|
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 (''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, (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. ==References== 〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hyperspecial subgroup」の詳細全文を読む スポンサード リンク
|