Unitarity preservation for lifting between orthogonal and metaplectic groups

Determine whether the lifting operator constructed between $G=\mathrm{SO}(n+1,n)$ and the nonlinear group $\widetilde{G'}=\mathrm{Mp}_{2n}(\mathbb{R})$ preserves unitarity in general.

Background

The paper surveys earlier lifting constructions relating stable virtual representations of real reductive groups to genuine representations of nonlinear covers of Langlands-dual groups. Adams and Renard established such a lifting between SO(n+1,n)\mathrm{SO}(n+1,n) and the metaplectic group, and proved that it preserves stability and maps stable sums of tempered representations to stable sums of tempered representations.

The unresolved issue is whether this lifting also maps unitary representations, or suitable stable combinations of them, to genuine unitary representations. The present paper addresses unitarity preservation for a substantial class of representations of complex classical groups, but does not resolve the general real-group case described here.

References

In this setting, however, it remains unclear in general whether this lifting preserves unitarity.

— Unitary Shimura Correspondence for Complex Classical Groups  (2608.26795 - Tsai et al., 27 Aug 2026) in Section 1, Introduction

As for whether $\mathrm{Lift}$ preserves unitarity or not, there are only partial results. Namely, for $\mathrm{GL}_n(\mathbb{R})$, proved that the lift of an irreducible unitary representation is either zero or an irreducible unitary representation, while no general result is known for other groups.

— Unitary Shimura Correspondence for Complex Classical Groups  (2608.26795 - Tsai et al., 27 Aug 2026) in Section 1, Introduction