Split Reductive Algebraic Group

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 $k$.

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.

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