Vanishing of Trace Forms in Low Characteristics

A finite-dimensional representation of an algebraic group G gives a trace symmetric bilinear form on the Lie algebra of G. We give a criterion in terms of the root system data for this form to vanish. As a corollary, we show that a Lie algebra of type E8 over a field of characteristic 5 does not have a so-called “quotient trace form”, answering a question posed in the 1960s. Let G be an algebraic group over a field F , acting on a finite-dimensional vector space V via a homomorphism ρ : G → GL(V ). The differential dρ of ρ maps the Lie algebra Lie(G) of G into gl(V ), and we put Trρ for the symmetric bilinear form Trρ(x, y) := trace(dρ(x) dρ(y)) for x, y ∈ Lie(G). We call Trρ a trace form of G. Such forms appear, for example, in the hypotheses for the Jacobson-Morozov Theorem [Ca, 5.3.1]. We prove: Theorem A. Assume G is simply connected, split, and almost simple. Then the following are equivalent: (a) The characteristic of F is a torsion prime for G. (b) Every trace form of G is zero. The set of torsion primes for G is given by the following table, cf. e.g. [St 75, 1.13]: type of G torsion primes An, Cn none Bn (n ≥ 3), Dn (n ≥ 4), G2 2 F4, E6, E7 2, 3 E8 2, 3, 5 A prime p is called a torsion prime for G if the corresponding group G(C) over C (or, equivalently, its compact form) is such that one of its homology groups, with coefficients in Z, contains an element of order p. We also prove a generalization of Theorem A that removes the hypotheses “simply connected” and “split”; it is somewhat more complicated, so we leave the statement until Th. D (and Remark 4.6). Replacing the simply connected group G with a nontrivial quotient G changes the situation in two ways: the group G has “fewer” representations and the Lie algebras of G and G may be different. These two changes are reflected in the integers N(G) and E(G) defined below. As a particular example of Th. A, for G of type E8 over a field of characteristic 2, 3, or 5, Trρ is zero for every representation ρ of G. One may ask whether the same is true for the representations of the Lie algebra Lie(G). That is, for a representation ψ of Lie(G), we write Trψ for the bilinear form (x, y) 7→ trace(ψ(x)ψ(y)), and ask 2000 Mathematics Subject Classification. 20G05 (17B50, 17B25).