The universal representation kernel of a Lie group

Abstract

Let G be a connected real Lie group. The universal representation kernel, Ko, oi G is defined as the intersection of all kernels of continuous finite dimensional representations of G. Evidently, Ko is a closed normal subgroup of G, and it is known from a theorem due to Goto (cf. [l, Theorem 7.1]) that G/Ko has a faithful continuous finite dimensional representation. Thus Kg is the smallest normal closed subgroup P of G such that G/P is isomorphic with a real analytic subgroup of a full linear group. The known criteria for the existence of a faithful representation lead to a determination oi Ko which we wish to record here. Suppose first that G is semisimple. Let @ denote the Lie algebra of G. Let C stand for the field of the complex numbers, and denote by @c the complexification of ®, i.e., the semisimple Lie algebra over C that is obtained by forming the tensor product, over the real field, of ® with C. Denote by 5(@) and 5(©c) the simply connected Lie groups whose Lie algebras are @ and ©c, respectively. The injection ©—>@c is the differential of a uniquely determined continuous homomorphism y of 5(@) into S(d$c). The kernel P of y is a discrete central subgroup of 5(@). Let (P), i.e., the universal representation kernel of the semisimple connected Lie group G is the image, under the universal covering epimorphism, of the kernel of the canonical homomorphism S(®)->S(®C).