Simply connected semisimple algebraic group

A connected reductive algebraic group is semisimple if its connected center is trivial, equivalently if it has no positive-dimensional central torus.

A central isogeny $f:H\to G$ of connected semisimple groups is a surjective morphism with finite central kernel. The group $G$ is simply connected in the algebraic sense if every central isogeny $H\to G$ is an isomorphism.

Root-datum criterion

For a maximal torus $T\subset G$ , semisimple $G$ is simply connected exactly when

X_*(T)=\mathbb Z\Phi^\vee,

the coroot lattice. Equivalently, its character lattice is the full weight lattice.

Every connected semisimple group has a simply connected central cover

G_{\mathrm{sc}}\longrightarrow G.

Not topological simple connectedness

This is a notion in the category of algebraic groups. Over $\mathbb C$ , it is closely related to simple connectedness of the associated complex Lie group, but over $\mathbb R$ or a nonarchimedean field it should not be defined from the topology of $G(F)$ .

For example, $\operatorname{SL}_n$ is algebraically simply connected, whereas $\operatorname{PGL}_n$ is adjoint and not simply connected.

Langlands duality

The Langlands dual of a simply connected semisimple group is adjoint, and the dual of an adjoint group is simply connected. This is the isogeny-form information carried by the root datum.

Relation to the letter

The letter chooses simply connected nonabelian factors so that the root, weight, and dual lattice constructions have their clean extremal form.

References

A. Borel, Linear Algebraic Groups, second edition, Springer, 1991.
Robert Steinberg, Lectures on Chevalley Groups, AMS, 2016.