Hyperspecial subgroup

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 ) 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.

References