Chevalley lattice and integral model

A Chevalley basis of a split semisimple Lie algebra defines a Lie lattice $\mathfrak g_{\mathbb Z}$ with integral structure constants. To integrate this to representations and groups, one uses the Kostant integral form of the universal enveloping algebra, including divided powers, and the Chevalley–Demazure group scheme attached to the full root datum.

Representation lattice

For a rational representation $V$ , a Chevalley or admissible lattice is a full $\mathbb Z$ -lattice $V_{\mathbb Z}\subset V$ stable under the appropriate Kostant $\mathbb Z$ -form, equivalently under the resulting integral group-scheme action when that action has been constructed.

Stability merely under the Lie lattice $\mathfrak g_{\mathbb Z}$ is not, by itself, equivalent in every representation to stability under the integral group scheme; divided-power integrality is the stronger condition.

Integral group model

A split reductive root datum determines a smooth reductive group scheme $\mathcal G/\mathbb Z$ . Base change gives $\mathcal G_{\mathbb Z_p}$ , and

\mathcal G(\mathbb Z_p) \subset \mathcal G(\mathbb Q_p)

is hyperspecial. For a reductive group originally defined over a number field, such a reductive integral model exists outside a finite set of bad places rather than canonically at every place.

Relation to the letter

The letter uses lattices to define integral points and hence the unramified spherical Hecke algebra at almost all primes. Modern language separates the Lie-algebra integral form, an integral representation lattice, and the reductive group scheme.

References

Claude Chevalley, “Sur certains groupes simples,” Tohoku Mathematical Journal 7 (1955), 14–66.
Michel Demazure and Alexander Grothendieck, eds., Schémas en groupes (SGA 3), Exposés XXII–XXV.