Extend the converse theorem to PGL2(C)

Establish that every nontrivial multiplicative representation of the Lie group PGL2(C) with discrete spectrum is isomorphic, as a multiplicative representation, to L^2(Γ \ G) for some cocompact lattice Γ ⊂ PGL2(C). This would generalize Theorem 1.1, proved here for PSL2(R), to PGL2(C).

Background

The paper proves a converse theorem for PSL2(R): any nontrivial multiplicative representation with discrete spectrum is isomorphic to L2(Γ \ G) for a cocompact lattice Γ. In Remark 1.5, the authors discuss extending this result to other semisimple Lie groups, specifically PGL2(C).

They note that the effectiveness of related bootstrap techniques for hyperbolic 3-manifolds (as studied in Bonifacio–Mazáč–Pal) suggests the analogous result should hold for PGL2(C). A proof would resolve Open Problem 8.1 in Bonifacio–Mazáč–Pal, linking this work to a prominent question in the literature.

References

Definition~\ref{def:mult_rep} extends essentially verbatim to other semisimple Lie groups, and one can ask if Theorem~\ref{thm:eq_Gelfand_duality} extends as well. Specifically for $G = \PGL_2(C)$, the effectiveness of suggests that Theorem~\ref{thm:eq_Gelfand_duality} holds. A proof of this would solve Open Problem~8.1.

— A converse theorem for hyperbolic surface spectra and the conformal bootstrap (2509.17935 - Adve, 22 Sep 2025) in Remark 1.5 (Other groups), Section 1.2