The question is contained in the title. I would guess that this question is already answered in the literature. Given the reductive group $GL(n)$ over a complete local field, how does the right ...
Let $G$ be a compact, simple, connected, simply connected (cscsc) Lie group, and let its smooth loop group $LG:=C^\infty(S^1,G)$. Given an interval $I\subset S^1$, we have the local loop group $$ L_IG ...