Hyperspecial subgroup
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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
- ΓF=G,
- Γk 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.
[edit] References
- ^ 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.
