prehomogeneous spaces for the coadjoint action of a parabolic group

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Journal of Algebra 279 (2004) 558–565

www.elsevier.com/locate/jalgebr

Prehomogeneous spaces for the coadjoint actiof a parabolic group

Simon Goodwin

School of Mathematics and Statistics, University of Birmingham, Birmingham, B15 2TT, UK

Received 12 November 2003

Available online 10 March 2004

Communicated by Jan Saxl

Abstract

Let k be an algebraically closed field and letG be a reductive linear algebraic group overk. LetPbe a parabolic subgroup ofG, Pu its unipotent radical andpu the Lie algebra ofPu. A fundamentalresult of R. Richardson says thatP acts onpu with a dense orbit (see [R.W. Richardson, BuLondon Math. Soc. 6 (1974) 21–24]). The analogous result for the coadjoint action ofP on p∗

u isalready known for chark = 0 (see [A. Joseph, J. Algebra 48 (1977) 241–289]). In this note we pthis result for arbitrary characteristic. Our principal result is thatb∗

u is a prehomogeneous spacea Borel subgroupB of G. From this we deduce that a parabolic subgroupP of G acts onn∗ with adense orbit for anyP -submodulen of P . Further, we determine when the orbit map for such an ois separable. 2004 Elsevier Inc. All rights reserved.

1. Introduction

Let G be a reductive linear algebraic group defined over the algebraically closed fik.Let P be a parabolic subgroup ofG, Pu its unipotent radical andpu = Lie(Pu) the Liealgebra ofPu. We consider the coadjoint action ofP onp∗

u. Our principal result is

Theorem 1.1. LetG be a reductive group and letB be a Borel subgroup ofG. Thenb∗u is

a prehomogeneous space forB.

E-mail address:[email protected]: http://www.mat.bham.ac.uk/S.M.Goodwin.

0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2004.01.007

S. Goodwin / Journal of Algebra 279 (2004) 558–565 559

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From which we deduce

Corollary 1.2. Let G be a reductive group, letP be a parabolic subgroup ofG and n

a P -submodule ofpu. Thenn∗ is a prehomogeneous space forP .

In [9], Richardson proved thatP always acts onpu with a dense orbit. Corollary 1.implies the analogous result for the coadjoint action ofP onp∗

u.Theorem 1.1 is already known for chark = 0 (see [5, 2.4]). The proof inloc. cit. deals

primarily with Lie algebras. The setM = {(f, x) ∈ b∗u × bu: f ([b, x]) = 0} is considered

and it is shown that there existsf ∈ b∗u such that{x ∈ bu: (f, x) ∈ M} = ∅. For suchf ,

B · f is shown to be dense inb∗u.

If chark �= 0, then forf ∈ b∗u the orbit mapB → B · f need not be separable. Th

implies that the proof in [5] is not valid for positive characteristic. Our proof uses a siminductive method for constructing representatives of a denseB-orbit on b∗

u and in factwe get the same representatives. However, our proof is group theoretic so is validcharacteristic.

In [1], the coadjoint action ofB on b∗ is considered fork = C. It is shown thatB actson b∗ with a dense orbit ifG is not of typeAr , Dr (r odd) orE6. Also in loc. cit. themaximal dimension of aB-orbit onb∗ is calculated, in the caseB does not act onb∗ witha dense orbit. These results are also given in [12, §4]. With more work, our methodsgive these results for arbitrary characteristic. However, we choose not to pursue this

For a Lie algebraq over an algebraically closed field of characteristic zero therebeen interest in the index ofq. This is defined by indq = minx∈q∗ qx whereqx denotesthe centraliser ofx and the definition goes back to J. Dixmier [4, 11.1.6]. For an algebLie algebraq, indq is zero precisely whenQ acts onq∗ with a dense orbit whereQ is aconnected algebraic group such that Lie(Q) = q. We refer the readerto the recent paper[6,7] of D. Panyushev for more details.

Let R be an algebraic group andV a rationalR-module. A result of V. Pyasetskii [8Corollary 2] says that ifR acts onV with finitely many orbits, thenR also acts onV ∗with finitely many orbits and there is a natural bijection between the set ofR-orbits onV

and the set ofR-orbits onV ∗. This was generalised by G. Röhrle in [11, Theorem 1which says that the modality of the action ofR onV is equal to the modality of the actioof R on V ∗. These results suggest that there may be a natural bijection between thof R-orbits onV andV ∗ in general. From considering the map in [8] one suspectssuch a bijection would reverse the closure order. However, there are many instances wa parabolic groupP does not act on aP -submodulen with a dense orbit, for example, se[10]. Therefore, a bijection between the sets ofR-orbits onV andV ∗ cannot reverse thclosure order in general.

In Section 2 we give the notation that we require. Then in Section 3 we provelemmas we need for the proof of Theorem 1.1. The proofs of our main results are conin Section 4, a lemma is required to prove Theorem 1.1 from which Corollary 1deduced. Table 1 in Section 4 gives a representative of the denseB-orbit onb∗

u for eachsimpleG.

560 S. Goodwin / Journal of Algebra 279 (2004) 558–565

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2. Notation

Let k be an algebraically closed field,R a linear algebraic group overk andV a rationalR-module. The dual spaceV ∗ of V has the structure of anR-module via the contragredieaction. For anR-submoduleU of V , we write AnnV ∗(U) = {φ ∈ V ∗: φ(u) = 0 ∀u ∈ U}.For r ∈ R andx ∈ V , we write r · x for the image ofx underr, R · x for theR-orbit ofx andRx for the stabiliser ofx in R. We note thatV is also a module for the Lie algebr = Lie(R) of R. For s ∈ r andx ∈ V we writes · x for the image ofx unders andrx forthe stabiliser ofx in r.

We recall thatV is called aprehomogeneous spacefor R providedR acts onV with adense orbit.

Let G be a reductive linear algebraic group overk. The rank ofG is denoted by rankG.We writeg or Lie(G) for the Lie algebra ofG; likewise for closed subgroups ofG. Fix amaximal torusT of G and letΨ be the root system ofG with respect toT . WhenG issimple we writeρ for the highest root ofΨ and when considering the (extended) Dyndiagram ofΨ we identify the vertices with the roots to which they correspond. For eaα ∈ Ψ we pick a generatoreα for the corresponding root subspacegα and an isomorphismuα : k → Uα to the corresponding root subgroup. For aT -regular closed subgroupH of G

we writeΨ (H) ⊆ Ψ for the roots ofH relative toT .Fix a Borel subgroupB of G containingT . ThenΨ (B) is a system of positive root

in Ψ . Let Σ = {α1, . . . , αr } be the set of simple roots defined byΨ (B).Let P be a parabolic subgroup ofG. We writePu for the unipotent radical ofP and

pu = Lie(Pu) for the Lie algebra ofPu. Let n be aP -submodule ofpu. ThenP acts onnvia the adjoint action and on the dual spacen∗ of n via the coadjoint action.

As a general reference for the theory of algebraic groups, we cite Borel’s booThroughout we use the labelling of the simple roots for simpleG from [3, Planches I–IX].

3. Preliminaries

We discuss the coadjoint action ofB on b∗u. We only consider the case whenG is

simple, the reductive case is similar. There is an isomorphism ofG-modulesφ :g ∼= g∗(asg andg∗ have the same highest weight, namely the highest root ofΨ ). We fix suchan isomorphismφ. Restriction fromG to B givesg andg∗ the structure ofB-modules.We haveb∗

u∼= g∗/Anng∗(bu) as B-modules, this gives an isomorphism ofB-modules

b∗u

∼= g/b via φ. In the sequel we identifyb∗u with g/b via the above isomorphism.

Next we prove a lemma which we require in Section 4. First we recall the followequation which is a consequence of [2, Theorem AG.10.1]:

dimR = dimRx + dimR · x, (3.1)

whereR is an algebraic group acting on an algebraic varietyX andx ∈ X.


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Lemma 3.1. Suppose the algebraic groupR = HN is the semi-direct product of the closesubgroupH and the closed normal subgroupN . LetV be anR-module and letx ∈ V . Then

dimR · x = dimH · x + dimN · x − dim(H · x ∩ N · x).

In particular, H · x ∩ N · x is finite if and only ifdimR · x = dimH · x + dimN · x.

Proof. Consider the morphism of algebraic varietiesψ :H × (N · x) → R · x, given by(h,n · x) → hn · x. By [2, Theorem AG.10.1], we have

dimR · x = dim(H × (N · x)

) − dimψ−1(x).

Elements ofψ−1(x) are of the form(hk−1, k · x) whereh ∈ Hx andk ∈ H is such thatk · x ∈ N · x. Consider the projection ofH × (N · x) ontoN · x. The image of its restrictionto ψ−1(x) is H · x ∩ N · x. Therefore, dimψ−1(x) = dimHx + dim(H · x ∩ N · x) (using[2, Theorem AG.10.1]). Then Eq. (3.1) implies that dimR · x = dimH · x + dimN · x −dim(H · x ∩ N · x).

Since dim(H · x ∩ N · x) = 0 if and only if H · x ∩ N · x is finite, the last part of thelemma follows. �

We also require the following easy lemma.

Lemma 3.2. Let R be an algebraic group and letS be a maximal torus ofR. Let V

be anR-module and letλ1, . . . , λj be linearly independent weights ofV with respectto S. Let v1, . . . , vj be eigenvectors ofS with weightsλ1, . . . , λj , respectively, and lex = v1 + · · · + vj . Then S · x = {t1v1 + · · · + tj vj : t1, . . . , tj ∈ k×}. In particular,dimS · x = j .

Proof. Sinceλ1, . . . , λj are linearly independent, we can find a cocharactersi : k× → S

for eachi such that

si (t) · vi′ ={

tvi if i ′ = i,

vi′ otherwise.

The result follows from considerings1(t1) · · · sj (tj ) · x. �The next result is about irreducible root systems and is a consequence of the fa

for N as in the statement,Ψ (N) consists of the positive roots which are not orthogoto ρ.

Lemma 3.3. AssumeG is a simple algebraic group and letN be the normal subgrouof B generated by the root subgroups ofG corresponding to the simple roots which aconnected to−ρ in the extended Dynkin diagram ofG. Then the mapα → ρ − α is abijection ofΨ (N) − {ρ}.


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Proof. Let β1, . . . , βr be the simple roots ofG, ordered so thatβ1, . . . , βl are connectedto −ρ in the extended Dynkin diagram ofG (so l is 1 or 2). Letα = b1β1 + · · · + brβr ∈Ψ (B). Now 〈βi, ρ〉 is nonzero if and only if 1� i � l. Therefore,〈α,ρ〉 �= 0 if and onlyif b1 + · · · + bl �= 0, i.e.α ∈ Ψ (N). Therefore, ifα ∈ Ψ (N) \ {ρ} then〈α,ρ〉 �= 0, whichin turn implies thatρ − α ∈ Ψ (N). Also ρ − α ∈ Ψ (N) means that〈α,ρ〉 �= 0 and soρ − α ∈ Ψ (N) \ {ρ}. The lemma follows. �

4. Proofs of Theorem 1.1 and Corollary 1.2

We are now ready to prove Theorem 1.1, the majority of work is done in the follolemma which we prove by induction. In the statement and proof of Lemma 4.1, wenotation by identifying root vectorseα with their imageeα +b in b∗

u and similarly for linearcombinations of root vectors.

Lemma 4.1. There existsj ∈ N andx ∈ b∗u which satisfy the following conditions.

(i) x is of the forme−β1 +· · ·+ e−βj whereβ1, . . . , βj ∈ Ψ (B) are linearly independent(ii) dim Bu · x � dimb∗

u − j .(iii) Bu · x ∩ T · x is finite.

Proof. We prove the result by induction on rankG, the base case whereG is of typeA1 istrivial. We proceed by induction first noting that for the inductive step we may assumG

is simple.We consider the extended Dynkin diagram ofG. Let I ⊆ Σ be the set of the simpl

roots not connected to−ρ in the extended Dynkin diagram. LetH be the closed subgrouof B generated by the maximal torusT and the root subgroups corresponding tosimple roots inI . We note thatH = T Cu is a semi-direct product whereC is a Borelsubgroup of the derived subgroup of a Levi subgroup of the standard parabolic subof G corresponding toI . Inductively we may findj ′ ∈ N and x ′ ∈ c∗u of the forme−β1 + · · · + e−βj ′ where theβi are linearly independent,

dimCu · x ′ � dimcu − j ′ (4.1)

andCu · x ′ ∩ S · x ′ is finite, whereS is a maximal torus ofC. We define

x = x ′ + e−ρ ∈ b∗u.

Let N be the normal subgroup ofB generated by the root subgroups ofG correspondingto the simple roots which are connected to−ρ in the extended Dynkin diagram ofG (as inLemma 3.3). We note thatΨ (B) is the disjoint union ofΨ (N) andΨ (C) andBu = CuN

is a semi-direct product. Therefore, ifh ∈ H , thenh · x is of the form

h · x = aρe−ρ +∑

aαe−α. (4.2)
α∈Ψ(B)\Ψ(N)


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et

We can write N = ∏α∈Ψ(N) Uα and an arbitrary element ofN in the form n =∏

α∈Ψ(N) uα(sα) where we order theα in a way so that the height ofα is non-decreasinfrom right to left. Supposen ·x ∈ H ·x and writen ·x = ∑

α∈Ψ(B) bαe−α . Then we note thabρ−β = sβ if β is one of the simple roots connected to−ρ in the extended Dynkin diagramof G. Lemma 3.3 implies thatρ − β ∈ Ψ (N) and thereforesβ = 0 sincen · x ∈ H · x, i.e.,n · x is of the form (4.2). Now considerγ ∈ Ψ (N) of height 2, sincesβ = 0 for the simplerootsβ ∈ Ψ (N), we see thatbρ−γ = sγ and sosγ = 0 (again using Lemma 3.3 and the fathatn · x ∈ H · x so is of the form (4.2)). Continuing this way we can show thatsβ = 0 forall β ∈ Ψ (N) \ {ρ}. Therefore,n · x = x andN · x ∩ H · x = {x}. Hence, by Lemma 3.1we have

dimB · x = dimH · x + dimN · x (4.3)

and

dimBu · x = dimCu · x + dimN · x (4.4)

sinceBu = CuN . An argument similar to that above gives

dimN · x �∣∣Ψ (N)

∣∣ − 1. (4.5)

We see thatCu · x = {e−ρ + z: z ∈ Cu · x ′} sinceCu centralisese−ρ . Also T · x ⊆{ae−ρ + z: a ∈ k×, z ∈ S · x ′} using Lemma 3.2. Therefore, sinceCu · x ′ ∩ S · x ′ is finite,we get thatCu · x ∩ T · x is finite. Thus, Lemma 3.1 gives

dimH · x = dimCu · x + dimT · x. (4.6)

From Eqs. (4.1), (4.4) and (4.5) and the fact thatCu centralisese−ρ , we get

dimBu · x � dimb∗u − j, (4.7)

wherej = j ′ + 1. Equations (4.3), (4.4) and (4.6) give

dimB · x = dimH · x + dimN · x = dimCu · x + dimT · x + dimN · x= dimBu · x + dimT · x. (4.8)

Then Lemma 3.1 implies thatBu · x ∩ T · x is finite. This completes the induction.�We now easily deduce Theorem 1.1.

Proof of Theorem 1.1. Let j and x be as in Lemma 4.1. By Lemma 3.2 we gdimT · x = j . Then using Eqs. (4.7) and (4.8), we deduce that dimB · x � dimb∗

u andtherefore dimB · x = dimb∗

u. Hence,B · x is dense inb∗u. �


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Table 1Representatives of denseB-orbits onb∗

u

Type Representative description

Ar∑r−j+1

i=j αi j = 1, . . . ,⌊

r2

⌋Br αj + ∑r

i=j+1 2αi , αj j = 1,3, . . . , r − 1 if r is even

αj + ∑ri=j+1 2αi , αj , αr j = 1,3, . . . , r − 2 if r is odd

Cr∑r−1

i=j 2αi + αr j = 1, . . . , r

Dr αj + ∑r−2i=j+1 2αi + αr−1 + αr , αj , αr−1, αr j = 1,3, . . . , r − 3 if r is even

αj + ∑r−2i=j+1 2αi + αr−1 + αr , αj ,

∑ri=r−2 αi , αr−2 j = 1,3, . . . , r − 4 if r is odd

G2 10, 32

F4 0100, 0120, 0122, 2342

E600100,

001110,

011111,

012321

2

E7000000,

1010000,

0000100,

0000001,

0012100,

1012221,

1234321

2

E80000000,

10100000,

00001000,

00000010,

10121000,

10122210,

12343210,

22465432

3

The proofs of Lemma 4.1 and Theorem 1.1 give a representative of a denseB-orbit onb∗

u of the forme−β1 + · · · + e−βj . In Table 1 we describe this representative by listingrootsβ1, . . . , βj .

Remark 4.2. Let x ∈ b∗u be as described in Table 1. We have calculated the dimensi

the centraliserbx of x in b. If chark �= 2 we have dimbx = dimb − dimb∗u. It then follows

from [2, Proposition 6.7] that the orbit mapB → B · x is separable. While in the caschark = 2 we have dimbx > dimb− dimb∗

u (for all simpleG) which implies that the orbimap is inseparable.

We now deduce Corollary 1.2. This requires the following two easy lemmas.

Lemma 4.3. LetP be a parabolic subgroup ofG and letm ⊆ n beP -submodules ofpu. Ifn∗ is a prehomogeneous space forP , then so ism∗.

Proof. We have an isomorphism ofP -modulesm∗ ∼= n∗/Annn∗(m). Therefore, a densP -orbit in n∗ induces a denseP -orbit in m∗. �Lemma 4.4. LetP ⊆ Q be parabolic subgroups ofG andn a Q-submodule ofqu. If n∗ isa prehomogeneous space forP , thenn∗ is a prehomogeneous space forQ.

Proof. If P · x = n∗, thenQ · x is dense inn∗. �Proof of Corollary 1.2. By Theorem 1.1B acts onb∗

u with a dense orbit. ThereforeLemma 4.3 impliesn∗ is a prehomogeneous space forB. Then by Lemma 4.4,P acts onn∗ with a dense orbit. �


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Remark 4.5. Let x ∈ b∗u be a representative for a denseB-orbit on b∗

u. The proofs ofLemmas 4.3, 4.4 and Corollary 1.2 show that we may take the image ofx in n∗ as arepresentative of a denseP -orbit on n∗ and in factB · x is dense inn∗. Therefore, theinformation given in Table 1 describes a representative of a denseP -orbit onn∗. Moreover,it follows from Remark 4.2 that the orbit mapP → P · x is separable precisely whechark �= 2.

Acknowledgments

I thank G. Röhrle for carefully reading through earlier versions of this note and gmany useful suggestions. I am also grateful to D. Panyushev for informing me of thein [5]. Finally, I thank EPSRC for their financial support.

References

[1] D.V. Alekseevsky, A.M. Perelomov, Poisson and symplectic structures on Lie algebras, J.Phys. 22 (3) (1997) 191–211.

[2] A. Borel, Linear Algebraic Groups, in: Grad. Texts Math., vol. 126, Springer-Verlag, 1991.[3] N. Bourbaki, Groupes et algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1975.[4] J. Dixmier, Algèbres enveloppantes, Gauthier–Villars, 1974.[5] A. Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra

(1977) 241–289.[6] D. Panyushev, Inductive formulas for the index of seaweed Lie algebras, Moscow Math. J. 1 (2)

221–241, 303.[7] D. Panyushev, The index of a Lie algebra, the centralizer of a nilpotent element, and the normalize

centralizer, Math. Proc. Cambridge Philos. Soc. 134 (1) (2003) 41–59.[8] V.S. Pyasetskii, Linear Lie groups acting withfinitely many orbits, Funct. Anal. Appl. 9 (1975) 351–353.[9] R.W. Richardson, Conjugacy classes in parabolicsubgroups of semisimple algebraic groups, Bull. Lond

Math. Soc. 6 (1974) 21–24.[10] G. Röhrle, A note on the modality of parabolic subgroups, Indag. Math. (N.S.) 8 (4) (1997) 549–559.[11] G. Röhrle, On the modality of parabolic subgroups oflinear algebraic groups, Manuscripta Math. 98 (19

9–20.[12] V.V. Trofimov, Semi-invariants of the coadjoint representation of Borel subalgebra of simple Lie algebr

Trudy Sem. Vektor. Tenzor. Anal. 21 (1983) 84–105, inRussian. English translation: Selecta MaSovietica 8 (1989) 31–56.

prehomogeneous spaces for the coadjoint action of a parabolic group

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