Automorphisms of ﬁnite order, periodic contractions, and Poisson-commutative subalgebras of S ( g )

Let g be a semisimple Lie algebra, ϑ ∈ Aut ( g ) a ﬁnite order automorphism, and g 0 the subalgebra of ﬁxed points of ϑ . Recently, we noticed that using ϑ one can construct a pencil of compatible Poisson brackets on S ( g ) , and thereby a ‘large’ Poisson-commutative subalgebra Z ( g , ϑ) of S ( g ) g 0 . In this article, we study invariant-theoretic properties of ( g , ϑ) that ensure good properties of Z ( g , ϑ) . Associated with ϑ one has a natural Lie algebra contraction g ( 0 ) of g and the notion of a good generating system (=g.g.s.) in S ( g ) g . We prove that in many cases the equality ind g ( 0 ) = ind g holds and S ( g ) g has a g.g.s. According to V. G.Kac’s classiﬁcation of ﬁnite order automorphisms (1969), ϑ can be represented by a Kac diagram, K (ϑ) , and our results often use this presentation. The most surprising observation is that g ( 0 ) depends only on the set of nodes in K (ϑ) with nonzero labels, and that if ϑ is inner and a certain label is nonzero, then g ( 0 ) is isomorphic to a parabolic contraction of g .


INTRODUCTION
1.1.Completely integrable Hamiltonian systems on symplectic algebraic varieties are fundamental objects having a rich structure.They have been extensively studied from different points of view in various areas of mathematics such as differential geometry, classical mechanics, algebraic and Poisson geometries, and more recently, representation theory.A natural choice for the underlying variety is a coadjoint orbit of an algebraic Lie algebra q.In this context, one may obtain an integrable system from a Poisson commutative (=PC) subalgebra of the symmetric algebra S(q).As is well-known, S(q) has the standard Lie-Poisson structure { , }.
In this paper, the base field k is algebraically closed, char k = 0, and g is the Lie algebra of a connected reductive algebraic group G. Let U(g) be the enveloping algebra of g.We are interested in PC subalgebras of S(g) h , where h = Lie(H) and H ⊂ G is a connected reductive subgroup.These subalgebras are closely related to commutative subalgebras of U(g) h and thereby to branching rules involving G and H, see [PY21,Sect. 6.1] for some examples.Note also that the centre of U(g) h is described in [Kn94,Theorem 10.1].
Whenever a PC subalgebra of S(g) h is large enough, one extends it to a PC subalgebra of S(g), which provides completely integrable systems on generic orbits.This idea is employed in [GS83, GS83'], where the foundation of a beautiful geometric theory has also been laid.
The Lenard-Magri scheme provides a method for constructing "large" PC subalgebras via compatible Poisson brackets.Let { , } ′ be another Poisson bracket on S(g) compatible with { , } and { , } t = { , } + t{ , } ′ .Using the centres of the Poisson algebras (S(g), { , } t ) for regular values of t, one obtains a PC subalgebra Z ⊂ S(g), see Section 2.1 for details.
Here the main questions are: • how to find/construct an appropriate compatible bracket { , } ′ ?
• what are the properties of PC subalgebras Z obtained?
(A) In [PY21], we prove that any involution of g yields a compatible Poisson bracket on S(g) and consider the related PC subalgebras of S(g).A generalisation of this approach to ϑ ∈ Aut(g) of arbitrary finite order is presented in [PY21"].The latter heavily relies on Invariant Theory of ϑ-groups developed by E.B. Vinberg in [V76].
(B) In [PY21'], we study compatible Poisson brackets related to a vector space sum g = r ⊕ h, where r, h are subalgebras of g.To expect some good properties of Z, one has to assume here that at least one of the subalgebras is spherical in g.
In case (A), the role of the Lie algebras g (0) and g (∞) is not symmetric.The algebra g (∞) is nilpotent, while a maximal reductive subalgebra of g (0) is g ϑ .Roughly speaking, the output of [PY21, PY21"] is that in order to expect some good properties of the PC subalgebra Z = Z(g, ϑ), one needs (at least) the following two properties of ϑ: (i) ind g (0) = ind g; (ii) the algebra S(g) g contains a good generating system (g.g.s.) with respect to ϑ, see Section 2.2 for details.(Then we also say that ϑ admits a g.g.s.) The Lie algebra g (0) is said to be the ϑ-contraction or a periodic contraction of g.

1.
2. This article is a sequel to [PY21"].It is devoted to invariant-theoretic properties of a Z m -graded simple Lie algebra g, which is motivated by our study of PC subalgebras of S(g).We concentrate on proving (i) and (ii) for various types of g and ϑ ∈ Aut(g).Accordingly, we establish some good properties of related PC subalgebras.Let Aut f (g) (resp.Int f (g)) be the set of all (resp.inner) automorphisms of g of finite order.For ϑ ∈ Aut f (g), we also say that ϑ is periodic.Let m = |ϑ| be the order of ϑ and ζ = m √ 1 a fixed primitive root of unity.If g i is the eigenspace of ϑ corresponding to ζ i , then g = m−1 i=0 g i is the Z m -grading of g associated with ϑ.A classification of periodic automorphisms of g is due to V. Kac [Ka69], and our results often invoke the Kac diagram of ϑ.We refer to [V76, § 8], [Lie3, Chap.3, § 3] and [Ka95,Ch. 8] for generalities on Kac's classification and the Kac diagrams.The Kac diagram of ϑ, K(ϑ), is an affine Dynkin diagram of g (twisted, if ϑ is outer) endowed with nonnegative integral labels.We recall the relevant setup and give an explicit construction of ϑ via K(ϑ), see Sections 2.3, 4, and 5. Actually, Kac's classification stems from the study of Z-gradings of "his" infinitedimensional Lie algebras [Ka69].Our recent results on g (0) and Z(g, ϑ) have applications to the infinite-dimensional case, too [PY21", Sect.8].However, in this article, we do not refer explicitly to Kac-Moody algebras, which agrees with the approach taken in [Lie3].
It is known that ind g (0) = ind g, if m = 2 [P07] or g 1 contains regular elements of g [P09].Here we prove equality (i) for ind g (0) in the following cases: (1) either m = 3 or m = 4, 5 and the G 0 -action on g 1 is stable, see Section 3; (2) ϑ is inner and a certain label on the Kac diagram of ϑ is nonzero, see Theorem 4.1 and Proposition 4.2; (3) ϑ is an arbitrary inner automorphism of g = sl n , see Proposition 4.10; (4) ϑ ∈ Aut f (sp 2n ) and m is odd, see Proposition 4.11; (5) ϑ is an arbitrary automorphism of G 2 (Example 4.9) or of so N , see Section 6.
Our proofs for (3)-(5) rely on a new result that g (0) depends only on the set of nodes in K(ϑ) with nonzero labels, i.e., having replaced all nonzero labels with '1', one obtains the same periodic contraction g (0) , see Theorem 4.7 (resp.5.2) for the inner (resp.outer) automorphisms of g.Another ingredient is that if ϑ is inner and a certain label on K(ϑ) is nonzero, then the ϑ-contraction g (0) is isomorphic to a parabolic contraction of g (Theorem 4.1).The theory of parabolic contraction is developed in [PY13], and an interplay between two types of contractions enriches our knowledge of PC subalgebras in both cases.For instance, we prove that Z(sl n , ϑ) is polynomial for any ϑ ∈ Int f (sl n ) (Theorem 4.14).
Frankly, we believe the equality ind g (0) = ind g holds for any ϑ ∈ Aut f (g), and it is a challenge to prove it in full generality.This equality can be thought of as a ϑgeneralisation of the Elashvili conjecture.For, a possible proof would require to check that, for a nilpotent element x ∈ g 1 , one has ind (g x ) (0) = ind g x , cf.Corollary 3.5.
We say that ϑ ∈ Aut f (g) is N-regular, if g 1 contains a regular nilpotent element of g.Properties of the N-regular automorphisms are studied in [P05, § 3].In particular, if a connected component of Aut(g) contains elements of order m, then it contains a unique G-orbit of N-regular elements of order m.That is, there are sufficiently many N-regular automorphisms of g.We prove that a g.g.s.exists for the N-regular ϑ, see Theorem 7.8.Furthermore, if ϑ and ϑ ′ belong to the same connected component of Aut(g), |ϑ| = |ϑ ′ |, dim g ϑ = dim g ϑ ′ , and ϑ is N-regular, then ϑ ′ also admits a g.g.s.(Theorem 7.12).
Another interesting feature is that if ϑ is inner and N-regular, then at most one label on K(ϑ) can be bigger that 1 (Theorem 7.10).Moreover, if |ϑ| does not exceed the Coxeter number of g, then all Kac labels belong to {0, 1}.

PRELIMINARIES ON PC SUBALGEBRAS AND PERIODIC AUTOMORPHISMS
2.1.Compatible Poisson brackets.Let q be an arbitrary algebraic Lie algebra.The index of q, ind q, is the minimal dimension of the stabilisers of ξ ∈ q * with respect to the coadjoint representation of q.If q is reductive, then ind q = rk q.Two Poisson brackets are said to be compatible if their sum is again a Poisson bracket.Suppose that { , } t = { , } ′ + t{ , } ′′ , t ∈ P 1 , is a pencil of compatible linear Poisson brackets on S(q), where P 1 = k ∪ {∞} and { , } 1 is the initial Lie-Poisson structure on q.
Let q (t) denote the Lie algebra structure on the vector space q corresponding to { , } t .The function (t ∈ P 1 ) → ind q (t) is upper semi-continuous and therefore is constant on a dense open subset of P 1 .This subset is denoted by P reg , and we set P sing = P 1 \ P reg .Then P sing is finite and Let Z t be the centre of the Poisson algebra (S(q), { , } t ) and Z the subalgebra of S(q) generated by all Z t with t ∈ P reg .We also write Then Z is Poisson commutative with respect to any bracket { , } t with t ∈ P 1 .In cases to be treated below, 1 ∈ P reg and all but finitely many algebras q (t) are isomorphic to q.Then one can prove that such a Z is a PC subalgebra of maximal transcendence degree in an appropriate class of subalgebras of S(q), see [PY21, PY21'].
2.2.Periodic automorphisms of g and related PC subalgebras of S(g).Suppose that g is reductive and ϑ ∈ Aut f (g).Using ϑ, one can construct a pencil { , of compatible linear Poisson brackets on S(g), see [PY21"] and Section 3.This pencil and the related PC subalgebra Z = Z(g, ϑ) have the following properties: • the Lie algebras g (t) , t ∈ k \ {0}, are isomorphic to g and hence P sing ⊂ {0, ∞}; • ∞ ∈ P reg if and only if If g 0 is abelian, then the right-hand side becomes (dim g + rk g)/2 =: b(g).
Recall that Z(g, ϑ) is generated by the centres Z t with t ∈ P reg .
, whose structure is easier to understand.Although Z × can be a proper subalgebra of Z(g, ϑ), this does not affect the transcendence degree, see [PY21",Cor. 3.8].Moreover, there are many cases in which the centre Z 0 can explicitly be described and one can check that Z 0 ⊂ Z × , see e.g.[PY21", Cor.4.7].Then Z(g, ϑ) is either equal to Z × (if g 0 is not abelian) or generated by Z × and Z ∞ (if g 0 is abelian).
Another notion, which is useful in describing the structure of Z × , is that of a good generating system in Z 1 = S(g) g .As is well known, S(g) g is a polynomial algebra in rk g generators.Let H 1 , . . ., H l (l = rk g) be a set of algebraically independent homogeneous generators of S(g) g such that each H i is a ϑ-eigenvector.Then we say that H 1 , . . ., H l is a set of ϑ-generators in S(g) g .If |ϑ| = m and g = m−1 i=0 g i is the associated Z m -grading, then we consider the 1-parameter group ϕ : k * → GL(g) such that ϕ(t)•x = t i x for x ∈ g i .(Note that ϕ(ζ) = ϑ.)This yields the natural Z-grading in S(g).If ϕ(t)•H j = i t i H j,i , then the nonzero polynomials H j,i are called the ϕ-homogeneous (or bi-homogeneous) components of H j .We say that i is the ϕ-degree of H j,i .Let H • j denote the ϕ-homogeneous component of H j of the maximal ϕ-degree.This maximal ϕ-degree is denoted by deg ϕ (H j ).Definition 1.A set of ϑ-generators H 1 , . . ., H l ∈ S(g) g is called a good generating system (=g.g.s.) with respect to ϑ, if H • 1 , . . ., H • l are algebraically independent.If there is g.g.s. with respect to ϑ, we also say that ϑ admits a g.g.s.
The following is the main tool for checking that a set of ϑ-generators forms a g.g.s.
Theorem 2.2 ([Y14, Theorem 3.8]).Let H 1 , . . ., H l be a set of ϑ-generators in S(g) g .Then Theorems 2.1 and 2.3 imply that under these hypotheses the total number of the nonzero bi-homogeneous components of all generators H j equals b(g, ϑ).

The Kac diagram of
sum of non-trivial ϑ-stable ideals.Otherwise (g, ϑ) is said to be indecomposable.A classification of finite order automorphisms readily reduces to the indecomposable case.The centre of g is always a ϑ-stable ideal and automorphisms of an abelian Lie algebra have no particular significance (in our context).Therefore, assume that g is semisimple.
If g is not simple and (g, ϑ) is indecomposable, then g = h ⊕n is a sum of n copies of a simple Lie algebra h and ϑ is a composition of a periodic automorphism of h and a cyclic permutation of the summands.
Below we assume that g is simple.By a result of R. Steinberg [St68, Theorem 7.5], every semisimple automorphism of g fixes a Borel subalgebra of g and a Cartan subalgebra thereof.Let b be a ϑ-stable Borel subalgebra and t ⊂ b a ϑ-stable Cartan subalgebra.This yields a ϑ-stable triangular decomposition g = u − ⊕ t ⊕ u, where u = [b, b].Let ∆ = ∆(g) be the set of roots of t, ∆ + the set of positive roots corresponding to u, and Π ⊂ ∆ + the set of simple roots.Let g γ be the root space for γ ∈ ∆.Hence u = γ∈∆ + g γ .
Clearly, ϑ induces a permutation of Π, which is an automorphism of the Dynkin diagram, and ϑ is inner if and only if this permutation is trivial.Accordingly, ϑ can be written as a product σ•ϑ ′ , where ϑ ′ is inner and σ is the so-called diagram automorphism of g.We refer to [Ka95, § 8.2] for an explicit construction and properties of σ.In particular, σ depends only on the connected component of Aut(g) that contains ϑ and ord(σ) equals the order of the corresponding permutation of Π.The index of ϑ ∈ Aut f (g) is the order of the image of ϑ in Aut(g)/Int(g), i.e., the order of the corresponding diagram automorphism.2.3.1.The inner periodic automorphisms.Set Π = {α 1 , . . ., α l } and let δ = l i=1 n i α i be the highest root in ∆ + .An inner periodic automorphism with t ⊂ g 0 is determined by an (l + 1)-tuple of non-negative integers (Kac labels) p = (p 0 , p 1 , . . ., p l ) such that gcd(p 0 , . . ., p l ) = 1 and p = (0, . . ., 0).Set m := p 0 + l i=1 n i p i and let p i denote the unique representative of {0, 1, . . ., m − 1} such that p i ≡ p i (mod m).The Z m -grading g = m−1 i=0 g i corresponding to ϑ = ϑ(p) is defined by the conditions that g α i ⊂ g p i for i = 1, . . ., l, g −δ ⊂ g p 0 , and t ⊂ g 0 .
For our purposes, it is better to introduce first the Z-grading of g defined by (p 1 , . . ., p l ) and then factorise ("glue") it modulo m, see Section 4 for details.
The Kac diagram K(ϑ) of ϑ = ϑ(p) is the affine (= extended) Dynkin diagram of g, D(g), equipped with the labels p 0 , p 1 , . . ., p l .In K(ϑ), the i-th node of the usual Dynkin diagram D(g) represents α i and the extra node represents −δ.It is convenient to assume that α 0 = −δ and n 0 = 1.Then (l + 1)-tuple (n 0 , n 1 , . . ., n l ) yields coefficients of linear dependence for α 0 , α 1 , . . ., α l .Set Π = Π ∪ {α 0 }.If n i = 1 for i 1, then the subdiagram without the i-th node is isomorphic to D(g) and Π \ {α i } is another set of simple roots in ∆.Hence any node of D(g) with n i = 1 can be regarded as an extra node, which merely corresponds to another choice of a Borel subalgebra containing our fixed Cartan subalgebra t.Practically this means that we consider these Kac diagrams modulo the action of the automorphism group of the graph D(g).

2.3.2.
The outer periodic automorphisms.Let σ be the diagram automorphism of g related to ϑ.The orders of nontrivial diagram automorphisms are: Therefore, σ defines either Z 2 -or Z 3 -grading of g.To avoid confusion with the ϑ-grading, this σ-grading is denoted as follows: and the latter occurs only for g = so 8 .In all cases, g σ = g (σ) 0 is a simple Lie algebra and each g as g σ -modules and g 1 ].Since b and t are σ-stable, b σ = t σ ⊕ u σ is a Borel subalgebra of g σ and t 0 = t σ is a Cartan subalgebra of both g σ and g 0 = g ϑ .Let ∆ + (g σ ) be the set of positive roots of g σ corresponding to u σ and let {ν 1 , . . ., ν r } be the set of simple roots in ∆ + (g σ ).
The subalgebras g σ and g σ -module g 1 are gathered in the following table, where V λ is a simple g σ -module with highest weight λ, and the numbering of simple roots and fundamental weights Some of the twisted affine diagrams are depicted below.We enhance these diagrams with the coefficients {a ′ i } over the nodes and the corresponding roots under the nodes.
In Section 5, we give a detailed description of this Z m -grading and use it to prove a modification result on K(ϑ) and the structure of g (0) .2.4.The description of g 0 and g 1 via the Kac diagram of ϑ.Let p 0 , p 1 , . . ., p l be the Kac labels of ϑ ∈ Int f (g).Then the subdiagram of nodes in D(g) such that p i = 0 is the Dynkin diagram of [g 0 , g 0 ], while the dimension of the centre of are the lowest weights of the simple g 0 -modules in g 1 , i.e., if V − µ stands for the g 0 -module with lowest weight µ, then The same principle applies to the outer periodic automorphisms, D(g) being replaced with the respective twisted affine Dynkin diagram.These results are contained in [V76, Prop.17].
It follows that the subalgebra of ϑ-fixed points, g 0 , is semisimple if and only if K(ϑ) has a unique nonzero label.At the other extreme, g 0 is abelian if and only if all p i are nonzero.Furthermore, if all p i 1, then the following conditions are equivalent: Example 2.4.Take the automorphism of D 4 of index 3 with Kac labels Here ϕ is the fundamental weight of SL 2 and ε is the basic character of T 1 .

ON THE INDEX OF PERIODIC CONTRACTIONS OF SEMISIMPLE LIE ALGEBRAS
In this section, we recall the structure of Lie algebras g (0) and g (∞) and then prove that ind g (0) = ind g for small values of m.Let ζ = m √ 1 be a fixed primitive root of unity.Then where the eigenvalue of ϑ on g i is ζ i .The Lie algebras g, g (0) , and g (∞) have the same underlying vector space, but different Lie brackets, denoted [ , ], [ , ] (0) , and [ , ] (∞) , respectively.More precisely, Hence vector space decomposition (3•1) is a Z m -grading for g, but it is an N-grading for g (0) .Then the (∞)-bracket can be defined as One readily verifies that g (∞) is also N-graded and its component of grade i is g m−i for i = 1, 2, . . ., m; in particular, the component of grade 0 is trivial.This implies that g we are interested now in the problem of computing ind g (0) .Let us recall some relevant results.
• By the semi-continuity of index under contractions, one has ind g (0) ind g; Proof.Write g = s ⊕ c, where c is the centre of g and s = [g, g].Then g (0) = s (0) ⊕ c (0) .Since c is an Abelian Lie algebra, then so is c (0) and ind c = ind c (0) .The result follows.
The Killing form κ on g induces the isomorphism τ : g → g * with τ (x)(y) := κ(x, y) for all x, y ∈ g.Clearly τ restricts to an isomorphism g i ≃ g * m−i for each i.Set ξ x := τ (x).Having identified g * and g * (0) as vector spaces, we may regard ξ x as an element of g * (0) .Then (g (0) ) ξx denotes the stabiliser of ξ x with respect to the coadjoint representation of g (0) .Proposition 3.4.Let x ∈ g 1 ⊂ g be arbitrary.
(i) Upon the identification of g and g (0) , the vector spaces g x and (g (0) ) ξx coincide.(ii) Moreover, the Lie algebra g x is ϑ-stable and its ϑ-contraction (g x ) (0) is isomorphic to (g (0) ) ξx as a Lie algebra.
Proof.(i) Since the Lie algebra g (0) is N-graded, (g (0) ) ξx is N-graded as well.On the other hand, g x inherits the Z m -grading from g.Let us show that the vector spaces g x ∩ g i and (g (0) ) ξx ∩ g i are equal for each i.Let ad * (0) denote the coadjoint representation of g (0) .For y ∈ g j , we have For any j, we then obtain (ii) This follows from (i) and the general relationship between the Lie brackets of the initial Lie algebra and a Z m -contraction of it, cf.(3•2).Corollary 3.5.If there is an x ∈ g 1 such that ind (g x ) (0) = ind g x , then ind g (0) = ind g.
These results yield the induction step for computing ind g (0) .If g is semisimple and x ∈ g 1 is a nonzero semisimple element, then g x g, g x is reductive, ind g x = ind g, and ϑ preserves g x .Hence it suffices to verify Conjecture 3.1 for the smaller semisimple Lie algebra [g x , g x ].One can perform such a step as long as g 1 contains semisimple elements.The base of induction is the case in which g 1 contains no nonzero semisimple elements.Then the existence of the Jordan decomposition in g 1 [V76, § 1.4] implies that all elements of g 1 are nilpotent.Actually, the 'base' can be achieved in just one step.Recall from [V76] that a Cartan subspace of g 1 is a maximal subspace c consisting of pairwise commuting semisimple elements.By [V76, § 3,4], all Cartan subspaces are G 0 -conjugate and dim c = dim g 1 / /G 0 .The number dim c is called the rank of (g, ϑ, m).We also denote it by rk(g 0 , g 1 ).
If x ∈ c is a generic element, then s = [g x , g x ] has the property that s 1 consists of nilpotent elements.
Thus, in order to confirm Conjecture 3.1, one should be able to handle the automorphisms ϑ of semisimple Lie algebras g such that g 1 ⊂ N. Using previous results, we can do it now for m = 3 and for m = 4, 5 (with some reservations, see Proposition 3.7).
Proof.By the inductive procedure above, we may assume that g 1 ⊂ N. Then G 0 has finitely many orbits in g 1 [V76, § 2.3].Take x ∈ g 1 from the dense G 0 -orbit.Then [g 0 , x] = g 1 and hence g x has the trivial projection to g 2 , i.e., g x = g x 0 ⊕g x 1 .This implies that [g x 1 , g x 1 ] = 0 and therefore the Lie algebras g x and g x (0) are isomorphic.Since ind g x = ind g by the Elashvili conjecture, the assertion follows from Corollary 3.5.
Recall that the action of a reductive group H on an irreducible affine variety X is stable, if the union of all closed H-orbits is dense in X.For x ∈ g 1 = X and H = G 0 , the orbit G 0 •x is closed if and only if x is semisimple in g [V76, § 2.4].Therefore, the linear action of G 0 on g 1 is stable if and only if the subset of semisimple elements of g is dense in g 1 .Proposition 3.7.Suppose that m = 4, 5 and the action (G 0 : g 1 ) is stable.Then ind g (0) = ind g.
Example 3.8.For g of type F 4 , the affine Dynkin diagram is . Take ϑ with the following Kac diagram ) as a g 0 -module.For the reader's convenience, we also provide the (numbering of the) fundamental weights of g 0 .
Since G 0 has a dense orbit in g 1 , we have g 1 ⊂ N and the induction step does not apply.Actually, our methods, including those developed in Section 4, do not work here, and the exact value of ind g (0) is not known yet.

INNER AUTOMORPHISMS, Z-GRADINGS, AND PARABOLIC CONTRACTIONS OF g
In this section, we prove that, for certain ϑ ∈ Int f (g), the ϑ-contraction g (0) is isomorphic to a parabolic contraction of g.Then comparing the results obtained earlier for parabolic contractions [PY13] and ϑ-contractions [PY21"] yields new knowledge in both instances.First, we need an explicit description of ϑ ∈ Int f (g) via a Z-grading of g associated with the Kac diagram K(ϑ).Recall that K(ϑ) is the affine Dynkin diagram of g, equipped with numerical labels p 0 , p 1 , . . ., p l , where p 0 is the label at the extra node.
For µ ∈ ∆, let d(µ) be the unique element of {0, 1, . . ., m − 1} such that g µ ⊂ g d(µ) .Then Using this description, we prove below that, for a wide class of inner automorphisms ϑ, the ϑ-contraction g (0) admits a useful alternate description as a semi-direct product.Recall the necessary setup.If h ⊂ g is a subalgebra, then h ⋉ (g/h) ab stands for the corresponding Inön ü-Wigner contraction of g, see [PY21', Sect.2].Here the superscript "ab" means that the h-module g/h is an abelian ideal of this semi-direct product.Let h = p be a standard parabolic subalgebra associated with Π.Then g/p can be identified with n − as a vector space, and In ön ü-Wigner contractions of the form p ⋉ (n − ) ab , which have been studied in [PY13], are called parabolic contractions of g.Theorem 4.1.Suppose that ϑ ∈ Int f (g) and p 0 = p 0 (ϑ) > 0. Let p and n − be the subalgebras associated with p 1 , . . ., p l as above.Then g (0) ≃ p ⋉ (n − ) ab .
Proof.Since p 0 > 0, we have g(0) = g 0 and d(µ) < m for any µ ∈ ∆ + .Hence Using this notation and the above relationship between Z and Z m -gradings, we now routinely verify that the Lie bracket in g (0) coincides with that in p ⋉ (n − ) ab .
Thus, for all x ∈ p and y ∈ n − , the Lie bracket [x, y] (0) is computed as the initial bracket [x, y] with the subsequent projection to n − (w.r.t. the decomposition g = p ⊕ n − ).This precisely means that g (0) and the semi-direct product p ⋉ (n − ) ab are isomorphic as Lie algebras.
Proof.If p i (ϑ) > 0 and n i = 1, then using an automorphism of D(g), i.e., making another choice of b, we can reduce the problem to the case i = 0, see Section 2.3.1.Hence g (0) is a parabolic contraction by Theorem 4.1.By [PY13, Theorem 4.1], the index does not change for the parabolic contractions of g, i.e., ind (p⋉(n − ) ab ) = ind g for any parabolic subalgebra p ⊂ g.
Remark 4.3.If p i = 0 for all i such that n i = 1, then the preceding approach fails and there seems to be no useful alternate description of g (0) .
The parabolic contractions of g are much more interesting than arbitrary In ön ü-Wigner contractions.Their structure is closely related to properties of the centralisers for the corresponding Richardson orbit.Since p admits a complementary subspace n − , which is a Lie subalgebra, the Lie-Poisson bracket associated with p⋉(n − ) ab is compatible with the initial bracket on g ([PY21', Lemma 1.2]).Then the Lenard-Magri scheme provides a PC subalgebra of S(g), which is denoted by Z(p, n − ).This leads to the following Corollary 4.4.If ϑ ∈ Int f (g) and p i > 0 for some i such that n i = 1, then Z(g, ϑ) = Z(p, n − ).
Example 4.5.Consider ϑ ∈ Int f (g) such that g 0 = g(0) = t.This is equivalent to that p i > 0 for all i = 0, 1, . . ., l.Then p = b is a Borel subalgebra and hence Z(g, ϑ) = Z(b, u − ).The advantage of this situation is that u − = [b − , b − ] is a spherical subalgebra, and our results for the PC subalgebra Z(b, u − ) are more precise and complete [PY21', Sect.4, 5].Namely, (i) tr.deg Z(b, u − ) = b(g), the maximal possible value for the PC subalgebras of S(g); (ii) Z(b, u − ) is a maximal PC subalgebra of S(g); (iii) Z(b, u − ) is a polynomial algebra, whose free generators are explicitly described.Thus, results on parabolic contractions provide a description of Z(g, ϑ) for a class of ϑ ∈ Int f (g).(And it is not clear how to establish (ii) and (iii) in the context of Z m -gradings!)Conversely, results on periodic contractions allow us to enrich the theory of parabolic contractions and give a formula for tr.degZ(p, n − ) with arbitrary p. Proposition 4.6.For any parabolic subalgebra p ⊂ g with Levi subalgebra l, we have Proof.Without loss of generality, we may assume that p ⊃ b and l ⊃ t.Let J ⊂ {1, . . ., l} correspond to the simple roots of [l, l], i.e., α j ∈ Π is a root of (l, t) if and only if j ∈ J. Take any ϑ ∈ Int f (g) with the Kac labels (p 0 , . . ., p l ) such that p j = 0 if and only if j ∈ J (in particular, p 0 = 0).Then the Z-grading corresponding to (p 1 , . . ., p l ) has the property that p = g( 0), l = g(0) = g 0 , and n − = g(<0).Hence g (0) ≃ p ⋉ (n − ) ab .On the other hand, since ind g (0) = rk g (Proposition 4.2), we have tr.degZ(g, ϑ) = b(g) − b(g 0 ) + rk g, see [PY21", Theorem 3.10].
Given ϑ with Kac labels p 0 , p 1 , . . ., p l , the subalgebra g 0 = g ϑ depends only on the set L(ϑ) := {i ∈ [0, l] | p i = 0}, see Section 2.4.(This also follows from the description of ϑ-grading given above.)Let us prove that the similar property holds for the whole ϑ-contraction g (0) .That is, having replaced all nonzero Kac labels p i with 1, one obtains another automorphism θ (of a smaller order), but the corresponding periodic contractions appear to be isomorphic.Note that it is not assumed now that p 0 > 0.
Proof.For sl l+1 , the affine Dynkin diagram A l is a cycle and n i = 1 for all i = 0, 1, . . ., l.The Kac diagram of an inner automorphism is determined up to a rotation of this cycle.Therefore, we may always assume that p 0 > 0. Hence g (0) is a parabolic contraction for every ϑ ∈ Int f (sl l+1 ) and thereby ind g (0) = ind g for all inner periodic automorphisms.Proposition 4.11.If g = sp 2l and ϑ ∈ Aut f (g) with |ϑ| odd, then ind g (0) = ind g = l.
To provide yet another illustration of the interplay between parabolic contractions and ϑ-contractions, we need some preparations.
If H ∈ S d (g), then one can decompose H as the sum of bi-homogeneous components Theorem 4.12 (cf.Theorem 5.1 in [PY13]).Let g be either sl l+1 or sp 2l .If q = p ⋉ (n − ) ab is any parabolic contraction of g, then S(q) q is a polynomial algebra.Moreover, there are free generators H 1 , . . ., H l ∈ S(g) g such that (H 1 ) • n − , . . ., (H l ) • n − freely generate S(q) q .
In the situation of Theorem 4.1, we have g (0) ≃ p ⋉ (n − ) ab and, for a homogeneous H ∈ S(g), there are two a priori different constructions: • First, one can take H • , the bi-homogeneous component of H with highest ϕ-degree.(Recall that this uses the Z m -grading g = m−1 i=0 g i and ϕ : k * → GL(g), see Section 2.2.) • Alternatively, one can take H • n − , which employs the direct sum g = p ⊕ n − .However, the two decompositions of g are related in a very precise way, and therefore the following is not really surprising.Lemma 4.13.Suppose that p 0 (ϑ) > 0, and let g = m−1 i=0 g i and g = p ⊕ n − be as above.If Proof.Recall that if p 0 > 0, then g 0 is a Levi subalgebra of p, i.e., p = g 0 ⊕ n.Take a basis for g that consists of the root vectors e γ , γ ∈ ∆, and a basis for t.Suppose that H ∈ S(g) t is a monomial in that basis and H ∈ S i (n − ) ⊗ S j (p).Then where γ 1 , . . ., γ i ∈ ∆(n), µ 1 , . . ., µ j ∈ ∆(p), f ∈ S j−j (t), and Hence the ϕ-degree of a t-invariant monomial depends only on its n − -degree.Thus, if H ∈ S(g) t is written in the basis above, then both H • and H • n − consist of the monomials of maximal n − -degree, and thereby H • = H • n − .The following is the promised "illustration".Theorem 4.14.For any ϑ ∈ Int f (sl n ), there is a g.g.s. in S(sl n ) sl n and the PC subalgebra Z(sl n , ϑ) is polynomial.
Proof.We assume below that n = l + 1.By Theorem 4.12, there is a set H 1 , . . ., H l of free homogeneous generators of S(g) g such that (H 1 ) • n − , . . ., (H l ) • n − freely generate S(q) q .Under the hypothesis on ϑ, we also have p ⋉ (n − ) ab ≃ g (0) (Theorem 4.1) and for each i (Lemma 4.13).This means that is a polynomial algebra and H 1 , . . ., H l is a g.g.s. with respect to ϑ.By Theorem 2.3, we conclude that Z 0 ⊂ Z × and that Z × is a polynomial algebra.
• If g 0 is not abelian, then ∞ ∈ P sing and hence Z × = Z(sl n , ϑ) is a polynomial algebra.
• If g 0 is abelian, then g 0 = t, p = b, and g (0) ≃ b ⋉ (u − ) ab .In this case, ∞ ∈ P reg and one has also to include Z ∞ in Z(sl n , ϑ).However, it was directly proved in [PY21', Theorem 4.3] that here Z(b, u − ) = Z(sl n , ϑ) is a polynomial algebra.

MODIFICATION OF KAC DIAGRAMS FOR THE OUTER AUTOMORPHISMS
Here we prove an analogue of Theorem 4.7 to the outer periodic automorphisms of simple Lie algebras.Let ϑ ∈ Aut f (g) be outer, with the associated diagram automorphism σ, see Section 2.3.Recall that r = rk g σ and Π (σ) = {ν 1 , . . ., ν r } is the set of simple roots of g σ .
Let p = (p 0 , p 1 , . . ., p r ) be the Kac labels of ϑ.Using p, we construct below the vector space sum g = j∈Z g(j).Unlike the case of inner automorphisms, this decomposition is not going to be a Lie algebra grading on the whole of g.Nevertheless, it will be compatible with the σ-grading (2•1), and it will provide a Lie algebra Z-grading on g σ .
-The Z-grading of g σ is given by the conditions: σ) , the root space (g σ ) ν i belongs to g σ (p i ) ⊂ g(p i ).
-For the lowest weight −δ 1 of g 1 , we set (g This defines a structure of a Z-graded g σ -module on g and the Z-grading on the latter is uniquely determined by the condition that [g i , the vector space sum obtained is compatible with the weight decomposition with respect to t σ .That is, for a t σ -weight space (g i , one can point out the integer j such that (g (σ) i ) γ ⊂ g(j).Then we write d i (γ) for this j.The preceding exposition shows that i .The Z m -grading of g associated with ϑ = ϑ(p) is obtained from the graded vector space decomposition of g by "glueing" modulo m = ord(σ) Lemma 5.1.For an outer ϑ ∈ Aut(g) with Kac labels (p 0 , p 1 , . . ., p r ), we have (ii) jp 0 d j (γ) m for any t σ -weight γ of g (σ) j , j = 1, 2.Moreover, the upper bound m is attained if and only if p 0 = 0.
We do this case-by-case.
(ii) For the weights of g 1 , the (Z, ϑ)-degrees range from d 1 (−δ 1 ) = p 0 , the degree of the lowest weight, until d 1 (δ 1 ) = p 0 + 2 r i=1 a ′ i p i , the degree of the highest weight.Since ord(σ) 2, we have then m 2(p 0 + r i=1 a ′ i p i ) and the result follows.In case ord(σ) = 3, the (Z, ϑ)-degrees for the weights of g i , then we also set d(x) = j.For an integer d, let d be the unique element of {0, 1, . . ., m − 1} such that d − d ∈ mZ.
Proof.With necessary alterations, we follow the proof of Theorem 4.7.The Lie algebra g 0 depends only on L(ϑ).If x ∈ g 0 and y ∈ g, then [x, y] (0) = [x, y].We always assume below that x, y ∈ g 0 .Furthermore, x and y are weight vectors of t σ in all cases.
For given x and y, it depends only on L(ϑ).
5. Now we consider the case, where both x, y ∈ m are weight vectors of t 0 and eigenvectors of σ.Set b 1 and [m, m] ⊂ g σ .By the construction, t 1 , then the (Z, ϑ)-degree of x, as well as of y, is larger than or equal to m/2, but smaller than m by Lemma 5.1(ii).Hence 1 .Write x ∈ m µ , y ∈ m β , where µ, β are weights of t σ , then µ + β ∈ ∆(g σ ), sinse [x, y] ∈ g 0 .Note that m −β = 0, since m is a self-dual g σ -module.This applies to every t σ -weight in m.
We have in type B l .Therefore the assumptions on the Kac labels imply that m is even and exactly m/2 labels are equal to 1.
If g is of type D l , then the Kac diagram of ϑ has l + 1 nodes and looks as follows: where k = (m/2)−1.By the assumption on Kac labels, we have b ′ , b ′′ 2. Hence g 0 has the non-trivial summands so 2b ′ , so 2b ′′ and (m/2) − 1 nonzero summands gl s i +1 .If A m = −I 2l , then neither 1 nor −1 is an eigenvalues of A, since m is even.Hence the centraliser of A in so 2l , i.e., g 0 , is a sum of m/2 summands gl dim V(λ) with λ m = −1.It has fewer summands that required by K(ϑ).Therefore A m = I 2l and the eigenvalues of A are m-th roots of unity.Arguing as in the proof of Lemma 6.1, we obtain that each m-th root of unity is an eigenvalue of A. In this case, the automorphism ϑ is again of type I in the sense of Vinberg [V76, § 7.2] and hence rk(g 0 , g 1 ) = min 0 j m/2 {b j } 1.Thus, g 1 contains nonzero semisimple elements.
If g is of type B l , then the argument is similar.The difference is that dim V = 2l + 1 and the Kac diagram of ϑ (having l + 1 nodes) looks as follows: where k = (m/2) − 1 and b ′ 2. Since dim V is odd, 1 or −1 has to be an eigenvalue of A.
Therefore A m = I 2l+1 and again we have b j 1 for all 0 j m/2.Theorem 6.3.If g = so N , then ind g (0) = rk g for any periodic automorphism ϑ.
Proof.We argue by induction on N + m with m = |ϑ|.If m 3, then the statement holds by Proposition 3.6 and [P07].Clearly, it holds also for N 3, cf.Proposition 4.10.If there is a Kac label of ϑ that is larger than 1, then we may replace it with '1' without changing the Lie algebra structure of g (0) , see Theorems 4.7 and 5.2.Clearly, m decreases under this procedure.Therefore we may assume that the Kac labels of ϑ belong to {0, 1}.
If ϑ is inner and at least one of the labels p 0 , p 1 , p l−1 , p l in type D l equals '1' or one of the labels p 0 , p 1 in type B l equals '1', then ind g (0) = rk g by Proposition 4.2.
Therefore, we may assume that either ϑ is outer or ϑ is inner with p 0 = p 1 = p l−1 = p l = 0 (in type D l ) and p 0 = p 1 = 0 (in type B l ).This implies that m is even and g 1 contains a nonzero semisimple element x, see Lemmas 6.1 and 6.2.By Corollary 3.5, it suffices to prove that ind (g x ) (0) = ind g x for some x ∈ g 1 .Let x = C i ∈ g 1 be one of the basis semisimple elements defined in [V76, § 7.2].As an endomorphism of V, it has the following properties: These properties imply that g x = so N −m ⊕ t m/2 , where t m/2 is an abelian Lie algebra of dimension m/2.Since [g x , g x ] is a smaller orthogonal Lie algebra, the induction hypothesis applies, which completes the proof.Remark 6.4.For g = sp 2l , we have Aut(g) = Int(g), but an analogue of Lemma 6.2 is not true.Here (n 0 , n 1 , . . ., n l−1 , n l ) = (1, 2, . . ., 2, 1) and it may happen that p 0 (ϑ) = p l (ϑ) = 0, but g 1 contains no nonzero semisimple elements, i.e., g 1 ⊂ N. In this case, m is necessarily even.The simplest example of such ϑ occurs if p i = p i+1 = 1 for certain i with 1 i l − 2 and all other p j are zero, see the Kac diagram below: Then m = 4, g 1 ⊂ N, and ind g (0) is not known.Here g 0 = sp 2i ⊕sp 2j ⊕t 1 , where j = l−i−1.

N-REGULAR AUTOMORPHISMS AND GOOD GENERATING SYSTEMS
In this section, we prove that if ϑ is an N-regular automorphism of g, then ϑ admits a good generating system and obtain some related results on the structure of the PC subalgebras Z × , Z(g, ϑ) ⊂ S(g) g 0 .Moreover, if θ is "close" to an N-regular automorphism (see Def. 3), then θ also admits a g.g.s.
Following [P05, Sect.3], we associate to ϑ the set of integers {k i } m−1 i=0 defined as follows: Then i k i = l.The eigenvalues {ε j } depend only on the image of ϑ in Aut(g)/Int(g) (denoted θ), i.e., on the connected component of Aut(g) that contains ϑ.Therefore, the vector k = k(m, θ) = (k 0 , . . ., k m−1 ) depends only on m and θ.We say that the tuple (|ϑ|, k) is the datum of a periodic automorphism ϑ.
As a modest contribution to the invariant theory of ϑ-groups, we record the following observation.Proposition 7.1.Let ϑ be an arbitrary periodic automorphism of g.Then Proof.(i) By [V76, § 2.3], N ∩ g 1 =: N 1 is the null-cone for the G 0 -action on g 1 .Therefore, the polynomials H 1 | g 1 , . . ., H l | g 1 have the same zero locus as the ideal in k[g 1 ] generated by the augmentation ideal k (For a short modern proof of Hilbert's result, we refer to [Ke87, Theorem 2].) (ii) If deg H j = d j and H j (x) = 0 for some x ∈ g 1 , then Hence m j + r j ≡ m − 1 (mod m).Therefore, there are at most k m−1 ϑ-generators {H j } that do not vanish on g 1 , and the assertion follows from (i).
Definition 2. A periodic automorphism ϑ is said to be N-regular, if g 1 contains a regular nilpotent element of g.
Basic results on the N-regular automorphisms are obtained in [P05, Section 3]: (ii) the dimension of a generic stabiliser for the G 0 -action on g 1 equals k 0 .
In particular, dim Hence the N-regular automorphism are distinguished by the properties that the restriction homomorphism ψ 1 is onto and dim g 1 / /G 0 has the maximal possible value among the automorphisms of g with a given datum.
Clearly, this yields formulae for dim g i with all i.
Lemma 7.5.In the N-regular case, we have Proof.Substitute the expression for either dim g 0 or dim g from (7•1) into (7•3).
Our next goal is to obtain an upper bound on the ϕ-degree of H j (Section 2.2).We recall the necessary setup, with a more elaborate notation.Using the vector space decomposition g = g 0 ⊕ g 1 ⊕ . . .⊕ g m−1 , we write H j as the sum of multi-homogeneous components: (m − 1)d j and d • j ≡ r j (mod m).For the m-tuple j = (0, . . ., 0, 1 i , 0, . . ., 0, m j ), we have p(j) = (m − 1)m j + i and p(j) − r j = mm j − (m j + r j − i) ≡ 0 (mod m), i.e., (H j ) j may occur in H j .Since (m − 1)m j p(j) (m − 1)d j and p(j) is the unique integer in this interval that is comparable with r i modulo m, we conclude that d • j p(j).
Proposition 7.7.For any ϑ ∈ Aut f (g) with |ϑ| = m, we have Since l j=1 m j = 1 2 (dim g−l) and l = i k i , the last expression is easily being transformed into the RHS in (7•5).
Since k = (k 0 , . . ., k m−1 ) depends only on m and θ, the upper bound in Proposition 7.7 depends only on the datum of ϑ.Let Y(m, k) denote this upper bound, i.e., the RHS in (7•5).
Furthermore, a set of ϑ-generators H 1 , . . ., H l is a g.g.s. with respect to ϑ if and only if D ϑ = l j=1 d • j , see again [Y14].Remark.The point of (3) is that if ϑ is N-regular, then any set of ϑ-generators is a g.g.s.If ϑ is not N-regular, then it may happen that the property of being g.g.s.depends on the choice of ϑ-generators.
Decomposition (7•4) provides the bi-homogeneous decomposition H j = i H j,i , where H j,i := i: p(i)=i (H j ) i .
Then d • j = max{i | H j,i = 0} and if H j,i = 0, then i ≡ r j (mod m).These bi-homogeneous decompositions have already been studied in [PY21"].In particular, the subalgebra of S(g) generated by all bi-homogeneous components {H j,i } is PC and it actually coincides with Z × , see [PY21", Eq. (4.1)].
Theorem 7.9.Let ϑ be an N-regular automorphism of order m.Then (i) all possible bi-homogeneous components of all H j are nonzero, i.e., H j,i = 0 if and only if 0 i d • j and i ≡ r j (mod m); (ii) all these bi-homogeneous components are algebraically independent and therefore Z × is a polynomial algebra; (iii) Remark 7.13.It can happen that l j=1 d • j (ϑ ′ ) < Y(m, k), but still D ϑ ′ = l j=1 d • j (ϑ ′ ), i.e., ϑ ′ admits a g.g.s..If this happens to be the case, then not every set of ϑ ′ -generators forms a g.g.s., and one has to make a right choice.It is known that all involutions of the classical Lie algebras admit a g.g.s.regardless of N-regularity [Y14], and there are exactly four involutions for exceptional Lie algebras of type E n that do not admit a g.g.s.[Y17].
The equality occurring in Theorem 7.12(ii) is not rare.Such non-conjugate pairs (ϑ, ϑ ′ ) do exist for m 3. Definition 3. We say that two non-conjugate automorphisms ϑ, θ form a friendly pair, if they have the same data, ϑ is N-regular, and dim g ϑ = dim g θ.
Together with presence of g.g.s., the members of a friendly pair share other good properties.To distinguish the Z m -gradings for ϑ and θ, we write g = m−1 i=0 g i for ϑ (which is N-regular) and g = m−1 i=0 gi for θ.
Proposition 7.14.Let (ϑ, θ) be a friendly pair.Then We assume below that H1 , . . ., Hl is a set of θ-generators.It is shown in Proposition 7.1 that if j ∈ P m−1 , then Hj | g1 = 0. On the other hand, since H1 , . . ., Hl is a g.g.s. with respect to θ, one has d • j = (m − 1)m j + m − 1 = (m − 1)d j for j ∈ P m−1 .Therefore, H• j = ( Hj ) i with i = (0, . . ., 0, d j ).Hence H• j ∈ S d j (g m−1 ), and the latter is the set of polynomial functions of degree d j on g 1 ≃ (g m−1 ) * .In other words, H• j is obtained as follows.We first take Hj | g 1 = ψ 1 ( Hj ) and then consider it as function on the whole of g via the projection g → g 1 .
Because H• 1 , . .(2) Although θ admits a g.g.s.(Theorem 7.12), we do not know in general whether the θ-contraction of g has the same index as g.

Conjecture 3. 1 .
For any periodic automorphism ϑ, one has ind g (0) = ind g.Let us record the following simple fact.Lemma 3.2.It suffices to verify Conjecture 3.1 for the semisimple Lie algebras.

we have Theorem 2.3. If
H l is a g.g.s.if and only if l i=1 deg ϕ H j = D ϑ .ind g (0) = l and H 1 , . . ., H l is g.g.s. with respect to ϑ, then Z × is a polynomial algebra, which is freely generated by the ϕ-homogeneous components of H 1 , . . ., H l .
the sum of all multi-homogeneous components of H j , where p(i) is maximal.Whenever we wish to stress that d • j is determined via a certain ϑ, we write d • j (ϑ) for it.Recall that a set of ϑ-generators H 1 , . . ., H l is called a g.g.s. with respect to ϑ, if H • 1 , . . ., H • l are algebraically independent.A ϑ-generator H j is said to be of type (i), if m j + r j ≡ i (mod m) for i ∈ {0, 1, . . ., m − 1}.Lemma 7.6.If H j is of type (i), then d • j (m − 1)m j + i. Proof.By definition, d • j