Topic Brylinski-Deligne K2-Extensions
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Brylinski-Deligne K2-Extensions

A Brylinski-Deligne extension is a central extension of a reductive group sheaf by the sheaf \(\mathbf{K}_2\) (Quillen's algebraic K-theory sheaf). Such extensions provide the canonical framework for metaplectic covers of reductive groups — the natural setting for the Langlands programme for covering groups.

Setup

Let \(X\) be a scheme (e.g.\ \(\mathrm{Spec}\, F\) for \(F\) a field). The sheaf \(\mathbf{K}_2 = \mathbf{K}_2^M\) is the sheafification of \(U \mapsto K_2^M(\Gamma(U, \mathcal{O}_U))\), with \(K_2^M\) the second Milnor K-theory group. On a smooth scheme this agrees with Quillen's \(K_2\) sheaf.

Let \(G\) be a connected reductive group over \(F\), regarded as an fppf sheaf of groups.

Definition

A Brylinski-Deligne extension of \(G\) by \(\mathbf{K}_2\) is a short exact sequence of fppf sheaves of groups on the small Zariski (or étale) site: \[ 1 \;\longrightarrow\; \mathbf{K}_2 \;\longrightarrow\; \tilde G \;\longrightarrow\; G \;\longrightarrow\; 1 \] with \(\mathbf{K}_2\) central in \(\tilde G\).

Such extensions form an abelian category \(\mathrm{CExt}(G, \mathbf{K}_2)\) under Baer sum.

Classification (BD theorem)

Theorem (Brylinski-Deligne, 2001). For a split connected reductive \(G\) over a field \(F\) with split maximal torus \(T\), root datum \((X, \Phi, X^\vee, \Phi^\vee)\):

The category of Brylinski-Deligne extensions of \(G\) by \(\mathbf{K}_2\) is equivalent to the category of triples \((Q, \mathcal{E}, \phi)\) where:

  1. \(Q : X^\vee \to \mathbb{Z}\) is a Weyl-invariant integer-valued quadratic form (\(Q(w \lambda) = Q(\lambda)\) for \(w \in W\)).
  2. \(\mathcal{E}\) is a central extension of \(T\) by \(\mathbf{K}_2\) (equivalently: a bilinear form \(D : X^\vee \times X^\vee \to F^\times\) compatible with \(Q\)).
  3. \(\phi\) is a compatibility datum linking the root subgroup extensions with \((Q, \mathcal{E})\).

For \(G\) semisimple simply connected, \(\mathcal{E}\) and \(\phi\) are determined, so the classification reduces to: a Weyl-invariant integer-valued quadratic form \(Q\) on \(X^\vee\).

Examples

\(G = \mathrm{SL}_n\)

\(X^\vee = \{(a_i) \in \mathbb{Z}^n : \sum a_i = 0\}\); the form is \(Q(\lambda) = c \cdot \sum a_i^2\) for some \(c \in \mathbb{Z}\). The extensions are classified by \(\mathbb{Z}\); the "fundamental" extension is the Steinberg-Matsumoto extension (the universal central extension of \(\mathrm{SL}_n\)).

\(G = \mathbb{G}_m\)

The classifying data is a single element of \(F^\times\) (a "tame symbol"); the resulting cover is the multiplicative analogue of the Heisenberg group.

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

The \(\mathbb{Z}/2\)-cover of \(\mathrm{SL}_2(F)\) (or \(\mathrm{Sp}_{2n}(F)\)) used in the theory of theta correspondence arises as a quotient of a Brylinski-Deligne extension by a suitable subgroup of \(\mathbf{K}_2\).

From BD extension to topological cover

Given a Brylinski-Deligne extension \(\tilde G\) of \(G\) by \(\mathbf{K}_2\), and a local field \(F\) with a chosen continuous character \(\chi : K_2(F) \to \mu_n\), one obtains a topological central extension \[ 1 \;\to\; \mu_n \;\to\; \overline G \;\to\; G(F) \;\to\; 1. \]

For \(F\) local, \(K_2(F)\) is computed via Moore's theorem; the relevant characters factor through the Hilbert symbol \((-, -)_n : F^\times \otimes F^\times \to \mu_n\).

This is the "geometric source" of metaplectic covers: BD extensions provide the universal target, and the Hilbert symbol gives the arithmetic realisation.

Functoriality

BD extensions are functorial in \(G\):

  • A morphism \(f : G \to H\) gives a pullback functor \(f^* : \mathrm{CExt}(H, \mathbf{K}_2) \to \mathrm{CExt}(G, \mathbf{K}_2)\).
  • Galois descent works: \(\mathrm{CExt}(G_{\bar F}, \mathbf{K}_2)^{\Gamma}\) contains the \(F\)-rational extensions.

Dual group for covers

For a BD extension classified by \((Q, \mathcal{E}, \phi)\), the associated dual group is constructed by Weissman and others. For the metaplectic \(n\)-cover of \(G\) at \((Q, \chi)\), the dual is a new reductive group \(\hat G^{(n, Q)}\) whose root datum is twisted by the bilinear form \(B_Q\) modulo \(n\).

For \(G = \mathrm{SL}_n\) and the standard \(Q\): the metaplectic dual of the double cover of \(\mathrm{SL}_2\) is again \(\mathrm{PGL}_2\) (self-dual case).

For higher-degree covers: the dual is a new group with rescaled coroots.

Role in Langlands

  • Langlands programme for covers (Gan, Weissman, Lysenko): the conjectural local and automorphic Langlands for metaplectic and general BD covers requires the BD-extension classification as input to construct the dual group.
  • Theta correspondence: the metaplectic double cover of \(\mathrm{Sp}_{2n}(F)\) is the natural setting for the Weil representation and the local theta correspondence.
  • Eisenstein series for covers: Eisenstein series on \(\overline G(\mathbb{A})\) (for global covers) generalise the classical metaplectic Eisenstein series; relevant to Kazhdan-Patterson generalisations.

References

  • Brylinski–Deligne, Central extensions of reductive groups by \(\mathbf{K}_2\) (Publ.\ IHES 2001).
  • Weissman, L-groups and parameters for covering groups (Asterisque 2018).
  • Gan–Gao–Weissman, L-groups and the Langlands program for covering groups (Asterisque 2018).
  • Kac, Infinite-dimensional Lie algebras (1990), §3.13 (the Kac-Peterson presentation of central extensions).

Prerequisite Topics

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