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.