1. ALGEBRAIC PRELIMINARIES
11
We now consider our setting to be that of §1.4, with an algebraic number field
as F. Given x G (Fv)™ with v G h, we can naturally define uo(x) to be an integral
gv-ideal, taking gv to be A. Then we put
(1.16) u{x) = N(v0{x)) = [fl„ : i/0(*)].
If x = d~1c is a left reduced expression for x, then v{x) = | det(d)|~1. Moreover,
if uxv =
n n
with u G GLm(gv), v G GLn(gv), and a = diag[ai, ... , a
r
], then
i/(x)
- 1
is the product of \ai\v for all i such that a* ^ gv (see [S97, Lemma 3.8
(2)])-
1.8. Take our setting to be the same as in Cases I and II in §1.1 with an algebraic
number field as F. Namely, K = F or K is a quadratic extension of F. We denote
by r the ring of algebraic integers in K and by k the set of all nonarchimedean
primes of K. (Thus r = g and k = h if K = F.) Given v G k, an tv-ideal a, and
a matrix x with entries in Kv, we write x - a if all the entries of x belong to a.
Similarly, for a matrix y with entries in KA and an r-ideal b, we write y - b if
all the entries of yv belong to bv for every v G k.
Take two positive integers m and n. For x
c d
G GLm+n(K)A with
a G
(.KA)™
and d G (K/i)n write a = ax, b = 6X, c = cx, and d = dx. With fixed
t-ideals rj and 3 such that 93 C r, we put
(1.17) Cfo, 3] = {x G GLm+n{K)A I det(x)h G F U h tf
ax ""
r
frx - t), cx - 3, dx - r | .
We easily see that this is a subgroup of GL
m
+
n
(if )A (see [S97, §9.1]). We also note
that if x G C[t), 3] and t/|t)3, then (det(arc) det(dx) det(#)) G t)v$v, and hence
we see that
(1.18) The map x »— ((ax, 6x)u) 1 defines a homomorphism of C[t), 3] into
FIvloj [CLm(^/9i;3v) X GLn^/t)^^] .
1.9. Lemma. Define a subgroup p(
m
'
n
) of GLm+n(K) by
p(m n) = {xeGLm+n{K)\cx = 0}.
Let C denote the group C[tj, 3] 0/ (1.16). Then
p
{jn,n)c
=
1 ^ ^
G j L m + n (
^ )
A
J ( ^

GLn(ift;) and
(d"
1
^),, - iv for every v\t)} } .
Moreover, assuming m = n, /e£ G denote GU{r)n), U(rjn), or SU(nn) with r\n of
(1.8); putP = GH P("'
n
), and D = GAnC. Then
PAD = GAnP£'n)C
= { x G GA I (dx)v G GLn(K)v and (d~1cx)v - 3^forevery i/|tj3 } .
in particular, GLrnjrn (K)A = P^n)C and GA = PAD if 93 = r.
PROOF .
The assertions for GLm+n(K) and U(rjn) are proved in [S97, Lemma
9.2]. Combining the result for U(rjn) with [S97, Lemma 9.10 (2)], we obtain the
assertion for SU(rjn). As for GU(rjn), let x G GU(nn)A, P = diag[ln, v(a)ln],
and y = p~xx. Then y G U(r)n). Suppose (dx)v G GLn(K)v and (d" 1 ^)^ - 3^
for every v|t)3- Then we easily see that y satisfies the same conditions, and so
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