Let G be a connected simple linear Lie group and \(H\subset G\) a symmetric subgroup such that the corresponding symmetric space \(G/H\) is non-compactly causal. We show that any irreducible unitary representation of G leads naturally to a net of standard subspaces on \(G/H\) that is isotone, covariant and has the Reeh–Schlieder and the Bisognano–Wichmann property. We also show that this result extends to the universal covering group of \( \mathop {{\mathrm {SL}}} \nolimits _2({\mathbb R})\) , which has some interesting application to intersections of standard subspaces associated to representations of such groups. For this, a detailed study of hyperfunction and distribution vectors is needed. In particular, we show that every H-finite hyperfunction vector is in fact a distribution vector.