Dual Groups for Covers
For a Brylinski-Deligne (BD) central extension of \(G\) by \(\mathbf{K}_2\), specialised by a character \(\chi : \mu_n \to \mathbb{C}^\times\), the resulting metaplectic cover \(\overline G\) has its own dual group \(\hat{\overline G}\) and L-group. The construction (due to McNamara, Weissman, Gan-Gao-Weissman) extends Langlands duality to the covering setting.
Setup
Let \(F\) be a local field with \(\mu_n \subseteq F^\times\), and let \(G\) be a connected reductive group over \(F\) with split maximal torus \(T\) (in the split case; the general case is similar via Galois descent).
A BD extension \(\tilde G \to G\) is classified by \((Q, \mathcal{E}, \phi)\) where \(Q : X_*(T) \to \mathbb{Z}\) is a Weyl-invariant quadratic form. A continuous character \(\chi : K_2(F) \to \mu_n\) gives an arithmetic cover: \[ 1 \;\longrightarrow\; \mu_n \;\longrightarrow\; \overline G \;\longrightarrow\; G(F) \;\longrightarrow\; 1. \]
Construction of the dual group
The dual group \(\hat{\overline G}\) is the connected complex reductive group whose root datum is constructed from that of \(G\) as follows:
Step 1: Rescale the cocharacter lattice
Let \(B_Q : X_*(T) \times X_*(T) \to \mathbb{Z}\) be the bilinear form associated to \(Q\): \(B_Q(\lambda, \mu) = Q(\lambda + \mu) - Q(\lambda) - Q(\mu)\). Define \[ Y \;:=\; \{\lambda \in X_*(T) : B_Q(\lambda, X_*(T)) \subseteq n\mathbb{Z}\}, \] the \(n\)-image-modular lattice for \(B_Q\).
Step 2: Rescale the coroots
For each root \(\alpha\) of \(G\) with coroot \(\alpha^\vee\), define the modified coroot \[ \alpha^{\vee, (n)} \;:=\; n_\alpha \cdot \alpha^\vee, \qquad n_\alpha \;:=\; \frac{n}{\gcd(n, Q(\alpha^\vee))}. \]
Step 3: Assemble the dual root datum
The root datum of \(\hat{\overline G}\) is \[ (Y, \;\{\alpha^{\vee, (n)}\}_\alpha, \;Y^\vee, \;\{\text{recover dual roots}\}). \]
This is well-defined (Weyl-invariance of \(Q\) ensures rescaling is compatible) and yields a connected reductive complex group.
Examples
Metaplectic double cover of \(\mathrm{SL}_2\)
\(G = \mathrm{SL}_2\), \(n = 2\), standard \(Q\): \(Q(\alpha^\vee) = 1\).
- Standard root datum: \(X_*(T) = \mathbb{Z}\), \(\alpha^\vee = 2\) (in normalisation where \(B_Q = \mathrm{id}\)).
- After rescaling: \(\alpha^{\vee, (2)} = \alpha^\vee = 2\); but \(Y = \mathbb{Z}\) still, so the dual root datum is the same shape.
- However the lattices switch: \(\hat{\overline{\mathrm{SL}_2}} = \mathrm{SL}_2(\mathbb{C})\), swapping the roles of \(\mathrm{SL}_2\) and \(\mathrm{PGL}_2\) as the dual of the cover.
(In effect: the metaplectic double cover of \(\mathrm{SL}_2\) has dual group \(\mathrm{PGL}_2\), the standard Langlands dual.)
\(n\)-fold cover of \(\mathrm{GL}_r\) (Kazhdan-Patterson)
The "standard" \(n\)-fold metaplectic cover \(\overline{\mathrm{GL}_r}^{(n)}\) has dual group depending on the "twist parameter" \(c\): \[ \hat{\overline{\mathrm{GL}_r}^{(n, c)}} \;=\; \mathrm{GL}_r(\mathbb{C}) \] when \(\gcd(n, r) = 1\) (no rescaling needed), and a non-trivial modification otherwise.
\(n\)-fold cover of \(\mathrm{SL}_n\)
Classical result (Kazhdan-Patterson, Suzuki, Banks-Levy-Sepanski): the dual is the dual of a different group, expressing a non-trivial \(n\)-modular structure on the dual side.
L-group for covers
The full L-group of \(\overline G\) is \[ {}^L \overline G \;=\; \hat{\overline G}(\mathbb{C}) \;\rtimes\; W_F, \] with Weil group action induced from the Galois action on the root datum (twisted by the BD data when relevant).
L-parameters for \(\overline G\) are continuous homomorphisms \(W_F' \to {}^L \overline G\) up to \(\hat{\overline G}\)-conjugacy.
Local Langlands for covers
The conjectural local Langlands for covers maps: \[ \Pi(\overline G(F)) \;\twoheadrightarrow\; \Phi({}^L \overline G), \] L-packets parametrised by L-parameters into the cover's L-group.
For genuine representations (those on which \(\mu_n\) acts by \(\chi\)), the parameters factor through a "covered" component of the dual.
Functoriality
Functoriality for covers compares L-parameters across different covers \(\overline G \leftrightarrow \overline H\):
- Theta correspondence (\(\mathrm{Sp}_{2n} \times \mathrm{O}_m\)): matches L-parameters between the metaplectic group and the orthogonal group.
- Shimura correspondence: \(\overline{\mathrm{SL}_2}\) ↔ \(\mathrm{PGL}_2\), the classical "half-integral weight" theta lift.
- Kazhdan-Patterson lift: \(\overline{\mathrm{GL}_r}^{(n)}\) ↔ \(\mathrm{GL}_r\), generalising Shimura.
Role in modern Langlands
- Langlands programme for covers: provides the conjectural framework for automorphic forms on covering groups.
- Theta correspondence and exceptional theta: rigorous foundation for the Howe dual pair theory.
- Special L-values: many special L-values are best understood via Eisenstein series on covering groups; the dual group recipe determines the natural L-functions.
References
- Weissman, Metaplectic tori over local fields (Pac.\ J.\ Math.\ 2009).
- McNamara, Principal series representations of metaplectic groups over local fields (Bull.\ AMS 2011).
- Gan–Gao, The Langlands-Weissman program for Brylinski-Deligne extensions (Asterisque 2018).
- Gan–Gao–Weissman (eds.), L-Groups and the Langlands Program for Covering Groups (Asterisque 2018).