Math. Z. 213, 425448 (1993) Mathematische Zeitschrift 9 Springer-Verlag1993 Unitary dilations of commutation relations associated to alternating bilinear forms Palle E.T. Jorgensen Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA Received 15 July 1991; in final form 22 May 1992 1 Introduction We study second quantization of symplectic spaces V with given alternating bilinear form. An algebraic framework is developed using the associated central extension Lie algebras. We then impose the further condition that V carries a reflection operator satisfying an Osterwalder-Schrader type positivity. Our construction of representations is based on induction from representations of associated (non-degenerate) Heisenberg algebras obtained as unitary dilations of given V. When Osterwalder-Schrader positivity is assumed, we show that V has a Heisenberg extension inducing representations of V if and only if V decomposes in a specific way such that the given symplectic form satisfies a certain a priori estimate relative to the subspaces of the decomposition. Our induction provides in particular representations of the free chiral quantum field on a higher genus Riemann surface. The metaplectic group is used extensively in harmonic analysis, see e.g. [KV], [-Sh] and ERa]. Its usefulness arises (to a large extent) from the fact that it may be realized as the automorphism group of a suitably defined Heisen- berg type Lie algebra. These Lie algebras are central extensions of Abelian Lie algebras. In the study of representations of the free chiral field on a Riemann surface, it appears useful to study homomorphisms (in the category of Lie algebras) from a certain class of more general central extensions into Heisenberg algebras. We shall think of these Lie homomorphisms as generalizations of automor- phisms [-H2], and we are motivated by the known harmonic analysis of the meta- plectic group. Our Lie algebras are infinite-dimensional. If V is a given vector space over a field ~, a central extension Lie algebra is then specified by a sym- plectic form, (., .): V x V~. The corresponding Lie bracket will be given by, [-u, v] .'=(u, v) c, u, v~ V, where c is a basis vector for the added central dimension. We shall study the representation theory of V in terms of the symplectic structure and associated Heisenberg Lie algebras [HI. Using the point of view of harmonic representations, we shall define induced representations of V. In the special case when ~ = ~, and V satisfies a certain reflection symmetry condition, we prove Work supported in part by NSF and NATO This paper is a revised version of an earlier preprint with the same title