It is well known that Fourier transform switches positive-definite functions with positive measures on a (locally compact topological) group. Further, the positive definite functions can be characterized as coefficients of the unitary representation of the group.

My question is: Is it possible to characterize Gaussian measures using this representation theory framework (even for obvious case like $\mathbb{R}$) for measures defined on the dual space of nuclear spaces? I thought about this for a while, but did not get anywhere. I also looked up standard reference book like Gelfand & Vilenkin and did not see any discussion there. I suspect the answer is something obvious and trivial. But I still do not know. In particular I am willing to restrict to LCA groups just to make matters simpler.