Complexification of real analytic groups
Gerhard P. Hochschild · Transactions of the American Mathematical Society · 1966
In dealing with the structure and representation theory of a given real analytic group G, one is led to ask whether or not there exists a continuous injective, or at least locally injective, homomorphism of G into a complex analytic group.The most convenient tool for attacking this question is the universal complexification of G.This is a continuous homomorphism y of G into a complex analytic group G+ with the property that, for every continuous homomorphism r¡ of G into a complex analytic group H, there is one and only one complex analytic homomorphism r¡ + of G+ into H such that tj+ ° y = -q.Clearly, any two universal complexifications of G are equivalent in the evident sense.The standard construction of y: G -> G+ is as follows.For any real or complex Lie algebra L, denote by £f(L) the simply connected real or complex analytic group whose Lie algebra is L. If ¿f(G) denotes the Lie algebra of G then ¿f(S£(G)) is the universal covering group of G, and we denote the kernel of the covering epimorphism £f(&(G)) -> G by L, i.e., L is the fundamental group of G. Let R and C stand for the fields of the real and the complex numbers, respectively, and let a denote the continuous homomorphism of £f(J£(G)) into 6^(^(G) B C) whose differential is the canonical injection of ¿f(G) into its complexification ¿?(G) ®B C. Considering the adjoint representation of £f(3?(G) £g¡B C), one sees immediately that a(F) lies in the center of Sf(Sâ(G) ®R C), whence the same is true for the smallest closed complex Lie subgroup, L* say, of £f(£?(G) <S)R C) that contains ct(L).We define G+ as the complex analytic group 9'(3?(G) ®fl C)jF* and y as the continuous homomorphism G^-G+ that is obtained from a by passing to the factor groups mod L and mod L*.It is easy to verify that this is actually a universal complexification of G.Our original question now becomes the question of whether or not y is injective or locally injective, i.e., of whether or not the kernel of y is trivial or discrete.Let us note first that ^^(^(G)))is always a closed real analytic subgroup of £f(¿¡?(G) ®b C).Indeed, the complex conjugation of ¿?(G) ®ß C induced from that of C is the differential of a real analytic involution a of ¡f(3?(G) ®R C), and o(£^(^C(G))) is evidently the connected component of the identity in the a-fixed subgroup.It follows that F* = o(F) whenever a(F) is discrete in o(¿f(Jí?(G))); for then ct(L) is a discrete central subgroup of íf(¿£(G) ®R C).