# adjoint orbit of (twisted) loop group

Let $G$ be the twisted loop group of $SL_2(\mathbb C)$ and let $g$ be its Lie algebra, where diagonal entries are even functions and off diagonal entries are odd functions (of loop parameter lambda).

What are the adjoint orbits in $g$?

Is the adjoint orbit structure known for all the (twisted) loop groups over complex simple Lie groups?

• What sort of structure on the adjoint orbits are you seeking? – Peter Crooks Mar 5 '14 at 2:05
• I am wondering if there is a normal form for an element in a loop algebra under formal adjoint action. For example in the ordinary sl_2(C) case, adjoint orbits are classified by determinant function, etc. – joe Mar 5 '14 at 21:30