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 [Tits, Jacques, Reductive Groups over Local Fields in [http://www.ams.org/online_bks/pspum331/ "Automorphic forms, representations and L-functions, Part 1"] , Proc. Sympos. Pure Math. XXXIII, 1979, pp. 29-69.] ) 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". [Milne, James, [http://www.jmilne.org/math/Preprints/Montreal.pdf 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.]
An interesting property of hyperspecial subgroups, is that among all compact subgroups of "G(F)", the hyperspecial subgroups have maximum measure.
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