Metaplectic covers of \(\text{GL}_n\) and the Gauss-Schering lemma (Q1826294)

This paper repeats results from [\textit{R. Hill}, Math. Ann. 310, No. 4, 735--775 (1998; Zbl 0908.11056)], but with more elementary proofs. Let \(k\) be a global field with group \(\mu\) of roots of unity and adèle ring \(\mathbb{A}\). Let \(G/K\) be the affine algebraic group \(\text{GL}_n/k\) (or \(\text{SL}_n/k\)). Metaplectic extensions of \(G\) by \(\mu\) are classified by elements of \(H^2(G(\mathbb{A}),\mu)\) which split when restricted to \(G(k)\). There is a canonical metaplectic extension for \(G= \text{SL}_n\) (compare [\textit{T. Kubota}, J. Math. Soc. Japan 19, 114--121 (1967; Zbl 0166.29603) or On automorphic functions and the reciprocity law in a number field. Lectures in Mathematics, Kyoto University 2, Tokyo, Japan: Kinokuniya (1969; Zbl 0231.10017)] and [\textit{H. Matsumoto}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 2, 1--62 (1969; Zbl 0261.20025)]). The embedding \(\text{GL}_n\subset \text{SL}_{n+1}\) yields a metaplectic extension of \(\text{GL}_n\). With \(S\) the set of places \(v\) of \(k\) with \(|\mu|_v\neq 1\), this paper derives a \(\mu\)-valued cocycle on \(\text{GL}_n(\mathbb{A}(S))\) from the Gauss-Schering lemma appearing in proofs of the quadratic reciprocity law. If \(k\) is a function field, then \(S=\emptyset\) and the cocycle is metaplectic and, on restriction, induces the canonical metaplectic extension of \(\text{SL}_n\). As a consequence, in positive characteristic a new proof of the power reciprocity law for \(k\) is obtained.