Remarks on Certain Metaplectic Groups

We study metaplectic coverings of the adelized group of a split connected reductive group G over a number field F. Assume its derived group G′ is a simply connected simple Chevalley group. The purpose is to provide some naturally defined sections for the coverings with good properties which might be helpful when we carry some explicit calculations in the theory of automorphic forms on metaplectic groups. Specifically, we1.construct metaplectic coverings of G(A) from those of G′(A);2.for any non-archimedean place v, show the section for a covering of G(Fv) constructed from a Steinberg section is an isomorphism, both algebraically and topologically in an open subgroup of G(Fv);3.define a global section which is a product of local sections on a maximal torus, a unipotent subgroup and a set of representatives for the Weyl group.

The Coset lattices of E. S. Barnes and G. E. Wall

John Conway's analysis in 1968 of the automorphism group of the Leech lattice and his discovery of three sporadic simple groups led to the immediate speculation that other Z-lattices might have interesting automorphism groups which give rise to (possibly new) finite simple groups. (The classification theorem for the finite simple groups has since told us that no new finite simple groups can arise in this or any other way.) For example in 1973, M. Broué and M. Enguehard constructed, in every dimension 2n, an even lattice (unimodular if n is odd) whose automorphism group is related to the simple Chevalley group of type Dn. This family of integral lattices received attention and acclaim in the subsequent literature. What escaped the attention of this literature, however, was the fact that these lattices had been discovered years earlier. Indeed in 1959, E. S. Barnes and G. E. Wall gave a uniform construction for a large class of positive definite Z-lattices in dimensions 2n which include those of Broué and Enguehard as special cases. The present article introduces an abstracted and generalized version of the construction of Barnes and Wall. In addition, there are some new observations about Barnes-Wall lattices. In particular, it is shown how to associate to each such lattice a continuous, piecewise linear graph in the plane from which all the important properties of the lattice, for example, its minimum, whether it is integral, unimodular, even, or perfect can be read off directly.

On the Automorphisms of Infinite Chevalley Groups

In (8, § 3.2) Steinberg proved the following result.THEOREM. Let K be a finite field, G′ a simple Chevalley group (“normal type1”) over K. Then every automorphism of G’ is the composite of inner, graph, field, and diagonal automorphisms.For the meaning of these notions, see (8). Our aim in this note is to indicate how the Theorem may be extended to arbitrary infinite fields K, provided we replace G′ by the group denoted G in (5) and Ĝ in (8). This amounts to proving the Theorem for automorphisms of G′ which are induced by automorphisms of G; when K is finite, Steinberg's results show that all automorphisms of G′ arise in this way. As Steinberg points out, the sole use made of the finiteness of K in his argument is in the proof of the following statement: Let U be the subgroup of G′ corresponding to the set of positive roots, and let σ be any automorphism of G′; then Uσ is conjugate to U in G′.