Hypotheses and general notation

Definitions

Let k be a global field and be the ring of adeles of k. For a finite place v of k, we write for the ring of integers. Let be the ring of finite adeles of k and, the product being over the archimedean places. If k is a function field, let q be the number of elements of its field of constants.

Let G be a connected reductive algebraic group defined over k. Fix an embedding into a linear group as follows. First choose an embedding, defined over k, with closed image. Then iG: G ↪ GL2n is defined by

There exists a finite set S of places of k, containing the archimedean places, such that the image of iG is defined and smooth over (see [Sp] §4.9). For v ∉ S, this allows us to define the group of points with values in. For almost all v ∉ S, this is a maximal compact subgroup of G(kv) (see [Sp] p.18, 1.3 and what follows). We fix a compact maximal subgroup K of G() such that K = ΠυKν product over all places of k, where Kν is a maximal compact subgroup of G(kυ). We suppose, as we may, that for almost all finite places. We will impose further properties on K in I.1.4.