Topic L-Groups for Covers
topic admitted

L-Groups for Covers

The L-group of a Brylinski-Deligne cover is the receptacle of L-parameters for the metaplectic representation theory of the cover. Built from the dual group of the cover plus a Galois action, it extends the classical L-group construction to all BD covers.

Setup

Let \(G\) be a connected reductive group over a local field \(F\), and let \(\tilde G\) be a Brylinski-Deligne extension of \(G\) by \(\mathbf{K}_2\) classified by data \((Q, \mathcal{E}, \phi)\). Pick a continuous character \(\chi : \mu_n \to \mathbb{C}^\times\) to specialise the BD extension to a finite-degree cover: \[ 1 \;\to\; \mu_n \;\to\; \overline G \;\to\; G(F) \;\to\; 1. \]

Dual group of the cover

The dual group \(\hat{\overline G}\) is constructed from the rescaled root datum (rescaling cocharacters and coroots according to \(Q\) modulo \(n\)); see dual groups for covers.

\(\hat{\overline G}\) is a connected reductive complex group, possibly different from \(\hat G\) — the rescaling can change Dynkin type.

L-group construction

The L-group of the cover is the semi-direct product: \[ {}^L \overline G \;:=\; \hat{\overline G}(\mathbb{C}) \;\rtimes\; W_F, \] where:

  • \(W_F\) is the Weil group of \(F\);
  • The action of \(W_F\) on \(\hat{\overline G}\) is the natural one coming from the Galois action on the rescaled root datum.

For the trivial cover (\(n = 1\)): \({}^L \overline G = {}^L G\) (the classical L-group).

For metaplectic covers (\(n \ge 2\)): \({}^L \overline G\) can be significantly different from \({}^L G\).

Examples

Metaplectic double cover of \(\mathrm{SL}_2\)

For \(\overline{\mathrm{SL}_2}\) (the genuine double cover): \[ \hat{\overline{\mathrm{SL}_2}} \;=\; \mathrm{SL}_2(\mathbb{C}). \] (Effectively swaps the dual: \(\widehat{\mathrm{SL}_2} = \mathrm{PGL}_2\) classically, but for the metaplectic cover the dual is \(\mathrm{SL}_2\).)

L-parameters for \(\overline{\mathrm{SL}_2}\): continuous maps \(W_F \to \mathrm{SL}_2(\mathbb{C}) \rtimes W_F\) — these correspond to 2-dimensional Galois representations of even determinant.

Kazhdan-Patterson cover of \(\mathrm{GL}_r\)

For the \(n\)-fold metaplectic cover \(\overline{\mathrm{GL}_r}^{(n)}\): \[ \hat{\overline{\mathrm{GL}_r}^{(n)}} \;=\; \mathrm{GL}_r(\mathbb{C}) \] when \(\gcd(n, r) = 1\); otherwise a more subtle rescaled group.

\(\overline{G}_2\) and exceptional covers

For the metaplectic cover of \(G_2\): the dual is again \(G_2\) (self- dual under reflection-coroot duality), but with a non-trivial central extension structure on the L-group reflecting the K_2 data.

L-parameters for covers

An L-parameter for \(\overline G\) is a continuous homomorphism \[ \varphi : W_F \times \mathrm{SL}_2(\mathbb{C}) \;\to\; {}^L \overline G \] satisfying:

  • Commuting with projection to \(W_F\);
  • Semisimple image of \(\varphi|_{W_F}\);
  • Algebraic restriction to \(\mathrm{SL}_2(\mathbb{C})\).

Up to \(\hat{\overline G}\)-conjugacy.

Genuine vs.\ non-genuine

Representations \(\pi\) of \(\overline G\) split into:

  • Non-genuine: \(\mu_n\) acts trivially; these are representations of \(G(F)\) lifted to \(\overline G\). They factor through the classical L-group.
  • Genuine: \(\mu_n\) acts by \(\chi\) (the chosen non-trivial character). These are the "honest" covering representations, with L-parameters into the genuine part of \({}^L \overline G\).

The L-parameter formalism distinguishes these via the central character of the L-parameter restricted to the centre of \(\hat{\overline G}\).

Functoriality

L-morphisms \({}^L \overline H \to {}^L \overline G\) between covers encode the functoriality phenomena specific to the metaplectic setting:

  • Shimura correspondence: \({}^L \overline{\mathrm{SL}_2} \to {}^L \mathrm{PGL}_2\) for half-integral weight modular forms.
  • Kazhdan-Patterson lifts: \({}^L \overline{\mathrm{GL}_r}^{(n)} \to {}^L \mathrm{GL}_r\).
  • Theta correspondence: \({}^L \overline{\mathrm{Sp}_{2n}} \times {}^L \mathrm{O}_m \to \ldots\) expressing the Howe duality at the L-group level.

L-functions for covers

For an L-parameter \(\varphi\) and a finite-dimensional representation \(r\) of \({}^L \overline G\): \[ L(s, \pi, r) \;:=\; L(s, r \circ \varphi) \] generalises the classical L-function to covers.

Role in modern Langlands

  • Metaplectic Langlands: organises automorphic forms on covering groups.
  • Special L-values: many special L-values are best expressed via Eisenstein series on covers; the dual-group recipe determines the natural L-functions to evaluate.
  • Theta correspondence and theta lifts: rigorous framework for Howe's dual pair theory.

References

  • Weissman, Metaplectic tori over local fields (Pac.\ J.\ Math.\ 2009).
  • Gan–Gao, The Langlands-Weissman program for Brylinski-Deligne extensions (Asterisque 2018).
  • Gan–Gao–Weissman, L-Groups and the Langlands Program for Covering Groups (Asterisque 2018).

Prerequisite Topics