Regularity of abelian linear actions

25 Pages·2013·0.38 MB·English

REGULARITY OF ABELIAN LINEAR ACTIONS
DIDIER ARNAL(1), BE´CHIR DALI(2), BRADLEY CURREY(3), AND VIGNON OUSSA(4)
Abstract. We study the regularity of orbits for the natural action of a Lie sub-
group G of GL(V), where V is a finite dimensional real vector space. When G is
connected, abelian, andsatisfiesacertainrationalitycondition, weshowthatthere
aretwopossibilities: eitherthereisaG-invariantZariskiopensetΩinwhichevery
orbit is regular, or there is a G-invariant conull G set in which every orbit is not
δ
regular. Moreover, under the rationality condition, an explicit characterization of
almost everywhere regularity is proved.
1. introduction
For the continuous action of a group on a topological space, the orbit of a point
is said to be regular if the relative topology on the orbit coincides with the quotient
topology that the orbit carries as a homogeneous space. Given a finite dimensional
realvectorspaceandasubgroupofGL(V)thatisdiscretelyorcontinuouslygenerated
by a finite subset of gl(V), we consider the broad and somewhat vague question: are
there general conditions on the generators that will ensure that the orbit of almost
every point is regular? In the case of one generator, when the group is expRA, or
expZA, then it is relatively easy to describe necessary and sufficient conditions on A
in order that the orbits are almost everywhere regular. In fact, one finds that
1. If A is not diagonalizable then almost all orbits are regular,
2. If A is diagonalizable, and there is at least one non-purely imaginary eigen-
valueorifalltheeigenvaluesarepurelyimaginaryandtheygenerateadiscrete
additive group, then almost all orbits are regular,
3. If A is diagonalizable, all the eigenvalues are purely imaginary and they gen-
erate a dense additive subgroup in iR, then almost all orbits are singular.
Moreover, in the case of one generator exactly one of the following obtains: either
there is a G-invariant Zariski open subset Ω in V in which all orbits are regular, or
there is a G-invariant, conull, G subset U of V in which every orbit is not regular.
δ
When there is more than one generator, the problem becomes much more difficult.
The motivation forthe studyof this questionarises in partfrom itsclose relation to
admissibility. LetGbeasubgroupofGL(n,R)andletτ betheunitaryrepresentation
Date: February 27, 2013.
2000 Mathematics Subject Classification. 57Sxx, 22Exx, 22E25, 22E27, 17B45, 17B08, 58E40.
Key words and phrases. regular orbit, Lie algebra root, linear Lie group action.
1
2 D. ARNAL, B. DALI, B. CURREY, AND V. OUSSA
of Rn (cid:111)G induced by the trivial character of G, acting in L2(Rn). Fix ψ ∈ L2(Rn)
and for f ∈ L2(Rn) define W f by
ψ
W f(x,s) = (cid:104)f,τ(x,s)ψ(cid:105), x ∈ Rn,s ∈ G.
ψ
We say that ψ is weakly admissible if W is a well-defined, bounded, injective map
ψ
from L2(Rn) into L2(Rn (cid:111) G), and ψ is admissible if W is an isometry. Now put
ψ
V = Rˆn; and (regarding G as a subgroup of GL(V)) for ψ ∈ L2(Rn) put
(cid:90)
c (v) = |ψˆ(s−1·v)|2ds, v ∈ V.
ψ
G
Then ψ is weakly admissible if and only if c is bounded and non-vanishing almost
ψ
everywhere, and ψ is admissible if and only c = 1 almost everywhere. We say that
ψ
G is (weakly) admissible if τ has a (weakly) admissible vector.
It is well-known that if G is weakly admissible then G is necessarily closed and
almost all of the stabilizers G(v),v ∈ V are compact. In [9] it is shown that if these
necessary conditions hold and in addition for almost every v ∈ V, the (cid:15)-stabilizer
G (v) = {s ∈ G : |s·v −v| ≤ (cid:15)}
(cid:15)
iscompactforsome(cid:15) > 0,thenGisweaklyadmissible. Butifforsomev thestabilizer
G(v) is compact, then v has a compact (cid:15)-stabilizer if and only if the orbit is regular
[5]. Thus, a sufficient condition for almost everywhere regularity provides a method
for finding weakly admissible groups. The necessity of the (cid:15)-stabilizer condition for
weak admissibility remains open, but in [6], it is shown that, if G is closed and G(v)
is compact almost everywhere, then G is weakly admissible if and only if there is a
conull, G-invariantBorelsubsetΩofV thatadmitsaBorelcross-sectionforitsorbits.
Existence of a Borel cross-section is only provisionally related to regularity: by the
fundamentalresults in[4,7], ifΩisalocally compact G-space, thenΩadmitsaglobal
Borel cross-section for its orbits if and only if all orbits in Ω are regular. We say that
a subgroup G of GL(V) is regular if there is a locally compact, G-invariant, conull,
BorelsubsetΩofV inwhicheveryorbitisregular. Bytheabove, ifaclosedsubgroup
of GL(V) is regular and has almost everywhere compact stabilizers, then it is weakly
admissible. We remark that it is trivial to check whether a weakly admissible group
is admissible: if G is weakly admissible, then it is admissible if and only if Rn(cid:111)G is
not unimodular.
In the present work we consider the case where G is the exponential of the real
linear span g of a finite subset of commuting elements in gl(V), or in other words,
G is a connected abelian Lie subgroup of GL(V). Here, as is described below and
also in [3], G can be identified with a group of complex, lower triangular matrices
in block form, so that the semisimple part of each block is a scalar multiple of the
identity. On each block these scalar values are given by characters Λ on G of the
form Λ(expX) = eλ(X) where λ is a linear form on g; we call such linear forms roots
and the corresponding characters root characters.
REGULARITY 3
Now suppose that we are given such a group G and we want to determine whether
G is regular. If the imaginary part of each root is a scalar multiple of the real part,
then (since G is connected and the action of G is of exponential type) all of its orbits
areregular[2]. Otherwise, letG bethesetofallelementsofGthatlieinthekernelof
1
all root characters. Since some of the roots have independent imaginary parts, G is
1
not necessarily connected, and so even though G is unipotent, it may not be regular.
1
Roughly speaking irregular orbits can arise as a result of two distinct characteristics:
either from an irrational relation between imaginary parts of the various roots, or
from an irregular action of the discrete part of the group G .
1
Much of the work in this paper is devoted to understanding the action of G and
1
its relation to that of G. We show that if G is regular, then G is regular, and that
1
if G is not regular then there is a G-invariant, conull, G subset U of V in which all
1 δ
G-orbits are not regular. Moreover, we present an explicit procedure for computing
a Zariski open subset Ω of V and for each v ∈ Ω , a subset ∆(v) of gl(V), so that
0 0
G is regular if and only if the function ∆ is constant and its value is a closed set.
1
We define precisely a rationality condition on (the imaginary parts of) the roots and
prove that if G satisfies the rationality condition, then G is regular if an only if G is
1
regular. Thus we obtain the following method for “finding” regular and admissible
connected abelian groups:
1. Check the rationality condition for G. If yes, then
2. compute G , Ω , and for v ∈ Ω compute ∆(v).
1 0 0
3. Determine whether ∆(v) is constant. If yes, then put ∆ = ∆(v), and
4. determine whether the set ∆ is closed. If yes, then G is regular.
5. Compute the stabilizers G(v). If they are compact almost everywhere and G
is closed, then G is weakly admissible.
In Section 2 we provide topological background and prove several characterizations
of regularity for an orbit of an analytic group action. The projective action of a Lie
group on a fiber bundle is considered in Section 3, and there we prove a crucial result
(Proposition 3.1) concerning the relations between regularity of actions on the base
space, fibers, and total space. We consider linear actions on a finite dimensional
vector space V in Section 4, where we define the rationality condition, and if this
condition is satisfied, we give an algorithm for computing an explicit presentation of
the unipotent group G . We then study the action of G by first studying a unipotent
1 1
connectedaction: weconstructthesmallestanalyticunipotentsubgroupN ofGL(V)
containing G , and using a standard algorithm, we compute an N-invariant Zariski
1
open subset Ω of V where the N-action is simply described. At this point the
0
G -orbits in Ω are embedded in the N-orbits in a very transparent way and this
1 0
permits the computation of ∆ (Lemma 4.13) and the main criterion for regularity of
G . It turns out that Ω is also G-invariant, and we use Proposition 3.1 to obtain an
1 0
explicit Zariski open set Ω in V, included in Ω , such that if G satisfies the rationality
0
4 D. ARNAL, B. DALI, B. CURREY, AND V. OUSSA
condition, then every G-orbit in Ω is regular if and only if every G -orbit in Ω is
1 0
regular.
2. analytic group actions and regular orbits
Much of this section is issued from [10], essentially Sections 1.1 and 2.9.
A smooth map ϕ : M −→ N from a m-dimensional C∞ manifold M into a n-
dimensional manifold N is an immersion if dϕ is injective for any x in N, ϕ is an
x
embedding if it is an one-to-one immersion, it is a regular embedding if it is an em-
bedding and a homeomorphism between M and ϕ(M), equipped with the relative
topology coming from the N topology.
A subset A in N is locally closed if it is open in its closure or if it is the intersection
of a closed and an open subset in N.
Proposition 2.1. If ϕ is a regular embedding, then ϕ(M) is locally closed in N.
Proof. First, since ϕ is an immersion, for any x in M, if y = ϕ(x), by the maximal
rank theorem, we can find open neighborhoods U for x, V for y, equipped with local
coordinates ξ : U (cid:51) z (cid:55)→ (x (z),...,x (z)) ∈] − a,a[m⊂ Rm and η : V (cid:51) z(cid:48) (cid:55)→
1 m
(y (z(cid:48)),...,y (z(cid:48))) ∈]−a,a[n⊂ Rn such that ξ(x) = 0, η(y) = 0 and
1 n
η ◦ϕ◦ξ−1(t ,...,t ) = (t ,...,t ,0,...,0).
1 m 1 m
Since ϕ is a homeomorphism, ϕ(U) is open in ϕ(M), for the relative topology.
There exists an open subset W in N such that ϕ(U) = ϕ(M) ∩ W, thus ϕ(U) =
ϕ(M) ∩ (W ∩ V). Now W ∩ V is open in N and contains y = ϕ(x), thus there is
0 < a(cid:48) < a such that V(cid:48) = η−1(]−a(cid:48),a(cid:48)[n) ⊂ W ∩V; by construction, V(cid:48) is open in
N. Put U(cid:48) = ξ−1(]−a(cid:48),a(cid:48)[m); U(cid:48) is open in M and, with the above expression of ϕ,
ϕ(U(cid:48)) = ϕ(M)∩V(cid:48).
It follows that ϕ(M) is locally closed: for each x ∈ M choose such open subsets U(cid:48),
x
V(cid:48) for any x in M as above, and put N = (cid:83) V(cid:48). Then ϕ(M) = (cid:83) ϕ(U(cid:48)) ⊂
x x∈M x x∈M x
(cid:83) V(cid:48) = N, and for any z in ϕ(M) ∩ N there is x such that z is in V(cid:48), and a
x∈M x x
sequence (z ) in ϕ(M) converging to z. If n is larger than some p, z is in the closed
n n
subset ϕ(M)∩V(cid:48) of V(cid:48), thus z also belongs to ϕ(M)∩V(cid:48), and hence
x x x
ϕ(M)∩N = ϕ(M).
(cid:3)
REGULARITY 5
Suppose now that G is a Lie group acting analytically on an analytic manifold N.
Given x ∈ N, we denote by G(x) the stability subgroup of x. It is closed in G and
we let G/G(x) have the quotient topology, which is Hausdorff, locally compact, and
second countable. The natural map ϕ : G/G(x) −→ N defined by gG(x) (cid:55)→ g·x is
x
well-defined, one-to-one, and G-equivariant. Since G/G(x) has the quotient topology,
ϕ is continuous. Moreover, there is a unique analytic structure on G/G(x) such that
x
it becomes an analytic manifold, the natural action of G on G/G(x) is analytic, and
ϕ is an immersion. We say that the orbit O = G·x of x is a regular orbit if the map
x
ϕ is a regular embedding. This property does not depend upon the choice of x in
x
O.
A topological space X is a Baire space if the Baire’s lemma holds for X: if (U ) is
n
a sequence of dense open subset of X, then (cid:84)∞ U is still dense in X. Any locally
n=1 n
compact space is a Baire space.
Theorem 2.2. Let O = G·x an orbit in N, equipped with the relative topology.
Then the following are equivalent:
1. O is a regular orbit,
2. O is locally closed,
3. O is locally compact,
4. O is a Baire space.
Proof.
1. =⇒ 2. By the preceding result, if O is regular, then O is locally closed.
2. =⇒ 3. If O is locally closed, O = O ∩ W, with W open, and hence O is a closed
subspace of the open subset W in N. Since W is locally compact, then for
the relative topology, O is locally compact.
3. =⇒ 4. Since O is locally compact, it is a Baire space.
4. =⇒ 1. Now let K be a compact subset in G/G(x), with non-empty interior. Thus
there is a sequence (g ) in G such that G/G(x) = (cid:83)∞ g K (since G/G(x)
n n=1 n
is second countable).
We get O = (cid:83)∞ g .ϕ (K) and each g .ϕ (K) is compact thus closed. By
n=1 n x n x
the Baire lemma, one of these subsets has a non-empty interior. Since, they
are all homeomorphic to ϕ (K) through the map z (cid:55)→ g−1.z, ϕ (K) has a non
x n x
empty interior W and we put V = ϕ−1(W). The restriction ϕ | of ϕ to V
1 x x V1 x 1
is a continuous bijection from V onto W, but V ⊂ K. Thus V is compact in
1 1 1
G/G(x) and ϕ | is a homeomorphism, and ϕ | is a homeomorphism from
x V1 x V1
V onto W.
1
6 D. ARNAL, B. DALI, B. CURREY, AND V. OUSSA
Let U be any open subset in G/G(x); then U is the union of a family of
subsets g ·U(cid:48), with g ∈ G and U(cid:48) open subset in V . Thus
i i i i 1
(cid:91)
ϕ (U) = g .ϕ (U(cid:48))
x i x i
i
is open in O. This shows that ϕ is an open map; hence ϕ is homeomorphism
x x
and O is regular.
(cid:3)
Let q : G −→ G/G(x) be the canonical mapping; in what follows we will also write
[g] = q(g),g ∈ G, and [K] = q(K) for K ⊆ G. The map q is not only continuous,
(cid:0) (cid:1)
but open: for any open subset V of G, q−1 [V] = V·G(x) = ∪ V·t is open in G
t∈G(x)
and hence [V] is open in G/G(x).
Since G(x) is a closed Lie subgroup of G, then G/G(x) has a structure of a dif-
ferentiable manifold, with a local chart around the base point [1] ∈ G/G(x) given as
follows. Let g(x) be the Lie algebra of G(x) and m a supplementary space for g(x) in
the Lie algebra g of G, there is a sufficiently small open neighborhood U of 0 in m, so
that V = expU·G(x) is open in G and for which the map θ : [expX] (cid:55)→ X (X ∈ U)
is well-defined on the open set [V] in G/G(x), and ([V],θ) is a local chart about [1].
As a consequence, any neighborhood of [1] in G/G(x) contains a subset of the form
[exp(M)] with M a compact neighborhood of 0 in m, included in U, and if K is a
x
compact neighborhood of 1 in G(x), then K = exp(M)K is a compact neighborhood
x
of 1 in G such that [K] = [exp(M)]. Since K contains an open neighborhood of 1 in
G and q is open, then [K] is a neighborhood of [1] in G/G(x).
To simplify a proof in the next section, we express the preceding proposition in a
little bit different way:
Corollary 2.3. The orbit O = G·x is regular if and only if, for any compact neigh-
borhood K of 1 in G, there is a neighborhood V of x in N such that O∩V ⊂ K·x.
Proof. Suppose O regular and let K be a compact neighborhood of 1 in G. Then [K]
is a neighborhood of [1] in G/G(x), K·x a neighborhood of x in O, in the relative
topology. There is V open in N, containing x such that O∩V ⊂ K·x.
Conversely, suppose that the condition of the corollary holds. To show that O is
regular it is enough (as in the preceding lemma) to show that ϕ : G/G(x) −→ O is
x
an open map. Indeed, let U be any open subset of G/G(x) and choose any s ∈ G
such that [s] ∈ U. Then s−1U is a neighborhood of [1], so by the remark preceding
the corollary, s−1U contains some subset [K] with K compact neighborhood of 1 in
G. Therefore we have a neighborhood V of x in N such that K·x = ϕ ([K]) contains
x
O∩V. Now ϕ ([s]) = s·x belongs to O∩s·V and O∩s·V is included in ϕ (U), and
x x
hence ϕ is open. (cid:3)
x
REGULARITY 7
Let Ω be a G-invariant subset of N. We say that Ω is G-regular if all the orbits
in Ω are regular. The following is a consequence of the fundamental work of Glimm
and Effros.
Theorem 2.4. [7, Theorem 1], [4, Theorem 2.9] Suppose that Ω is locally compact;
then the following are equivalent.
1. Ω is regular.
2. Ω/G is countably separated.
3. Ω/G possesses the T separation property.
0
4. There is a Borel subset Σ of Ω that meets each G-orbit in exactly one point.
3. Group action on a fiber bundle
Suppose that π : X −→ W is a fibre bundle with fiber Z, and that G is a Lie
group acting smoothly and projectively on X, through (s,x) (cid:55)→ s·x. Then there is a
smooth G-action (s,w) (cid:55)→ b(s)w such that, for any s ∈ G and x ∈ X,
π(s·x) = b(s)π(x).
For each w ∈ W, denote by i the homeomorphism from π−1(w) onto Z; the group
w
action on X gives rise to a collection of mappings φ (s) : Z −→ Z,s ∈ G, such that,
w
for any s and x, i (s·x) = φ (s)(i (x)), or
b(s)w w w
(3.1) φ (s)z = i (s·i−1z), s ∈ G, z ∈ Z.
w b(s)w w
Clearly (s,w,z) (cid:55)→ φ (s)z is smooth, φ (1) = Id, and for s ,s ∈ G,
w w 1 2
φ (s s ) = φ (s )◦φ (s ).
w 1 2 b(s2)w 1 w 2
In the present paper, we consider only two simple cases of this situation:
1. Smooth action on product manifold.
In this case, X is the trivial bundle W × Z, and the action has the form
(s,w,z) (cid:55)→ (b(s)w,φ(s)z), where φ is a G-action on Z. The typical example
is the action of R on C× = C \ {0}, defined by (t,z) (cid:55)→ Λ(t)z = eatz (a ∈
C). Identifying C× with the product manifold (R×)×T where T is the one
+
dimensional torus, through z = reiθ (cid:55)→ (r,eiθ) = (|z|,sign(z)), the action
becomes
(cid:0)
(t,z) = (t,r,eiθ) (cid:55)→ (etRe(a)r,ei(θ+Im(a)t)) = |Λ(t)||z|,Λε(t)sign(z)),
where Λε(t) is the character sign(Λ(t)).
8 D. ARNAL, B. DALI, B. CURREY, AND V. OUSSA
2. Linear action with an invariant subspace.
In this case, the total space is a finite dimensional vector space V, and we
suppose that G acts linearly on V, and that V contains a subspace Z which
is G-invariant. Then there is a natural linear G-action on the quotient space
W = V/Z, defined by b(s)(v + Z) = s·v + Z. Let us fix a supplementary
0
space W for Z in V. We can identify W with W and the canonical map
0
π : V −→ W becomes the projection onto W parallel to Z. If p is Id − π,
0
the projection onto Z parallel to W, then G-action on v = w+z is
s·(w+z) = s·w+s·z = b(s)w+p(s·w)+s·z,
and the map φ (s) is the affine map
w
z (cid:55)→ φ (s)z = p(s·w)+s·z.
w
In these two cases the fiber bundle is trivial: the total space is identified with a
productW×Z andtheactioniswrittenas(s,w,z) (cid:55)→ (b(s)w,φ (s)z). Forsimplicity
w
of notation, we assume that the bundle is trivial in the following proposition.
For w ∈ W, put G(w) = {s ∈ G : b(s)w = w}; then the relation (3.1) shows that
the action of G(w) on X naturally induces an action on Z by
(s,z) (cid:55)→ φ (s)z, s ∈ G(w), z ∈ Z.
w
Fix x = (w ,z ) ∈ X, put O = G·x , O = b(G)w , and ω = φ (G(w ))z . We
0 0 0 0 0 0 w0 0 0
have the following:
Proposition 3.1. If O is regular, then ω is regular. On the other hand if O and ω
0
are both regular, then O is regular.
Proof. Suppose O regular. Let g(w ) be the Lie algebra of G(w ), and m a supple-
0 0
mentary space to g(w ) in the Lie algebra g of G. Fix a compact neighborhood M of
0
0 in m, sufficiently small such that the map M −→ expMG(w ) ⊂ G/G(w ) defines
0 0
a local chart for the manifold G/G(w ).
0
Let K be any compact neighborhood of 1 in G(w ), denote K = expMK .
w0 0 w0
Then K·x ∩({w }×Z) = K ·x = {w }×φ (K )z . Indeed, for any X ∈ M,
0 0 w0 0 0 w0 w0 0
k ∈ K , we have x = expXk·x is in {w } × Z if and only if b(expX)w = w ,
w0 0 0 0 0
if and only if X = 0. Similarly, O ∩ ({w } × Z) = G(w )·x = {w } × ω, since
0 0 0 0
s·x ∈ {w }×Z if and only if s ∈ G(w ).
0 0 0
Since O is regular, there is a neighborhood V of x in X such that O∩V ⊂ K·x ,
0 0
thus{w }×φ(K )z = K·x ∩({w }×Z)containsO∩({w }×Z)∩V = V∩({w }×ω).
0 w 0 0 0 0 0
Since W = p(V ∩({w }×Z)) is a neighborhood of z in Z, the corollary implies that
0 0
ω is regular.
Now suppose that both O and ω are regular. Let K be any compact neighborhood
0
of 1 in G; K contains a compact neighborhood of 1 of the form exp(M)K (M and
w0
K are as above).
w0
REGULARITY 9
Since ω is regular, we have a neighborhood Z of z in Z such that
0
Z ∩ω ⊂ φ (K )z .
w0 w0 0
Define f : m×W ×Z −→ Z by
f(X,w,z) = φ (expX)z.
w
By continuity of f, we have a compact neighborhood M(cid:48) of 0 in m, a neighborhood
W of w in W, and a neighborhood Z(cid:48) of z in Z such that
0 0
f(M(cid:48) ×W ×Z(cid:48)) ⊂ Z.
We may assume that M(cid:48) ⊂ M, and moreover, −M(cid:48) = M(cid:48).
Since O is regular, we have a neighborhood W(cid:48) of w in W such that
0 0
W(cid:48) ∩O ⊂ b(expM(cid:48))w .
0 0
We can choose W(cid:48) ⊂ W.
We claim that (W(cid:48) × Z(cid:48)) ∩ O ⊆ K·x . Let x = (w,z) ∈ (W(cid:48) × Z(cid:48)) ∩ O. Then
0
w ∈ W(cid:48) ∩O , and so we have X ∈ M(cid:48) such that w = b(expX)w . Now (−X,w,z)
0 0
belongs to M(cid:48) ×W ×Z(cid:48), and hence
exp−X·x = (b(exp−X)w,φ (exp−X)z) = (w ,φ (exp−X)z) = (w ,f(−X,w,z))
w 0 w 0
belongs to O∩({w }×Z). But
0
O∩({w }×Z) = {w }×(ω ∩Z) ⊆ {w }×φ (K )z .
0 0 0 w0 w0 0
Hence exp−X·x belongs to {w }×φ (K )z and so
0 w0 w0 0
(cid:0) (cid:1)
x ∈ expX {w }×φ (K )z = expXK ·x ⊆ K·x .
0 w0 w0 0 w0 0 0
Thus the claim is proved. Now by Corollary 2.3, O is regular.
(cid:3)
4. Linear action of a connected abelian group
Let V be a finite dimensional real vector space and let G be a Lie subgroup of
GL(V); we say that G is regular if there is a locally compact, invariant, conull Borel
subset Ω of V such that Ω is G-regular for the natural action of G on V. Now suppose
that G is connected and abelian; we are looking for those G which are regular. We
remark that we do not assume that G is closed in GL(V).
Example 4.1. Let us recall a very well-known example: the Mautner group is the
semidirect product C2 (cid:111)R, where the action of s ∈ R is given by the complex 2×2
matrix
(cid:20) (cid:21)
i 0
exps √ .
0 2i
10 D. ARNAL, B. DALI, B. CURREY, AND V. OUSSA
In this example V = C2 (cid:39) R4 and G is the one-parameter group above. The group G
is therefore not closed in GL(V) (it is dense in the set of unitary diagonal matrices).
Let v = [v ,v ]t be in V, if v v (cid:54)= 0, then G·v is also a non locally closed, dense
1 2 1 2
subset inside the 2-dimensional torus {u = [u ,u ]t : |u | = |v |, j = 1,2}. Since the
1 2 j j
union of these orbits is conull, G is not regular.
Throughout the paper, we use the following notation: given e ,e ,...,e ele-
1 2 q
ments of a vector space over a field K, denote the K-span of these elements by
(e ,e ,...,e ) .
1 2 q K
Let g be a commutative Lie subalgebra of gl(V) and G = exp(g). Using the com-
plexificationV ofV andtheconjugationσ : u (cid:55)→ u, wecanwriteaconvenientmatrix
C
representation for elements of g.
A complex form λ is a root for the action of g if, for each X ∈ g, λ(X) is an
eigenvalue for X. If λ is a root, the corresponding generalized eigenspace for λ is
E = ∩ ker (X −λ(X)Id)dimVC.
λ X∈g VC
Since g is abelian, V = ⊕ E and for each λ, XE ⊆ E for all X ∈ g. If λ is real,
C λ λ λ λ
σ(E ) = E ; otherwise there is λ(cid:48) such that λ(cid:48) = λ and E = σ(E ). From now on,
λ λ λ(cid:48) λ
fix an ordering for the roots such that λ ,...,λ are real, λ ,...,λ are not real,
1 r r+1 p
and
λ = λ ,...,λ = λ .
p+1 r+1 2p−r p
where it is understood that if there are no real roots, then r = 0. Put E = E for
j λj
all j.
Now V is the following real subspace in V :
C
r p
(cid:77) (cid:77) (cid:16) (cid:17)
V = (E ∩V)⊕ (E +σ(E ))∩V
j j j
j=1 j=r+1
(cid:40) (cid:41)
p
(cid:88) (cid:88)
= v + (w +w ), v ∈ E ∩V (1 ≤ j ≤ r), w ∈ E (r < j ≤ p) .
j j j j j j j
j≤r j=r+1
(cid:80) (cid:80)
Then the map ι : V −→ V , defined by ι(v) = v + w , is a bijection
C j≤r j r<j≤p j
(cid:76) (cid:76)
between V and the space (E ∩V)⊕ E . Put
j≤r j r<j≤p j
V = E ∩V (1 ≤ j ≤ r), and V = E (r < j ≤ p),
j j j j
then ι(V) = (cid:76)p V . Since ι is a homeomorphism, the orbit of x in V is regular if
j=1 j
and only if the orbit of ι(x) is regular in ι(V). Therefore we identify V with (cid:76)p V
j=1 j
and each X in gl(V) with ι◦X ◦ι−1.