1. ALGEBRAIC PRELIMINARIES 9

This settles our problem if K = F, since U(r)n) = SU(rjn); so assume K ^ F. Put

q — det(/3). Then qqp = 1 by (1.12), and hence q = r/rp with some r G Kx.

Put 7 = diag[a*, a - 1 ] with any diagonal matrix a such that det(a) = r. Then

7 G U(r]n) and det(7/?) = 1. This proves (3) when K ^ F. Finally, consider the

homomorphism v : GU(ip) — F x . We easily see that F x [/(£) is the inverse image

of { a2 | a G F x }, and hence we obtain (4).

1.4. Let F be the field of quotients of a Dedekind domain g. By a g-lattice in a

finite-dimensional vector space W over F we mean a finitely generated g-module in

W that spans W over F. Every fractional ideal in F with respect to g is a g-lattice

in F and vice versa; we call it a g-ideal. A g-ideal is called integral if it is contained

in g.

We now assume that F is an algebraic number field of finite degree. We denote

by a and h the sets of archimedean primes and nonarchimedean primes of F; we

put v = a U h. Further we denote by g the maximal order of F. For every v G v

we denote by Fv the ^-completion of F. In particular, for v G h and a g-ideal a

we denote by av the ^-closure of a in Fv, which coincides with the g^-linear span

of a in Fv. We denote by N(a) and N(av) the norm of a and av as usual. They

are positive rational numbers with the standard multiplicative property such that

N(a) = [g : a] if a is integral and N(av) — [gv : a„] if av is integral.

Given a finite-dimensional vector space X over F and a g-lattice L in X, we put

Xv = X 0 F Fv for every v G v, and denote by Z^ the gv-linear span of L in Xv if

t G h. Clearly Lv is a gv-lattice in Xv, and is the closure of L in Xv. Notice also

that every g^-lattice in Xv is an open compact subgroup of Xv.

1.5. Lemma. With F and X as above, let L be an arbitrarily fixed g-lattice in

X. Then the following assertions hold:

(1) If M is a g-lattice in X, then Lv = Mv for almost all v. Moreover, L C M

(resp. L = M) if Lv c Mv (resp. Lv = Mv) for every v G h.

(2) Given a gv-lattice Nv in Xv for each v € h such that Nv = Lv for almost

all v, there exists a g-lattice M in X such that Mv = Nv for every v eh.

These assertions are well-known. For the proof, see [S97, Lemma 8.2].

1.6. Given an algebraic group G over F, we denote by G A the adelization of G

and by Gv for v G v the localization of G at v. (The reader is referred to [S97,

Section 8] for basic definitions and elementary facts on this topic.) We consider G

a subgroup of G A as usual. In particular, F A and F£ denote the adele ring and

the idele group of F, respectively. The archimedean and nonarchimedean factors of

G A are denoted by G

a

and Gh- Namely Ga = FLea ^

an(^

^ h = G A H Ylveh ^v

For x G G A we denote by x

a

, #h, and xv the projection of x to Ga, Gh, and Gv,

respectively. If G C GL(V) with a vector space V over F, then for a G G A and a

g-lattice L in V, we denote by La the g-lattice in V determined by {La)

for every v G h. The existence of such a lattice La is guaranteed by Lemma

1.5 (2). In particular, for x G F£ we denote by xg the fractional ideal such

that (xg)i; = xvgv. Also we put |X|A = FLev lxrk where | |v is the normalized

valuation at v. To emphasize that this is defined on F£ , we shall also write \X\F

for

\X\A-

Given algebraic groups G and Gr over F and an F-rational homomorphism /

of G into G', we can extend / naturally to a homomorphism of G A to G'A, which

we shall denote by the same letter /. For example, we employ Tr^/ p even for the