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Central extensions of reductive groups by \(K_2\). (English) Zbl 1093.20027

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If a group \(G\) is perfect (i.e., \(G\) is equal to its commutator subgroup), it has a universal central extension \(\pi\to E_G\to G\). Suppose that \(G\) is the group \(G(k)\), where \(G\) is a split simple simply connected algebraic group over a field \(k\). Then the adjoint group \(G^{ad}(k)\) acts on \(G(k)\), hence on the corresponding universal central extension, and the group of co-invariants \(\pi_{G^{ad}(k)}\) is the Milnor group \(K_2(k)\) (if \(G\) is not of type \(C_n\), we even have \(\pi\simeq K_2(k)\)). The universal central extension hence gives rise to a central extension \(K_2(k)\to G(k)^{\sim}\to G(k)\). If moreover \(k\) is a global field, this yields in turn a topological \(G(k)\)-split central extension of the adelic group \(G({\mathbf A})\) by the group \(\mu\) of roots of unity in \(k\): \[ \begin{tikzcd} & & G(k)\ar[d]\\ \mu\ar[r] & G(\mathbf A)^{\sim}\ar[r] & G(\mathbf A)\rlap{\,.}\end{tikzcd} \] In this paper, the authors consider (connected) reductive groups \(G\) over \(k\), for which they define objects called central extensions by \({\mathbf K}_2\), which give rise to a central extension \(G(k)\) by \(K_2(k)\) and, for a global field \(k\), to a \(G(k)\)-split adelic central extension as above. More specifically, an algebraic group \(G\) over \(k\) defines a sheaf \(S\mapsto G(S)=\operatorname{Hom}_{\text{Spec}(k)}(S,G)\) on the big Zariski site \(\text{Spec}(k)_{\text{Zar}}\) of \(\text{Spec}(k)\), and a central extension of \(G\) by \({\mathbf K}_2\) is a central extension, on \(\text{Spec}(k)_{\text{Zar}}\), of the sheaf of groups \(G\) by the sheaf of Abelian groups \({\mathbf K}_2\). By SGA7, it can be viewed as a \({\mathbf K}_2\)-torsor \(P\) on \(G\), provided with a certain multiplicative structure. In the particular case where \(k\) is infinite and \(G\), as an algebraic variety over \(k\), is connected and unirational, there is an alternate, heuristically useful description by 2-cocycles in a canonical complex \[ K_2k\to K_2k(G)\to K_2k(G\times G)\to K_2k(G\times G\times G)\to\cdots. \] The main goal of the authors is to classify the central extensions of \(G\) by \({\mathbf K}_2\) and to determine their functoriality in \(G\) and \(k\). To this end, instead of determining the set of isomorphism classes, they rather determine the category of central extensions by \({\mathbf K}_2\). They show that this category is naturally equivalent to a more down-to-earth category \(\mathcal C\). More precisely, let \(G\) be a reductive group over \(k\), with maximal torus \(T\) split over a Galois extension \(k'/k\), and let \(Y\) be the dual of the character group of \(T\) over \(k'\). Then \(\mathcal C\) is the category of triples \((Q,\xi,\varphi)\), where \(Q\) is a Weyl and Galois invariant integer-valued quadratic form on \(Y\); \(\xi\) is a Galois equivariant central extension of \(Y\) by \(k^{\prime\ast}\) such that the commutator of liftings \(\widetilde y_1,\widetilde y_2\in\xi\) of \(y_1,y_2\in Y\) is given by \((-1)^{B(y_1,y_2)}\in k^\ast\), \(B\) denoting the bilinear form associated to \(Q\); and finally \(\varphi\) is a certain Galois equivariant “covering” \(\xi_{sc}\to\xi\), where \(\xi_{sc}\) a Galois equivariant central extension \(k^{\prime\ast}\to\xi_{sc}\to Y_{sc}\) deduced from \(\xi\) and from the simply connected covering of the derived group of \(G\). Technically, Galois descent is proved for pointed \({\mathbf K}_2\)-torsors, and the categorical approach allows to apply it to reduce the classification problem to the split case. Note that the classification theorem obtained here includes earlier particular results of Deligne (1976, 1996). The authors give many examples in the last paragraph of their paper, and they express hope that, for a global field, their study will prove useful in the harmonic analysis of functions on \(G(A)^{\sim}/G(k)\).

MSC:

20G15 Linear algebraic groups over arbitrary fields
19C09 Central extensions and Schur multipliers
20G30 Linear algebraic groups over global fields and their integers
20G10 Cohomology theory for linear algebraic groups
20E22 Extensions, wreath products, and other compositions of groups
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