# A note on étale representations from nilpotent orbits

aa r X i v : . [ m a t h . R T ] F e b A NOTE ON ÉTALE REPRESENTATIONS FROM NILPOTENT ORBITS

HEIKO DIETRICH, WOLFGANG GLOBKE, AND MARCOS ORIGLIA

Dedicated to the memory of Professor È. B. Vinberg A BSTRACT . A linear étale representation of a complex algebraic group G is given by a complex algebraic G -module V such that G has a Zariski-open orbit on V and dim G = dim V . A current line of researchinvestigates which étale representations can occur for reductive algebraic groups. Since a complete classiﬁ-cation seems out of reach, it is of interest to ﬁnd new examples of étale representations for such groups. Theaim of this note is to describe two classical constructions of Vinberg and of Bala & Carter for nilpotent orbitclassiﬁcations in semisimple Lie algebras, and to determine which reductive groups and étale representa-tions arise in these constructions. We also explain in detail the relation between these two constructions.

1. I

NTRODUCTION

Let G be a complex linear algebraic group. A prehomogeneous module ( G, ̺, V ) is a complex algebraicrepresentation ̺ : G → GL( V ) such that V is ﬁnite-dimensional and G has a Zariski-open orbit in V .The points of the open orbit are said to be in general position in V . In this case, V is a prehomogeneousvector space and dim G > dim V . If, in addition, dim G = dim V , then ( G, ̺, V ) is an étale module ,and, accordingly, ̺ is an étale representation of G . Clearly, for étale modules, the stabiliser in G is aﬁnite subgroup for any point in the open orbit. In terms of Lie algebras g , an étale representation is onewhere the action of g on a point in general position yields a vector space isomorphism of g and V , inparticular dim g = dim V , and the stabiliser subalgebra at a generic point is trivial. The existence ofétale representations implies the existence of left-symmetric products on Lie algebras and thereby alsothat of left-invariant ﬂat and torsion-free afﬁne connections on the corresponding Lie groups, see Burde[4] for details and additional references. Due to this relationship, Lie groups or Lie algebras admittingétale representations are also called (locally) afﬁnely ﬂat .We are interested in studying étale representations for complex reductive algebraic groups. It is well-known that many reductive groups do not admit étale representations, for example, this is true for semi-simple groups. Burde [3] shows that if a reductive G with simple commutator subgroup S has an étalerepresentation, then S = SL( n, C ) with n > . If G is reductive with 1-dimensional centre and S isnot simple but has only pairwise isomorphic simple factors, then there are no étale modules for G , seeBurde & Globke [5]. A complete classiﬁcation of étale modules for reductive algebraic groups seemsfar away, and so the current aim is to ﬁnd further examples. Some can be directly obtained by inspectingSato & Kimura’s [10] classiﬁcations of prehomogeneous modules for reductive algebraic groups, cf. [5].Additional examples with interesting properties were constructed by Burde et al. [6]. Results.

In the present note, we take a look at the étale representations for reductive algebraic groupsarising in the classiﬁcation of nilpotent orbits in semisimple Lie algebras, in particular, the classiﬁcationsof Vinberg [11] and Bala & Carter [1, 2]. We show in Proposition 2.1 that these groups are subject to cer-tain restrictions, notably that all their simple factors are either special linear or orthogonal groups. In lightof known examples for groups with symplectic groups as simple factors, e.g. [6], this shows that étalemodules of this type are a proper subclass of the étale modules for general reductive algebraic groups. (H. Dietrich, M. Origlia) M

ONASH U NIVERSITY , S

CHOOL OF M ATHEMATICS , C

LAYTON , VIC 3800, A

USTRALIA (W. Globke) F

ACULTY OF M ATHEMATICS , U

NIVERSITY OF V IENNA , 1090 V

IENNA , A

USTRIA

E-mail addresses : [email protected], [email protected],[email protected] .Dietrich and Origlia were supported by Australian Research Council grant DP190100317; Globke was supported by anAustrian Science Fund FWF grant I 3248. Vinberg’s and Bala & Carter’s classiﬁcation methods are very similar, and the second aim of this noteis to provide concise descriptions of these methods and to explain how they are related. In Section 2,we ﬁrst take a look at Vinberg’s construction of carrier algebras for nilpotent elements in a graded Liealgebra. From the classiﬁcation of simple carrier algebras we determine the types of reductive groupsfor which étale modules arise by this method. In Section 3, we show how Bala & Carter ﬁnd minimalLevi subalgebras for a given nilpotent element in a semisimple Lie algebra, and explain how it relates toVinberg’s carrier algebras, see Proposition 3.2 and its corollary. In fact, for Z -graded algebras, the twoapproaches coincide. Gyoja [9] described how to construct, given a prehomogeneous module ( G, ̺, V ) for a reductive algebraic group, an étale module ( G ′ , ̺ ′ , V ′ ) for a reductive subgroup G ′ G anda quotient module V ′ of V . In Proposition 4.1 we show how this generalises Vinberg’s and Bala &Carter’s constructions. Notation.

All Lie algebras g we consider here are deﬁned over the ﬁeld of complex numbers. Thecentraliser of a subset X ⊆ g in g is z g ( X ) = { y ∈ g : [ y, X ] = { }} , and the normaliser is n g ( X ) = { y ∈ g : [ y, X ] ⊆ span C ( X ) } . An element x ∈ g is nilpotent (or semisimple ) if its adjoint representation ad( x ) on g is nilpotent (or semisimple). An algebraic group G is reductive if its maximal unipotentnormal subgroup is trivial. A Lie algebra g is reductive if it is the Lie algebra of a reductive algebraicgroup. In this case g = z ⊕ s , where s is the semisimple commutator subalgebra of g and z = z ( g ) is thecentre of g . Let n > be an integer and Z n = { , . . . , n − } , or n = ∞ and Z ∞ = Z . A Lie algebra g is Z n -graded if g = L i ∈ Z n g i , where each g i g is a subspace and [ g i , g j ] ⊆ g i + j for all i, j ; here g k = g k mod n for all k ∈ Z . Note that g is a subalgebra of g .2. V INBERG ’ S CARRIER ALGEBRAS

Vinberg [11] studied complex semisimple Lie algebras graded by an arbitrary abelian group. However,the ﬁrst step in his analysis is to restrict to a subalgebra graded by a cyclic group, so we will only considerthis case. Let g be a Z n -graded semisimple Lie algebra, where n > is an integer or n = ∞ . If n isﬁnite, then such a grading is the eigenspace decomposition of a Lie algebra automorphism of order n . If n = ∞ , then the Z -grading of g comes from a derivation ϕ that acts as multiplication by i on each g i .For semisimple g , this derivation is inner, that is, ϕ = ad( h ) for a unique deﬁning element h ∈ g .Carrier algebras for g are constructed as follows. For a nonzero nilpotent e ∈ g choose an sl -triple ( h, e, f ) where h ∈ g and f ∈ g − ; this means [ h, e ] = 2 e , [ h, f ] = − f , and [ e, f ] = h . Let t be amaximal toral subalgebra of the centraliser of ( h, e, f ) in g and deﬁne t = C h ⊕ t . Equivalently, t is amaximal toral subalgebra of the normaliser of C e in g , cf. [7, Lemma 30]. Now let λ : t → C such that [ t, e ] = λ ( t ) e for all t ∈ t , and deﬁne the Z -graded algebra g ( t , e ) by(2.1) g ( t , e ) = M k ∈ Z g ( t , e ) k with g ( t , e ) k = { x ∈ g k : [ t, x ] = kλ ( t ) x for all t ∈ t } ; the derived subalgebra of g ( t , e ) is the carrier algebra of e , denoted(2.2) c ( e ) = [ g ( t , e ) , g ( t , e )] . It is Z -graded with the induced grading; note that e ∈ c ( e ) . This carrier algebra of e is unique up toconjugacy under the adjoint group G of g ; one therefore also speaks of the carrier algebra of e in g .Moreover, two nonzero nilpotent elements of g are G -conjugate if and only if their carrier algebras are G -conjugate, which makes carrier algebras a useful tool for classifying nilpotent orbits. We will notgo into the details of this classiﬁcation, as they are not required for our purposes here, but an outline isfound in Vinberg [11, Section 4]. For details on the classiﬁcation of nilpotent orbits in real semisimpleLie algebras using carrier algebras deﬁned over the real ﬁeld we refer to Dietrich et al. [7].Vinberg [11, Theorem 4] showed that every carrier algebra is semisimple Z -graded with c ( e ) k g k for each k ∈ Z , and that carrier algebras are characterized by the following three conditions: Vinberg called c ( e ) locally ﬂat and complete if it satisﬁes (V1) and (V3), respectively. We avoid this terminology, as thesemisimple Lie algebra c ( e ) does not admit an afﬁnely ﬂat structure; only its reductive subalgebra c ( e ) is locally afﬁnely ﬂat. NOTE ON ÉTALE REPRESENTATIONS FROM NILPOTENT ORBITS 3 (V1) dim c ( e ) = dim c ( e ) ;(V2) c ( e ) is normalised by a maximal toral subalgebra of g ;(V3) c ( e ) is not a proper subalgebra of a reductive Z -graded subalgebra of g of the same rank.Moreover, [11, Theorem 2] shows that e is in generic (or general ) position in c ( e ) , that is, [ c ( e ) , e ] = c ( e ) , so (V1) states that the adjoint action of c ( e ) on c ( e ) yields an étale representation for c ( e ) .Only property (V1) is intrinsic to c ( e ) , whereas (V2) and (V3) are determined by its embedding inthe ambient Lie algebra g . Thus, to describe the Lie algebras that can appear as carrier algebras fornilpotent elements in semisimple Lie algebras, one must merely classify Z -graded Lie algebras with(V1); we call such an algebra an abstract carrier algebra . Every abstract carrier algebra is a direct sum ofsimple abstract carrier algebras, so to describe the possible étale modules ( c ( e ) , ad , c ( e ) ) coming fromsemisimple carrier algebras, it is sufﬁcient to focus on simple abstract carrier algebras in Lie algebras. Inthe next section we follow Djokoviˇc’s description [8] (based on work of Vinberg [11]) of the classiﬁcationof all simple abstract carrier algebras. Using a different terminology, Bala & Carter [1] have also obtaineda classiﬁcation for the classical case. With these classiﬁcations, we determine the following. Proposition 2.1.

A reductive Lie algebra g admitting an étale representation coming from the adjointaction of a nilpotent element is a direct sum of the degree -components of j > simple abstract carrieralgebras. As such, the semisimple part of g , if non-trivial, has simple factors of type A and at most j factors of type B or D . The centre of g has dimension > j , unless all the simple abstract carrieralgebras involved have weighted Dynkin diagrams of types in { A , E (11)8 , F (4)4 , G (2)2 } as deﬁned in [8,Table II] , in which case the centre of g has dimension j . From Burde et al. [6] we know that there exist étale representations for reductive algebraic groupswith a simple factor of type C ; this shows the following: Corollary 2.2.

There are étale representations for reductive Lie algebras that do not come from theadjoint action of a nilpotent element.

Simple abstract carrier algebras Lie algebras.

Recall that the grading of a semisimple Z -gradedLie algebra g with deﬁning element h ∈ g is determined as g k = { x ∈ g : [ h, x ] = kx } for k ∈ Z . Two Z -graded Lie algebras g and g ′ with deﬁning elements h and h ′ are isomorphic if there is a Lie algebraisomorphism ϕ : g → g ′ with ϕ ( h ) = h ′ . Djokoviˇc [8] has classiﬁed, up to isomorphism, semisimple Z -graded Lie algebras in terms of weighted Dynkin diagrams: Let h g be a maximal toral subalgebracontaining h , with corresponding root system Φ . Let Π be a basis of simple roots such that α ( h ) > for every α ∈ Π . Let ∆( g ) be the Dynkin diagram of g with respect to h , with vertices labeled by Π ,and to each vertex α ∈ Π attach the integer α ( h ) . The resulting weighted Dynkin diagram is denoted ∆( g , h ) . It is proved in [8, Theorem 1] that there is a bijection between (isomorphism classes of) Z -graded semisimple Lie algebras ( g , h ) and (isomorphism classes of) weighted Dynkin diagrams ∆( g , h ) .In the following, let ( g , h ) be simple Z -graded, and deﬁne deg α ∈ Z for α ∈ Φ by x α ∈ g deg α ,where x α ∈ g is a root vector corresponding to α . If deg α = k , then α ( h ) x α = [ h, x α ] = kx α , hence α ( h ) = k ; this shows that deg α = α ( h ) . If r k is the number of roots with degree k , then g is anabstract carrier algebra if and only if dim h + r = r . It is shown in [8, p. 374] that if g is an abstractcarrier algebra, then deg α ∈ { , } for every simple root α ∈ Π . So for the classiﬁcation it remains todetermine the weighted Dynkin diagrams with weights { , } such that dim h + r = r . The reductivesubalgebra g is then given by the subdiagram consisting of the vertices with weight . To illustrate themethod, we include the full proof for type A. To keep the exposition short, for the other types we onlydescribe the results and refer to [8, Section 4], [1, Section 3], and [11, p. 30] for more details.If g = h is a maximal toral subalgebra, then r = 0 and all labels in the weighted diagram are ; onesays that g is principal . In this case the -component of the carrier algebra is abelian. NOTE ON ÉTALE REPRESENTATIONS FROM NILPOTENT ORBITS 4

Type A.

Let g = sl ( n + 1 , C ) . Let the diagonal matrix h = diag( λ , . . . , λ n +1 ) be the deﬁningelement with λ > . . . > λ n +1 . Consider the root system Φ = {± ( ε i − ε j ) : 1 i < j < n } andbasis Π = { α , . . . , α n } where each ε i maps h to λ i , and α i = ε i − ε i +1 . Let k = λ − λ n +1 and for i = 0 , . . . , k let d i be the number of λ r with λ r = λ − i . A root ± ( ε i − ε j ) has degree if and only if λ i = λ j = λ − r for some r , and for each r there are d r ( d r − possibilities for ε i and ε j . This impliesthat r = P kj =0 d j ( d j − . In a similar way, r = P k − j =0 d j d j +1 , and now a direct calculation showsthat n + r = r if and only if ( d − d ) + ( d − d ) + · · · + ( d k − − d k ) + ( d −

1) + ( d k −

1) = 0; to see the latter, note that d + . . . + d k = n + 1 . In conclusion, g is an abstract carrier algebra if andonly if d = . . . = d k = 1 and k = n , which is equivalent to g being principal. Type B and D.

Let g = so ( m, C ) be realised as g = { X ∈ gl ( m, C ) : X ⊺ J = − J X } where J isthe matrix that only has s on its anti-diagonal and s elsewhere, and either m = 2 n + 1 (with n > ) or m = 2 n (with n > ). Write the deﬁning element as h = diag( λ , . . . , λ m ) with λ > . . . > λ m . Since hJ = − J h , each − λ i = λ m +1 − i . It has been shown that there are s > and integers k > . . . > k s > such that, as multisets, { λ , . . . , λ m } = { k i , k i − , . . . , − k i , − k i : 1 i s } , (2.3)and m = (2 k +1)+ . . . +(2 k s +1) ; note that occurs s times in { λ , . . . , λ m } , and occurs at least s − times, etc. Conversely, for any such integers k > . . . > k s > with m = (2 k + 1) + . . . + (2 k s + 1) there is a deﬁning element h whose eigenvalues satisfy (2.3). To determine the labeled Dynkin diagrams,one chooses the diagonal matrices in g as maximal toral subalgebra, and then the following holds:If m = 2 n + 1 , then s is odd and λ n +1 = 0 . The corresponding simple abstract carrier algebra B ( k , . . . , k s ) of type B n has a weighted Dynkin diagram with labels λ − λ , . . . , λ n − − λ n , λ n ,where λ n is the label of the shorter root, see [8, Figure 5]. In that ﬁgure the last label is given as λ n ,which is a typo; cf. [1, pp. 410–412]. If s = 1 , then { λ , . . . , λ m } = { n, n − , . . . , − n, − n } and B ( n ) is principal. If s = 3 , then λ n +1 , λ n = 0 and < λ n − , implying that semisimple part of g is adirect sum of algebras of type A. If s > , then λ n +1 , λ n , λ n − = 0 and that semisimple part is a directsum of algebras of type A and one algebra of type B.If m = 2 n , then s is even and λ n = 0 . The corresponding abstract carrier algebra D ( k , . . . , k s ) of type D n has a weighted Dynkin diagram with labels λ − λ , . . . , λ n − − λ n − , λ n − , λ n − , where λ n − − λ n − is the label of the vertex of degree connected to the two vertices of degree 1 with label λ n − , see [8, Figure 6]. If s > , then λ n , λ n − , λ n − = 0 and the semisimple part of g is a direct sumof algebras of type A and one algebra of type D. If s = 2 and k > , or s = 4 , then that semisimplepart is a direct sum of algebras of type A; if s = 2 and k = 0 , then D ( n − , is principal. Type C.

Let g = sp (2 n, C ) be realised as g = { X ∈ gl (2 n + 1 , C ) : X ⊺ S = − SX } where S has theidentity matrix I n and the negative − I n on its anti-diagonal. The simple abstract carrier algebras havethe form C ( k , . . . , k s ) and the construction is similar to those for type B and D: here we can assume thedeﬁning element is h = diag( λ , . . . , λ n , − λ n , . . . , − λ ) with λ > . . . > λ n > and, as multisets, {± λ , . . . , ± λ n } = { k i − , k i − , . . . , − k i , − k i : 1 i s } for some k > k > · · · > k s > with n = k + . . . + k n . If one chooses the diagonal matrices in g as maximal toral subalgebra, then theDynkin diagram of C ( k , . . . , k s ) has labels λ − λ , . . . , λ n − − λ n , λ n , where λ n is attached to thelonger root, see [8, Figure 7]. Since λ n = 0 , we have λ n = 1 , and so the semisimple part of g is adirect sum of algebras of type A. If s = 1 , then C ( n ) is principal; Exceptional types.

A direct calculation yields the abstract carrier algebras g of exceptional types G , F , E , E , E ; the semisimple part of g is always a sum of Lie algebras of type A, see [11, Table 1]. NOTE ON ÉTALE REPRESENTATIONS FROM NILPOTENT ORBITS 5

The centre of g . Let g be as before and write g = z ⊕ s where z is the centre and s is semisimple. Itfollows from [11, p. 19] that dim z = rk g − rk s = rk g − rk s . Since rk s equals the number of labels in the weighted Dynkin diagram of g , the dimension of z equals the number of labels . For example, if g = B (5 , , with rank , then λ , . . . , λ = 5 , , , , , , , , , yielding labels , , , , , , , , ;thus, dim z = 5 . From the above classiﬁcation, it follows that dim z = 1 if and only if g has type A orif g is the Z -graded algebra E (11)8 , F (4)4 , or G (2)2 as deﬁned in [8, Table II].3. B ALA AND C ARTER ’ S CONSTRUCTION

Bala & Carter [1, 2] classiﬁed the nilpotent orbits in a complex simple Lie algebra using a constructionvery similar to Vinberg’s, without the assumption that the Lie algebras are graded. Just like Vinberg’sconstruction, this yields an étale representation for a certain reductive subalgebra of g . In this section, wereview some of these results and show how the approaches by Vinberg and by Bala & Carter are related.First, we recall a few deﬁnitions. Let g = z ⊕ s be a reductive Lie algebra with s semisimple and z = z ( g ) the centre. A Borel subalgebra of g is a maximal solvable subalgebra of g , and a subalgebraof g is a parabolic subalgebra if it contains a Borel subalgebra of g . Every parabolic subalgebra p of g is a semidirect product p = m ⋉ n of a nilpotent ideal n of p , all of whose elements are nilpotent, and areductive subalgebra m . A parabolic subalgebra is distinguished if dim n / [ n , n ] = dim m . Any reductivesubalgebra m of g arising in the above way for some parabolic subalgebra of g is a Levi subalgebra in g .Its commutator c = [ m , m ] is semisimple of parabolic type . Bala & Carter deﬁned the terms above forsemisimple g , but they carry over without change to reductive g .For semisimple g , it is shown in [2, Theorem 6.1] that the classiﬁcation of nilpotent orbits is equivalentto the classiﬁcation of conjugacy classes of pairs ( c , q c ) , where c is semisimple subalgebra of parabolictype in g , and q c is a distinguished parabolic subalgebra of c : For a nonzero nilpotent e ∈ g with sl -triple ( h, e, f ) deﬁne g k = { x ∈ g : [ h, x ] = kx } for k ∈ Z ; this furnishes g with a Z -grading. With thisgrading, e ∈ g . The element e is distinguished in g if ad( e ) : g → g is an isomorphism, that is, if e isin generic position. If e is not distinguished in g , then [2, Propositions 5.3 & 5.4] tell us how to constructa semisimple subalgebra c of g in which e is distinguished: if h is a maximal toral subalgebra of thecentraliser of ( h, e, f ) in g , then(3.1) m = z g ( h ) and c = [ m , m ] are a minimal Levi subalgebra of g containing e and a semisimple subalgebra of parabolic type, respec-tively, such that e is distinguished in c . The pair corresponding to e can be chosen to be ( c , q c ) , where q c is the Jacobson-Morovoz parabolic, see [1, Proposition 4.3] and [2, Theorem 6.1]. If G is a semisimplealgebraic group with Lie algebra g , then c is determined uniquely up to the action by Z G ( e ) , the cen-traliser of e in Ad g ( G ) . Since e is distinguished in c , the Z -grading of its sl -triple in c yields an étalerepresentation for the adjoint action of the reductive subalgebra c on the subspace c by evaluation at e .The adjoint action of g integrates to that of G , and thus we obtain an étale representation of the reductivegroup with Lie algebra c on the space c .The Borel and parabolic subalgebras of a reductive g = s ⊕ z as above are precisely z ⊕ b and z ⊕ p ,respectively, with b s a Borel subalgebra and p s parabolic. It follows that the Levi subalgebrasof g are precisely z ⊕ m , where m is a Levi subalgebra of s ; moreover, m is a minimal Levi subalgebracontaining e in s if and only if z ⊕ m is a minimal Levi subalgebra containing e in g .3.1. Relation to carrier algebras.

We compare the Bala & Carter construction with Vinberg’s carrieralgebras. Vinberg starts with a semisimple Z n -graded Lie algebra g = L i ∈ Z n g i ; recall that we allow Z ∞ = Z here. Let e ∈ g be nonzero nilpotent with sl -triple ( h, e, f ) such that h ∈ g and f ∈ g − ,and deﬁne g ( t , e ) as in (2.1); as mentioned before, t is a maximal toral subalgebra of the normaliser n g ( e ) and λ : t → C is deﬁned by [ t, e ] = λ ( t ) e . Let h = h deﬁne the following Z -graded algebra(3.2) g ( h ) = M k ∈ Z g ( h ) k with g ( h ) k = { x ∈ g k : [ h , x ] = kx } ; NOTE ON ÉTALE REPRESENTATIONS FROM NILPOTENT ORBITS 6 note that t g ( h ) . It follows from [11, Lemmas 1 & 2] that g ( h ) and g ( t , e ) are both reductive.Recall that t = C h ⊕ t , where t is a maximal toral subalgebra of z g ( h, e, f ) . More precisely: Lemma 3.1.

We have t = ker λ = z ( g ( t , e )) and g ( t , e ) = z g ( h ) ( t ) .Proof. Write z = z ( g ( t , e )) . Recall that t = C h ⊕ t , so ker λ = t follows from [ h, e ] = 2 e . Clearly,if t ∈ ker λ , then [ t, y ] = 0 for each y ∈ g ( t , e ) k , so t ∈ z . Since z commutes with the deﬁning elementof g ( t , e ) , we have z g ( t , e ) . Thus, z n g ( e ) , and therefore z t , This implies z ker λ , hence z = ker λ . Suppose x ∈ g ( h ) centralises t and write x = L k ∈ Z x k with each x k ∈ g ( h ) k . Since t g ( h ) , it follows from x, t ] = L k ∈ Z [ x k , t ] that each x k centralises t . By assumption, [ h, x k ] = 2 kx k , so x k ∈ g ( t , e ) k , and hence z g ( h ) ( t ) g ( t , e ) . Conversely, if x ∈ g ( t , e ) k , then [ h, x ] = 2 kx and, if t ∈ t , then [ t, x ] = kλ ( t ) x = 0 since t = ker λ . Thus, x ∈ z g ( h ) ( t ) k . (cid:3) The next proposition shows that Vinberg’s construction (2.1) of g ( t , e ) and its carrier algebra is thesame as applying Bala & Carter’s approach (3.1) to the Z -graded Lie algebra g ( h ) . Below, let G be thesemisimple algebraic group with Lie algebra g ( h ) . The conjugacy up to the centraliser Z G ( e ) reﬂectsthe freedom in choosing an sl -triple ( h, e, f ) for a given nonzero nilpotent element e . Proposition 3.2.

Let g be a Z n -graded complex semisimple Lie algebra, where n ∈ N ∪ {∞} , and let e , h , and t be as above. Then g ( t , e ) is a minimal Levi subalgebra of g ( h ) containing e and, up to Z G ( e ) -conjugacy, the carrier subalgebra c ( e ) = [ g ( t , e ) , g ( t , e )] is the unique semisimple subalgebra ofparabolic type in g ( h ) in which e is distinguished.Proof. Note that h stabilizes each g k , and g ( h ) k is the intersection of g k with the k -eigenspace of h .Lemma 3.1 shows that g ( t , e ) g ( h ) , and the Z -gradings of both algebras are determined by theeigenvalues of ad( h ) . The semisimple part s of g ( h ) is a semisimple ideal in g ( h ) containing ( h, e, f ) ;let a be the subalgebra generated by { h, e, f } . Note that for every subset X ⊆ g ( h ) we have( ∗ ) z g ( h ) ( X ) = z ( g ( h )) ⊕ z s ( X ) . We claim that t is a maximal toral subalgebra of z g ( h ) ( a ) : recall that t is deﬁned as a maximal toralsubalgebra of z g ( a ) , which is reductive by [11, p. 21]. Since t g ( h ) g , we know that t isalso a maximal toral subalgebra in z g ( h ) ( a ) . On the other hand, we have z g ( h ) ( a ) = z g ( h ) ( a ) becauseelements of degree = 0 in g ( h ) do not commute with the deﬁning element h ∈ a ; thus, t z g ( h ) ( a ) is a maximal toral subalgebra. We can write t = z ( g ( h )) ⊕ t ′ , where t ′ is a maximal toral subalgebraof z s ( a ) , and so for every subset X ⊆ g ( h ) we have( ∗∗ ) z X ( t ) = z X ( t ′ ) . The construction in (3.1) shows that m ′ = z s ( t ′ ) is a minimal Levi subalgebra of s containing e , so m = z ( g ( h )) ⊕ m ′ is a minimal Levi subalgebra of g ( h ) containing e . Now ( ∗ ), ( ∗∗ ), and Lemma 3.1 show m = z ( g ( h )) ⊕ z s ( t ′ ) = z g ( h ) ( t ′ ) = z g ( h ) ( t ) = g ( t , e ) , so g ( t , e ) is a minimal Levi subalgebra in g ( h ) containing e . The construction in (3.1) shows that e isdistinguished in [ m , m ] , and the latter is semisimple of parabolic type. Since [ g ( t , e ) , g ( t , e )] = [ m , m ] isthe carrier algebra, the claim follows; [2, Proposition 5.3] shows uniqueness up to Z G ( e ) -conjugacy. (cid:3) For Z -graded semisimple Lie algebras g , we have g = g ( h ) where h = 2 h is the deﬁning element,see [7, Remark 33], so the two approaches by Vinberg and Bala & Carter coincide. Corollary 3.3. If g is a Z -graded complex semisimple Lie algebra, then up to Z G ( e ) -conjugacy, thesubalgebra g ( t , e ) obtained by Vinberg’s construction and the subalgebra m obtained by Bala & Carter’sconstruction coincide. Even in the situation where g is given without a grading and e is a nonzero nilpotent element in g , achoice of h induces a Z -grading on g to which Vinberg’s approach can be applied; this is then equivalentto Bala & Carter’s approach; note that Bala & Carter use the element h rather than h = h to deﬁnetheir grading, which leads to an additional factor two in the degrees. NOTE ON ÉTALE REPRESENTATIONS FROM NILPOTENT ORBITS 7

4. G

YOJA ’ S CONSTRUCTION

Gyoja [9] described constructions of étale modules out of a given prehomogeneous module. Let G be acomplex reductive algebraic group with algebraic representation ̺ : G → GL( V ) on a ﬁnite-dimensionalcomplex vector space V such that ( G, ̺, V ) is a prehomogeneous module. Let v ∈ V be the point ingeneric position. The construction in [9, Theorem A] proceeds as follows: Let G v be the stabilizer of v in G with Cartan subgroup T G v , deﬁne G ′ = N G ( T ) /T , and let V ′ = V T be the set of ﬁxed pointsin V under T ; then V ′ is an étale module for the induced action of G ′ . Arising from a normaliser of atorus, G ′ is a reductive algebraic group. The second construction [9, Theorem B] yields a procedure toobtain a super-étale module from an étale module. (This means that the stabiliser of the point in genericposition is trivial and not just ﬁnite.) Given an étale module ( G, ̺, V ) with v ∈ V in generic positionand stabilizer G v , choose = h ∈ G v , and let G ′′ = N G ( h ) and V ′′ = V h , the set of ﬁxed points for h in V . Then V ′′ is an étale module for the induced action of G ′′ , and | G ′′ v | < | G v | . Since h h i is ﬁnite, G ′′ is also reductive. After ﬁnitely many iterations (with G ′′ instead of G ), one obtains a super-étale module.4.1. Relation to carrier algebras.

Gyoja’s construction was formulated for groups, but can just aswell be formulated for the corresponding Lie algebras. We show that this covers some of Vinberg’sconstructions. For this let g be a semisimple Lie algebra with nonzero nilpotent e ∈ g , let ( h, e, f ) be an sl -triple in g , and furnish g with the Z -grading induced by ad( h ) , where h = h . Proposition 4.1.

Gyoja’s construction, applied to the reductive Lie algebra g , the nilpotent element e ,and the adjoint g -module (ad , g ) , produces Vinberg’s étale representation associated with e ∈ g .Proof. The stabiliser algebra of e is z g ( e ) . We have shown that t = ker λ (as introduced in Lemma 3.1)is a maximal toral subalgebra of z g ( a ) , where a is the subalgebra spanned by ( h, e, f ) . By [11, p. 21], wehave z g ( a ) = z g ( e, h ) , and since h = 2 h ∈ g , it follows that z g ( a ) = z g ( e ) . Thus, a maximal toralsubalgebra of z g ( e ) is t = ker λ , which takes the place of Gyoja’s t . Now Lemma 3.1 shows that theﬁxed point set of t in g is V ′ = z g ( t ) ∩ g = g ( t , e ) , with t = C h ⊕ t ; moreover, z g ( t ) = g ( t , e ) is reductive with centre t . Hence, g ′ = z g ( t ) / t satisﬁes g ′ ∼ = [ g ( t , e ) , g ( t , e ) ] , so Gyoja’s g ′ is the -component of the carrier algebra of e in g . Lastly, V ′ = g ( t , e ) = [ g ( t , e ) , g ( t , e )] is the 1-componentof that carrier algebra: this follows since the centre of the reductive g ( t , e ) is contained in g ( t , e ) . (cid:3) Gyoja’s result encapsulates what is interesting to us in Vinberg’s and Bala & Carter’s theory: thefocus on the étale action of the reductive subalgebra of degree on the subspace of degree , ignoringthe subspaces of higher degree. R EFERENCES [1] P. Bala, R. W. Carter.

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