The Shimura–Waldspurger correspondence for $\mathrm {Mp}_{2n}$

Abstract

We generalize the Shimura–Waldspurger correspondence, which describes the generic part of the automorphic discrete spectrum of the metaplectic group $\mathrm {Mp}_2$, to the metaplectic group $\mathrm {Mp}_{2n}$ of higher rank. To establish this, we transport Arthur’s endoscopic classification of representations of the odd special orthogonal group $\mathrm {SO}_{2r+1}$ with $r \gg 2n$ by using a result of J.-S. Li on global theta lifts in the stable range.