THE MOD p REPRESENTATION THEORY OF p - ADIC GROUPS
نویسنده
چکیده
Exercise 1 (Maximal compact subgroups of G). A lattice in Qp is a finitelygenerated Zp-submodule of Qp that generates Qp as vector space. In particular, it’s free of rank n. Note that G acts transitively on the set of lattices in Qp . (i) Show that K = StabG(Zp ). (ii) Suppose that K ′ is a compact subgroup of G. Show that K ′ stabilises a lattice. (Hint: show that the K ′-orbit of Zp is finite and note that a finite sum of lattices is a lattice.) (iii) Deduce that every compact subgroup is contained in a maximal compact subgroup and that any maximal compact subgroup is conjugate to K.
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