CHAPTER I
ALGEBRAIC AN D LOCA L THEORIE S O F
GENERALIZED UNITAR Y GROUP S
1. Elementar y propertie s o f hermitia n form s an d unitar y group s
1.1. Le t K b e a n associativ e rin g with identit y element . Give n lef t if-module s
V an d W, w e denote b y Hom(W , V ; if ) th e se t o f al l If-linea r map s o f W int o V.
We then pu t End(F , if ) = Hom(F, V; if) , GL(V, if ) = End(F, if) x , an d
(1.1.1) Inj(W , V;K) = {fe Hom(W , V ; if ) | Ker(/) = {0 } } ,
(1.1.2) SL{V, K) = {ae GL(V, if ) | det(a) = 1 } .
The latte r i s define d onl y whe n K i s commutativ e an d V i s a fre e if-modul e
of finite rank . I f K i s fixed an d clea r fro m th e context , w e denot e thes e b y
Hom(W, V), End(V
r),
GL(V), Inj(W , V), an d SL(V). W e let Hom(W, V ) act on W
on the right; namely we denote by wa th e image of w G W unde r a G Hom(W, V) .
If (3 Hom(X , W) , th e composit e ma p o f ^ S and a ca n b e writte n f3a, s o tha t
x((3a) = (xyS) a fo r x E X. Th e identit y elemen t o f End(F ) i s denote d b y ly.
Whenever w e spea k o f a if-modul e i n thi s section , w e assum e tha t i t i s finitely
generated an d th e identit y elemen t o f i f act s a s the identit y map .
Hereafter unti l th e en d o f §2.11w e assum e tha t i f i s a divisio n rin g wit h a n
involution p, b y whic h w e mea n a n additiv e bijectio n o f i f ont o itsel f suc h tha t
(xy)p
= y
pxp
an d (x
p)p
= x fo r ever y x , y G if. Fo r a left if-modul e V w e denote
by dim(V ) th e dimensio n o f V ove r if . Fo r brevit y w e call a if-submodul e simpl y
a subspace.
Given V an d e = ±1, b y a n e-hermitian form on V w e understan d a ma p
(p : V x V if suc h tha t
(1.1.3a) (p(x, y) p = ep(y, x), ip{x + x\ y) = (p(x, y) + ip(x\ y),
(1.1.3b) p(ax, by) = ap(x, y)b p for every a , b G if .
For a G Hom(W, V ) w e can defin e a n £-hermitia n for m a (p on W b y
(1.1.4) (a ip)(u, v) = p(ua, va) (U, V G W).
Then ((3a) if = (5 (a (p) fo r fie Hom(X , W). W e cal l p isotropic (o n F ) i f
(/?(x, x) = 0 for som e x G V, ^ 0 ; we call p anisotropic (o n V' ) if ip(x, x) = 0 only
for x = 0 . For a subspace U of V we pu t
(1.1.5) R^iU) = {xeU\ p(U, x) = 0}
l
http://dx.doi.org/10.1090/cbms/093/01
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