Group Representations and Lattices

This paper began as an attempt to understand the Euclidean lattices that were recently constructed (using the theory of elliptic curves) by Elkies [E] and Shioda [Sh I, Sh2], from the point of view of group representations. The main idea appears in a note of Thompson [Th2]: if one makes strong irreducibility hypotheses on a rational representation V of a finite group G, then the G-stable Euclidean lattices in V are severely restricted. Unfortunately, these hypotheses are rarely satisfied when V is absolutely irreducible over Q. But there are many examples where the ring EndG ( V) is an imaginary quadratic field or a definite quaternion algebra. These representations allow us to construct some of the Mordell-Weillattices considered by Elkies and Shioda, as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves. In § I we discuss lattices and Hermitian forms on T/, and in §§2-4 the strong irreducibility hypotheses we wish to make. In § 5 we show how our hypotheses imply the existence of a finite number (up to isomorphism) of Euclidean Z[G]lattices L in V with EndG(L) a maximal order in EndG(V). We give some examples with dim V :5 8 in §6, and in §§7-9 discuss the invariants of L, such as the dual lattice and theta function. The rest of the paper is devoted to examples: in most, G is a finite group of Lie type and V is obtained as an irreducible summand of the Weil representation of G. Some of the representation theoretic problems left open by this paper are: to find all examples of pairs (G, V) satisfying the strong irreducibility hypotheses of §§2-4, and to determine the invariants (shortest nonzero vector, theta function, Thompson series, ... ) of the G-Iattices L so effortlessly constructed inside V.