The existence of a semialgebraic continuous factorization map for some compact linear groups

0.09 MB·English

by  O. G. Styrt

UDK 512.815.1+512.815.6+512.816.1+512.816.2
MSC 14L24+22C05
O.G.Styrt
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The existence
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n of a semialgebraic continuous
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factorization map

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for some compact linear groups

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It is proved that each of compact linear groups of one special type admits
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h a semialgebraic continuous factorization map onto a real vector space.
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a Key words: Lie group, factorization map of an action, semialgebraic map.
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§1. Introduction
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v
8 In this paper, we prove that each of compact linear groups of one certain type admits
2 a semialgebraic continuous factorization map onto a real vector space. This problem arose
3 fromthequestionwhenthetopologicalquotientofacompactlineargroupishomeomorphic
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to a vector space that was researchedin [1, 2, 3, 4, 5].
0
. Forconvenience,denotebyPR (resp.byPC)theclassofallfinite-dimensionalEuclidean
1 (resp. Hermitian) spaces. Let F be one of the fields R and C. We will write Fk (k > 0) for
0 the space Fk ∈ PF whose standard basis is orthonormal. For any spaces W1,W2 ∈ PF, we
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1 will write W1⊕W2 for their orthogonal direct sum W1⊕W2 ∈PF (unless they are linearly
: independent subspacesofthe samespace).ForanyspaceW ∈PF,denote byS(W)the real
v subspace A ∈ EndF(W): A = A∗ ⊂ EndF(W), by O(W) the compact Lie group C ∈
Xi ∈GLF(W(cid:8)): CC∗ =E ⊂GLF(W),(cid:9)and by SO(W) its compact subgroup O(W)∩SLF(cid:8)(W).
(cid:9)
r The following theorem is the main result of the paper and will be proved in §3.
a
Theorem1.1. Consideranumberk ∈{1,2},aspaceW ∈PF suchthatdimFW >2−k,
and the representation
O(W): S(W)/(RE) ⊕HomF(Fk,W), C: (A+RE,B)→(CAC−1+RE,CB). (1.1)
(cid:0) (cid:1)
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1)If k = 1, then there exist a real vector space V and, in terms of the representation
O(W): V⊕F, C: (v,λ)→(v,detFC·λ),asemialgebraiccontinuous O(W) -equivariant
(cid:0) (cid:1)
surjective map S(W)/(RE) ⊕HomF(Fk,W) → V ⊕F whose fibres coincide with the
(cid:0) (cid:1)
SO(W) -orbits.
(cid:0) (cid:1)
2)If k =2, then there exist a real vector space V and a semialgebraic continuous surjective
map S(W)/(RE) ⊕HomF(Fk,W)→V whose fibres coincide with the O(W) -orbits.
(cid:0) (cid:1) (cid:0) (cid:1)
§2. Auxiliary facts
Consider a space W ∈PF and the subsets S+(W):= A∈S(W): A>0 ⊂S(W) and
(cid:8) (cid:9)
S0(W):=(cid:0)S+(W)(cid:1)/(cid:0)GLF(W)(cid:1)⊂S+(W)⊂S(W).
Set M(W):= (A,λ)∈S+(W)×F: detFA=|λ|2 ⊂S(W)⊕F.
(cid:8) (cid:9)
The following statement is well-known.
Statement 2.1. The fibres of the map π: EndF(W)→S(W)⊕F, X →(XX∗,detFX)
coincide with the orbits of the action
SO(W): EndF(W), C: X →XC−1, (2.1)
and π EndF(W) =M(W)⊂S(W)⊕F.
(cid:0) (cid:1)
Lemma2.1. ThemapM(W)→ S(W)/(RE) ⊕F, (A,λ)→(A+RE,λ)isabijection.
(cid:0) (cid:1)
(cid:3) See Lemma 5.1 in [3]. (cid:4)
Corollary 2.1. The map S0(W)→S(W)/(RE), A→A+RE is a bijection.
Corollary 2.2. The map EndF(W) → S(W)/(RE) ⊕F, X → (XX∗+RE,detFX)
(cid:0) (cid:1)
is surjective, and its fibres coincide with the orbits of the action (2.1).
(cid:3) Follows from Statement 2.1 and Lemma 2.1. (cid:4)
Now consider arbitrary spaces W,W1 ∈PF over the filed F.
Statement 2.2. The fibres of the map π: HomF(W1,W)→S(W), X →XX∗ coincide
with the orbits of the action
O(W1): HomF(W1,W), C1: X →XC1−1, (2.2)
and π HomF(W1,W) = A ∈ S+(W): rkFA 6 dimFW1 ⊂ S(W). In particular, in the
(cid:0) (cid:1) (cid:8) (cid:9)
case dimFW1 =dimFW −1, we have π HomF(W1,W) =S0(W)⊂S(W).
(cid:0) (cid:1)
We omit the proof since it is clear.
Corollary 2.3. Suppose that dimFW1 =dimFW −1. Then the map HomF(W1,W)→
→S(W)/(RE), X →XX∗+RE is surjective, and its fibres coincide with the orbits of the
action (2.2).
(cid:3) Follows from Statement 2.2 and Corollary 2.1. (cid:4)
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§3. Proof of the main result
In this section, we will prove Theorem 1.1.
Firstofall,letusprovetheclaimforapair(k,W)(k ∈{1,2},W ∈PF,dimFW =2−k).
If k = 2 and W = 0, then the space of the representation (1.1) is trivial. If k = 1 and
dimFW = 1, then the representation (1.1) is isomorphic to the tautological representation
of the group O(W) and, consequently, to the representation O(W): F, C: λ → detFC ·λ,
and the subgroup SO(W)⊂O(W) is trivial.
Thiscompletelyprovestheclaimforapair(k,W)(k ∈{1,2},W ∈PF,dimFW =2−k).
Take an arbitrary pair (k,W) (k ∈ {1,2}, W ∈ PF, n := dimFW > 2−k) and assume
that the claim of Theorem 1.1 holds for any pair (k,W1) (W1 ∈PF, dimFW1 =n−1). We
will now prove it for the pair (k,W).
Sincen>2−k >0,thereexistsaspaceW1 ∈PFsuchthatdimFW1 =n−1.Considerthe
embedding R: O(W1)֒→O(W1⊕Fk), R(C1): (x1,x2)→(C1x1,x2). The representations
O(W1): S(W1⊕Fk), C1: A→R(C1)AR(C1−1) (3.1)
and O(W1): S(W1)⊕HomF(Fk,W1)⊕S(Fk), C1: (A1,B,A2) → (C1A1C1−1,C1B,A2) are
isomorphic via the O(W1) -equivariant R-linear isomorphism
(cid:0) (cid:1)
θ: S(W1)⊕HomF(Fk,W1)⊕S(Fk)→S(W1⊕Fk),
∗
θ(A1,B,A2): (x1,x2)→(A1x1+Bx2,B x1+A2x2).
Therefore, the representation (3.1) is isomorphic to the direct sum of some trivial real
representation of the group O(W1) and the representation
O(W1): S(W1)/(RE) ⊕HomF(Fk,W1), C1: (A1+RE,B)→(C1A1C1−1+RE,C1B)
(cid:0) (cid:1)
′
withtrivialfixedpointsubspace.Thus,thereexistarealvectorspaceV andanisomorphism
ϕ: S(W1⊕Fk)/(RE)→V′⊕ S(W1)/(RE) ⊕HomF(Fk,W1) of the representations
(cid:0) (cid:1)
O(W1): S(W1⊕Fk)/(RE), C1: A+RE →R(C1)AR(C1−1)+RE;
O(W1): V′⊕ S(W1)/(RE) ⊕HomF(Fk,W1),
C1: (v′,A1+RE(cid:0),B)→(v′,C1(cid:1)A1C1−1+RE,C1B).
In terms of the representations
O(W)×O(W1): HomF(W1⊕Fk,W), (C,C1): Z →CZR(C1−1);
O(W)×O(W1): S(W)/(RE) ⊕HomF(Fk,W),
(C,C1): (A+RE(cid:0),B)→(CA(cid:1)C−1+RE,CB);
O(W)×O(W1): V′⊕ S(W1)/(RE) ⊕HomF(Fk,W1),
(C,C1): (v′,A1+RE,B(cid:0) )→(v′,C1A(cid:1)1C1−1+RE,C1B),
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the maps
π0: HomF(W1⊕Fk,W)→ S(W)/(RE) ⊕HomF(Fk,W),
ψ: HomF(W1⊕Fk,W)→ZV→′⊕(cid:0)(ZS|(WW1)1()Z/(cid:0)(|RWE1))∗+⊕RHEo,mZ(cid:1)F|(FFkk(cid:1);,W1), Z →ϕ(Z∗Z+RE)
(cid:0) (cid:1)
are O(W)×O(W1) -equivariant. By Corollary 2.3, the map π0 is surjective, and its fibres
(cid:0) (cid:1)
coincide with the O(W1) -orbits.
(cid:0) (cid:1)
Case 1). k =1.
Consider the representation
O(W)×O(W1): V′⊕ S(W1)/(RE) ⊕HomF(Fk,W1)⊕F,
(C,C1): (v′,A1+RE,B,µ)→(v(cid:0)′,C1A1C1−1+(cid:1)RE,C1B,detFC·detFC1−1·µ).
SincedimFW1 =n−1,thereexistarealvectorspaceV1 and,intermsoftherepresentation
O(W)×O(W1): V′⊕V1⊕F2, (C,C1): (v′,v1,λ,µ)→(v′,v1,detFC1·λ,detFC·detFC1−1·µ),
a semialgebraic continuous O(W)×O(W1) -equivariant surjective map
(cid:0) (cid:1)
π1: V′⊕ S(W1)/(RE) ⊕HomF(Fk,W1)⊕F→V′⊕V1⊕F2
(cid:0) (cid:1)
whose fibres coincide with the SO(W1) -orbits. Since dimF(W1⊕Fk)=(n−1)+k=n=
=dimFW, we can identify W a(cid:0)nd W1⊕(cid:1)Fk as spaces of the class PF. The map
ψ: HomF(W1⊕Fk,W)→V′⊕ S(W1)/(RE) ⊕HomF(Fk,W1)⊕F,
e Z → ψ(Z),detFZ =(cid:0) ϕ(Z∗Z+R(cid:1)E),detFZ
(cid:0) (cid:1) (cid:0) (cid:1)
is O(W)×O(W1) -equivariant. Also, by Corollary 2.2, the map ψ is surjective, and its
(cid:0) (cid:1)
fibres coincide with the SO(W) -orbits. Hence, the map π1 ◦ψ: HeomF(W1 ⊕Fk,W) →
→V′⊕V1⊕F2 is O(W)(cid:0)×O(W1(cid:1)) -equivariant and surjective, aend its fibres coincide with
(cid:0) (cid:1)
the SO(W)×SO(W1) -orbits. If we consider the representation
(cid:0) (cid:1)
O(W)×O(W1): V′⊕V1⊕R⊕F, (C,C1): (v′,v1,t,λ)→(v′,v1,t,detFC·λ),
then the map γ: V′⊕V1⊕F2 →V′⊕V1⊕R⊕F, (v′,v1,λ,µ)→ v′,v1,|λ|2−|µ|2,λµ is
(cid:0) (cid:1)
O(W)×O(W1) -equivariant. Denote by T the multiplicative group c ∈ F: |c| = 1 . We
(cid:0) (cid:1) (cid:8) (cid:9)
have n>2−k =1, dimFW1 =n−1>0. Consequently,
detFC1: C1 ∈O(W1) =T. (3.2)
(cid:8) (cid:9)
One caneasily see thatthe map F2 →R⊕F, (λ,µ)→ |λ|2−|µ|2,λµ is surjective,andits
fibres coincide with the orbits of the representation T:(cid:0)F2, c: (λ,µ)→(cid:1)(cλ,c−1µ). By (3.2),
themapγ issurjective,anditsfibrescoincidewiththe O(W1) -orbits.Itfollowsfromabove
(cid:0) (cid:1)
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thatthemapγ◦π1◦ψ: HomF(W1⊕Fk,W)→V′⊕V1⊕R⊕Fis O(W)×O(W1) -equivariant
(cid:0) (cid:1)
and surjective, and iets fibres coincide with the SO(W)×O(W1) -orbits.
(cid:0) (cid:1)
Recall that the map π0 is O(W)×O(W1) -equivariant and surjective, and its fibres
(cid:0) (cid:1)
coincide with the O(W1) -orbits. Therefore,there exists an O(W) -equivariantsurjective
(cid:0) (cid:1) (cid:0) (cid:1)
map π: S(W)/(RE) ⊕HomF(Fk,W) → V′ ⊕V1 ⊕R⊕F satisfying π◦π0 ≡ γ ◦π1 ◦ψ
(cid:0) (cid:1)
whosefibrescoincidewiththe SO(W) -orbits.Sincethemapsγ,π1,ψ,π0aresemialgebraice,
(cid:0) (cid:1)
continuous, and surjective, so is the map π. e
This completely proves the claim in the case k =1.
Case 2). k =2.
SincedimFW1 =n−1,thereexistarealvectorspaceV1 andasemialgebraiccontinuous
surjective map π1: V′⊕ S(W1)/(RE) ⊕HomF(Fk,W1) → V′⊕V1 whose fibres coincide
(cid:0) (cid:1)
withthe O(W1) -orbits.Further,dimF(W1⊕Fk)=(n−1)+k=n+1=dimFW+1,and,by
(cid:0) (cid:1)
Corollary2.3,the mapψ is surjective,andits fibres coincide withthe O(W) -orbits.Since
themapψis O(W)×O(W1) -equivariant,themapπ1◦ψ: HomF(W1⊕(cid:0)Fk,W(cid:1))→V′⊕V1 is
(cid:0) (cid:1)
surjective,anditsfibrescoincidewiththe O(W)×O(W1) -orbits.Recallthatthemapπ0 is
(cid:0) (cid:1)
O(W)×O(W1) -equivariantandsurjective,anditsfibrescoincidewiththe O(W1) -orbits.
(cid:0)Hence,there exi(cid:1)stsa surjectivemap π: S(W)/(RE) ⊕HomF(Fk,W)→V(cid:0)′⊕V1 sa(cid:1)tisfying
(cid:0) (cid:1)
π◦π0 ≡π1◦ψ whose fibres coincide with the O(W) -orbits. Since the maps π1,ψ,π0 are
(cid:0) (cid:1)
semialgebraic, continuous, and surjective, so is the map π.
This completely proves the claim in the case k =2.
Thus,wehavecompletelyprovedTheorem1.1(usingmathematicalinductionondimFW
separately for each number k ∈{1,2}).
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