prehomogeneous spaces for parabolic group actions in classical groups

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Journal of Algebra 276 (2004) 383–398

www.elsevier.com/locate/jalgebr

Prehomogeneous spaces for parabolic group actin classical groups

Simon Goodwin and Gerhard Röhrle

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

Received 7 May 2003

Communicated by Alexander Premet

Abstract

Let G be a reductive linear algebraic group,P a parabolic subgroup ofG andPu its unipotentradical. We consider the adjoint action ofP on the Lie algebrapu of Pu. Richardson’s dense orbtheorem says that there is a denseP -orbit in pu. We consider some instances whenP acts with a

dense orbit on termsp(l)u of the descending central series ofpu. In particular, we show (in good

characteristic) that a Borel subgroupB of a classical group acts onb(l)u with a dense orbit for alll.

Further we give some families of parabolic subgroupsP such thatp(l)u contains a denseP -orbit for

all l. 2004 Elsevier Inc. All rights reserved.

1. Introduction

We look at the adjoint action of a parabolic subgroupP of a reductive linear algebraigroupG on its Lie algebrap = Lie(P ). Richardson proved in [12] that for any parabosubgroupP there is a Zariski denseP -orbit on the Lie algebra of the unipotent radicof P , pu = Lie(Pu). Subsequently there has been interest in the adjoint action ofP on pu,and its action on higher termsp(l)

u of the descending central series ofpu. ForG classical,there is a classification of the instances whenP acts onp

(l)u with a finite number of

orbits (see [2,4,8,9]). By general theory of linear algebraic groups, if the number ofP -or-bits onp

(l)u is finite, then one of the orbits is dense. In [10] Hille and the second au

E-mail addresses:[email protected] (S. Goodwin),[email protected] (G. Röhrle).URLs:http://web.mat.bham.ac.uk/S.M.Goodwin (S. Goodwin), http://web.mat.bham.ac.uk/G.E.Roeh

(G. Röhrle).

0021-8693/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jalgebra.2003.11.005

384 S. Goodwin, G. Röhrle / Journal of Algebra 276 (2004) 383–398

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considered whether a parabolic subgroupP of G acts onp(l)u with a dense orbit. For certai

parabolic subgroups (including Borel subgroups) of GLn they showed this was true for al � 0. However it is possible to find parabolic subgroupsP of GLn such thatP fails to actonp′

u with a dense orbit (see [6] or [10]). It was conjectured in [10] that a Borel subg

B of a classical group acts on each memberb(l)u of the descending central series ofbu with

a dense orbit.In this paper we investigate whenP acts on termsp(l)

u of the descending central seriof pu with a dense orbit. Our main result is

Theorem 1.1. LetB be a Borel subgroup of a classical groupG and assumechark is zeroor good forG. Then each memberb(l)

u of the descending central series ofbu contains adenseB-orbit.

In [10] it is shown thatB fails to act onb(2)u with a dense orbit ifG is an exceptiona

group not of type G2. The only other instance when a Borel subgroupB of a simplealgebraic group fails to act onb(l)

u is whenG is of type E8 andl = 4 (see [5]).Further, for the general linear groups we exhibit a family of parabolic subgroupsP such

thatP acts onp(l)u with a dense orbit for eachl. This family is given in [7], but the proo

here uses only elementary techniques. We also deduce some analogous results abouother classical groups.

In Section 2 we recall some elementary results which allow us to develop astrategy to show that a linear algebraic groupH acts on anH -submodule ofh with adense orbit. We also introduce the notation that we will need. In Section 3 we explaiour results about the Lie algebras imply analogous results about the groups. A generesult is given in Section 4 which gives us a reduction technique when investigwhen P acts onP -submodules ofpu with a dense orbit. We prove Theorem 1.1Section 5. In Section 6 we give results about when certain parabolic subgroupsP act onp(l)u with a dense orbit. Finally, in the appendix we prove the technical lemma requir

Section 5.

2. Preliminaries

Throughout,k is an algebraically closed field. LetH be a linear algebraic group overk

and Lie(H) = h the Lie algebra ofH . SupposeH acts morphically on an algebravarietyX, for x ∈ X we writeH · x for theH -orbit of x. Let V be a rationalH -module.We sayV is aprehomogeneous spacefor H if H acts onV with a dense orbit, i.e., theris somex ∈ V such thatH · x = V .

H acts on itself by conjugation and on its Lie algebrah via the adjoint action. Thecentralisers ofx ∈ h are defined byZH(x) = {y ∈ H : Ady(x) = x} andzh(x) = {y ∈ h:ady(x) = [y, x] = 0}. We have the following well-known results, which follow fro[1, Proposition 6.7].

S. Goodwin, G. Röhrle / Journal of Algebra 276 (2004) 383–398 385

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Lemma 2.1. Let x ∈ h.

(i) dimH = dimH · x + dimZH(x).(ii) Lie(ZH (x)) ⊆ zh(x), so in particulardimZH(x) � dimzh(x).

Let n be anH -submodule ofh. Suppose we wish to show thatn is a prehomogeneouspace forH . Using Lemma 2.1(i) we see that it suffices to show that dimZH(x) = dimH −dimn for somex ∈ n. Then by Lemma 2.1(ii) it suffices to show dimzh(x) = dimh−dimn

for somex ∈ n.

Remark 2.2. We make the observation that if we can find such anx, then we havedimzh(x) = dimZH(x), which forces Lie(ZH (x)) = zh(x). Then by [1, Proposition 6.7the orbit maph �→ h · x from H to H · x is separable.

Remark 2.3. We choose a faithful representationh → gln(k) for somen so that we havenatural vector space isomorphismsh ∼= kdimh andn ∼= kdimn. Then considerh ⊆ gln(k).Let x ∈ n, to find zh(x) we need to look at thosey ∈ h for which [y, x] = 0. Lety = (yij ) ∈ gln(k) and consider the non-zeroyij as variables. We see that the condit[y, x] = 0 is equivalent to a system of dimn linear equations in the dimh variablesyij .The dimension of their solution space is dimzh(x). To prove thatn contains a denseH -or-bit it therefore suffices to findx for which these equations are independent.

We now introduce the notation that we shall require. LetG be a reductive lineaalgebraic group overk. Let T be a maximal torus ofG and letB be a Borel subgrouof G containingT . Let Φ be the root system ofG with respect toT andΠ = {α1, . . . , αr }the base ofΦ corresponding toB. For β ∈ Φ+ write β = ∑

α∈Π cαβα with cαβ ∈ N0.A primep is said to bebad for G if it divides cαβ for someα andβ , else it is calledgoodfor G. (We remark that all primes are good for GLn and SLn and thatp = 2 is the only badprime for Sp2m and SOn.)

LetP be a parabolic subgroup ofG, Pu the unipotent radical ofP andpu the Lie algebraof Pu. The descending central series ofpu is defined byp(0)

u = pu andp(l+1)u = [pu,p

(l)u ]

(l � 0). As usual we writep′u = p

(1)u . For eachl, p

(l)u is aP -submodule ofpu.

In this paragraph we assume that chark �= 2. We consider On to consist of the matricex ∈ GLn such thatxtJ x = J whereJ is the matrix whose(i, j)th entry is 1 ifi +j = n+1and 0 otherwise. Then we take SOn to be the subgroup of On consisting of matrices withdeterminant 1. We consider Sp2m to consist of the matricesx ∈ GL2m such thatxtJ x = J

whereJ is the matrix whose(i, j)th entry is 1 if i + j = 2m + 1 and i � m, −1 ifi + j = 2m + 1 andi � m + 1, and 0 otherwise.

Let Θ be a semisimple automorphism ofG. We write θ for the derivative ofΘ atthe identity. For aΘ-stable subsetS of G we denote the fixed points ofΘ in S bySΘ = {x ∈ S: Θ(x) = x}. Similarly for θ -stableS ⊆ g we writeSθ = {x ∈ S: θ(x) = x}.We note that ifΘ is a semisimple automorphism ofG with finite order|Θ|, then charkdoes not divide|Θ|.

In this paragraph we assume that chark �= 2 and discuss some semisimple automphisms of classical groups, which we require in the sequel. These automorphisms


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quently referred to as graph automorphisms. We refer the reader to [16, §11, p. 169]more details. For the reader’s convenience we give explicit descriptions of the derivatthese automorphisms. There exists a semisimple automorphismΘ of GLn with GLΘ

n = On,its derivativeθ is given byθ(xij ) = (−xn+1−j,n+1−i ). There is a semisimple automorphisΘ of GL2n such that GLΘ2n = Sp2n, its derivativeθ is given byθ(xij ) = (εxn+1−j,n+1−i )

whereε = (−1)�|i−j |/n+1. Further there exists a semisimple automorphismΨ of O2n suchthat OΨ

2n = O2n−1, its derivativeψ is given byψ(xij ) = (yij ), whereyij = xin if j = n+1,yij = xi,n+1 if j = n, yij = xnj if i = n + 1, yij = xn+1,j if i = n andyij = xij otherwise.

As general references for algebraic groups we cite [1,14].

3. From Lie algebras to groups

We consider the link betweenP -conjugacy classes inPu and adjointP -orbits inpu. Wedenote byN the set of nilpotent elements ing and byU the set of unipotent elements inG.

Assume the derived subgroup ofG is simply connected. A slightly strengthentheorem of Springer says that if chark is zero or good forG, then there existsG-equivariant isomorphismφ :U → N (see [11, 6.20]). Suchφ is called a Springer mapUsing such a map one can deduce the following result (see [13, Theorem 4.1]).

Lemma 3.1. Supposechark is zero or good forG. Let P be a parabolic subgroup ofG,andN a closed, connected, normal subgroup ofP contained inPu. Then there is aP -equi-variant isomorphismφ :N → n.

In fact the proof of this lemma inloc. cit.shows we can takeφ to be a Springer map. LeN ⊆ Pu be a closed, connected, normal subgroup ofP so thatn is aP -submodule ofpu.It follows from Lemma 3.1 thatP acts onn with a dense orbit if and only ifP acts onNwith a dense orbit.

Now suppose thatπ :G → H is an epimorphism of algebraic groups such that kerπ ⊆Z(G) and kerdπ ⊆ z(g), whereZ(G) andz(g) denote the centres ofG andg respectively.Let O be aP -orbit onN . Thenπ(O) is aπ(P )-orbit onπ(N) and we see thatO is densein N if and only if π(O) is dense inπ(N). This discussion implies the following resuwhere we no longer assume the derived subgroup ofG to be simply connected.

Theorem 3.2. Supposechark is zero or good forG. LetP be a parabolic subgroup ofGandN a closed, connected, normal subgroup ofP contained inPu. SupposeP acts onn

with a dense orbit. ThenP acts onN with a dense orbit.

This theorem means the results we prove for the Lie algebra imply analogous rabout the group in good characteristic. Further in [15, III, 3.14], explicit Springer mare given whenG is GLn, Sp2m or SOn. Therefore, if we have a representative of a deP -orbit onp

(l)u , we can calculate a representative of a denseP -orbit onP

(l)u .


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4. Reduction strategy

We now present a tool which allows us to make reductions when investigprehomogeneous spaces for parabolic groups. The tool is a consequence of the followitheorem.

Theorem 4.1. LetR be an algebraic group,Θ a semisimple automorphism ofR with finiteorder andn a θ -stableR-submodule ofr. Suppose there existsx ∈ nθ such thatR · x isdense inn and the orbit mapR → R · x is separable. ThenRΘ · x is dense innθ and theorbit mapRΘ → RΘ · x is separable.

Proof. It follows from [1, Proposition 6.7] that the separability of the orbit mapR → R ·ximplies thatTx(R · x) = [r, x]. We prove the following series of inclusions

Tx

(RΘ · x) ⊆ Tx

((R · x)θ

) ⊆ (Tx(R · x)

)θ = [r, x]θ ⊆ [rθ , x

] ⊆ Tx

(RΘ · x)

.

The last inclusion is clear. To show[r, x]θ ⊆ [rθ , x] we let [r, x] ∈ [r, x]θ . One easilychecks that ifs = 1/|Θ|∑|Θ|−1

i=0 θ i(r) then θ(s) = s and [r, x] = [s, x] ∈ [rθ , x]. FromTx(R · x) = [r, x] it follows immediately that(Tx(R · x))θ = [r, x]θ . Since(R · x)θ ⊆ R · xwe get thatTx((R · x)θ) ⊆ Tx(R · x). Therefore, to showTx((R · x)θ ) ⊆ (Tx(R · x))θ itsuffices to show thatTx((R · x)θ ) is fixed by θ . But (R · x)θ ⊆ nθ , so Tx((R · x)θ) ⊆Tx(n

θ ) = nθ . Sincenθ is RΘ -stable, we have thatRΘ · x ⊆ (R · x)θ which implies the firstinclusion.

SinceR · x is dense inn we haveTx(R · x) = n. The series of inclusions above thimplies thatTx(R

Θ · x) = nθ and thus thatRΘ · x is dense innθ . The series of inclusionalso implies thatTx(R

Θ · x) = [rθ , x] which by [1, Proposition 6.7] implies the orbit maRΘ → RΘ · x is separable. �

The following corollary is a trivial consequence of Theorem 4.1, we state itconvenience.

Corollary 4.2. Let G be a reductive algebraic group and letΘ be a semisimpleautomorphism ofG. Let P be a Θ-stable parabolic subgroup ofG and n a θ -stableP -submodule ofpu. Let Q = PΘ andm = nθ . Suppose there existsx ∈ m such that theorbit mapP → P · x is separable andP · x = n. ThenQ · x = m.

We shall use Corollary 4.2 to deduce thatm is a prehomogeneous space forQ if n is aprehomogeneous space forP andm contains a representative of the denseP -orbit onn.

5. Borel subgroups

In this section we prove Theorem 1.1. We require the technical lemma whoseis given in the appendix. We begin with the following result which follows easily fr[10, Proposition 2.1].


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Lemma 5.1. LetG = GLn and letB be a Borel subgroup ofG. For eachl � 0 there existsx ∈ b

(l)u with B · x = b

(l)u and such that the orbit mapB → B · x is separable.

Proof. We note that the conjugacy of Borel subgroups implies that we only need tothe result for one particular Borel subgroup.

We takeB to consist of the upper triangular matrices inG. From [10, Proposition 2.1we know we can takex ∈ b

(l)u defined byxi,i+l+1 = 1 for 1 � i � n − l − 1 and

xij = 0 otherwise. The separability of the orbit map follows from consideringzb(x) andRemark 2.2. �Remark 5.2. We note that sincek is algebraically closed, we get the analogous resulSLn.

The following remark is required in the proof of Theorem 1.1.

Remark 5.3. We use the notation of Lemma 5.1. Supposen = 2m is even and letΘ be thesemisimple automorphism ofG such thatGΘ = Sp2m. We see that by acting onx by themaximal torus of diagonal matrices inG we may assume thatx ∈ (b

(l)u )θ .

We now restate and prove Theorem 1.1.

Theorem 5.4. LetB be a Borel subgroup of a classical groupG and assumechark is zeroor good forG. Then each memberb(l)

u of the descending central series ofbu contains adenseB-orbit.

Proof. As in Lemma 5.1, we note that the conjugacy of Borel subgroups implies that wonly need to prove the result for one particular Borel subgroup. Also we note that wehave to consider one isogeny class for each type.

The type A case is covered in [10, Proposition 2.1].Next we consider the type C case. LetG = Sp2m, H = GL2m and let Θ be the

semisimple automorphism ofH such thatHΘ = G. Let C be the Borel subgroup ofHconsisting of the upper triangular matrices inH andB = CΘ a Borel subgroup ofG. Wenote that(c(l)u )θ = b

(l)u for eachl. We may now use Corollary 4.2 and Lemma 5.1 w

Remark 5.3 to deduce that for eachl there existsx ∈ b(l)u such thatB · x = b

(l)u .

Now we consider the type D case. LetG = SO2m, H = GL2m and let Θ be thesemisimple automorphism ofH such thatHΘ = O2m. Let C be a Borel subgroup oH consisting of the upper triangular matrices inH and B = CΘ a Borel subgroupof G. We require the technical Lemma A.1 from the appendix. We emphasize thC-submodulesnl of cu from Lemma A.1 are such thatnθ

l = b(l)u for eachl � 0. Therefore,

using Lemma A.1 and Corollary 4.2, we deduce that for eachl there existsx ∈ b(l)u such

thatB · x = b(l)u .

Finally, we consider the type B case. LetG = SO2m+1, H = SO2m+2 and letΨ be thesemisimple automorphism of O2m+2 such that OΨ2m+2 = O2m+1. LetC be a Borel subgrouof H consisting of the upper triangular matrices inH andB = CΨ a Borel subgroup ofG.


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We note that we have(c(l)u )ψ = b(l)u for eachl. From the proof of the type D case abo

and Remark A.2 we see there isx ∈ b(l)u such thatC · x = c

(l)u . Further using Theorem 4.

we see that the orbit mapC → C · x is separable so we can apply Corollary 4.2 toB · x = b

(l)u as required. �

Remark 5.5. We note we could have proved the type B case forl even using Lemma 5.and Corollary 4.2. Further we remark that Theorem 1.1 could be proved directly usinstrategy from Remark 2.3. However, the calculations involved are messy.

6. Parabolic subgroups

In this section we exhibit a family of parabolic subgroupsP in GLn such thatp(l)u

contains a denseP -orbit for eachl. Further we generalize some of our results to the oclassical groups using Corollary 4.2.

We adopt the notation for parabolic subgroups of GLn used in [10], which we brieflyrecall. A parabolic subgroupP of GLn is determined up to conjugacy by an ordered tu(d1, . . . , dt ) of positive integers whered1 + · · · + dt = n. If e1, . . . , en is the standard basof kn, then we takeP = P(d1, . . . , dt ) to be the stabiliser of the flag{0} = V0 ⊆ V1 ⊆ · · · ⊆Vt = kn whereVi is thek-span of{e1, . . . , ed1+···+di } (so dimVi = d1 + · · · + di).

Consider the parabolic subgroupP = P(d1, . . . , dt) ⊆ GLn. View elementsx ∈ gln asblock matrices witht2 blocks, the(i, j)th blockXij being adi ×dj matrix. The Lie algebra

p of P consists of thex ∈ gln for which Xij = 0 for i > j . Elementsx ∈ p(l)u are the

matrices such thatXij = 0 for i � j − l.We aim to show that ifd1 � d2 � · · · � ds � ds+1 � · · · � dt for somes, thenP has a

dense orbit onp(l)u for eachl. This result is given in [7, 1.4.5]. The proof inloc. cit. uses

representation theoretic methods from [3]. Our proof uses only elementary techniquusing the strategy of Remark 2.3.

Theorem 6.1. LetP = P(d1, . . . , dt ) be a parabolic subgroup ofGLn such thatd1 � d2 �· · · � ds � ds+1 � · · · � dt for somes. Thenp

(l)u is a prehomogeneous space forP for

eachl.

Proof. We definex ∈ p(l)u by saying what the matrixXij is for each i and j . For

j = i + l + 1 anddi � dj defineXij to have adi × di identity matrix in its firstdi columnsand zeros elsewhere. Forj = i + l + 1 anddi � dj defineXij to have adj × dj identitymatrix in its firstdj rows and zeros elsewhere. OtherwiseXij is zero.

We use the strategy of Remark 2.3, so we takey ∈ p arbitrary and consider the equatiofor theyij in [y, x] = 0. To show that these equations are independent, we use induonn, the base casen = 0 being trivial.

We look at these equations in a particular order. We look at the((∑u

i=1 di) + 1)throws of [y, x] = 0, for u = 0,1, . . . , t − l − 2. First we look at the equations in th((

∑t−l−2i=1 di) + 1)th row. Each such equation involves a distinctyn−dt+1,j and may

also involve ayn−dt−dt−1+1,j . The yn−dt+1,j ’s do not occur elsewhere. Therefore, t


er thens

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roup

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w arend 1st

the

dering

equations in the((∑t−l−2

i=1 di) + 1)th row of [y, x] = 0 must be independent of thother equations in[y, x] = 0. So we can neglect these equations. Next we consideequations in the((

∑t−l−3i=1 di) + 1)th row of [y, x] = 0. We see that these equatio

involve yn−dt−dt−1+1,j ’s. The only other equations involvingyn−dt−dt−1+1,j ’s are in

the ((∑t−l−2

i=1 di) + 1)th row of [y, x] = 0. Having neglected the equations in t((

∑t−l−2i=1 di) + 1)th row of [y, x] = 0, we see that we may neglect the equations in

((∑t−l−3

i=1 di) + 1)th row. We move up the((∑u

i=0 di) + 1)th rows of [y, x] = 0 and seethat we can continue to neglect equations in them.

Next we consider the((∑u

i=1 di) + 1)th columns of[y, x] = 0 for u = l + 1, l + 2,

. . . , t − 1. We can use arguments analogous to those above to see that we can negequations in these columns.

We are left with a system of equations which are equivalent to those we getconsidering the action ofP(d1 − 1, . . . , dt − 1) on the lth member of the descendincentral series of the Lie algebra of its unipotent radical. These equations are indepby induction. It now follows thatP · x = p

(l)u by Lemma 2.1 and Remark 2.3.�

The following remark is similar to the remark after Theorem 1.4.1 in [7].

Remark 6.2. The assumption thatd1 � d2 � · · · � ds � ds+1 � · · · � dt is required whenapplying the inductive hypothesis. Supposedi − 1 = 0 for some 1< i < t but d1 − 1 �= 0anddt − 1 �= 0. We can still neglect the same equations as in the proof of TheoremHowever, the remaining equations are not ingeneral equivalent to the equations wewhen considering the action ofP(d1 − 1, . . . , dt − 1) on thelth member of the descendincentral series of the Lie algebra of its unipotent radical. This is not a problemconsidering the casel = 0 and so we get an alternative proof of Richardson’s densetheorem for GLn.

Remark 6.3. We note by Remark 2.2 that ifx is as in the proof of Theorem 6.1, then torbit mapP → P · x is separable.

Example 6.4. We illustrate the proof of Theorem 6.1 by looking at the parabolic subgP = P(2,3,4,2,1) � GL12 acting onpu. The matricesy, x, and[y, x] are given in Fig. 1(where dots represent zeros).

We look at the equations for theyij ’s in [y, x] = 0. First we consider the 10th row o[y, x] = 0. We see thaty12,12 does not occur elsewhere, so we may neglect the equon the 10th row. Next we consider the 6th row of[y, x] = 0 we see that the only otheoccurrences of they10,j ’s have already been neglected. So the equations in the 6th roindependent of the other equations. Similarly we see that the equations in the 3rd arows can be neglected.

Next we consider the 3rd column of[y, x] = 0. We see that theyi1’s do not occurelsewhere so these equations may be neglected. Then we see that we can neglectequations in the 6th, 10th and 12th columns.

Now we are left with a system of equations equivalent to those we get when consithe action ofP(1,2,3,1) ⊆ GL7 on the Lie algebra of its unipotent radical.

S.G

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.Röhrle

/Journ

alo

fAlgeb

ra276

(2004)

383–398

391 y1,1 y1,2 y1,3 y1,4 y1,5 y1,6 y1,7 y1,8 y1,9 y1,10 y1,11 y1,12

x =

· · 1 · · · · · · · · ·· · · 1 · · · · · · · ·· · · · · 1 · · · · · ·· · · · · · 1 · · · · ·· · · · · · · 1 · · · ·· · · · · · · · · 1 · ·· · · · · · · · · · 1 ·· · · · · · · · · · · ·· · · · · · · · · · · ·· · · · · · · · · · · 1· · · · · · · · · · · ·· · · · · · · · · · · ·

y3,9 y1,6 − y3,10 y1,7 − y3,11 y1,10 − y3,12y4,9 y2,6 − y4,10 y2,7 − y4,11 y2,10 − y4,12y6,9 y3,6 − y6,10 y3,7 − y6,11 y3,10 − y6,12y7,9 y4,6 − y7,10 y4,7 − y7,11 y4,10 − y7,12y8,9 y5,6 − y8,10 y5,7 − y8,11 y5,10 − y8,12· y6,6 − y10,10 y6,7 − y10,11 y6,10− y10,12· y7,6 − y11,10 y7,7 − y11,11 y7,10− y11,12· y8,6 y8,7 y8,10· y9,6 y9,7 y9,10· · · y10,10− y12,12· · · y11,10· · · ·

y =

y2,1 y2,2 y2,3 y2,4 y2,5 y2,6 y2,7 y2,8 y2,9 y2,10 y2,11 y2,12· · y3,3 y3,4 y3,5 y3,6 y3,7 y3,8 y3,9 y3,10 y3,11 y3,12· · y4,3 y4,4 y4,5 y4,6 y4,7 y4,8 y4,9 y4,10 y4,11 y4,12· · y5,3 y5,4 y5,5 y5,6 y5,7 y5,8 y5,9 y5,10 y5,11 y5,12· · · · · y6,6 y6,7 y6,8 y6,9 y6,10 y6,11 y6,12· · · · · y7,6 y7,7 y7,8 y7,9 y7,10 y7,11 y7,12· · · · · y8,6 y8,7 y8,8 y8,9 y8,10 y8,11 y8,12· · · · · y9,6 y9,7 y9,8 y9,9 y9,10 y9,11 y9,12· · · · · · · · · y10,10 y10,11 y10,12· · · · · · · · · y11,10 y11,11 y11,12· · · · · · · · · · · y12,12

[y, x] =

· · y1,1 − y3,3 y1,2 − y3,4 −y3,5 y1,3 − y3,6 y1,4 − y3,7 y1,5 − y3,8 −· · y2,1 − y4,3 y2,2 − y4,4 −y4,5 y2,3 − y4,6 y2,4 − y4,7 y2,5 − y4,8 −· · · · · y3,3 − y6,6 y3,4 − y6,7 y3,5 − y6,8 −· · · · · y4,3 − y7,6 y4,4 − y7,7 y4,5 − y7,8 −· · · · · y5,3 − y8,6 y5,4 − y8,7 y5,5 − y8,8 −· · · · · · · ·· · · · · · · ·· · · · · · · ·· · · · · · · ·· · · · · · · ·· · · · · · · ·· · · · · · · ·

Fig. 1.


le torate thedense

,

g

1

g

or

6 for

or

sume

f

We now give two easy corollaries of Theorem 6.1. With more work it is possibgive better results. However, we choose not to pursue this here and just demonstuse of Corollary 4.2. There is a lack of symmetry in the given representatives of theP -orbits so we only deduce results for parabolics with equal size blocks. Also ifΘ is thesemisimple automorphism of GLn such that GLΘn = On and if P is aΘ-stable parabolicsubgroup of GLn, then ifn is even, we do not have(p(l)

u )θ = (pθu)

(l) in general. Thereforewe only give the following two results.

Corollary 6.5. Assumechark �= 2. Let Θ be the semisimple automorphism ofGL2m suchthat GLΘ

2m = Sp2m. Let P = P(d, d, . . . , d) � GL2m. Let Q = PΘ be the correspondinparabolic subgroup ofSp2m. Thenq

(l)u is a prehomogeneous space forQ for eachl.

Proof. We note that we have(p(l)u )θ = (pθ

u)(l) for eachl. Let l � 0, then by Theorem 6.

there existsx ∈ p(l)u such thatP · x = p

(l)u . Moreover looking at thex given in the proof

of Theorem 6.1 we note that by using the action of the maximal torus ofP (consisting ofdiagonal matrices) we may assume thatx ∈ (p

(l)u )θ = q

(l)u . We now deduce the result usin

Corollary 4.2 and Remark 6.3.�The next result is more limited as we requirex ∈ (p

(l)u )θ to apply Corollary 4.2. This is

not the case whenl is odd.

Corollary 6.6. Assumechark �= 2. Let Θ be the semisimple automorphism ofGL2m+1such thatGLΘ

2m+1 = O2m+1. Let P = P(d, d, . . . , d) � GL2m+1. Let Q = PΘ be thecorresponding parabolic subgroup ofO2m+1. Thenq

(l)u is a prehomogeneous space f

Q for each evenl.

Proof. The proof is analogous to the proof of Corollary 6.5.�Remark 6.7. Let Q be a parabolic subgroup of O2m+1 thenQ̂ = Q ∩ SO2m+1 has index 2in Q and q̂

(l)u = q

(l)u . Therefore, we may deduce the analogous result of Corollary 6.

SO2m+1.

Acknowledgments

We are grateful to the referee for making anumber of useful comments. The first authacknowledges support from EPSRC.

Appendix A

In this appendix we prove the technical lemma required in Section 5. We asthroughout this section that chark �= 2.

Let G = GL2n, T the maximal torus of diagonal matrices andB the Borel subgroup oupper triangular matrices. LetΦ be the root system ofG with respect toT andΠ the base


f

of Φ corresponding toB. Write Π = {α1, . . . , α2n−1}. For i � j , we denoteαi + · · · + αj

by ij . We describe aB-submodulen of bu by giving the minimal set of rootsα in Φ

such thatn is generated by thegα ⊆ n as aB-module. For examplebu is denoted by{11, . . . , (2n − 1)(2n − 1)} andb

(l)u is denoted by{1(l + 1), . . . , (2n − 1− l)(2n − 1)}.

For eachl � 0 we define aB-submodulenl of bu. For l evennl is denoted by the set ominimal roots

{i(i + l): 1 � i � n − l − 1

} ∪ {i(i + l + 1): n − l � i � n − 1

}∪ {

i(i + l): n + 1 � i � 2n − l − 1}.

For example forn = 5 andl = 2, nl consists of matrices of the form

· · · ∗ ∗ ∗ ∗ ∗ ∗ ∗· · · · ∗ ∗ ∗ ∗ ∗ ∗· · · · · · ∗ ∗ ∗ ∗· · · · · · · ∗ ∗ ∗· · · · · · · · ∗ ∗· · · · · · · · ∗ ∗· · · · · · · · · ∗· · · · · · · · · ·· · · · · · · · · ·· · · · · · · · · ·

For l oddnl is denoted by the set of minimal roots

{i(i + l): 1 � i � n − l − 1

} ∪ {i(i + l + 1): n − l � i � n − (l + 3)/2

}∪ {

i(i + l + 1): n − (l − 1)/2 � i � n − 1} ∪ {

i(i + l): n + 1 � i � 2n − l − 1}.

For example forn = 7 andl = 3, nl consists of matrices of the form

· · · · ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗· · · · · ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗· · · · · · ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗· · · · · · · · ∗ ∗ ∗ ∗ ∗ ∗· · · · · · · · · · ∗ ∗ ∗ ∗· · · · · · · · · · ∗ ∗ ∗ ∗· · · · · · · · · · · ∗ ∗ ∗· · · · · · · · · · · ∗ ∗ ∗· · · · · · · · · · · · ∗ ∗· · · · · · · · · · · · · ∗· · · · · · · · · · · · · ·· · · · · · · · · · · · · ·· · · · · · · · · · · · · ·· · · · · · · · · · · · · ·


3.

rowey also

refor

of

Lemma A.1. Let Θ be the semisimple automorphism ofG such thatGΘ = O2n. For eachl � 0 there existsx ∈ nθ

l such thatB · x = nl and the orbit mapB → B · x is separable.

Proof. Let l � 0. To simplify notation in this proof we writea = n − l − 1, b = n +(l + 3)/2, c = n − (3l + 3)/2, andd = n − (l + 1)/2. We use the strategy of Remark 2.

First we consider the case whenl is even. We definex ∈ nl as follows:

xi,i+l+1 = 1 if 1 � i � a,

xi,i+l+2 = 1 if a � i � n,

xi,i+l+1 = 1 if n + 1 � i � 2n − l − 1,

xij = 0 otherwise.

For example forn = 5 andl = 2 we have

x =

· · · 1 · · · · · ·· · · · 1 1 · · · ·· · · · · · 1 · · ·· · · · · · · 1 · ·· · · · · · · · 1 ·· · · · · · · · 1 ·· · · · · · · · · 1

· · · · · · · · · ·· · · · · · · · · ·· · · · · · · · · ·

We lety ∈ b be arbitrary and consider the equations for theyij in [y, x] = 0. We showthat these equations are independent by induction onn the base casen = 0 being trivial.

First we consider the case wheren � l + 1. We consider the occurrences of they1j ’s inthe equations in[y, x] = 0. They occur only in the top row and each entry of the topof [y, x] = 0 contains a distincty1j . Therefore, these equationsmust be independent of thother equations, so we may neglect the equations in the top row. By symmetry we maneglect the equations in the rightmost column of[y, x] = 0. The remaining equations aequivalent to the analogous equations we get when considering the corresponding caseGL2n−2 which are independent by induction.

Now supposen � l + 2. Again we consider the equations of the top row of[y, x] = 0.Each such equation contains ay1j but y1a occurs twice. Further, the only occurrencesthe y1j ’s are in the top row. Now the occurrences ofy1a are asy1a − yl+2,n = 0 in the(1, n)th entry of[y, x] = 0 andy1a − yl+2,n+1 = 0 in the(1, n + 1)th entry of[y, x] = 0.The only other occurrence ofyl+2,n andyl+2,n+1 is in the (l + 2, n + l + 2)th entry of[y, x] = 0 where we haveyl+2,n + yl+2,n+1 − ∗ = 0 where∗ does not involveyl+2,n oryl+2,n+1. As chark �= 2 it follows that the equations on the top row of[y, x] = 0 must


apply

e

ed byin

be independent of the other equations and so we may neglect them. We may nowinduction as in the previous case.

Therefore, by induction the equations in[y, x] = 0 are independent.Next we consider the case wherel is odd. We definex ∈ nl as follows:

xi,i+l+1 = 1 if 1 � i � a,

xi,i+l+2 = 1 if a � i � n − (l + 3)/2,

xi,i+l+2 = 1 if n − (l − 1)/2 � i � n,

xi,i+l+1 = 1 if n + 1 � i � 2n − l − 1,

xcb = 1,

xd,n+(3l+5)/2 = 1,

xij = 0 otherwise.

For example, forn = 7 andl = 3 we have

x =

· · · · 1 · · · · 1 · · · ·· · · · · 1 · · · · · · · ·· · · · · · 1 1 · · · · · ·· · · · · · · · 1 · · · · ·· · · · · · · · · · · · · 1

· · · · · · · · · · 1 · · ·· · · · · · · · · · · 1 · ·· · · · · · · · · · · 1 · ·· · · · · · · · · · · · 1 ·· · · · · · · · · · · · · 1

· · · · · · · · · · · · · ·· · · · · · · · · · · · · ·· · · · · · · · · · · · · ·· · · · · · · · · · · · · ·

We lety ∈ b be arbitrary and consider the equations for theyij in [y, x] = 0. As in thel

even case, we show that these equations are independent by induction onn, the base casn = 0 being trivial.

First we consider the case wheren � l + 1. We look at the top row of[y, x] = 0. Apartfrom the(1, b)th entry each equation in the top row contains ay1j . Moreover, there is onlyone occurrence of eachy1j . The(1, b)th entry isyl+3,b = 0 and this is the only occurrencof yl+3,b in [y, x] = 0. Therefore, we may neglect the equations in the top row ansymmetry those in the rightmost column of[y, x] = 0. Thus we may apply induction asthe proof of thel even case.


ion

and

of

apply

3.ssume

alaen

tions int

Now we consider the casel +2� n � (3l + 3)/2. As in the previous case each equatin the top row of[y, x] = 0 contains ay1j apart from the one in the(1, b)th entry. Againthis entry isyl+2,b = 0 and this is the only occurrence ofyl+2,b. We see thaty1a occurstwice in the top row and each othery1j occurs once. We may deal with they1a as inthe proof of thel even case. Therefore, we may neglect the equations in the top rowrightmost column and apply induction.

Now we consider the casen � (3l + 5)/2. We look at the equations in the top row[y, x] = 0, we see that each of these contains ay1j . Bothy1a andy1c occur twice. They1a

can be dealt with as in the proof of thel even case. We see thaty1c occurs in the(1, d)thentry of [y, x] = 0 asy1c − yl+2,d = 0 and in the(1, b)th entry asy1c − yl+2,b = 0. Nowthere is only one other occurrence ofyl+2,d andyl+2,b in the(l +2, n+ (3l + 5)/2)th entryof [y, x] = 0 where they occur asyl+2,d + yl+2,b − ∗ = 0 where∗ does not involveyl+2,d

or yl+2,b. As chark �= 2 it follows that the equations on the top row of[y, x] = 0 mustbe independent of the other equations and so we may neglect them. We may nowinduction.

Therefore, by induction the equations in[y, x] = 0 are independent.In both cases these arguments show thatB · x = nl by Lemma 2.1 and Remark 2.

We note that using the action of the maximal torus of diagonal matrices, we may ax ∈ nθ

l . The separability of the orbit map follows from Remark 2.2.�Remark A.2. Let Ψ be the semisimple automorphism of O2n such that OΨ2n = O2n−1. Wenote thex ∈ nθ

l we get from the proof of Lemma A.1 are elements of(nθl )

ψ .

Example A.3. It seems more natural to tryx with 1 at entries corresponding to the minimgenerating set ofnl for a representative for a denseB-orbit onnl . In this example we givecalculation which illustrates why thisx does not work in general. In fact we show that whl is odd it is necessary to setxcb = 1 andxd,n+(3l+5)/2 = 1 as well asxa,a+l+2 = 1 andxn,n+l+2 = 1. We consider the casen = 4, l = 1. If we set the entries ofx correspondingto the minimal generating set ofnl to 1 and also putxa,a+l+2 = 1 andxn,n+l+2 = 1, thenwe get

x =

0 0 1 0 0 0 0 0

0 0 0 1 1 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

and[y, x] as in Fig. 2. We see that there is a linear dependence between the equathe (1,6)th and(3,8)th entries of[y, x] = 0. Therefore, it follows from Remark 2.3 thaB · x cannot be dense innl .

S.G

oodw

in,G

.Röhrle

/Journ

alo

fAlgeb

ra276

(2004)

383–398

397

y1,4 + y1,5 − y3,7 y1,6 − y3,8y2,4 + y2,5 − y4,7 − y5,7 y2,6 − y4,8 − y5,8

y3,4 + y3,5 y3,6y4,4 + y4,5 − y7,7 y4,6 − y7,8

y5,5 − y7,7 y5,6 − y7,80 y6,6 − y8,80 00 0

[y, x] =

0 0 y1,1 − y3,3 y1,2 − y3,4 y1,2 − y3,5 −y3,60 0 0 y2,2 − y4,4 y2,2 − y4,5 − y5,5 −y4,6 − y5,60 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

Fig. 2.


t

he

ch.

th.,

s,

ift,

nt

ent

43,

on

99)

s,

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prehomogeneous spaces for parabolic group actions in classical groups

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