In particular, let F be a nonarchimedean local field, O its ring of integers, k its residue field and G a reductive group over F. A subgroup K of G(F) is called hyperspecial if there exists a smooth group scheme Γ over O such that
- Γk is a connected reductive group, and
The original definition of a hyperspecial subgroup (appearing in section 1.10.2 of ) was in terms of hyperspecial points in the Bruhat-Tits Building of G. The equivalent definition above is given in the same paper of Tits, section 3.8.1.
Hyperspecial subgroups of G(F) exist if, and only if, G is unramified over F.
An interesting property of hyperspecial subgroups, is that among all compact subgroups of G(F), the hyperspecial subgroups have maximum measure.
- Tits, Jacques, Reductive Groups over Local Fields in Automorphic forms, representations and L-functions, Part 1, Proc. Sympos. Pure Math. XXXIII, 1979, pp. 29-69.
- Milne, James, The points on a Shimura variety modulo a prime of good reduction in The zeta functions of Picard modular surfaces, Publications du CRM, 1992, pp. 151-253.