Hyperspecial subgroup

In the theory of reductive groups over local fields, a hyperspecial subgroup of a reductive group G is a certain type of compact subgroup of G.

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

• Γ_F=G,
• Γ_k is a connected reductive group, and
• Γ(O)=K.

The original definition of a hyperspecial subgroup (appearing in section 1.10.2 of ^[1]) 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.^[2]

An interesting property of hyperspecial subgroups, is that among all compact subgroups of G(F), the hyperspecial subgroups have maximum measure.

References

1. 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.
2. 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.