Central Extensions of Algebraic Groups
A central extension of an algebraic group \(G\) by an abelian group \(A\) is a short exact sequence with \(A\) in the centre. Such extensions classify covering groups, metaplectic covers, and the arithmetic phenomena studied in the Langlands programme for covers.
Definition
Let \(G\) be a group scheme (or fppf sheaf of groups) over a base \(S\), and \(A\) a sheaf of abelian groups on \(S\). A central extension of \(G\) by \(A\) is a short exact sequence \[ 1 \;\longrightarrow\; A \;\stackrel{\iota}{\longrightarrow}\; \tilde G \;\stackrel{\pi}{\longrightarrow}\; G \;\longrightarrow\; 1 \] of fppf sheaves of groups such that \(\iota(A)\) is contained in the centre of \(\tilde G\).
Equivalence: two extensions are equivalent if there is an isomorphism \(\tilde G \to \tilde G'\) of fppf sheaves of groups making the obvious diagram commute. Equivalence classes form an abelian group under Baer sum.
Classification by cohomology
For \(G\) a sheaf of groups and \(A\) a sheaf of abelian groups (with \(G\) acting trivially): \[ \mathrm{CExt}(G, A) \;\cong\; H^2_{\mathrm{fppf}}(BG, A), \] the second cohomology of the classifying stack \(BG\).
For \(G\) abelian: equivalence classes are \(\mathrm{Ext}^1(G, A)\) in the abelian category of fppf sheaves; this is computable by standard homological methods.
For \(G\) non-abelian: \(\mathrm{CExt}(G, A)\) is still an abelian group (under Baer sum), but the description via \(H^2\) requires care.
Examples
\(\mathbb{G}_m\) by \(\mu_n\)
The Kummer sequence \[ 1 \to \mu_n \to \mathbb{G}_m \xrightarrow{[n]} \mathbb{G}_m \to 1 \] gives a central extension of \(\mathbb{G}_m\) by \(\mu_n\). Equivalence classes are \(\mathrm{Ext}^1(\mathbb{G}_m, \mu_n)\), computed via Kummer theory.
Heisenberg group
The integer Heisenberg group is a central extension \[ 1 \to \mathbb{Z} \to H_3(\mathbb{Z}) \to \mathbb{Z}^2 \to 1. \] Its algebraic analogue (for additive groups) is \(1 \to \mathbb{G}_a \to H \to \mathbb{G}_a^2 \to 1\), classified by \(\mathrm{Ext}^1(\mathbb{G}_a^2, \mathbb{G}_a)\), which is non-trivial.
Universal central extension of \(\mathrm{SL}_n\)
For \(n \ge 3\) and an algebraically closed field \(k\), \(\mathrm{SL}_n(k)\) is perfect but has a non-trivial universal central extension: \[ 1 \to K_2(k) \to \mathrm{St}_n(k) \to \mathrm{SL}_n(k) \to 1, \] where \(\mathrm{St}_n(k)\) is the Steinberg group and \(K_2(k)\) is the (Milnor) \(K_2\) group of \(k\). This is Matsumoto's theorem and the prototype for Brylinski-Deligne extensions.
Spin extension of \(\mathrm{SO}_n\)
\[ 1 \to \mu_2 \to \mathrm{Spin}_n \to \mathrm{SO}_n \to 1, \] the algebraic-group version of the topological double cover.
Metaplectic group
For \(F\) a local field of residue characteristic \(\ne 2\): \[ 1 \to \mu_2 \to \overline{\mathrm{Sp}_{2n}(F)} \to \mathrm{Sp}_{2n}(F) \to 1, \] the metaplectic double cover. Topologically it does not split; in BD-language it arises from the standard K_2-extension by specialising via the quadratic Hilbert symbol.
Brylinski-Deligne extensions
The most important non-classical case: central extensions of a reductive group \(G\) by the K-theory sheaf \(\mathbf{K}_2\): \[ 1 \to \mathbf{K}_2 \to \tilde G \to G \to 1. \]
These are classified by combinatorial data on the root datum (quadratic forms, bilinear pairings). See BD K2-extensions.
Universal central extension
For a perfect group \(G\) (i.e.\ \(G = [G, G]\)), the universal central extension is the initial object in \(\mathrm{CExt}(G, -)\): \[ 1 \to H_2(G, \mathbb{Z}) \to \tilde G^{\mathrm{univ}} \to G \to 1, \] where \(H_2\) is the Schur multiplier. The universal central extension exists uniquely up to isomorphism for any perfect group.
For \(G\) a semisimple algebraic group: the Schur multiplier is related to \(K_2\) of the base field.
Splittings and obstructions
For a central extension \(1 \to A \to \tilde G \to G \to 1\), a splitting is a homomorphism \(s : G \to \tilde G\) with \(\pi \circ s = \mathrm{id}\).
The obstruction to splitting lies in \(H^2(G, A)\). For trivial \(A\): the extension splits iff it is "lifted from \(G\)."
In the metaplectic case, the extension does not split as algebraic groups, but it splits over arithmetically interesting subgroups: unipotent radicals of parabolics, integral subgroups (for global metaplectic covers), and parahoric subgroups (for local covers).
Role in Langlands
- Langlands programme for covers: requires functorial central extensions to construct L-groups for covering groups.
- Theta correspondence: the metaplectic cover of \(\mathrm{Sp}_{2n}\) is essential for Weil representations and the theta dual pair.
- Eisenstein series for covers: integration over Eisenstein series on the metaplectic group recovers special \(L\)-values.
- Kazhdan-Patterson: their "generalised theta series" arise from Eisenstein series on \(\overline{\mathrm{GL}_n}\) (\(n\)-fold cover).
References
- Moore, Group extensions of p-adic and adelic linear groups (Publ.\ IHES 1968).
- Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés (Ann.\ Sci.\ ENS 1969).
- Deligne, Extensions centrales de groupes algébriques simplement connexes (CRAS 1996).
- Brylinski–Deligne, Central extensions of reductive groups by \(\mathbf{K}_2\) (Publ.\ IHES 2001).