Geometric construction of metaplectic covers of $\GL_{n}$ in characteristic zero

Richard Hill

Published 2007-08-01Version 1

This paper presents a new construction of the m-fold metaplectic cover of $\GL_{n}$ over an algebraic number field k, where k contains a primitive m-th root of unity. A 2-cocycle on $\GL_{n}(\A)$ representing this extension is given and the splitting of the cocycle on $\GL_{n}(k)$ is found explicitly. The cocycle is smooth at almost all places of k. As a consequence, a formula for the Kubota symbol on $\SL_{n}$ is obtained. The construction of the paper requires neither class field theory nor algebraic K-theory, but relies instead on naive techniques from the geometry of numbers introduced by W. Habicht and T. Kubota. The power reciprocity law for a number field is obtained as a corollary.