Which p-adic groups are simply-connected?

Question

Suppose that we are working over a nonarchimedean local field $F$, for instance $\mathbb{Q}_p$. Which semisimple algebraic groups (or Lie groups) over $F$ are simply-connected? In particular, I am interested in when the theorem of Iwahori-Matsumoto on a double coset decomposition with respect to an Iwahori subgroup applies.

Answer 1

This doesn't really have anything to do with working over a nonarchimedean field. Any semi-simple group (say over a field of char. zero, to be safe --- although it's probably not necessary) has a finite centre. In its isogeny class, there is a group with largest possible centre, and one with trivial centre (the adjoint group in the given isogeny class).

For classical groups, $SL_n$ and $Sp_n$ are simply connected. The groups $SO_n$ are not; they have spin double covers which are simply connected. The groups $SU_n$ are twists of $SL_n$, and so are also simply connected.