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.


  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.