Topic Local and Global Metaplectic Covers
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Local and Global Metaplectic Covers

A metaplectic cover is a central extension of a reductive group \(G(F)\) (local) or \(G(\mathbb{A}_F)\) (global) by \(\mu_n\) obtained by specialising a Brylinski-Deligne extension via a character of \(K_2\). The local-global compatibility is a major aspect of the modern Langlands programme for covers.

Local metaplectic covers

For \(F\) a local field with \(\mu_n \subseteq F\) and a connected reductive \(G/F\) with BD extension \(\tilde G\):

A continuous character \(\chi : K_2(F) \to \mu_n\) (where \(K_2(F)\) is Milnor's K_2) gives a topological central extension: \[ 1 \;\to\; \mu_n \;\to\; \overline G \;\to\; G(F) \;\to\; 1. \]

By Moore's theorem, continuous characters of \(K_2(F)\) for \(F\) local are given by the \(n\)-th Hilbert symbol \((-, -)_n : F^\times \otimes F^\times \to \mu_n\), so cover data is encoded in Hilbert symbols.

Classical examples

Metaplectic double cover \(\overline{\mathrm{Sp}_{2n}(F)}\)

For \(F = \mathbb{Q}_p\) (\(p\) odd) or \(F = \mathbb{R}\), the \(\mu_2\)-cover of \(\mathrm{Sp}_{2n}(F)\) via the quadratic Hilbert symbol. This is the Weil-metaplectic group, foundational for the Weil representation and the theta correspondence.

Higher metaplectic covers \(\overline{\mathrm{SL}_n(F)}^{(k)}\)

For \(\mu_k \subseteq F\): the \(k\)-fold metaplectic cover of \(\mathrm{SL}_n(F)\). For \(F\) a global field with class number 1, the global cover gives the Kazhdan-Patterson family of generalised theta series.

Global metaplectic covers

For \(F\) a global field and \(G/F\) with BD extension, the global metaplectic cover is \[ 1 \;\to\; \mu_n \;\to\; \overline{G(\mathbb{A}_F)} \;\to\; G(\mathbb{A}_F) \;\to\; 1, \] constructed by:

  1. At each place \(v\): the local cover \(\overline{G(F_v)}\) determined by the Hilbert symbol \((-, -)_n^{(v)}\).
  2. Take the restricted product over all \(v\): \(\overline{G(\mathbb{A}_F)} = \prod_v' \overline{G(F_v)}\) (modulo the diagonal \(\mu_n\)).
  3. The global cover descends to \(G(F)\) via the product formula for Hilbert symbols: \[ \prod_v (-,-)_n^{(v)} \;=\; 1 \] (the global reciprocity law for the \(n\)-th Hilbert symbol).

Splittings and rational sections

A splitting of the global cover over \(G(F)\) is a section \(G(F) \to \overline{G(\mathbb{A}_F)}\). The product formula ensures this exists, since each local Hilbert symbol's contribution cancels.

This is the foundation of automorphic forms on metaplectic groups: they are functions on \(\overline{G(F) \backslash G(\mathbb{A}_F)}\) of fixed central character \(\chi\) under the \(\mu_n\)-action.

Automorphic forms on covers

\(\overline{G(F) \backslash G(\mathbb{A}_F)}\) has a natural left-action by \(G(\mathbb{A}_F)\); automorphic forms on \(\overline G\) are functions on this space that:

  • Are smooth (in archimedean variables) or locally constant (in non-archimedean variables);
  • Transform by the central character \(\chi : \mu_n \to \mathbb{C}^\times\);
  • Satisfy growth and admissibility conditions.

The space \(\mathcal{A}(\overline G)\) of automorphic forms on \(\overline G\) decomposes into automorphic representations.

Examples

Modular forms of half-integral weight

For \(G = \mathrm{SL}_2 / \mathbb{Q}\) and the double cover \(\overline{\mathrm{SL}_2(\mathbb{A}_\mathbb{Q})}\): genuine automorphic representations are precisely the half-integral weight modular forms in the classical sense (Shimura, Niwa).

Generalised theta series

For \(G = \mathrm{GL}_r / \mathbb{Q}\) and the \(n\)-fold metaplectic cover: Kazhdan-Patterson constructed automorphic forms via residues of Eisenstein series, generalising the classical theta series.

Patterson's generalised \(L\)-functions

The metaplectic Eisenstein series produce \(L\)-functions with unusual Euler product structure (involving Hilbert symbols at all places), studied via integral representations.

Role in Langlands

  • Metaplectic Langlands: organises automorphic forms on metaplectic covers parallelly to the classical Langlands programme.
  • Theta correspondence: the metaplectic double cover of \(\mathrm{Sp}_{2n}\) realises the Weil representation; the theta lift is local-global.
  • Symmetric power lifts: the modular form symmetric power lifts (Kim-Shahidi, Newton-Thorne) bridge the classical and metaplectic worlds.
  • Bump-Friedberg-Hoffstein: special L-values via integrals on metaplectic Eisenstein series.

References

  • Weil, Sur certains groupes d'opérateurs unitaires (Acta Math.\ 1964).
  • Kubota, On automorphic functions and the reciprocity law in a number field (1969).
  • Kazhdan–Patterson, Metaplectic forms (Publ.\ IHES 1984).
  • Brylinski–Deligne, Central extensions of reductive groups by \(\mathbf{K}_2\) (Publ.\ IHES 2001).
  • Gan–Gao–Weissman, L-Groups and the Langlands Program for Covering Groups (Asterisque 2018).

Prerequisite Topics

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