Real Orthogonal Representation of Real Semisimple Lie Group

Yan Zhi-da and Zhang Da-gan classified finite dimensional real representations of real semisimple Lie group in Ref. [1]. In this note, we shall discuss real orthogonal representations (infinite dimension), mainly using lowest K-type theory that D. Vogan described in Ref. [2]. Let G be a connected real semisimple Lie group, K the maximal compact subgroup of G and , the Lie algebras of G, K respectively. Let V be a real Hilbert space and