Chevalley Lattice and Integral Model

For split semisimple $G$, a Chevalley basis yields a Chevalley $\mathbb{Z}$-form $\mathfrak g_\mathbb{Z}\subset \mathfrak g$.

A Chevalley lattice in a representation $V$ is a $\mathbb{Z}$-lattice $V_\mathbb{Z}\subset V$ stable under the action of $\mathfrak g_\mathbb{Z}$ (equivalently, stable under the corresponding group scheme over $\mathbb{Z}$).

For a prime $p$, base change gives a group scheme $G_{\mathbb{Z}_p}$ and a canonical compact subgroup $G(\mathbb{Z}_p)$ (hyperspecial when $p$ is good; see hyperspecial ).

In the letter: this is the “lattice-stabilizer” definition of $G_{\mathbb{Z}_p}$ used to define the spherical Hecke algebra.