Split Reductive Algebraic Group - Imported Knowlpedia Content

A (connected) reductive algebraic group over a field $k$ is a smooth connected affine $k$ -group $G$ whose unipotent radical $R_u(G)$ (largest connected normal unipotent $k$ -subgroup) is trivial.

$G$ is split over $k$ if it contains a split maximal torus $T \simeq \mathbb{G}_m^r$ defined over $k$ .

Remarks

Key properties (used in the letter):

Splitness lets one define roots and weights inside $X^*(T)$ and build the Langlands dual group.
"Almost all primes $p$ " are "good" places where $G$ is unramified and has a canonical hyperspecial compact.

Examples

$G=\mathrm{GL}_n$ over $k$ is split reductive.