Siegel domains and representations of Jordan algebras

In the analysis of infinitesimal automorphisms of arbitrary Siegel domains, a certain class of nonsemisimple Jordan algebras occurs. The description of all the infinitesimal automorphisms of the domain may be based on a study of representations of the associated Jordan algebra satisfying a certain "strange identity". In this paper, all the possibilities for the Jordan algebra and representations satisfying the identity are given. 0. The infinitesimal automorphisms of arbitrary Siegel domains are now rather well understood as a result of the investigations in [2, 4-9], save for some technical questions concerning representations of Jordan algebras satisfying a "strange" identity [8, 9]. In this paper we will elucidate these remaining questions, drawing freely on the definitions, notations, and results of our papers [6-8]. U is a real finite dimensional vector space, Uc= U ® C its complexification. Q is a regular convex cone in U. F is a finite dimensional complex vector space, and F(-, •), complex linear in the first variable, is an Í2-Hermitian form on V. The set of points D = {(u,v) E Uc X V\lmu — F(v, v) E fi} is a Siegel domain of Type II. Let g denote the Lie algebra of the group G of holomorphic automorphisms of D. As is well known [4, 5, 8], g is graded. g = g-i ® g-i/2 ® So ® g\/2 ® £i> and anY x G g\ may be written X = R(u)u^+ S(u)v^au ov (there is a slight notational change from that in [8]). For an X of the above form, the necessary and sufficient conditions that X E gx are that: (1) The multiplication u ° u' = R(u)u' endows U with structure 6E of real Jordan algebra, (2) The map u -» S(u) is a representation of <3, in End V; i.e. S(u o u') = {S(u)S(u') + {S(u')S(u). (When necessary, we extend the representation to the complexification of &.) (3) (&, ñ) is a Jordan pair, and 5 is 77-related to 6B, i.e. a ° F(v, w) = i-F(S(a)v, w) + {F(v, S(a)w), Received by the editors November 30, 1979 and, in revised form, December 9, 1980. 1980 Mathematics Subject Classification. Primary 17C15, 32A07; Secondary 17B60, 32M05. 'This research was supported in part by NSF grants MCS-7810470 and MCS-7718723. ©1982 American Mathematical Society 0002-9947/82/0000-1008/S05.25 197 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use