IWAMOTO, TAKASHI

Iwamoto, Takashi

Osaka University Knowledge Archive

Density properties of complex Lie groups

Osaka City University Repository

Iwamoto, T.
Osaka J. Math.
23 (1986), 859-865
DENSITY PROPERTIES OF COMPLEX LIE GROUPS
TAKASHI IWAMOTO
(Received January 24, 1985)
0. Introduction
Let G be a locally compact group. A subgroup H of G is called of finite
covolume if H is closed and G/H has a finite G-invariant Radon measure.
A. Borel studied properties of subgroups of finite covolume in semi-simple
Lie groups without compact factors [1] and proved:
Theorem (BoreΓs density theorem). Let G be a semi-simple Lie group
without compact factors and H a subgroup of finite covolume in G. Let f be a
finite dimensional linear representation of G. Then every f(H)-ίnvariant vector
subspace is f(G)-invariant.
H. Furstenberg showed that BoreΓs theorem holds for minimally almost
periodic groups and those subgroups of finite covolume [2].
In this paper, applying Furstenberg's idea to some more general situa-
tions, we shall prove complex Lie group version of BoreΓs theorem, that is:
Theorem. Let G be a complex analytic group and H a subgroup of finite
covolume in G. Let f be a holomorphic representation of G on a finite dimensional
complex vector space. Then every f(H}-invariant vector subspace is f(G)-ίnvarίant.
Using this theorem, we obtain properties of subgroups of finite covolume
in a complex analytic group [see Section 3].
1. Preliminary results
Let G be a locally compact group and V a finite dimensional vector space
over the field K, where K is the real number field R or the complex number
field C. Let / be a continuous representation of G on V.
DEFINITION 1. (f?,/) is said to have property (A) if the following condi-
tions are satisfied:
(1) G has no closed subgroup of finite index.
(2) For any/(G)-invariant subspace Wof V,f(G)\
w
dK l
w
 or
860 T. IWAMOTO
is unbounded in GL(W), where f ( g ) \
 w
 and det f ( g ) \
 w
 are the restriction oϊf(g)
to W and its determinant, respectively.
Let P(V] denote the projective space corresponding to V. For a subset
AdVy A denotes the canonical image of A in P(V). For a vector subspace
Wd V, W is called a linear subvariety. Following Furstenberg's terminology,
we call a finite union of linear subvarieties a quasi-linear subvariety. For sim-
plicity we denote a quasi-linear subvariety by "q.l.v.". By the descending chain
condition for all the algebraic sets, we have that for any subset BdP(V) there
exists a minimal q.l.v. containing B. In this case this q.l.v. is determined unique-
ly. We denote it by q(B). For a linear map t of a subspace Wd V to V, t
denotes the map of T^\ker t to P(V) corresponding to t. The following lemma
is essentialy due to Furstenberg.
Lemma 1. Let fe}Γ=ι be in GL(V) such that
|det f 4 | / | | f j | -»0 as k-»oo
where //—dim W and \\ \\ is a suitable norm on End(F).
Then there exist a transformation T on P(V) whose range is a proper q.l.v. g:
P(V) and a suitable subsequence {^}Γ=ι of fe}Γ=ι such that
ϊl(x) -> T(x) as k -> oo
for any
Proof. Let W be a subspace of V. Passing to a subsequence and taking
suitable constants ak's such that ||tf f t £Λ | | = l (where || || is a suitable norm on
Hom(ίF, K)), we may assume that ak tk converges to a nonzero linear map h of
W into V with respect to the natural topology in Hom(W, V). We note that
for v e W^\ker h
tk(v) -> h(v) as k -> oo .
Set W0=V. There exist a subsequence fe(0)} of {tk} and a linear map hQ
of WQ=V to F" such that {^ (0)} converges pointwise to h0 on P(F)\ker h0.
We shall define for /=!, 2, 3, ••-, subspaces Wiy subsequences {tk(i)} of {tk}
and linear maps /?t of W{ to F, inductively. Set Wt=ker A .j. Take a subse-
quence fe(z)} of {ίA(z — 1)} and a linear map hi such that for
as
Since dim F< + oo) there exists an integer m such that dim W^+j^dim ker hm
=0.
Set T(v)=h^) for z eWΓΛ^+i, where i=0, 1, 2, — . Let #ί> = {**(»!)}.
DENSITY PROPERTIES OF COMPLEX LIE GROUPS 861
Then the range of T is U hi(Wi) and {t'k} converges pointwise to T on P(V). In
order to show that the range of T is proper, it is sufficient to prove det h0=0. By
the assumption we have that
I det AO I = I det lίm ak
where || || is a norm on End (V)=Hom(W0, V) and ak is a scalar constant, q.e.d.
Lemma 2. Let Wifor ί=l, 2, 3, ••-, k, be a subspace of V. If a subspace
Wd V is contained in U Li Wiy there exists an integer l^i'^k such that Wd Wy.
Proof. Suppose that W is not contained in any Wit Then for every /=
1, 2, 3, •••, k, there exists a nonzero vector v^W which is not contained in Wf.
We shall prove that fory=l, 2, 3, •••, k, there exist j real numbers ί/s such
that Σί=ι ti Vi is not contained in U ί-i Wi9 by induction onj. By the assump-
tion of induction we can find (/—I) real numbers ί/s such that u= Σί-ί t^Vi is
not contained in Ull i Wf. If u^Wjy set -^ — 0. Assume that u^Wj. Since
Uί'li W7; is closed, we can find a sufficiently small number tj such that
Φ Uί;} Wi. Since weίFy and t^v^W^ we have that Σi=ι ti Vi=
Wj. Consequently we can find k real numbers £f 's such that Σί-i ίj ^iΦ U *.ι
However Σ?-ι t^v^Wd. U *»ι W7,- leads to a contradiction. q.e.d.
Lemma 3. Assume that (G,f] has a property (A). Let f be a representa-
tion of G on P(V) induced by f and μ a finite f(G}-invariant Radon measure on
P(V). Then the support of μ consists of f(G)-fixed points.
Proof. If f(G)dK l
v
, there is nothing to prove. Hence we may assume
that there exists a sequence {gk} C G such that
{|det/(A) I -1/dim" •/(&); k = 1, 2, 3, -}
is unbounded in End(F). If necessary taking a subsequence we may assume,
by Lemma 1, that there exists a transformation T on P(V] whose range is a
proper q.l.v. Q and that f(gk) converges pointwise to T.
Let D(x) be the distance from x to Q for some metric on P(V). By the
bounded convergence theorem, we have that for any
0 = ( D(T(x)) dμ
Jp(F)
862 T. IWAMOTO
= ( D(x) dμ
JP(V)
This implies that supp
Let X be the unique minimal q.l.v. containing supp μ. X can be denoted
by X= U ?=ι T ί^ where W7,- is a subspace of F. We may assume that there is no
inclusion relation among W^s. Remark that X is also proper in P(V). Since
supp μ is /(G)-invariant and X is the smallest q.l.v. containing supp μ, X is /(G)-
invariant. For every g^G, f(g) W^d U f=ι W7,. By Lemma 2, there exists for
every /= 1, 2, 3, •••, m, there exists an integer s(i) such that/(^) PF, C W
s
a). Since
there is no inclusion relation among PF/s, U f=ι Wi=f(g) U ?=ι W^ C U JLi W^co C
Uf.i Wi implies that /(#) W
έ
=W
s(i) for ι=l, 2, 3, •••, m. Thus G permutes
PF/s. Since G has no closed subgroup of finite index, G leaves each Wi in-
variant.
We shall show that/(G) \
 WidK lWi for /=!, 2, 3, ••-, w.
Suppose that there exists W^ such that f(G)\Wi,<f.K lws. The same
argument as above with respect to W^9 f \ Wi*y and μ \ W^ shows that there exists a
q.l.v. X' contained properly in W^ such that supp μ \w.s(ΞX'. Thus X contains
( U i&'Wi) \JX' properly. This contradicts the definition of X.
Therefore /(G) \
 WidK lWi for /=!, 2, 3, •••, m. q.e.d.
2. Main theorem
Let G be a locally compact group and / a continuous representation of G
on a finite dimensional vector space V over K.
Lemma 4. Assume that (G,f) has property (A). Let H be a subgroup of
finite covolume in G. Then for \-dimensionalsubspace W of Vy Wis f(G)-invarίant
if and only if W is f(H)-invariant.
Proof. In order to prove the lemma it is sufficient to show "if" part.
Set p= W^P(V). Define the map π of G/H to P(V) by
Then π carries a finite G-invariant measure on G/H to a finite /(G)-invariant
measure on P(V}. Since p is contained in the support of this measure, by
Lemma 3, p is a /(G)-fixed point. q.e.d.
For a representation / of G on a vector space F, the &-th exterior product
representation Akf of/ on AkV is defined by
λv2 Λ.v3 Λ ••• Avk)
DENSITY PROPERTIES OF COMPLEX LIE GROUPS 863
where g e G and Vi e V.
DEFINITION 2. (G,/) is said to have property (B) if for k=l, 2, 3, •••, dim
V, (G, Akf) has property (A).
Theorem 1. Assume that (G,/) has property (B). Let H be a subgroup of
finite covolume in G. Then for any subspace W of V, W is f(G)-invariant if and
only if W is f(H)-invarίant.
Proof. In order to prove the theorem it is sufficient to show "if" part.
Let k be dim W. Taking k-th exterior product of /, we can reduce the proof
to Lemma 4.
Proposition 1. Let G be a complex analytic group and f a holomorphic re-
presentation of G on a finite dimensional complex vector space V. Then (G,/)
has property (B).
Proof. If / is holomorphic, so is Λft/. Thus, in order to prove the pro-
position, it is sufficient to show that (G,/) has property (A). Since G is con-
nected G has no closed subgroup of finite index. In order to show the condition
(2) of Definition 1 we may assume that W=V, for the restriction / 1
 w
 of /to
a invariant subspace W is also holomorphic.
Let G' denote the complex linear group /(G) and ό' its Lie algebra. For
^=(α
ίy)eEnd(F), we define the norm of A by ||^IMΣuKΊ2)1/2
For nonzero X €Ξ G', set
where n=dim V and #eC. We note that/χ (#) can be written the form;
Λ(*) = (l/ι(*)l2+l/,(*)l'+ + 1 /.(«) I1)
where m=n2 and/)(#) is a holomorphic function of #. Thus there exist only two
possible cases:
Case 1. There exists a nonzero X^G' such that fχ(%) is unbounded. Since
/c(^)=||exρ nz X\\l\ det exp z X\ ^  ||exρ * X\\n/ \ det exp * X \ , we have that
{ I det exp z X \ "1/n exp * X\ z e C}
is unbounded.
Case 2. For every X^G', f
x
(%) is constant. In this case each element of the
matrix (exp n z X)/(det exp % X) is constant. Substituting 0 for #, we have that
(exp n % X)/(det exp z X) = l
v
for all z^C and all X^G. Consequently we have that
864 T. IWAMOTO
expsXeC ly for all X(ΞO .
Since f(G)=G' is connected, /(G) C C 17. q.e.d.
From Theorem 1 and Proposition 1, it follows that:
Theorem 2. Let G be a complex analytic group and H a subgroup of finite
covolume in G. Let f be a holomorphίc representation of G on a finite dimensional
complex vector space. Then every f(H)-invarίant subspace is f(G)-invariant.
REMARK. There are several other cases in which property (B) holds. If
G is minimally almost periodic and / is an arbitrary representation, or if G is
an analytic group and / is a unipotent representation (G,/) has property (B).
In both the cases Theorem 1 holds [2, 4].
3. Density properties
In this section G always denotes a complex analytic group, H a subgroup
of finite covolume in G and / a holomorphic representation of G on a finite di-
mensional complex vector space V.
Corollary 1. Every element of /(G) is a linear combination of elements of
f(H).
Proof. Let IF be the subspace spanned by the elements off(H) in End(F).
The action of G on End(F)
GxEnd(F)3(£, A) H>/(^)o^eEnd(F)
defines a holomorpnic representation of G on End(iF). Since W is H-invariant
under this action Theorem 2 concludes that W is G-invariant. Thus we have
that for
r<^W. q.e.d.
From Corollary 1, it follows immediately that:
Corollary 2. The centralizer of f(H) in GL(V) coincides with the cen-
tralizer of f(G).
Corollary 3. /(G) and f(H) has the same Zariskί closure in GL,(V).
Proof. Let G' and H' be the Zariski closures of /(G) and/(H), respectively.
Clearly JFΓ'cG'. By Chevalley's theorem we can find a rational representation
r of GL(F) on a complex vector space E and a nonzero vector v^E such that
H' =
DENSITY PROPERTIES OF COMPLEX LIE GROUPS 865
Since rof is a holomorphic representation of G and C v is ro/(jfϊ)-invariant, by
Theorem 2, C v is ro/(G)-invariant. Thus we have that/(G)C/Γ. q.e.d.
Appendix. Professor Goto pointed a criterion for property (A). This
criterion seems to make the meaning of property (A) clear.
Let V be a finite dimensional complex vector space. An endmorphism A
on V is called conformed if A is semi-simple and the real part of every eigen
value of A is equal to each other. Let the totality of the conformal endomorphi-
sms on V be denoted by c(V). For a representation / of a Lie group by df
we denote the associated representation of its Lie algebra.
Proposition (M. Goto). Let G be an analytic group and ό its Lie algebra.
Let f be a representation of G on a finite dimensional complex vector space V. As-
sume that for every f(G)-invaήant subspace W of V df(ό)\
w
dc(W) implies that
df(0) I
 w
 C C l
w
. Then (GJ) has property (A).
References
[1] A. Borel: Density properties of certain subgroups of semi-simple groups without com-
pact components, Ann. of Math. 72 (1960), 179-188.
[2] H. Furstenberg: A note on Borel's density theorem, Proc. Amer. Math. Soc. 55
(1976), 209-212.
[3] J. von Neumann and E.P. Wigner: Minimally almost periodic groups, Ann. of
Math. 41 (1940), 746-750.
[4] A.I. MaΓcev: On a class of homogeneous spaces, Amer. Math. Soc. Transl. 39
(1951).
Department of Mathematics
Kyushu University 33
Hakozaki, Fukuoka 812
Japan



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