times \rho, \psi) = \mu (s, \sigma \times \rho, \psi)\), \(L(s, \pi \times \chi) = L_{\psi}(s, \sigma \times \chi)\) and \(\varepsilon(s, \pi \times \chi, \psi)= \varepsilon(s, \sigma \times \chi, \psi)\). Additionally, if \(\pi\) is generic we have \(L(s, \pi \times \rho) = L_{\psi}(s, \sigma \times \rho)\) and \(\varepsilon(s, \pi \times \rho, \psi)= \varepsilon(s, \sigma \times \rho, \psi)\).
There are two extensions of \(\pi\) to \(\mathrm{O}(V)\), and we denote them by \(\pi^{+}\) and \(\pi^{-}\). Then \(\pi^{\varepsilon}\) participates in the theta correspondence with \(\mathrm{Mp}(W)\), with respect to \(\psi\), if and only if \(\varepsilon = \varepsilon(V) \cdot \varepsilon(1/2, \pi)\), where \(\varepsilon(V) = 1\) if \(V\) is split and \(\varepsilon(V) = -1\) if \(V\) is non-split.