Let us consider a Lie group $G$ with Lie algebra $\mathfrak{g}$ and let $L\mathfrak{g} = C^\infty(S^1, \mathfrak{g})$ the Lie algebra of the loop group $LG$. My question is about continuous Lie algebra 2-cocycles on $L\mathfrak{g}$.

It is well-known (see, e.g., Prop. 4.2.4 in Pressley-Segal "Loop groups") that if $G$ is semisimple and compact, the only continuous $G$-invariant 2-cocycles are of the form $$\omega(X, Y) = \int_{S^1} b(X(t), Y^\prime(t))dt, $$ where $b$ is some (necessarily symmetric) $G$-invariant bilinear map on $\mathfrak{g}$.

If I understand correctly, the proof in Pressley-Segal does not use the compactness assumption (which is a general assumption throughout the entire chapter there), but the compactness is used in the proof that all continuous 2-cocycles on $L\mathfrak{g}$ are cohomologous to a $G$-invariant one (this follows from averaging over $G$, which only works in the compact case).

**Question: What is known if $G$ is a non-compact semisimple Lie group?**

Still, all continuous $G$-invariant cocycles are of the form given above, but now there may be non-trivial cohomology classes that are not represented by a $G$-invariant one.

**More precisely: What are examples of non-trivial classes in $H^2_c(L\mathfrak{g}, \mathbb{R})$ that are not represented by a $G$-invariant one, where $G$ is some semisimple Lie group?**