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Could anyone point me to a reference for the following fact :

Let $G$ denote the orthogonal or symplectic complex group and $H$ be a complex Lie group, then if $\rho_1\,:H\, \rightarrow G$, $\rho_2\,:H\, \rightarrow G$ are two equivalent representations (conjugated by the linear group), they are conjugated by the group $G$.

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See Theorem 2.1 (b) in arXiv:math/0505268 [math.SG] for the symplectic case. The orthogonal case should be similar.

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