On Jordan classes for Vinberg's theta-groups
aa r X i v : . [ m a t h . R T ] J u l ON JORDAN CLASSES FOR VINBERG’S θ -GROUPS GIOVANNA CARNOVALE, FRANCESCO ESPOSITO, AND ANDREA SANTIAbstract. Popov has recently introduced an analogue of Jordan classes (packets, or decom-position classes) for the action of a θ -group ( G , V ) , showing that they are finitely-many, locally-closed, irreducible unions of G -orbits of constant dimension partitioning V . We carry out alocal study of their closures showing that Jordan classes are smooth and that their closure is aunion of Jordan classes. We parametrize Jordan classes and G -orbits in a given class in termsof the action of subgroups of Vinberg’s little Weyl group, and include several examples andcounterexamples underlying the differences with the symmetric case and the critical issuesarising in the θ -situation. C ontents1. Introduction 12. Preliminaries on Vinberg’s θ -groups and Jordan classes 32.1. Graded Lie algebras 32.2. The Jordan decomposition 42.3. The Cartan subspace 52.4. Dimensions of centralizers and regularity conditions 52.5. Jordan classes and sheets for θ -groups 73. Closure of a G -Jordan class 83.1. Closure of G -Jordan classes: the semisimple parts 83.2. A local study of the closure of a G -Jordan class 93.3. Regularity questions 134. Slice-induction and parametrization of orbits and classes 174.1. Slice-induction 174.2. Parametrization of orbits and classes 18Appendix A. Cartan, Levi and parabolic subalgebras in Z m -graded Lie algebras 21Acknowledgments 23References 231. Introduction
Theta groups (or, equivalently, periodically graded reductive Lie algebras) were deeplystudied in [31, 32] as a natural generalisation of symmetric spaces, [11, 12]. In all situations g is a Z m -graded complex reductive Lie algebra, its degree 0 part g is again reductive andthe focus is on the action of the corresponding connected algebraic group G on the otherhomogeneous components g i of g . As observed by Vinberg, there is no loss of generalityin studying the action on the degree 1 component V = g only. Key results in [31] concerninvariant theory and include: the introduction of a little Weyl group and the analogue of theSteinberg map and Chevalley’s restriction theorem and the proof that the little Weyl groupsare complex reflection groups. These results were confirmed also in positive characteristic,[13], where an alternative description of the little Weyl group in terms of the usual Weylgroup is proposed. Many interesting examples in representation theory can be interpretedin terms of graded Lie algebras: for instance, if g is the Lie algebra of a classical group G , agrading on the defining representation of G induces a grading on g and the G -action on V can be seen as a representation of a cyclic quiver with additional structure, [17, § § A structural feature of theta groups is that they are observable groups, that is, connec-ted reductive algebraic groups for which each fiber of the Steinberg map consists of finitelymany orbits. This property almost characterizes the theta groups: more precisely, a con-nected simple irreducible observable linear group is either a (commutant of a) theta groupor it is isomorphic to Spin ( ) or Spin ( ) [10]. Various explicit descriptions of the orbitsand invariants for theta groups of order m = m > p -adic field F . Indeed, the classification of positive rank gradings [13, 14, 24]over the residue field k of a maximal unramified extension L of F leads to the classificationof non-degenerate K -types, and stable G -orbits in V ∗ are stricly related to supercuspidalrepresentations of the rational points of G over F attached to elliptic Z-regular elements ofthe Weyl group, [25]. Also, in the context of a graded version of the generalized Springercorrespondence, the block decomposition of the G -equivariant derived category suppor-ted on the nilpotent part of each g i leads to the construction of representations of variousgraded double affine Hecke algebras with possibly unequal parameters, one for each block,[17, 18]. It emerges from these constructions that parabolic induction is no longer the rightinstrument in the graded setting, leading to the introduction of spirals. This shows that eventhough many results in the classical symmetric case have an analogue in the graded setting,generalisations to the case of m > G -stable stratifications in V . In thesetting of the ungraded generalized Springer correspondence, one of the relevant stratifica-tions is given by the decomposition into Jordan classes (packets, or decomposition classes)in a reductive group G , or Lie algebra g . In the Lie algebra setting Jordan classes were in-troduced in [4] and were crucial in the construction of sheets for the adjoint action of asemisimple group G on its Lie algebra. These classes are G -stable, disjoint, finitely-many,locally-closed, smooth and irreducible. The decomposition into Jordan classes in a Lie al-gebra turns out to coincide with the decomposition into orbit-types, i.e., into the subsets ofelements with same stabilizer up to conjugation. Borho and Kraft proved that sheets areeasily described as regular closures of those Jordan classes which satisfy some maximalityproperty with respect to closure inclusion, and it was shown in [3] that the closure and reg-ular closure of Jordan classes can be described in terms of Lusztig-Spaltenstein’s parabolicinduction of adjoint orbits. The symmetric analogue of Jordan classes and sheets has beenstudied by Tauvel and Yu, (see [29] and references in there) and their closures were studiedin [6, 7]. In the latter it is again observed that parabolic induction is no longer efficient, andslice induction is proposed: one of the difficulties in working with parabolic induction is thefact that many homogenous Levi subalgebras do not necessarily lie in a homogeneous para-bolic subalgebra, see the Appendix A for an example of this phenomenon. An analogue ofJordan classes for theta groups when g is semisimple has been recently introduced by Popovin [23], generalizing the classical and symmetric ones. As in these cases, Jordan classes forma partition of V into finitely-many, locally-closed, irreducible unions of G -orbits of constantdimension, and so sheets for the G -action on V are regular closures of some Jordan class.In this paper we introduce a local study of such Jordan classes and their closures leadingus to prove that any Jordan class is smooth and that its closure is a union of Jordan classes.In order to characterize the closure relation, we provide an analogue of the results in [7]on slice induction. For our inductive arguments, we needed to extend slightly the notionof Jordan classes to the case of reductive Lie algebras. Our local approach differs from [7]because we rely on Luna’s fundamental Lemma and use the Slodowy slice only after reduc-tion to neighbourhoods of nilpotent points; Luna’s slice theorem is also used for the proofof smoothness.It is also worthwhile to notice that a different, coarser, notion of Jordan equivalence re-lation could have been introduced, by using regularity for the G -action rather than for the ORDAN CLASSES FOR θ -GROUPS 3 action of the full group G . In the symmetric setting these two notions coincide in virtue of[12, Proposition 5], but they might differ for m >
2. Popov’s choice of Jordan classes in V ensures that each of them is contained in a usual Jordan class in g . We devote § § §
2, we recall the basics on periodically graded com-plex reductive Lie algebras, introduce the relevant notions of regularity and extend to thereductive case the general treatment in [23] of Jordan classes and sheets in V . We then focusin § V , the main results here are The-orem 3.9 and Proposition 3.10. We conclude § V and the G -orbits contained in a class. Thepaper finishes with Example 4.11 on trivectors in 9-dimensional space and with AppendixA, dealing with obstructions to the existence of homogeneous parabolic subalgebras in g .During completion of this paper we were informed that Professor `E. B. Vinberg hadpassed away. Without his work in [31] this manuscript would never have been written, sowe would like to dedicate it to his memory.2. Preliminaries on Vinberg’s θ -groups and Jordan classes Graded Lie algebras.
Let g be a complex reductive Lie algebra which is Z m -graded,that is, it admits a direct sum decomposition of vector spaces g = M l ∈ Z m g l (2.1)with [g i , g l ] ⊂ g i + l for all i , l ∈ Z m . We note that the subspaces of (2.1) can be recovered as theeigenspaces of the automorphism θ : g → g of g associated to the primitive m th root of unity ω = e πim , that is, the one for which θ ( x ) = ω l x for all x ∈ g l . Conversely, any automorphism θ of g of period m defines a Z m -grading. Due to this, we will denote a Lie algebra g with a Z m -grading by the triple { g , θ , m } , or often simply by { g , θ } . Whenever a subspace A ⊂ g ishomogeneous, i.e., it satisfies A = ⊕ l ( A ∩ g l ) , we will write A l = A ∩ g l and A = ⊕ l A l .The Lie algebra g has a decomposition into homogeneous ideals g = z(g) ⊕ s , where s := [g , g] . (2.2)We denote by κ a bilinear form on g that is non-degenerate, g -invariant, θ -invariant and suchthat z(g) and s are orthogonal. We call any such bilinear form adapted . Lemma 2.1.
There exists an adapted bilinear form κ on g if and only if z(g) is symmetrically graded,i.e., dim z(g) l = dim z(g) − l for all l ∈ Z m . In this case dim g l = dim g − l for all l ∈ Z m .Proof. If κ is adapted, then κ (g l , g i ) = i + l =
0, hence g − l and g l are dual spaces,and so are s − l and s l and also z(g) − l and z(g) l . In particular z(g) is symmetrically graded.Conversely, it is enough to consider an appropriate extension of the Killing form of s . (cid:3) With the term reductive Z m -graded Lie algebra { g , θ } , we will always mean a complex reductive Liealgebra g = z(g) ⊕ s together with a Z m -grading such that the center z(g) is symmetrically graded. This is also the class of graded Lie algebras considered in [31], since they allow for adaptedbilinear forms. By Lemma 2.1 we may assume κ to be an extension of the Killing form of s .Let G be any connected algebraic group with Lie algebra Lie ( G ) = g , let S be the connectedsubgroup of G with Lie ( S ) = s , and let ◦ denote the identity component of a closed subgroup,so G = Z ( G ) ◦ S . Let G be the connected subgroup of G with Lie ( G ) = g . Unless otherwisestated, for Lie subalgebras of g we will use a gothic letter, the corresponding Roman capitalletter will indicate the connected subgroup of G with that Lie algebra, a lower index 0 itsintersection with G . So, the decomposition g = z(g) ⊕ s , gives an almost direct product G = ( Z ( G ) ◦ ) ◦ S ◦ , where S ◦ is the semisimple, connected subgroup of S with Lie ( S ◦ ) = s . By GIOVANNA CARNOVALE, FRANCESCO ESPOSITO, AND ANDREA SANTI restricting the adjoint representation, G and S ◦ act on g l , for any l ∈ Z m , with trivial actionof ( Z ( G ) ◦ ) ◦ . The reduction process in [31, pag. 467] shows that it is enough to focus on thecase of l =
1; we set V := g . The linear group of transformations of V associated to G iscalled the θ -group of the graded Lie algebra { g , θ } and it does not depend on the choice of G inthe class of locally isomorphic groups. However, by abuse of notation, we will directly referto G as the θ -group of { g , θ } . The decomposition (2.2) in degree 1 gives a decomposition of V into G -stable subspaces V = z(g) ⊕ s with trivial G -action on z(g) . Observe that z(g) = θ is not inner.Let x ∈ g and m be a Lie subalgebra of g with associated subgroup M ⊂ G . The orbit of x for the action of M is denoted by O Mx , and the stabilizer of x in M by M x . The centralizer of x in m is denoted by m x , with center z(m x ) . If x ∈ V , then g x , z(g x ) and [g x , g x ] are θ -stable,in other words homogeneous. We recall that if x ∈ g is semisimple, then G x is a connectedsubgroup of G , the Levi subgroup of a parabolic subgroup of G [28, 7.3.5]. In this case, therestriction of κ to g x = z(g x ) ⊕ [g x , g x ] is an adapted bilinear form, so z(g x ) = z(g) ⊕ z(s x ) issymmetrically graded. We stress that G x = G x ∩ G is not connected in general.We recall the following general results on centralizers, that we will later apply when x ∈ V . Lemma 2.2. [29, Proposition 35.3.1, Corollary 35.3.2]
Let x ∈ g . Then [g , x ] ⊥ = [g , z(g x )] ⊥ = g x , and [g , g x ] ⊥ = z(g x ) . (2.3) The following conditions are equivalent for any x , y ∈ g : ( i ) y ∈ z(g x ) ; ( ii ) g x ⊂ g y ; ( iii ) [g , y ] ⊂ [g , x ] ; ( iv ) z(g y ) ⊂ z(g x ) ; Corollary 2.3.
Let g ∈ G and x , y ∈ V . Then the following conditions are equivalent: ( i ) g · g x = g y ; ( ii ) g · z(g x ) = z(g y ) ; ( iii ) g · z(g x ) = z(g y ) .Proof. Clearly ( i ) ⇔ ( ii ) by Lemma 2.2 since g · g x = g g · x and g · z(g x ) = z(g g · x ) , and ( ii ) ⇒ ( iii ) .If ( iii ) holds, then y ∈ z(g y ) = z(g g · x ) and x ∈ z(g g − · y ) , hence g · g x = g y by Lemma 2.2. (cid:3) The Jordan decomposition.
Let { g , θ } be a reductive Z m -graded Lie algebra. For ele-ments x , y , z in g , lower indices s and n will always indicate semisimple and nilpotent partsin the Jordan decomposition, i.e., they stand for x = x s + x n with x s ∈ g semisimple, x n ∈ g nilpotent, and [ x s , x n ] =
0. Elements of z(g) are always intended to be semisimple.Let S (resp. N ) be the set of semisimple (resp. nilpotent) elements of g . We note that θ preserves both S and N , so semisimple and nilpotent parts of any x ∈ g l also belong to g l .We set S V = S ∩ V , N V = N ∩ V , and stress that the number of G -orbits in N V is finite [31]. Lemma 2.4.
The Lie algebra g is reductive and its action on g is completely reducible.Proof. Since g = z(g) ⊕ s , it is sufficient to prove the claim for s . Now κ restricted to s isnon-degenerate and s contains the semisimple and nilpotent parts of any of its elements.The claim then follows from, e.g., [29, Proposition 20.5.12]. (cid:3) We emphasize that g is not a subalgebra of maximal rank of g in general, that is, it mightnot contain any Cartan subalgebra of g . Let x ∈ V . A direct consequence of Lemma 2.2 is: Lemma 2.5.
The tangent space T x O G x to O G x at x is given by the subspace [g , x ] of V . Its orthogonalin g − coincides with g x − . ORDAN CLASSES FOR θ -GROUPS 5 The Cartan subspace. A Cartan subspace of { g , θ } is an abelian subspace c of V whichconsists of semisimple elements and it is maximal in the class of such subspaces. Theorem 2.6. [31, pag. 472]
Any two Cartan subspaces of { g , θ } are conjugate by the action of anelement in G . As a consequence, if x ∈ S V , then O G x meets any Cartan subspace of { g , θ } . The dimension of a Cartan subspace of a graded Lie algebra { g , θ } is called the rank of { g , θ } .It is clear that { g , θ } has zero rank if and only if V ⊂ N V . For any set R of commuting elementsof S V , the centralizer c g ( R ) = ∩ x ∈ R g x of R in g is a homogeneous Levi subalgebra of g , so c g ( R ) = z(c g ( R )) M [c g ( R ) , c g ( R )] (2.4)and these summands are also homogeneous. We recall a useful characterization of a Cartansubspace in terms of its centralizer [31, pag. 471]. Proposition 2.7.
A subspace c ⊂ V consisting of commuting semisimple elements is a Cartan sub-space if and only if z(c g (c)) = c and the graded Lie algebra { [c g (c) , c g (c)] , θ } has zero rank. Let c be a Cartan subspace. By the previous result and equation (2.4) for R = c , we have adecomposition c g (c) = c L [c g (c) , c g (c)] , with c ⊂ S V and [c g (c) , c g (c)] ⊂ N V . In other words,this decomposition gives the Jordan components of any element of c g (c) . Corollary 2.8.
For any x ∈ c , we have z(g x ) ⊂ c .Proof. Since z(g x ) consists of semisimple elements, it follows that z(g x ) ⊂ c g (c) ∩ S V = c . (cid:3) Before turning to the next subsection, we recall that the Weyl group in the sense of Vinbergis the group W Vin = W Vin (g , θ ) of linear transformations of c given by W Vin ∼ = N G (c) /Z G (c) ,where N G (c) (resp. Z G (c) ) is the normalizer (resp. centralizer) of c in G . Theorem 2.9. [31, pag. 473]
The group W Vin is finite and for x , y ∈ c we have y ∈ O G x if andonly if y ∈ W Vin · x . There is a geometric counterpart to this result [31, § C [ V ] → C [c] of poly-nomial functions from V to c induces a “Chevalley-type” isomorphism C [ V ] G ∼ = C [c] W Vin and each fiber of the “Steinberg quotient map” ϕ : V → V//G ∼ = c /W Vin consists of finitelymany G -orbits. Here V//G is the GIT quotient of V , and two elements of V fail to be sep-arated by the invariants if and only if their semisimple parts lie in the same G -orbit. Recallthat semisimple (resp. nilpotent) orbits can also be characterized as the closed orbits (resp.orbits whose closure contains 0). Hence, each fiber of ϕ contains exactly one closed orbit.2.4. Dimensions of centralizers and regularity conditions.
This subsection deals with somegeneral observations, which encompas a classical result of Kostant and Rallis (see [12] andalso [20]), and motivates the introduction of two distinct notions of regularity.
Proposition 2.10.
Let { g , θ } be a reductive Z m -graded Lie algebra (with symmetrically graded center,as usual). Then dim g l − dim g xl = dim g − l − − dim g x − l − for all x ∈ V and l ∈ Z m .Proof. Let κ be an adapted bilinear form on g . The bilinear form given by κ x ( y , z ) := κ ( x , [ y , z ]) is skew-symmetric for all y , z ∈ g and its radical is the centralizer g x , which is homogeneous.It induces a non-degenerate bilinear form on the quotient g / g x = L l ∈ Z m g l / g xl with theproperty that g i / g xi ⊥ g l / g xl if i + l + =
0, in particular g l / g xl ∼ = (g − l − / g x − l − ) ∗ . (cid:3) Corollary 2.11. ( i ) For all x ∈ V we have dim O Gx = O G x + P l =− (cid:0) dim g l − dim g xl (cid:1) ; ( ii ) If x ∈ S V , then dim g l − dim g xl = dim g l + − dim g xl + is independent of l ∈ Z m and wehave dim O Gx = m dim O G x ; ( iii ) Let x ∈ V , then g x = g if and only if x ∈ z(g) . GIOVANNA CARNOVALE, FRANCESCO ESPOSITO, AND ANDREA SANTI
Proof.
Claim (i) is immediate from Proposition 2.10. If x ∈ S V , then the restriction of κ to g x is non-degenerate and dim g xl = dim g x − l for all l ∈ Z m , so (ii) follows from Proposition 2.10and (i). If x ∈ z(g) , then clearly g x = g . Conversely, if g x = g then x ∈ c g (h ) , where h isa Cartan subalgebra of g , and x is semisimple by a classical result, see e.g. [33, pag. 116].Then g x = g by (ii) and x ∈ z(g) . (cid:3) If x , y ∈ V are two elements with dim O G x = dim O G y , then dim g xl = dim g yl for l = − x , y ∈ S V is indeed necessary fordim g xl = dim g yl to hold also for l = − Example 2.12.
Let g be of type E and θ be the automorphism of g of order 3 extensivelystudied in [34]. Here g ≃ Λ C , g = sl( ) and g − = Λ ( C ) ∗ . The orbits of SL ( ) on V = Λ C have been classified in loc. cit . Let e i , for 1 i
9, be the canonical basis vectorsof C and let e ijl := e i ∧ e j ∧ e l . The trivector x s = e + e + e is semisimple, withcentralizer g x s a reductive Lie algebra with semisimple part r of type E . More precisely r = r − ⊕ r ⊕ r with r = X ⊗ Y ⊗ Z , r = sl( X ) ⊕ sl( Y ) ⊕ sl( Z ) , r − = X ∗ ⊗ Y ∗ ⊗ Z ∗ ,where X = span { e , e , e } , Y = span { e , e , e } , Z = span { e , e , e } and where we identi-fied tensor products with subspaces of g ± by mapping pure tensors to the correspondingantisymmetrizations. Since g x s has maximal rank, its center is two-dimensional and it is notdifficult to see that it consists of x s ∈ g and x ∗ s ∈ g − .Now x s is the semisimple part of trivectors x = x s + x n in the VI family, cf. [34, Table 5].We consider those trivectors for which dim O G x =
76, i.e., x = x s + x n with nilpotent part: Class x n = e + e + e + e + e ; Class x n = e + e + e + e ; Class x n = e + e + e + e .In all the three cases dim g x = g x − = g x = (cid:8) y ∈ g x s | y ∧ x n = (cid:9) has dimension 6, 8 and 10, respectively.Corollary 2.11 and Example 2.12 motivate the following. Definition 2.13.
For any subset A ⊂ V , we set(i) A reg = (cid:10) x ∈ A | dim g x dim g y for all y ∈ A (cid:11) ;(ii) A • = (cid:10) x ∈ A | dim g x dim g y for all y ∈ A (cid:11) .The subset A reg (resp., A • ) is called the regular part (resp., the G -regular part) of A .Note that A • = (cid:10) x ∈ A | dim g x − dim g y − for all y ∈ A (cid:11) due to Lemma 2.5. A simplerelation between the two notions is given by the following. Lemma 2.14.
Let A ⊂ V be irreducible. Then A reg = \ l ∈ Z m (cid:8) x ∈ A | dim g xl dim g yl for all y ∈ A (cid:9) (2.5) and so A reg ⊂ A • as a Zariski open subset.Proof. Clearly each subset on the R.H.S of (2.5) is non-empty and Zariski open in A . Since A is irreducible, the (finite) intersection of all such subsets is non-empty, so equal to A reg . (cid:3) Example 2.15.
A semisimple element y ∈ S V belongs to S regV if and only if dim g y = dim c g (c) .Indeed y is G -conjugated to some x ∈ c , whose centralizer has the form g x = c g (c) ⊕ M σ ∈ Σ ( x ) g σ (2.6)with Σ ( x ) = { σ ∈ Σ | σ ( x ) = } , and x ∈ S regV if and only if σ ( x ) = σ ∈ Σ , i.e., g x = c g (c) .Here the abelian subalgebra c acts semisimply on g and g x , and Σ is the set of restricted ORDAN CLASSES FOR θ -GROUPS 7 roots, that is, the non-zero linear functions on c occurring in the weight space decomposition g = c g (c) ⊕ L σ ∈ Σ g σ of g . Example 2.16.
Contrarily to the ungraded case, an element x s ∈ c ∩ S regV is not necessarilyin V • (let alone V reg or g reg , since g reg ∩ V ⊂ V reg ⊂ V • ). In general, x s extends to anelement x = x s + x n ∈ V • where x n is an element in general position in [c g (c) , c g (c)] (recallthat [c g (c) , c g (c)] consists of nilpotent elements). Then g x ( g x s due to (iii) of Corollary 2.11applied to the reductive Lie algebra g x s . The G -orbits in V • have codimension in V equal tothe rank of { g , θ } , hence dim g x = dim g − dim V + dim c , see [31, Theorem 5].2.5. Jordan classes and sheets for θ -groups. V. L. Popov has recently generalized the notionof a Jordan class to the case of semisimple Z m -graded Lie algebras { g , θ } and studied its maingeometric properties in [23]. For m =
1, 2, the notion coincides with that studied in [7, 29].We here briefly extend his general treatment to the reductive case, which is more suitablefor our inductive and local arguments of § §
4, and directly refer to [23, §
3] for more details.(We warn the reader that the symbol “reg” in [23] is replaced by “ • ” in the present paper.)Let { g , θ } be a reductive Z m -graded Lie algebra. Two elements x = x s + x n and y = y s + y n of V are G -Jordan equivalent if there exists g ∈ G such that g y s = g · g x s , y n = g · x n . (2.7)This is an equivalence relation x G ∼ y on V , the equivalence class J G ( x ) of x ∈ V is called the G -Jordan class of x in V . Evidently the union of all G -Jordan classes in V is a partition of V . Remark 2.17. (1) By construction any G -Jordan class is a G -stable set consisting of G -orbits of the same dimension. For example S regV constitutes a G -Jordan class, as itcan be easily seen from Theorem 2.6 and Example 2.15.(2) The equality g x s = g z + x s for any z ∈ z(g) and x ∈ g implies that z + x G ∼ x , so theadditive group underlying z(g) acts on each G -Jordan class J G ( x ) by translations.(3) Since G = ( Z ( G ) ◦ ) ◦ S ◦ , the element g from (2.7) can always be chosen in S ◦ . Then, for x = z + x ′ ∈ z(g) ⊕ s and y = w + y ′ ∈ z(g) ⊕ s , the statement x G ∼ y holds if and onlyif x ′ S ◦ ∼ y ′ holds and the decomposition of V = z(g) ⊕ s induces a decomposition J G ( x ) = J G ( x ′ ) = z(g) × J S ◦ ( x ′ ) (2.8)where J S ◦ ( x ′ ) is the S ◦ -Jordan class of x ′ ∈ s as introduced in [23].(4) Equality (2.8) applied to x ′ ∈ N V ⊂ s gives J G ( x ′ ) = z(g) × O G x ′ = z(g) × O S ◦ x ′ . For z ∈ z(g) we then get J G ( z ) = J G ( ) = z(g) .Observe that if x = x s + x n ∈ V , then x n lies in the degree 1 component of the homogeneoussemisimple subalgebra [g x s , g x s ] . Lemma 2.18.
We have z(g x ) = z(g x s ) ⊕ z(g x n ∩ [g x s , g x s ]) and the components of an element in z(g x ) with respect to this decomposition coincide with its semisimple and nilpotent parts, respectively. Thus, z(g x ) = z(g x s ) ⊕ z(g x n ∩ [g x s , g x s ]) .Proof. The first claim is [29, Proposition 39.1.1], the second follows since z(g x ) and its sum-mands are homogeneous. (cid:3) Lemma 2.2 tells us that (z(g x ) ) reg = (cid:8) y ∈ z(g x ) | g y = g x (cid:9) = (cid:8) y ∈ z(g x ) | z(g y ) = z(g x (cid:1) } = (cid:8) y ∈ z(g x ) | rk ( ad g ( y )) = rk ( ad g ( x )) (cid:9) , (2.9)which is a Zariski open subset of z(g x ) , hence irreducible. We note that this is also the setof all y ∈ V such that g y = g x and that x ∈ (z(g x ) reg ) , so (z(g x ) reg ) = (z(g x ) ) reg and wewill omit the parentheses in the sequel. GIOVANNA CARNOVALE, FRANCESCO ESPOSITO, AND ANDREA SANTI
The proof of the following result is as in [29, Lemma 39.1.2 & Proposition 39.1.5], once thelast claim of Lemma 2.18 is taken into account. See also [23, Proposition 3.10].
Proposition 2.19.
Let x = x s + x n ∈ V . Then the decomposition in Lemma 2.18 induces a decom-position z(g x ) reg = z(g x s ) reg × z(g x n ∩ [g x s , g x s ]) reg and the G -Jordan class of x is the irreduciblesubset of V given by J G ( x ) = G · (z(g x s ) reg + x n ) . We will need the following results from [23] which readily generalize to the reductive casein virtue of (2.8).
Proposition 2.20. ( [23, Proposition 3.9 and Proposition 3.17] ). Let { g , θ } be a reductive Z m -graded Lie algebra and x , y ∈ V . Then the following conditions are equivalent: ( i ) x G ∼ y ; ( ii ) there exists g ∈ G such that g y = g · g x ; ( iii ) there exists g ∈ G such that z(g y ) = g · (z(g x )) .Moreover the number of G -Jordan classes in V is finite. Corollary 2.21.
The G -Jordan class of x ∈ V coincides also with J G ( x ) = G · z(g x ) reg , it is locallyclosed in V (hence a subvariety of V ) and dim J G ( x ) = dim g − dim g x + dim z(g x s ) .Proof. The first two statements can be proved as in [29, Corollary 39.1.7], for the last one see[23, Proposition 3.13]. (cid:3)
It follows from Corollary 2.21 that any G -Jordan class J G ( x ) = G · z(g x ) reg is contained inthe G -Jordan class J G ( x ) = G · z(g x ) reg . However it is well-known that two elements x , y ∈ V in the same G -Jordan class are not G -Jordan equivalent in general (see, e.g., [29, 38.7.18]).We conclude this subsection recalling the relationship between the sheets for the G -actionon V and the G -Jordan classes.Let H be a connected algebraic group acting on a variety X and let d ∈ N . We set X ( d ) = { x ∈ X | dim O Hx = d } and for any subset A ⊂ X we set A ( d ) = A ∩ X ( d ) . Each X ( d ) is locallyclosed and its irreducible components are called sheets for the H -action on X . We observethat X ( d ) := S j d X ( j ) is closed so X ( d ) ⊂ X ( d ) [29, Proposition 21.4.4].If A ⊂ V , and p is the largest integer with A ( p ) = ∅ then, according to Definition 2.13, wehave A ( p ) = A • , which is a Zariski open subset of A . In particular, the set V • is a Zariskiopen subset of V , hence it is irreducible, and it is called the G -regular sheet of V . Proposition 2.22. ( [23, Proposition 3.19] ) For any sheet S in V there exists a unique G -Jordanclass J ⊂ S such that S = J • . Moreover we have S = J . Closure of a G -Jordan class Closure of G -Jordan classes: the semisimple parts. In virtue of Proposition 2.22, itis important to understand the closure and G -regular closure of a G -Jordan class and tosee which classes are dense in a sheet. We start with a preliminary result and then describewhich semisimple parts occur in the closure of a G -Jordan class.Let J = J G ( x ) ⊂ V ( d ) be a G -Jordan class in V . Then its closure J is a union of G -orbits andif O G y ⊂ J , then O G y ⊂ J . Let M J be the set of G -orbits contained in J which are maximal withrespect to the partial order given by inclusion of orbit closures. By construction J = S O ∈ M J O . Proposition 3.1.
Let J = J G ( x ) be a G -Jordan class in V . Then J • = S O ∈ M J O .Proof. We may assume without loss of generality that x = x s + x n with x s ∈ c . First ofall J ⊂ V ( d ) ⊂ V ( d ) , so dim O d for any O ∈ M J . We then consider the restriction ψ = ϕ | J : J −→ ϕ ( J ) to J of the Steinberg map ϕ : V → V//G ∼ = c /W Vin and set to show thatits image is ϕ ( J ) = W Vin · z(g x s ) W Vin , (3.1)
ORDAN CLASSES FOR θ -GROUPS 9 where z(g x s ) ⊂ c , cf. Corollary 2.8.First of all ϕ ( J ) = ϕ (z(g x s ) reg ) by Proposition 2.19, G -equivariance and [31, Theorem 3],hence ϕ ( J ) ⊂ ϕ ( J ) = ϕ (z(g x s ) reg ) = W Vin · z(g x s ) reg W Vin = W Vin · z(g x s ) W Vin . (3.2)On the other hand, if y s ∈ z(g x s ) reg , then y = y s + x n ∈ O G y ⊂ J and so y s ∈ O G y ⊂ J , [31,Proposition 4], giving z(g x s ) reg ⊂ J . It follows that z(g x s ) = z(g x s ) reg ⊂ J , (3.3)hence W Vin · z(g x s ) ⊂ J and W Vin · z(g x s ) W Vin = ϕ ( W Vin · z(g x s ) ) ⊂ ϕ ( J ) ,proving our first claim. We stress that (3.1) is a closed subset of c /W Vin , i.e., an affine variety.Let z ∈ O for O ∈ M J . By [31, Theorem 4] the irreducible component of the fiber ψ − ψ ( z ) containing z is the closure of a G -orbit in J , i.e., it is O . Since ψ is a dominant morphism ofirreducible affine varieties, we may argue as in [31, Corollary 2] and the fibers of ψ are allof the same dimension, which is the maximum dimension of an orbit in J , namely d . Hencedim O = d , O ⊂ J • and [ O ∈ M J O ⊂ J • .The other inclusion follows because O \ O is always a union of G -orbits of dimension < d . (cid:3) Lemma 3.2.
Let J = J G ( x ) be a G -Jordan class and y = y s + y n ∈ J . Then: ( i ) y s ∈ J ; ( ii ) y s is G -conjugate to an element of z(g x s ) ; ( iii ) For any y ′ s ∈ z(g x s ) there exists a y ′ n ∈ g y ′ s ∩ N V such that y ′ s + y ′ n ∈ J • . ( iv ) If z ∈ z(g) , then z + y ∈ J , and in that case z + y ∈ J • if and only if y ∈ J • .Proof. Since J is G -invariant, claim (i) follows from [31, Proposition 4] because y s ∈ O G y ⊂ J .We now turn to (ii). We may assume y s ∈ c by Theorem 2.6. Claim (ii) is then an immediateconsequence of the following identity J ∩ c = W Vin · z(g x s ) , (3.4)which we now establish.First of all W Vin · z(g x s ) ⊂ J by (3.3) and W Vin · z(g x s ) ⊂ c by Corollary 2.8, so oneinclusion is clear. Conversely ϕ ( J ∩ c) ⊂ ϕ ( J ) = ϕ ( W Vin · z(g x s ) ) , where the last equality hasbeen established in the proof of Proposition 3.1. It follows that J ∩ c ⊂ W Vin · z(g x s ) , sincethe restriction of ϕ to c is just the natural projection to c /W Vin and both sets are W Vin -stable.We prove (iii). By Proposition 2.19 we have that z(g x s ) reg + x n ⊂ J , so z(g x s ) + x n ⊂ J and y ′ s + x n ∈ J . Therefore the orbit O G y ′ s + x n is contained in the closure of an orbit O in M J . Sincethe fibers of the Steinberg map are closed and [31, Theorem 3] is in force, O is representedby an element of the form y ′ s + y ′ n for some y ′ n ∈ g y ′ s ∩ N V . Clearly O ⊂ J • by Proposition 3.1.Finally, (iv) follows from the action of z(g) on J , cf. Remark 2.17 (2). (cid:3) Corollary 3.3.
The G -regular closure J • of a G -Jordan class J contains at least a nilpotent G -orbit. A local study of the closure of a G -Jordan class. We start with a local characterizationof the closure of a G -Jordan class. Lemma 3.4.
The following statements are equivalent for a G -Jordan class J : ( i ) J is a union of G -Jordan classes; ( ii ) For every y ∈ J there exists an analytic or Zariski open neighbourhood U y of y in J G ( y ) suchthat U y ⊂ J . Proof.
The implication ( i ) ⇒ ( ii ) is immediate, since G -Jordan classes are disjoint and wemay take U y = J G ( y ) . Assume now that ( ii ) holds. Let y ∈ J and set J ′ = J G ( y ) . Then J ′ ∩ J is a non-empty closed subset of J ′ . On the other hand, condition ( ii ) implies that any pointof J ′ ∩ J has an open neighbourhood of J ′ therein, therefore J ′ ∩ J is also open in J ′ . Since J ′ is a Zariski irreducible variety, it is connected both in the Zariski and analytic topology [26,pag. 321], thus J ′ ⊂ J and ( i ) holds. (cid:3) In virtue of Lemma 3.4 we shall apply a local approach and look at the closure of a G -Jordan class in the neighbourhood of a point of V . For the rest of this subsection for any y s ∈ S V we will use the following notation: m := g y s ; M := G y s G ; and M := M ∩ G with identity component M ◦ . For any subset X ⊂ m , we will write X reg , M to indicate theregular part of X for the action of M . We also recall that for any GIT quotient π : X → X//H ofa reductive algebraic group H acting on a variety X , a subset U of X is called π -saturated or H -saturated if U = π − π ( U ) . Saturated implies H -stable, the converse is not necessarily true.For m as above, we consider the M -stable subset of m defined as follows: U m = { z ∈ m | g z ⊂ m } . Lemma 3.5.
With notations as above: ( i ) U m is M -saturated; ( ii ) U m is open in m ; ( iii ) For all z = z s + z n ∈ U m we have z(g z s ) reg + z n = (cid:0) z(m z s ) reg , M + z n (cid:1) ∩ U m ; ( iv ) For any G -Jordan class J such that J ∩ U m = ∅ , we have J ∩ U m = [ i ∈ I J J M , i ∩ U m , (3.5) where { J M , i | i ∈ I J } is the (finite) set of M ◦ -Jordan classes in m such that J M , i ∩ U m ∩ J = ∅ .In addition, dim J M , i = dim J M , j for any i , j ∈ I J ; ( v ) Let y = y s + y n for y n ∈ N V ∩ m . Then J G ( y ) ∩ U m = z(m) reg + S n i ∈ N G (m) /M ◦ n i · O M ◦ y n .Proof. For m =
2, parts (i)-(ii) are [7, Lemma 2.1]. We propose a slightly different proof for(i). Saturation is equivalent to say that g z ⊂ m if and only if g z s ⊂ m , for any z = z s + z n ∈ m .As g z = g z s ∩ g z n , one implication is immediate. We will now show that g z s g s implies g z g s for any semisimple element s ∈ g and any z ∈ g s , independently of the Z m -grading.Since z s and s commute, we can always find a Cartan subalgebra h of g containing both.Then g s = h ⊕ M α ∈ Φ ( s ) g α , g z s = h ⊕ M α ∈ Φ ( z s ) g α ,where Φ ( h ) is the set of roots vanishing on an element h ∈ h . Since ( Φ ( s ) + ( Φ \ Φ ( s ))) ∩ Φ ⊂ Φ \ Φ ( s ) , the reductive subalgebra g s ∩ g z s = h ⊕ (cid:16)L α ∈ Φ ( s ) ∩ Φ ( z s ) g α (cid:17) stabilizes the subspace X = L α ∈ Φ ( z s ) \ Φ ( s ) g α . As z n ∈ g s ∩ g z s acts nilpotently on X , there is a non-zero ξ in theresuch that [ z n , ξ ] =
0. In other words ξ ∈ g z \ g s .To prove ( ii ) we use the argument in [5, Lemma 2.1]. We may assume y s ∈ c and that h is aCartan subalgebra of g , hence of m , containing c . The product f = Q α ∈ Φ \ Φ ( y s ) α is a homo-geneous polynomial on h that is invariant for the Weyl group of m . By Chevalley’s restrictiontheorem f extends to an M -invariant polynomial F on m . By (i), U m = { z ∈ m | g z s ⊂ m } , andit is not hard to verify that this is equal to { z ∈ m | F ( z ) = } , hence it is open in m .Since U m is M -saturated, it is enough to prove ( iii ) for z = z s ∈ U m . We have g z s = m z s ,so z(g z s ) = z(m z s ) . If x ∈ z(g z s ) reg then g x = g z s = m z s ⊂ m , so x ∈ z(m z s ) reg , M ∩ U m .Conversely, if x ∈ z(m z s ) reg , M ∩ U m , then g x ⊂ m , so g x = m x = g z s and x ∈ z(g z s ) reg .We prove (iv). Clearly J ∩ U m ⊂ S i ∈ I J J M , i ∩ U m , and we now show the other inclusion.Let z = z s + z n ∈ J ∩ J M , i ∩ U m for some i ∈ I J , so J = J G ( z ) and J M , i = J M ◦ ( z ) . Combining ORDAN CLASSES FOR θ -GROUPS 11 the fact that U m is M ◦ -stable with (iii) gives J M , i ∩ U m = (cid:0) M ◦ · (z(m z s ) reg , M + z n ) (cid:1) ∩ U m = M ◦ · (cid:0) (z(m z s ) reg , M + z n ) ∩ U m (cid:1) = M ◦ · (z(g z s ) reg + z n ) ⊂ G · (z(g z s ) reg + z n ) = J ,establishing (3.5). Corollary 2.21 then givesdim J M , i = dim M ◦ − dim m z + dim z(m z s ) = dim M ◦ − dim g z + dim z(g z s ) ,which is independent of i ∈ I J . This proves (iv).Finally, we prove (v). By construction, z(m) reg + [ n i ∈ N G (m) /M ◦ n i · O M ◦ y n = U m ∩ (cid:0) z(m) reg + N G (m) · y n (cid:1) = U m ∩ (cid:0) N G (m) · (z(m) reg + y n ) (cid:1) ⊂ U m ∩ J G ( y ) .Conversely, let z ∈ J G ( y ) ∩ U m . Then, there is g ∈ G such that g z s = g · m and z n = g · y n .Saturation of U m gives g z s ⊂ m , hence g · m ⊂ m and z ∈ N G (m) · (z(m) reg + y n ) = z(m) reg + N G (m) · y n = z(m) reg + [ n i ∈ N G (m) /M ◦ n i · O M ◦ y n ,and the proof is completed. (cid:3) Let H and L be reductive algebraic groups acting on an affine variety X , with H ⊂ L .Then H acts with trivial stabilizers on the product L × X via h · ( l , x ) = ( lh − , h · x ) : we set L × H X := ( L × X ) /H ∼ = ( L × X ) //H and note that L acts on L × H X by multiplication from theleft. The class of ( l , x ) ∈ L × X will be denoted by the symbol l ∗ x ∈ L × H X .We consider the natural action maps˜ µ : G × m → g , ˜ µ : G × m → V , (3.6)and the induced maps µ : G × M m → g and µ : G × M m → V . We will also consider theGIT quotient maps π : G × M m → (cid:0) G × M m (cid:1) //G , π : G × M m → (cid:0) G × M m (cid:1) //G , (3.7)associated to multiplication from the left by G and G , respectively. We will invoke a variantof Luna’s ´etale slice Theorem [15] and its consequences to deduce properties of the closureof G -Jordan classes. Lemma 3.6.
Let y = y s + y n for y n ∈ N V ∩ m . Then there exist: ( i ) an affine open neighbourhood U of y s in m , which is M -saturated and such that its intersection U = U ∩ V with V is contained in U m . For any G -Jordan class J meeting U , we have J ∩ U = [ i ∈ I J ( J M , i ∩ U ) , (3.8) where { J M , i | i ∈ I J } is the (finite) set of M ◦ -Jordan classes in m such that J M , i ∩ U ∩ J = ∅ ; ( ii ) an affine M ◦ -stable open neighbourhood U ′ of y s in m such that M · U ′ ⊂ U and J G ( y ) ∩ U ′ = J M ◦ ( y ) ∩ U ′ . (3.9) Proof.
The differential of the map ˜ µ at ( y s ) maps any element ( x ′ , y ′ ) ∈ g ⊕ m to [ x ′ , y s ] + y ′ ,therefore it is surjective by (2.3) and since [g , y s ] ∩ m = y s .Therefore, the differential of the induced map µ at 1 ∗ y s is also surjective, hence it is anisomorphism by dimensional reasons. The orbit O G ∗ y s is closed and so is the semisimpleorbit O Gy s . It is not hard to verify that the restriction of µ to O G ∗ y s is injective.By [15, Lemme Fondamental, § II.2] applied to X = G × M m and Y = g there exists an affine π -saturated open neighbourhood of 1 ∗ y s in G × M m such that the restriction of µ to it is´etale and the image is an affine open subset of g , saturated for p : g → g //G . In fact, being G -stable, this open neighbourhood is of the form G × M U for an affine open neighbourhood U of y s in m , which is M -saturated. It is then easy to see that U = U ∩ V is an M -saturatedaffine open neighbourhood of y s in m . Let z = z s + z n ∈ U . By saturation z s ∈ U . As µ is G -equivariant and ´etale, G z s = ( G z s ) ◦ = ( G ∗ z s ) ◦ . By construction G ∗ z s = M z s , so U ⊂ U m .Then, (3.8) follows from Lemma 3.5 (iv).We prove (ii). Lemma 3.5 (v) and (iii) give J G ( y ) ∩ U = (cid:0) z(m) reg + [ n i ∈ N G (m) /M ◦ n i · O M ◦ y n (cid:1) ∩ U = (cid:0) z(m) + [ n i ∈ N G (m) /M ◦ n i · O M ◦ y n (cid:1) ∩ U ,since z(m) reg , M = z(m) . The orbits n i · O M ◦ y n are finitely-many and of the same dimension,so we may replace U by a smaller M ◦ -stable Zariski open neighbourhood U ′ of y s in m toensure that J G ( y ) ∩ U ′ = (z(m) + O M ◦ y n ) ∩ U ′ = J M ◦ ( y ) ∩ U ′ . Finally M · U ′ ⊂ M · U = U since U is M -stable. (cid:3) Lemma 3.7.
Let y = y s + y n for y n ∈ N V ∩ m . Then there exist:(i) an affine open neighbourhood U of y s in m , which is M -saturated and such that the restric-tion of µ to G × M U is ´etale with Zariski open image G · U in V ;(ii) an M -stable analytic open neighbourhood V of y s in m such that the restriction of µ to G × M V is an analytic diffeomorphism with analytic open image G · V in V .Proof. The restriction of the differential of the map ˜ µ at ( y s ) to the degree 1 terms readilyimplies surjectivity of the differential of ˜ µ at ( y s ) , whence the differential of µ at 1 ∗ y s isbijective. As before, the remaining hypotheses of [15, Lemme Fondamental, § II.2] are easilyverified for X = G × M m and Y = V and give the existence of U . As observed in [15, § III.1,Remarques 3 ◦ ], U may be further reduced to an M -stable analytic open neighborhood V sothat the restriction of µ to G × M V is an analytic diffeomorphism with open image. (cid:3) Proposition 3.8.
Let J be a G -Jordan class in V . Then J is a union of G -Jordan classes and it isdecomposable, i.e., it contains the semisimple and nilpotent components of all its elements.Proof. We will show that condition ( ii ) in Lemma 3.4 is satisfied for any y = y s + y n ∈ J .Let U , U ′ be as in Lemma 3.6 and V as in Lemma 3.7 (ii). We consider the M -stable opensubset U ′′ = M · U ′ ⊂ U of m and apply M to both sides of (3.9) to get J G ( y ) ∩ U ′′ = M · (cid:0) J G ( y ) ∩ U ′ (cid:1) = M · (cid:0) J M ◦ ( y ) ∩ U ′ (cid:1) ⊂ (cid:0) M · J M ◦ ( y ) (cid:1) ∩ U ′′ . (3.10)We then intersect U ′′ with V and obtain an M -stable analytic open neighbourhood of y s in m . For simplicity of exposition, we still denote this intersection by V and note that therestriction of µ to G × M V is a diffeomorphism with analytic open image G · V in V .Since Jordan classes are locally closed (in the Zariski topology), their Zariski and analyticclosures coincide, and all closures in the sequel are meant in the analytic topology. As aconsequence, y s ∈ J by Lemma 3.2, so J ∩ G · V = ∅ and J ∩ V = ∅ . Then J ∩ G · V = J ∩ G · V G · V ∼ = G × M ( J ∩ V V )= G × M [ i ∈ I J ( J M , i ∩ V ) V = G × M [ i ∈ I J ( J M , i ∩ V ) , (3.11)where the first and last equalities follow from elementary topology, the second from the ana-lytic diffeomorphism and the bundle structure of G × M V and the third from (3.8) appliedto J = J and followed by restriction to V ⊂ U .As y s ∈ O M ◦ y = y s + O M ◦ y n , ORDAN CLASSES FOR θ -GROUPS 13 any M ◦ -stable neighbourhood of y s in m meets y , so y ∈ J ∩ V . We then have y ∈ J M , l ∩ V for l ∈ I J by (3.11). Now y s ∈ z(m) , so combining (3.10), Remark 2.17 (2) and Lemma 3.2(iv) yields J G ( y ) ∩ V ⊂ (cid:0) M · J M ◦ ( y ) (cid:1) ∩ V = (cid:16) M · (cid:0) z(m) + O M ◦ y n (cid:1)(cid:17) ∩ V ⊂ (cid:16) M · J M , l (cid:17) ∩ V = M · (cid:16) J M , l ∩ V (cid:17) ⊂ J ∩ V ,where in the last step we again used (3.11). Arguing as we did for (3.11) we finally arrive at J G ( y ) ∩ G · V ∼ = G × M ( J G ( y ) ∩ V ) ⊂ G × M ( J ∩ V ) ∼ = G · ( J ∩ V ) ⊂ J , so J G ( y ) ∩ G · V is thesought open neighbourhood of J G ( y ) . This proves that J is the union of G -Jordan classes.We finally prove that J is decomposable. Let y = y s + y n ∈ J and J G ( y ) the corresponding G -Jordan class. Then y s ∈ J by Lemma 3.2 (i) and y n ∈ z(m) + y n = z(m) reg + y n ⊂ J G ( y ) ⊂ J ,where we used our previous result J G ( y ) ⊂ J . (cid:3) Theorem 3.9.
Let J be a G -Jordan class and let S be a sheet in V . Then J • , J reg and S are unions of G -Jordan classes.Proof. By Proposition 3.8, the closure J is a union of G -Jordan classes. Since all such classesare of constant G - and G -orbit dimension, it follows that also J • and J reg are unions of G -Jordan classes. The statement for S is a direct consequence of Proposition 2.22. (cid:3) We conclude this subsection with the following important consequence of the local studyof the closure of a G -Jordan class. Proposition 3.10. G -Jordan classes are smooth.Proof. Let J = J G ( y ) be a G -Jordan class in V and m = g y s . We will show that y has a smoothZariski open neighbourhood in J . Let U and U be the M -saturated open neighbourhoods of y s in m as in Lemma 3.6 and Lemma 3.7, respectively. By construction y , y s ∈ U ∩ U ⊂ U m .By Lemma 3.5 (v), the intersection J ∩ U m is smooth, therefore J ∩ U ∩ U is non-emptyand smooth as well. Since M acts on G × J with trivial stabilizer, p : G × J → G × M J is aprincipal M -bundle [15, III.1, Corollaire 1]. In other words, there is a surjective ´etale map f : Y → G × M J such that the base change X → Y of G × J → G × M J is isomorphic to theprojection ˜ p : M × Y → Y . Being the base change of an ´etale and smooth map, the inducedmorphism ˜ f : M × Y → G × J is again so. By [1, ´Exp 1, Corollaire 9.2], G × ( J ∩ U ∩ U ) issmooth if and only if ˜ f ˜ p − f − ( G × M ( J ∩ U ∩ U )) = p − (cid:0) G × M ( J ∩ U ∩ U ) (cid:1) is so. One mayverify that the scheme-theoretic fiber of G × M ( J ∩ U ∩ U ) through p is G × ( J ∩ U ∩ U ) hence G × M ( J ∩ U ∩ U ) is smooth. Invoking again [1, ´Exp 1, Corollaire 9.2] we concludethat µ (cid:0) G × M ( J ∩ U ∩ U ) (cid:1) is smooth and it is a smooth open neighbourhood of y in J . (cid:3) Regularity questions.
Let J = J G ( x s + x n ) be a G -Jordan class. Then J reg ⊂ J • since J is irreducible, hence J too, and Lemma 2.14 is in force. Note that J • = J reg whenever x s = J = z(g) × O G x n and orbits are locally closed, so J = J • = J reg . The equality J • = J reg is always satisfied in the symmetric case m = J • = J reg also for m >
3, by combining Theorem 3.9 and the fact that G -Jordan classes aredefined in terms of regular parts for the action of G , cf. Corollary 2.21.However, this is not the case. The reason is that open G -orbits O G in irreducible compon-ents of the fibers of the Steinberg map ϕ : V → V//G ∼ = c /W Vin do not give rise in general toopen G -orbits G · O G in the irreducible components of the Steinberg map p : g → g //G ∼ = h /W .To make this more precise, we need some notions and results from [20, 21] and, for simplicityof exposition, we restrict to the case where g is semisimple. Definition 3.11.
A complex semisimple Z m -graded Lie algebra { g , θ } is called:(i) S -regular if S V ∩ g reg = ∅ ; (ii) N -regular if N V ∩ g reg = ∅ ;(iii) very N -regular if each irreducible component of N V intersects g reg non-trivially.Clearly (iii) implies (ii). It is an important result of L. V. Antonyan and D. I. Panyushevin [20] that if a connected component of Aut (g) contains automorphisms of order m , then itcontains a unique N -regular automorphism of that order (up to conjugation by the group ofinner automorphisms of g ). Moreover, as mentioned in the introduction of [20], the condi-tion of S -regularity is equivalent to N -regularity in the symmetric case m =
2, but for m > S -regular grading that is not N -regular is given in [20, Example 4.5]. Here g is of type E with the inner automorphism oforder m = ✉ ❡ ❡ ❡ ✉✉❡ (3.12)This is the affine Dynkin diagram of g of type E , where the white and black nodes corres-pond to roots subspaces of degree 0 and 1, respectively. The semisimple part of g is givenby the subdiagram consisting of white nodes and the dimension of the centre of g is thenumber of black nodes minus 1. We have G ∼ = SL ( ) × SL ( ) × ( C x ) up to local isomorph-ism, acting on V = g ∼ = C ⊕ ( C ) ∗ ⊕ ( Λ C ⊠ C ) . The reader is referred to e.g. [33, Chapter 3, §
3] for a detailed treatment of periodic automorphisms and their associated Kac diagrams.Now G -Jordan classes form a finite partition of V , which is irreducible, so there is oneclass J that is open in V . We call it the G -regular Jordan class of V and note that it is theunique G -Jordan class that is dense in the G -regular sheet S = V • of V . (See Example 2.16for an explicit description of representatives of the G -orbits in the G -regular Jordan class.)Since the grading (3.12) is S -regular, we have J reg = V reg = g reg ∩ V in this case. Let O G be the nilpotent G -orbit that is open in one of the irreducible components of N V . We have O G ⊂ J • = V • by [31, Corollaries 1 and 2], but O G J reg since the grading is not N -regular.The cone N V is often reducible and a larger class of examples for which J reg = J • comesfrom N -regular gradings that are not very N -regular: the G -regular Jordan class J satisfies J reg = g reg ∩ V and, by an argument as above, there is a nilpotent G -orbit contained in J • but not in J reg . Exceptional N -regular gradings whose nodes are not all black are classifiedin [8], and very N -regular gradings appear to occur very rarely. Inner exceptional gradingswith all nodes black are N -regular but not very N -regular [20, Example 4.4] and the same istrue for the outer grading of E with all nodes black (W. A. de Graaf, 05-05-2020, personalcommunication). The following result is a consequence of these observations, and the tablesare a specialization of Tables 2-7 of [8]. Proposition 3.12.
Let { g , θ , m } be an exceptional complex simple Z m -graded Lie algebra, m > .Then { g , θ , m } is N -regular but not very N -regular if and only if the associated Kac diagram has allthe nodes black or is one in the following tables. In all these cases we have that J reg ⊂ J • properly,where J is the G -regular Jordan class of V . Table N -regular but not very N -regular automorphisms of G . m Kac diagram N V N V dim N V dim c ✉ ✉ ❡ > Table 2. N -regular but not very N -regular automorphisms of F . m Kac diagram N V N V dim N V dim c ✉ ❡ ✉ ❡ ❡ >
29 3 12 2
ORDAN CLASSES FOR θ -GROUPS 15 N -regular but not very N -regular inner automorphisms of F .6 ✉ ❡ ✉ ❡ ✉ >
35 6 8 28 ✉ ✉ ✉ ❡ ✉ >
30 4 6 1
Table 3. N -regular but not very N -regular inner automorphisms of E . m Kac diagram N V N V dim N V dim c ✉ ❡ ✉ ❡ ❡❡❡
43 3 18 26 ✉ ❡ ✉ ❡ ✉❡✉
133 9 12 28 ✉ ❡ ✉ ✉ ✉❡✉
70 4 9 19 ✉ ✉ ❡ ✉ ✉✉✉
118 6 8 1
Table 4. N -regular but not very N -regular outer automorphisms of E . m Kac diagram N V N V dim N V dim c ✉ ❡ ❡ ✉ ❡ <
34 5 12 38 ✉ ❡ ❡ ✉ ✉ <
22 3 9 110 ✉ ✉ ❡ ✉ ❡ <
25 2 8 112 ✉ ✉ ❡ ✉ ✉ <
30 4 6 1
Table 5. N -regular but not very N -regular automorphisms of E . m Kac diagram N V N V dim N V dim c ✉ ❡ ❡ ✉ ❡❡ ❡ ✉
233 10 21 37 ✉ ❡ ❡ ✉ ❡❡ ✉ ❡
112 3 18 18 ✉ ❡ ❡ ✉ ❡❡ ✉ ✉
163 2 17 19 ✉ ✉ ❡ ✉ ❡❡ ✉ ❡
132 4 14 1 N -regular but not very N -regular automorphisms of E .10 ✉ ❡ ✉ ❡ ✉✉ ❡ ✉
199 4 13 112 ✉ ❡ ✉ ❡ ✉✉ ✉ ✉
217 5 11 114 ✉ ✉ ✉ ❡ ✉✉ ✉ ✉
238 7 9 1
Table 6. N -regular but not very N -regular automorphisms of E . m Kac diagram N V N V dim N V dim c ❡ ❡ ❡ ❡ ✉❡ ❡ ❡ ❡
144 2 60 46 ❡ ❡ ❡ ✉ ❡❡ ❡ ❡ ✉
270 7 40 48 ❡ ❡ ✉ ❡ ❡❡ ❡ ✉ ❡
219 2 30 29 ❡ ❡ ✉ ❡ ❡❡ ❡ ✉ ✉
206 2 28 110 ❡ ❡ ✉ ❡ ❡❡ ✉ ❡ ✉
300 7 24 212 ✉ ❡ ✉ ❡ ❡❡ ✉ ❡ ✉
398 10 20 214 ✉ ❡ ✉ ❡ ❡❡ ✉ ✉ ✉
333 4 18 115 ✉ ❡ ✉ ❡ ✉❡ ❡ ✉ ✉
354 5 16 118 ✉ ✉ ❡ ✉ ❡✉ ✉ ❡ ✉
397 5 14 120 ✉ ✉ ❡ ✉ ❡✉ ✉ ✉ ✉
438 7 12 124 ✉ ✉ ❡ ✉ ✉✉ ✉ ✉ ✉
478 8 10 1
ORDAN CLASSES FOR θ -GROUPS 17 Slice-induction and parametrization of orbits and classes
Slice-induction.
Proposition 3.8 shows that the closure of a G -Jordan class in V is aunion of G -Jordan classes, generalising results of [4, 7]. We aim at detecting which G -Jordan classes lie in the closure of a given one. In the classical case, this can be described interms of Lusztig-Spaltenstein’s parabolic induction of adjoint orbits, [16, 3]. Slice inductionis introduced in [7] to deal with the m = m the construction in [7], by combiningsome of its general arguments with our local approach.Let m be a θ -stable reductive subalgebra of g and M the connected subgroup of G withLie ( M ) = m . For a nilpotent element e ∈ m we consider a graded sl( ) -triple { e , h , f } in m ,so that h ∈ m and f ∈ m − , and the corresponding Slodowy slice S m , e = e + m f ⊂ m . Since m f is homogeneous, we can consider its intersection with V , obtaining S m , e = e + m f ⊂ m .If e =
0, we consider the trivial triple as an sl( ) -triple, so S m ,0 = m . We start with twopreliminary results in the case m = g . Lemma 4.1.
Let { e , h , f } be a graded sl( ) -triple in g and let X ⊂ V be an irreducible locally closed G -stable subset such that X ∩ S g , e = ∅ . Then the action morphism ψ : G × S g , e → V is smooth, itsrestriction ψ X : G × ( S g , e ∩ X ) → X is smooth and dominant, more precisely ψ X ( G × C ) is densein X for any irreducible component C of S g , e ∩ X .Proof. For m =
2, this is part of [7, Proposition 2.4 (i)], we record the proof for completeness.The action morphism ψ is G -equivariant with smooth domain and codomain, hence it suf-fices to verify that the differential is surjective at any point of the form ( y ) ∈ G × S g , e . Wenote that dψ | ( y ) : g × g f → V ( x , z ) → [ x , y ] + z and by sl( ) -representation theory g = [g , e ] ⊕ g f , which in degree 1 becomes V = [g , e ] ⊕ g f , so the differential at ( e ) is surjective. The contracting C ∗ -action argument in [27, 7.4,Corollary 1] carries over to the Z m -graded case because { e , h , f } is a graded sl( ) -triple and h ∈ m , so ψ is smooth at any point ( y ) , hence everywhere. Thus, the dimension of anynon-empty fiber F of ψ is dim g + dim ( S g , e ) − dim V .The restriction ψ X is again smooth, by [27, III.5, Lemma 2] applied to the G -equivariantmorphism given by the inclusion of X in V . We now prove that ψ X ( G × C ) is dense in X for any irreducible component C of S g , e ∩ X , from which the dominance of ψ X follows. Thedensity condition is obtained by comparing the estimate dim C > dim X + dim ( S g , e ) − dim V from dimension properties of intersections with the estimatedim ( G · C X ) > dim G + dim C − dim F coming from smoothness. It follows that dim ( G · C X ) > dim X , hence the claim. (cid:3) Lemma 4.2.
Let J be a G -Jordan class in V and e ∈ N V . Then e ∈ J if and only if J ∩ S g , e = ∅ ifand only if J ∩ S g , e = ∅ .Proof. We note that J is a locally closed G -stable cone by Proposition 2.20 and Corollary 2.21,so when m = ( i ) = ( iv ) = ( v ) in [7, Theorem 2.6]. The proof of[7, Lemma 2.3] shows the existence of a contracting C ∗ -action on S g , e and it carries over tothe m > J ∩ S g , e = ∅ then each irreducible component of J ∩ S g , e is non-empty andstable under the C ∗ -action, so e lies in each of them. As a consequence, e ∈ J .Clearly e ∈ J gives J ∩ S g , e = ∅ , so it remains to show that J ∩ S g , e = ∅ implies J ∩ S g , e = ∅ .We follow the proof of [7, Proposition 2.5], establishing that J ∩ S g , e is dense in J ∩ S g , e .Since J is open in J , the subset J ∩ S g , e is open in J ∩ S g , e and therefore it is enough toprove that it meets every irreducible component C of J ∩ S g , e . The latter follows then fromthe density of G · C in J , guaranteed by Lemma 4.1 applied to X = J . (cid:3) Theorem 4.3.
Let J , J be G -Jordan classes in V . Then the following conditions are equivalent: ( i ) J ⊂ J ; ( ii ) J ∩ J = ∅ ; ( iii ) There exist x ∈ J , y ∈ J such that g x s ⊂ m and J M ◦ ( x ) ∩ S m , y n = ∅ , where m = g y s and M ◦ is the identity component of M = G y s = G ∩ G y s ; ( iv ) There exist x ∈ J , y ∈ J such that g x s ⊂ m and y ∈ J M ◦ ( x ) , where m and M ◦ are as in ( iii ) .Proof. This is the generalization of [7, Theorem 3.5] to the m > ( i ) ⇔ ( ii ) is immediate from Proposition 3.8. We prove the other ones. Claim ( iii ) ⇔ ( iv ) . Lemma 4.2 applied to m , y n and J M ◦ ( x ) says that J M ◦ ( x ) ∩ S m , y n = ∅ ifand only if y n ∈ J M ◦ ( x ) . Since y s ∈ z(m) , the latter condition is equivalent to y ∈ J M ◦ ( x ) byLemma 3.2 ( iv ) . Claim ( iv ) ⇒ ( ii ) . Let x , y be as in ( iv ) . Since g x ⊂ g x s , we have x ∈ U m and hence J G ( x ) ∩ U m ∩ J M ◦ ( x ) = ∅ . Lemma 3.5 ( iv ) gives J G ( x ) ∩ U m = [ i ∈ I J M , i ∩ U m (4.1)and J M ◦ ( x ) is, by construction, one of the M ◦ -Jordan classes occurring in the R.H.S. Let V beas in Lemma 3.7. Without loss of generality assume that V ⊂ U m . Then J G ( x ) ∩ G · V ∼ = G × M [ i ∈ I ( J M , i ∩ V ) , (4.2)arguing as we did for (3.11). We also recall that y s ∈ O M ◦ y , so y ∈ V .By hypothesis y ∈ J M ◦ ( x ) so (4.2) gives y ∈ J G ( x ) , therefore J ∩ J = ∅ . Claim ( ii ) ⇒ ( iv ) . Assume now y ∈ J ∩ J . Then (4.2) gives y ∈ S i ∈ I ( J M , i ∩ V ) ⊂ S i ∈ I ( J M , i ∩ U m ) , so that y ∈ J M , i for some i ∈ I . Let e x be a representative of J M , i ∩ U m , which is also arepresentative of J G ( x ) due to (4.1). We have y ∈ J M ◦ ( e x ) by construction and g e x s ⊂ m since e x ∈ U m and U m is M -saturated. In summary, the points y ∈ J and e x ∈ J satisfy ( iv ) . (cid:3) Comparing dimensions of orbits in J and J we readily get: Corollary 4.4.
Let J , J be G -Jordan classes in V . Then J ⊂ J • if and only if there exist x ∈ J , y ∈ J such that g x s ⊂ m , J M ◦ ( x ) ∩ S m , y n = ∅ and dim O M x = dim O M y n . Remark 4.5.
Condition ( iii ) from Theorem 4.3 is called weak slice-induction in [7]. If J isweakly slice-induced from J and satisfies the dimension condition in Corollary 4.4, then it iscalled slice-induced from J . Slice-induction is shown to coincide with parabolic inductionin the ungraded case m = Corollary 4.6. A G -Jordan class J = J G ( y ) contained in V ( d ) is dense in a sheet if and only if J M ◦ ( x ) ∩ S m , y n = ∅ for any x ∈ V ( d ) \ J such that g x s ⊂ m .Proof. First of all, the irreducible subset J is contained in some sheet S in V ( d ) and there isa unique G -Jordan class J ′ ⊂ V ( d ) such that S = J ′• by Proposition 2.22. The condition J M ◦ ( x ) ∩ S m , y n = ∅ for any x ∈ V ( d ) \ J such that g x s ⊂ m is equivalent to say that there areno G -Jordan classes J = J such that J ⊂ J • , in other words, that J = J ′ . (cid:3) Parametrization of orbits and classes.
We aim at a parametrization of the G -orbitscontained in a G -Jordan class J G ( x ) = G · (z(g x s ) reg + x n ) . By Theorem 2.6, we may assumethat x = x s + x n ∈ V with x s ∈ c , so Corollary 2.8 ensures that z(g x s ) ⊂ c . Let Γ := N W Vin (z(g x s ) ) ,the stabilizer of z(g x s ) in W Vin . ORDAN CLASSES FOR θ -GROUPS 19 Remark 4.7. (1) Observe that x s ∈ c implies Z G (c) ⊂ G x s ⊂ N G (g x s ) . Corollary 2.3gives also N G (g x s ) = N G (z(g x s ) ) = N G (z(g x s )) , so Γ ∼ = (cid:0) N G (c) ∩ N G (g x s ) (cid:1) /Z G (c) .In other words, if w ∈ Γ , then any of its representatives ˙ w ∈ N G (c) lies in N G (g x s ) .(2) The group N G (c) ∩ N G (g x s ) normalizes G x s and g x s and thus acts on the set of G x s -orbits in g x s . Since Z G (c) ⊂ G x s , this action factors through an action of Γ on the setof G x s -orbits in g x s which preserves the set of nilpotent ones.We shall need the stabilizer Γ n in Γ of O G xs x n with respect to the action defined above: Γ n = Stab Γ ( O G xs x n ) . Proposition 4.8.
Let x = x s + x n ∈ V with x s ∈ c . The assignment ϕ from z(g x s ) reg to the orbitset J G ( x ) /G given by y s O G ( y s + x n ) induces a homeomorphism ϕ : z(g x s ) reg /Γ n −→ J G ( x ) /G ,where the orbit set is endowed with the quotient topology.Proof. The map ϕ is well-defined and surjective by Proposition 2.19. We prove injectivity.Let y s , z s ∈ z(g x s ) reg be such that g · ( y s + x n ) = z s + x n for some g ∈ G , i.e., g · y s = z s , (4.3) g · x n = x n , (4.4)and consider w ∈ W Vin such that w · y s = z s , cf. Theorem 2.9. Any representative ˙ w ∈ N G (c) of w satisfies ˙ w · g x s = ˙ w · g y s = g z s = g x s , so w ∈ Γ by Remark 4.7. Moreover,˙ wg − ∈ G z s ∩ G = G x s by (4.3). It follows from (4.4) that˙ w · x n ∈ O G xs x n so ˙ w · O G xs x n = O G xs x n ,in other words w ∈ Γ n and ϕ is injective.Let p : J G ( x ) → J G ( x ) /G be the quotient map and U an open subset in J G ( x ) /G . Then p − ( U ) is a G -stable open subset in J G ( x ) and its intersection p − ( U ) ∩ (cid:0) z(g x s ) reg × O G xs x n (cid:1) is an open Γ n -stable subset of z(g x s ) reg × O G xs x n . Its projection onto z(g x s ) reg is again an open Γ n -stable subset, and so is its image through the quotient map by the finite group Γ n . Wehave therefore proved that ϕ is a continuous bijection, and it remains to show that is open.By Corollary 2.21 and Proposition 3.10, the action morphism G × (z(g x s ) reg + x n ) → J G ( x ) is a morphism of smooth varieties whose induced map on the tangent spaces is surjective.Hence it is smooth, and an open morphism in the Zariski topology (see [2, VII, Remark 1.2]and [2, V, Theorem 5.1 and VII, Theorem 1.8]). From this, it is straightforward to see that ϕ is open. (cid:3) We briefly turn to the parametrization of G -Jordan classes. Thanks to Theorem 2.9 andExample 2.15 describing the centralizer of an element of c , we easily establish the following. Lemma 4.9.
Let x s and y s be two elements in c . Then the centralizers g x s and g y s are G -conjugateif and only if there exists w ∈ W Vin such that w · Σ ( x s ) = Σ ( y s ) . The hyperplane arrangement on c determined by the restricted roots σ ∈ Σ admits anaction of W Vin and it induces a stratification on c , where two elements lie in the same stratum z(g x s ) reg = { y s ∈ c | Σ ( y s ) = Σ ( x s ) } if and only if their centralizers coincide. Equivalently, thestratum associated to a closed and symmetric subset e Σ ⊂ Σ (as in of [33, pag. 182]) is S e Σ = (cid:10) x ∈ c | Σ ( x ) = e Σ (cid:11) and the collection of S e Σ ’s is a finite partition of c . Already in the ungraded case, where theclass of centralizers of semisimple elements coincides with the class of Levi subalgebras, notall closed and symmetric subsets e Σ of Σ give rise to a non-empty stratum. In the graded case,some information on stabilizers of generic elements in S V is to be found in [31] under the assumption that V is a simple G -module. We refer to [23, Proposition 3.4] for an alternativegeneral description of centralizers of semisimple elements. In view of Lemma 4.9, two strata S e Σ and S e Σ ′ are equivalent if w · e Σ = e Σ ′ for some w ∈ W Vin . Given e Σ ⊂ Σ , we set m( e Σ ) to be the θ -stable Levi subalgebra of g constructed as in (2.6) and M ( e Σ ) ⊂ G , M ( e Σ ) = M ( e Σ ) ∩ G ⊂ G as usual. Proposition 4.10.
Jordan classes in V are in one-to-one correspondence with W Vin -classes of pairs ( e Σ , O ) where e Σ ⊂ Σ satisfies S e Σ = ∅ and O is a nilpotent orbit in m( e Σ ) for the action of M ( e Σ ) .Proof. Observe that N G (c) acts on the set of pairs ( e Σ , O ) as above and that if m( e Σ ) is thecentralizer of some x s ∈ c , then Z G (c) ⊂ M ( e Σ ) , hence it acts trivially on ( e Σ , O ) . Thus theaction of N G (c) induces an action of W Vin .Now recall that, for x ∈ V the assignment J G ( x ) (g x s , O G xs x n ) establishes a one-to one-correspondence between G -Jordan classes in V and G -classes of pairs (l , O ) where l is thestabilizer of a semisimple element in V and O a nilpotent orbit in l for the action of L .Theorem 2.6 guarantees that we can always find a pair in the G -orbit where l = m( e Σ ) forsome e Σ ⊂ Σ . Assume that for two pairs (m( e Σ ) , O ) and (m( e Σ ′ ) , O ′ ) of this form there is g ∈ G such that ( g · m( e Σ ) , g · O ) = (m( e Σ ′ ) , O ′ ) . By Lemma 4.9 we can decompose g = g ′ ˙ w ,where g ′ ∈ N G (m( e Σ ′ )) and ˙ w ∈ N G (c) . In addition, g ′ = l ˙ σ with l ∈ M ( e Σ ′ ) and ˙ σ ∈ N G (m( e Σ ′ )) ∩ N G (c) . In other words, we may replace g by an element in N G (c) , so (m( e Σ ) , O ) and (m( e Σ ′ ) , O ′ ) lie in the same W Vin -orbit. (cid:3)
The results of [34] encompass a parametrization of the G -Jordan classes, where θ is theautomorphism of order m = g = E for which g = Λ C , g = sl( ) and g − = Λ ( C ) ∗ as in Example 2.12. This is shown in the following: Example 4.11.
By the discussion in [34, § §
1] para-metrize the Levi subalgebras l = g x s that arise from elements x s ∈ S V up to G -conjugation,and the “classes” in Tables 1-6 of [34, §
1] parametrize the nilpotent orbits in l for the actionof G x s . (If x s is in family I then g x s = h , there is no non-trivial nilpotent orbit and only oneclass.) By Proposition 4.10, our G -Jordan classes almost coincide with the classes of [34]:the finite group N G (l) /G x s acts on the set of nilpotent G x s -orbits in l , possibly glueing someof them.Hence, some of the 164 classes of [34] may correspond to the same G -Jordan class. A lookat Tables 1-6 tells us that this may happen only in a few cases, since centralizers of elementsof a G -Jordan class are G -conjugate by Proposition 2.20 and N G (l) /G x s = in the VIIfamily: III family : Classes 2-3, 4-6, and 7-8;
V family : Classes 7-8, and 10-11;
VI family : Classes 5-6, 8-9, 11-12, and 17-18.Recall that the support of a trivector ϕ ∈ Λ C is the unique minimal subspace E ⊂ C such that ϕ ∈ Λ E . Its dimension is the rank of ϕ , one of the simplest discrete G -invariants ofa trivector. The nilpotent G x s -orbits associated to the classes 7-8 in V family have differentrank, so they are not G -related. Thus, they correspond to different G -Jordan classes. Asimilar observation works in all the remaining cases, except those of the III family and theclasses 5-6 of VI family, but it is not difficult to see that the nilpotent G x s -orbits of these lasttwo classes are not G -related. It remains therefore to deal with the III family.First of all, the rank of the nilpotent orbit in class 4 is strictly smaller than the rank of thosein classes 5 and 6. However, the permutation matrix g = − Id × × × (4.5) ORDAN CLASSES FOR θ -GROUPS 21 is an element of N G (l) and it does relate the nilpotent G x s -orbits associated to classes 5-6,which then correspond to a single G -Jordan class. The same is true for classes 2-3 and 7-8.In summary, the space Λ C is partitioned into 161 G -Jordan classes.The quotient Γ/W x s of Γ with the stabilizer W x s of x s ∈ c in W Vin was found in [34, § G -Jordan class III.5, represented by x = x s + x n . A simple check shows that g as in (4.5)normalizes also c , so g ∈ N G (z(g x s )) ∩ N G (c) and, by our previous discussion, it is not in Γ n .The G -orbits in the G -Jordan class III.5 are then parametrized by the quotient z(g x s ) reg /Γ n of z(g x s ) reg by a group Γ n of order 36.We conclude with an application of Theorem 4.3. Let J = J G ( y ) be the G -Jordan classnumbered III.7, i.e., the one with the representative y = y s + y n given by y s = (cid:0) e + e + e (cid:1) + i (cid:0) e + e + e (cid:1) , y n = e . (4.6)The centralizer m = g y s is a reductive Lie algebra with semisimple part r of type A ⊕ A .More precisely, the center of m is 4-dimensional and sits in degrees ±
1: it consists of thetwo components in brackets that defines y s in (4.6) and of their duals. The semisimple part r = r − ⊕ r ⊕ r is graded as follows [34, § r = span { e , e , e } ⊕ span { e , e , e } , r = span { d , d , d } ⊕ span { d , d , d } , r − = span { e , e , e } ⊕ span { e , e , e } , (4.7)where e i , for 1 i
9, is the dual basis of ( C ) ∗ , e ijl := e i ∧ e j ∧ e l and the elements d ijk = [ e ijk , e ijk ] satisfy d + d + d = d + d + d =
0. The direct sums of vectorspaces in (4.7) correspond to the Lie algebra decomposition of r .Let J = J G ( x ) be any of the G -Jordan classes in the II family, i.e., one of II.1, II.2 or II.3.The choice of representative x = x s + x n given by x s = y s + (cid:0) e + e + e (cid:1) , x n = e + e for II.1 , e for II.2 ,0 for II.3 , (4.8)easily allows to check that J ⊂ J . First of all z(g x s ) is generated by the 3 vectors in bracketsin (4.6) and (4.8), hence y s ∈ z(g x s ) and g x s ⊂ m . A graded sl( ) -triple { e , h , f } in m with e = y n is provided by f = e and h = d , and the required Slodowy slice S m , e = e + m f is the affine subspace in m modeled on m f = span { e , e } ⊕ span { e , e , e } ⊕ z(m) .It is evident that x ∈ S m , e , so J ⊂ J thanks to Theorem 4.3 (iii). Appendix A. Cartan, Levi and parabolic subalgebras in Z m -graded Lie algebras Let { g , θ } be a reductive Z m -graded Lie algebra and c ⊂ V a fixed Cartan subspace. Theexistence of a homogeneous Cartan subalgebra h of g containing c is a result probably knownto experts by a long time; the proof in [24, § g simple, but its proof carriesover for any reductive g . Proposition A.1.
There exists a homogeneous Cartan subalgebra h = L l ∈ Z m h l of g that satisfies h ⊃ z(c g (c)) and h = c . Remark A.2.
By [31, § c is not an algebraic subalgebra in general,unless m
2. On the other hand h and z(c g (c)) are algebraic, hence h ⊃ z(c g (c)) ⊃ c , where c is the algebraic closure of c . It is clear that h = z(c g (c)) if and only if [c g (c) , c g (c)] = z(c g (c)) = c . We will call adapted any Cartan subalgebra h of g as in Proposition A.1. For such an h , let g = h ⊕ M α ∈ Φ g α (A.1)be the root space decomposition of g with respect to h , with associated set of roots Φ ⊂ h ∗ .The automorphism θ : g → g permutes the root spaces in (A.1): Lemma A.3.
For any α ∈ Φ , we have α ◦ θ ∈ Φ and θ − (g α ) = g α ◦ θ . We note that any root α ∈ Φ can be decomposed as α = α + α + · · · + α m − + α m − ,where α l = α | h l for any l ∈ Z m . Repeatedly applying Lemma A.3, we see that α ◦ θ l = α + ω l α + · · · + ( ω l ) m − α m − + ( ω l ) m − α m − is a root too, for any l ∈ Z m . In other words, we may consider the equivalence class of rootsgiven by [ α ] = (cid:8) α ◦ θ l | l ∈ Z m (cid:9) for any α ∈ Φ . We let [ Φ ] = { [ α ] | α ∈ Φ } be the collection ofsuch equivalence classes and note that the direct sum of root spaces g [ α ] = M l ∈ Z m g α ◦ θ l is a homogeneous subspace of g , whence g = h ⊕ L [ α ] ∈ [ Φ ] g [ α ] is a decomposition of g intohomogeneous subspaces.Now, the centralizer g x of any x ∈ c is a homogeneous Levi subalgebra containing c g (c) . Anatural question is whether there exists a parabolic subalgebra of g with Levi factor g x that isalso homogeneous: we will now see that this is rarely the case. For simplicity of exposition,we restrict to the case where g is semisimple.Let p be a parabolic subalgebra of g with a homogeneous Levi factor l that contains c g (c) .Then, there exists a Z -grading g = M j ∈ Z g( j ) (A.2)of g such that p = g( > ) = L j > g( j ) and l = g( ) . We let Z ∈ g be the grading element of(A.2), the unique element in g that satisfies [ Z , X ] = jX for all X ∈ g( j ) , j ∈ Z , see, e.g., [30].Now Z ∈ z(c g (c)) , so it belongs to the adapted Cartan subalgebra h = L l ∈ Z m h l of g ofProposition A.1. We will write Z = Z + · · · + Z m − , where Z l ∈ h l for all l ∈ Z m . Definition A.4.
Let α = α + · · · + α m − ∈ Φ be a root with respect to h and l ∈ Z m . The l th mode of α is the complex number λ l = α l ( Z l ) .We remark that α ( Z ) = P l ∈ Z m λ l . Since the adjoint action of Z has integer eigenvalues,we may apply Lemma A.3 repeatedly to the roots α ◦ θ l ∈ Φ and get: Proposition A.5.
The modes of α satisfy a system of linear equations of the form · · · ω ω · · · ω m − ω ( ω ) · · · ( ω m − ) ... ... ... ... ω m − ( ω ) m − · · · ( ω m − ) m − λ λ λ ... λ m − = n n n ... n m − , (A.3) where n l = α ( θ l ( Z )) ∈ Z for any l ∈ {
0, . . . , m − } . The m × m matrix on the L.H.S. of (A.3) is a symmetric matrix of Vandermonde type withcoefficients in the cyclotomic field Q ( ω ) . We denote it by M ( ω ) and compactly rewrite (A.3)as M ( ω ) ~ λ = ~ n , where ~ λ ∈ C m is the vector of modes and ~ n ∈ Z m . Clearly all modes areelements of Q ( ω ) , but we have the following stronger result for λ . Proposition A.6.
The identity mλ = P l ∈ Z m n l is always satisfied, therefore λ ∈ m Z . If h = ,then p is not θ -stable. ORDAN CLASSES FOR θ -GROUPS 23 Proof.
Let W = { ~ y ∈ C m | P l ∈ Z m y l = } . All columns of M ( ω ) but the first one lie in W , so ~ n − λ ∈ W . (A.4)Adding all entries on the left and the right hand side of (A.4) gives mλ = P l ∈ Z m n l ∈ Z . If h =
0, then Z =
0, so λ = ~ n ∈ W ∩ Z m .Now, h ⊂ g( ) and p = h ⊕ L α ∈ Φ , α ( Z ) > g α . If g α ⊂ L j> g( j ) , then α ( Z ) = n > h =
0, there exists l ∈ Z m such that n l <
0, i.e., θ − l g α = g α ◦ θ l p . (cid:3) Example A.7.
The Z m -graded Lie algebra { g , θ , m } = { E , θ , 3 } as in Examples 2.12 and 4.11satisfies h =
0. By Theorem 2.6 and Proposition A.6, all centralizers g x of non-zero x ∈ S V do not extend to θ -stable parabolic subalgebras. Acknowledgments
We thank Michael Bulois for useful email exchanges on sheets in symmetric Lie algeb-ras and slice induction, Willem de Graaf for a helpful discussion on N -regular gradings,Dmitri Panyushev for information on generic stabilizers of semisimple elements in V , andVladimir Popov for pointing us reference [23]. This research was partially supported byDOR1898721/18 and BIRD179758/17 funded by the University of Padova and FFABR2017funded by MIUR. This project started when the third named author was holding a Type BPost doc Fellowship at the University of Padova. References [1]
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