I have an infinite dimensional Lie group $G$ of the form $\varinjlim G_n$, for example $SU(9,\infty)$ or $Sp(\infty,R)$. I also have a closed subgroup $P$ of $G$ and a factor representation $\chi$ of $P$ (say on the Hilbert space $H_\chi$). I can use the universal enveloping algebra $U(\mathfrak g)$ to form the induced representation $\pi = Ind_P^G(\chi)$ of $G$ on $U(\mathfrak g) \otimes_P\, H_\chi$.
Has anyone studied the properties of $\pi$? For example: if $\chi$ is factorial of type $II_1$, can $\pi$ be described in terms of factor representations? Any pointers or references would be appreciated.