Daily Archives: 2026/11/09

On the unitary dual of the metaplectic cover of SL(2), revisited

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Over 25 years ago, I wrote a paper on the unitary dual of the n-fold metaplectic cover of SL(2,F), over a p-adic field (pdf of original), under some assumptions (such as p>2 and p does not divide n). It was mostly expository and never published, but I referenced it in other works. Although it boasted the overly pretentious title above, it does not give a full description of the unitary dual for arbitrary covering degree and wild residue characteristic.

I asked AI agents Argon-Gemini and Sol-ChatGPT to look it over and suggest corrections, each acting as referees. They found several errors, some serious, but they easily corrected them all. In addition, the paper was revised for readability, adding a literature review and glossary. The AI-aided corrected version is available (pdf here).

Here is an AI-generated summary of the corrected version:

This manuscript studies genuine representations of the n-fold metaplectic cover \widetilde{G} of G=\mathrm{SL}(2,F), where F is a nonarchimedean local field containing the nth roots of unity. The central complication is that the inverse image of the diagonal torus can be nonabelian. This affects its irreducible representations, orbital integrals, and principal-series characters.

Four main results

  1. Representations of the covering torus. Write \widetilde{A} for the covering torus, Z_A for its center, and d=n/\gcd(n,2). In the tame setting, each genuine character \lambda of Z_A determines a unique irreducible genuine representation \chi_\lambda of \widetilde{A}, of dimension d. Its character satisfies \rm{tr}\chi_\lambda(a)=d\lambda(a) for a\in Z_A, and \rm{tr}\chi_\lambda(a)=0 for a\notin Z_A. Thus the trace vanishes away from the center of the covering torus.
  2. Split orbital integrals. The corrected construction uses a subgroup that actually centralizes the element being integrated. For residue characteristic prime to 2n, the normalized split orbital-integral transform maps genuine compactly supported test functions onto the compactly supported, genuine, Weyl-invariant locally constant functions on Z_A.
  3. Principal-series character formula. For \Pi=\rm{Ind}_{\widetilde B}^{\widetilde G}(\chi\otimes\delta_B^{1/2}), the character on regular split elements is \Theta_\Pi(a)=\frac{\rm{tr}\chi(a)+\rm{tr}\chi^{\widetilde w}(a)}{|D(a)|^{1/2}}. The second term involves Weyl conjugation, not naive inversion. This distinction is essential for the character to transform correctly under the central roots of unity.
  4. A restricted character transfer. Suppose \gcd(p,2n)=1 and either n is odd or 4\mid n. A power map \nu(h(t))=(h(t^d),1) gives an ordinary torus character \eta_\lambda=\lambda\circ\nu. For test functions with matching normalized split orbital integrals, the corrected paper proves \rm{tr}\Pi(\chi_\lambda)(\bar f)=\rm{tr}\rm{Ind}_{B}^{G}(\eta_\lambda\delta_B^{1/2})(f). It also establishes the existence of a split-matching function f for each genuine test function \bar f on the cover.

What remains unresolved?

Despite its title, the revised paper does not claim a complete classification of the genuine unitary dual. It surveys known complementary-series and supercuspidal results, but does not establish a uniform complementary-series theorem in arbitrary degree. Nor does the transfer theorem cover n\equiv2\pmod4 (including the double cover), elliptic orbital integrals, or a full correspondence of all admissible representations.

The revision therefore identifies a precise, provable core of the original project while making clear where further work is needed.