Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of S(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}({\mathfrak g})$$\end{document}

Let g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}$$\end{document} be a semisimple Lie algebra, ϑ∈Aut(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta \in \textsf{Aut}({\mathfrak g})$$\end{document} a finite order automorphism, and g0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}_0$$\end{document} the subalgebra of fixed points of ϑ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document}. Recently, we noticed that using ϑ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} one can construct a pencil of compatible Poisson brackets on S(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}({\mathfrak g})$$\end{document}, and thereby a ‘large’ Poisson-commutative subalgebra Z(g,ϑ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {Z}}({\mathfrak g},\vartheta )$$\end{document} of S(g)g0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}({\mathfrak g})^{{\mathfrak g}_0}$$\end{document}. In this article, we study invariant-theoretic properties of (g,ϑ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathfrak g},\vartheta )$$\end{document} that ensure good properties of Z(g,ϑ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {Z}}({\mathfrak g},\vartheta )$$\end{document}. Associated with ϑ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} one has a natural Lie algebra contraction g(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}_{(0)}$$\end{document} of g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}$$\end{document} and the notion of a good generating system (=g.g.s.) in S(g)g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}({\mathfrak g})^{\mathfrak g}$$\end{document}. We prove that in many cases the equality indg(0)=indg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{ind\,}{\mathfrak g}_{(0)}=\mathrm{ind\,}{\mathfrak g}$$\end{document} holds and S(g)g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}({\mathfrak g})^{\mathfrak g}$$\end{document} has a g.g.s. According to V. G. Kac’s classification of finite order automorphisms (1969), ϑ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} can be represented by a Kac diagram, K(ϑ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}(\vartheta )$$\end{document}, and our results often use this presentation. The most surprising observation is that g(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}_{(0)}$$\end{document} depends only on the set of nodes in K(ϑ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}(\vartheta )$$\end{document} with nonzero labels, and that if ϑ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta $$\end{document} is inner and a certain label is nonzero, then g(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}_{(0)}$$\end{document} is isomorphic to a parabolic contraction of g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak g}$$\end{document}.

1 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 [19,Sect. 6.1] for some examples. Note also that the centre of U(g) h is described in [10, 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 [4,5], 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  • what are the properties of PC subalgebras Z obtained?
• is it possible to quantise Z, i.e., lift it to U(g)?
In recent articles [19][20][21], we develop new methods for constructing { , } and for studying the corresponding PC subalgebras Z.
(A) In [19], 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 [21]. The latter heavily relies on Invariant Theory of ϑ-groups developed by Vinberg in [24].
(B) In [20], 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 both cases, we get two compatible linear Poisson brackets { , } and { , } such that { , } = { , } + { , } is the initial Lie-Poisson structure and study the pencil of Poisson brackets where { , } ∞ = { , } . Each bracket { , } t provides a Lie algebra structure on the vector space g, denoted by g (t ) . The brackets with t ∈ k * :=k\{0} comprise Lie algebras isomorphic to g = g (1) , while the Lie algebras g (0) and g (∞) are different. Since both are contractions of the initial Lie algebra g, we have ind g (0) ind g and ind g (∞) ind 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 [19,21] 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 Sect. 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 [21]. It is devoted to invariant-theoretic properties of a Z m -graded simple Lie algebra g. Interest in these properties 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. I nt 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 Kac [7], and our results often invoke the Kac diagram of ϑ. We refer to [24, §8], [25,Chap. 3,§ 3] and [8,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 Sects. 2.3, 4, and 5. Actually, Kac's classification stems from the study of Z-gradings of "his" infinitedimensional Lie algebras [7]. Our recent results on g (0) and Z(g, ϑ) have applications to the infinite-dimensional case, too [21,Sect. 8]. However, in this article, we do not refer explicitly to Kac-Moody algebras, which agrees with the approach taken in [25].
It is known that ind g (0) = ind g, if m = 2 [16] or g 1 contains regular elements of g [17]. 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 Sect. 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 Sect. 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 [18], 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 ∈ I nt 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 ϑ-generalisation 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 [15, § 3]. In particular, if a connected component of Aut(g) contains elements of order m, then it contains a unique G-orbit of Nregular 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}.

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. Quite often 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 [19,20].
Recall that Z(g, ϑ) is generated by the centres Z t with t ∈ P reg .
It is convenient to introduce the PC subalgebra Z × = alg Z t | t ∈ k\{0} ⊂ Z(g, ϑ), 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 [21,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. [21,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 ). . . , 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.  . . , H l be a set of ϑ-generators in S(g) g . Then  . . , 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 . 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 # ∈ Aut f (g)
A pair (g, ϑ) is decomposable, if g is a direct 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 Steinberg [23,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 ⊂ + 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 , 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 [8, § 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 . The index of ϑ ∈ Aut f (g) is the order of the image of ϑ in Aut(g)/I nt(g), i.e., the order of the corresponding diagram automorphism. . . , 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

The inner periodic automorphisms
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 Sect. 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 = ∪ {α 0 }. If n i = 1 for i 1, then the subdiagram without the i-th node is isomorphic to D(g) and \{α i } is another set of simple roots in . Hence any node ofD(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 graphD(g).
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: 1) and the latter occurs only for g = so 8 . In all cases, 0 is a simple Lie algebra and each g 2 as g σ -modules and g (σ ) 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 Kac diagrams of outer periodic automorphism are supported on the twisted affine Dynkin diagrams of index 2 and 3, see [24, § 8] and [25, Table 3]. Such a diagram has r + 1 nodes, where r = rk g σ , certain r nodes comprise the Dynkin diagram of the simple Lie algebra g σ , and the additional node represents the lowest weight −δ 1 of the g σ -module g (σ ) 1 . Write δ 1 = r i=1 a i ν i and set a 0 = 1. Then the (r + 1)-tuple (a 0 , a 1 , . . . , a r ) yields coefficients of linear dependence for −δ 1 , ν 1 , . . . , ν r .
The subalgebras g σ and g σ -module g (σ ) 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 {ϕ i } for g σ follows [25, Table 1].
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. Let p = ( p 0 , p 1 , . . . , p r ) be an (r + 1)-tuple such that p = (0, 0, . . . , 0) and gcd( p 0 , p 1 . . . , p r ) = 1. The Kac diagram of ϑ = ϑ( p) is the required twisted affine diagram equipped with the labels ( p 0 , p 1 , . . . , p r ) over the nodes. Then m = |ϑ( p)| = ord(σ )· r i=0 a i p i . Similar to the inner case, the Z m -grading g = m−1 i=0 g i corresponding to ϑ = ϑ( p) is defined by the conditions that In Sect. 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) .

The description of g 0 and g 1 via the Kac diagram of #
Let p 0 , p 1 , . . . , p l be the Kac labels of ϑ ∈ I nt f (g). Then the subdiagram of nodes inD(g) 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 [24,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: • g 0 = g ϑ is semisimple; • g 1 is a simple g 0 -module; • K(ϑ) has a unique nonzero label.

Example 2.4 Take the automorphism of D 4 of index 3 with Kac labels
Then |ϑ| = 3(1 + 1) = 6, G 0 = SL 2 × T 1 , and 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 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 , 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;
Let us record the following simple fact.

Lemma 3.2 It suffices to verify Conjecture 3.1 for the semisimple Lie algebras.
Proof Write g = s ⊕ c, where c is the centre of g and s = [g, g]. Then 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 : . Having identified g * and g * (0) as vector spaces, we may regard ξ x as an element of g * (0) . Then (g (0) ) ξ x 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) ) ξ x coincide.
(ii) Moreover, the Lie algebra g x is ϑ-stable and its ϑ-contraction (g x ) (0) is isomorphic to (g (0) ) ξ x as a Lie algebra.
Proof (i) Since the Lie algebra g (0) is N-graded, (g (0) ) ξ x 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) ) ξ x ∩ 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).
where the last equality is the celebrated Elashvili conjecture proved via contributions of many people, see [2]. Then Lemma 3.3 applies.
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 [24, § 1.4] implies that all elements of g 1 are nilpotent. Actually, the 'base' can be achieved in just one step. Recall from [24] that a Cartan subspace of g 1 is a maximal subspace c consisting of pairwise commuting semisimple elements. By [24, § 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 [24, § 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 [24, § 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 .
Proof If x ∈ g 1 is semisimple, then the action (G x 0 : g x 1 ) is again stable. Therefore, for a generic semisimple x ∈ c ⊂ g 1 , the induction step provides the semisimple Lie algebra .
Take ϑ with the following Kac diagram 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 Sect. 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 ∈ I nt f (g), the ϑ-contraction g (0) is isomorphic to a parabolic contraction of g. Then comparing the results obtained earlier for parabolic contractions [18] and ϑ-contractions [21] yields new knowledge in both instances. First, we need an explicit description of ϑ ∈ I nt 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.
As in Sect. 2.3, l = rk g, = {α 1 , . . . , α l }, δ = l i=1 n i α i ∈ + is the highest root, n 0 = 1, and m = |ϑ| = l i=0 p i n i = p 0 + l i=1 p i n i . The labels ( p 1 , . . . , p l ) determine the Z-grading g = j∈Z g( j) such that t ⊂ g(0) and we see that the root space g γ belongs to g(d(γ )). We say that d(γ ) is the (Z, ϑ)-degree of the root γ . For this Z-grading, we have is the nilradical of an opposite parabolic subalgebra, and g = p ⊕ n − . In this setting, one has d(β) d(δ) for any β ∈ + and The Z m -grading associated with ( p 0 , p 1 , . . . , p l ) is obtained from this Z-grading by "glueing" modulo m. That is, for j = 0, 1, . . . , m −1, we set g j = k∈Z g( j +km). The resulting decomposition g = m−1 j=0 g j is the Z m -grading associated with ϑ = ϑ( p 0 , . . . , p l ). It follows from (4.1) that g i = g(i) ⊕ g(i − m) for i = 1, 2, . . . , m − 1 (the sum of at most two spaces) and g 0 = g(−m) ⊕ g(0) ⊕ g(m) (at most three spaces). Moreover, g(0) = g 0 if and only if d(δ) < m, i.e., 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 Inönü-Wigner contraction of g, see [20,Sect. 2]. Here the superscript "ab" means that the hmodule g/h is an abelian ideal of this semi-direct product. Let h = p be a standard parabolic subalgebra associated with . Then g/p can be identified with n − as a vector space, and Inönü-Wigner contractions of the form p (n − ) ab , which have been studied in [18], are called parabolic contractions of g.

Theorem 4.1
Suppose that ϑ ∈ I nt 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 is the set of roots of p (resp. n − ).
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 .

Proposition 4.2
If ϑ ∈ I nt f (g) and p i (ϑ) > 0 for some i such that n i = 1, then g (0) is a parabolic contraction of g and ind g (0) = rk g.
Proof If p i (ϑ) > 0 and n i = 1, then using an automorphism ofD(g), i.e., making another choice of b, we can reduce the problem to the case i = 0, see Sect. 2  This leads to the following Corollary 4. 4 If ϑ ∈ I nt 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 ∈ I nt 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 [20,Sect. 4,5]. Namely, (i) tr.deg Z(b, u − ) = b(g), the maximal possible value for the PC subalgebras of S(g); 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 ∈ I nt 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.deg Z(p, n − ) with arbitrary p. 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 ∈ is a root of (l, t) if and only if j ∈ J . Take any ϑ ∈ I nt 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.deg Z(g, ϑ) = b(g) − b(g 0 ) + rk g, see [21,Theorem 3.10].
Thus, this condition refers only to (g 0 ), which is the same for ϑ andθ.

Proposition 4.10
If g = sl l+1 and ϑ ∈ I nt f (g), then g (0) is a parabolic contraction of g and ind g (0) = ind g = l.
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 ∈ I nt f (sl l+1 ) and thereby ind g (0) = ind g for all inner periodic automorphisms.
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   [18]) 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 Sect. 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.
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 γ 1 + · · · + γ i = μ 1 + · · · + μ j . Let 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 ∈ I nt 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 [20,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 for the outer periodic automorphisms of simple Lie algebras. Let ϑ ∈ Aut f (g) be outer, with the associated diagram automorphism σ , see Sect. 2.3. Recall that r = rk g σ and (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: -For the lowest weight −δ 1 of g This defines a structure of a Z-graded g σ -module on g 2 and the Z-grading on the latter is uniquely determined by the condition that [g (σ )

For each g (σ )
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 . 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 )·( p 0 + r i=1 [δ 1 : ν i ]· p i ) = ord(σ )·d 1 (0).

. Moreover, the upper bound m is attained if and only if p
It then suffices to check the inequality d 0 (β) m only for β = δ σ , the highest root in + (g σ ). We do this case-by-case. In all cases, a i ord(σ )·a i = 2a i for all i, whence the assertion.
Here we have where the inequality holds by Lemma 5.1(i). Hence [x, y] (0) in this case. 4. Suppose that x ∈ (g σ ) μ with μ ∈ − (g σ ), while y ∈ m γ is a weight vector of t 0 and an eigenvector of σ . Here we have and [x, y] (0) = 0 by (5.1). 5. Now we consider the case, where both x, y ∈ m are weight vectors of t 0 and eigenvectors of σ . Set b (σ ) 1 and [m, m] ⊂ g σ . By the construction, t (σ ) 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 Suppose that x ∈ b (σ ) 1 and y ∈ u − ∩ g (σ ) 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.

The index of periodic contractions of the orthogonal Lie algebras
In this section, we prove that ind g (0) = ind g for any ϑ ∈ Aut f (g), if g = so N . To this end, we need Vinberg's description of the periodic automorphisms for the classical Lie algebras and related Cartan subspaces in g 1 [24, § 7].
In the rest of the section, we work with g = so N = so(V, B), where V = k N and B is a symmetric non-degenerate bilinear form on V.
If ϑ ∈ Aut(so N ) and |ϑ| = m, then ϑ = ϑ A is the conjugation with a matrix Suppose that A m = I N . Then S = {1, ζ, . . . , ζ m−1 }, and we set b j = dim V(ζ j ) for j = 0, 1, . . . , m − 1. Note that b j = b m− j for j 1. If ϑ A is outer, then N = 2l is even, m is also even, and det(A) = −1. The latter implies that dim V(−1) is odd, hence V(−1) = 0. We see that A m = I N . Since dim V(−1) is odd and dim V is even, b 0 = dim V(1) is also odd and hence b 0 = 0 as well as b m/2 = dim V(−1).

Lemma 6.1
Let ϑ be an outer periodic automorphism of g = so 2l such that the Kac labels of ϑ are zeros and ones. Then g 1 contains a nonzero semisimple element.

Proof
We have ϑ = ϑ A with A ∈ O 2l and det(A) = −1; as above, A m = I N . In [24, § 7.2], Vinberg gives a formula for rk (g 0 , g 1 ) (i.e., the dimension of a Cartan subspace in g 1 ) in terms of the A-eigenspaces in V. In the present setting, we have the so-called automorphism of type I, and then rk (g 0 , g 1 ) = min{b 0 , b 1 , . . . , b m/2 }. We already know that b 0 , b m/2 1.
The spectrum of A in V shows that the centraliser of A in so 2l 2 V is On the other hand, we can use the Kac diagram K(ϑ) and the hypothesis that the labels does not exceed 1. Here g σ = so 2l−1 , r = l − 1, and the twisted affine Dynkin diagram D Since m = |ϑ| = ord(σ ) l−1 i=0 p i (ϑ)a i = 2 l−1 i=0 p i (ϑ) is even and p i (ϑ) 1, the Kac diagram contains m/2 nonzero labels. This implies that K(ϑ) is of the following form: e e e e e e e e e e e e · · · · · · · · · · · · · · · < > where the zero Kac labels are omitted and k = (m/2) − 1. According to the description of g 0 via the Kac diagram (Sect. 2.4), we obtain here Hence Thus, b j 1 for all j and hence rk(g 0 , g 1 ) 1, i.e., g 1 contains nonzero semisimple elements.

Lemma 6.2
Let ϑ be an inner periodic automorphism of g = so N such that p i (ϑ) ∈ {0, 1} for all i. Furthermore, assume that p i (ϑ) = 0 for all i such that n i = 1, i.e., Then g 1 contains a nonzero semisimple element.
We have (n 0 , n 1 , . . . , n l−1 , n l ) = (1, 1, 2, . . . , 2, 1, 1) in type D l , (1, 1, 2, . . . , 2) 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: · · · · · · · · · · · · · · · 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 [24, § 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: e e e e e e e e e e · · · · · · · · · · · · · · · > Proof We argue by induction on N + m with m = |ϑ|. If m 3, then the statement holds by Proposition 3.6 and [16]. 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 [24, § 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.

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, ifθ is "close" to an N-regular automorphism (see Def. 3), thenθ also admits a g.g.s.
As before, we assume that ϑ ∈ Aut f (g), |ϑ| = m, and ζ = m √ 1 is a primitive root of unity. Let H 1 , . . . , H l be a set of ϑ-generators in S(g) g and deg H j = d j . We have ϑ(H j ) = ε j H j and ε j = ζ r j for a unique r j ∈ {0, 1, . . . , m − 1}.
Following [15,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)/I nt(g) (denotedθ), 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 andθ. We say that the tuple (|ϑ|, k) is the datum of a periodic automorphism ϑ.
If F ∈ k[g] G , then F| g 1 ∈ k[g 1 ] G 0 . However, the restriction homomorphism is not always onto. 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 [24, § 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[g 1 ] G 0 + in k[g 1 ] G 0 . By a result of Hilbert (1893), this implies that (For a short modern proof of Hilbert's result, we refer to [9, Theorem 2].) (ii) If deg H j = d j and H j (x) = 0 for some x ∈ g 1 , then 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 [15, Section 3]: Theorem 7.2 If ϑ is N-regular and |ϑ| = m, then (ii) the dimension of a generic stabiliser for the G 0 -action on g 1 equals k 0 .
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 (Sect. 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: where i = (i 0 , i 1 , . . . , i m−1 ), i 0 + i 1 + · · · + i m−1 = d j , and 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 andθ, 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 [26]. (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   There is a strong constraint on the Kac labels of N-regular inner automorphisms.

Theorem 7.10
Suppose that ϑ ∈ I nt f (g) is N-regular. Then for all i such that n i > 1; (ii) if p i (ϑ) > 1 for some i such that n i = 1, then p j (ϑ) = 1 for all other j.
Proof Let O reg be the G-orbit of regular nilpotent elements. By hypothesis, O reg ∩ g 1 = ∅.
(i) Suppose that p j (ϑ) > 1 for some j. Then g 1 ⊂ N [24, § 8.3] (this also follows from the construction of the Z m -grading in Sect. 4). The subdiagram ofD(g) without the j-th node gives rise to the regular semisimple subalgebraḡ ⊂ g with a set of simple roots ( \{α j }) ∪ {−δ}. Since p j (ϑ) > 1, the induced Z m -gradingḡ = i∈Z mḡ i has the property thatḡ 1 = g 1 . Hence O reg ∩ḡ = ∅. On the other hand,ḡ is the fixed-point subalgebra ofθ ∈ I nt f (g), whereθ is defined by the Kac labels p j (θ) = 1 and p i (θ) = 0 for all other i. Hence |θ| = n j . If n j > 1, thenθ is a non-trivial automorphism of g such that O reg ∩ gθ = ∅, which is impossible. Indeed,θ = I nt(x) for some non-central semisimple x ∈ G and x ∈ G e for e ∈ O reg ∩ gθ . But G e (e ∈ O reg ) contains no non-central semisimple elements. Thus, if p j (ϑ) > 1, then n j = 1 andḡ = g.
We assume below thatH 1 , . . . ,H l is a set ofθ-generators. It is shown in Proposition 7.1 that if j / ∈ P m−1 , thenH j |g 1 = 0. On the other hand, sinceH 1 , . . . ,H l is a g.g.s. with respect toθ, one has Therefore,H • j = (H j ) i with i = (0, . . . , 0, d j ). HenceH • 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 takeH j | g 1 = ψ 1 (H j ) and then consider it as function on the whole of g via the projection g → g 1 .
(2) Althoughθ admits a g.g.s. (Theorem 7.12), we do not know in general whether thẽ ϑ-contraction of g has the same index as g.

How to determine K(#) for N-regular inner automorphisms
We provide some hints that are sufficient in most cases.
• Suppose that m < h.
-Since p i (ϑ) ∈ {0, 1} (Corollary 7.11), it suffices to determine the subset J ⊂ {0, 1, . . . , l} such that p j = 1 if and only if j ∈ J . The obvious condition is that j∈J n j = m. If there are several possibilities for such J , then one can compare dim g 0 and dim g 1 obtained from these J with those required by Proposition 7.4. -Foranym ∈ N, there is an explicit construction of an N-regular inner ϑ with |ϑ| = m.
Let g = i∈Z g(i) be the standard Z-grading. This means that t ⊂ g(0) and g(1) = α∈ g α . Then g γ ⊂ g(ht(γ )) for any γ ∈ , where ht(γ ) = α∈ [γ : α]. Here O reg ∩ g(1) is dense in g(1). Hence glueing this Z-grading module m yields the unique, up to G-conjugacy, N-regular ϑ of order m. For m < h, this construction does not allow us to see the Kac labels of ϑ. Nevertheless, one easily determines g 0 , because the root system of [g 0 , g 0 ] is (m) = {γ ∈ | ht(γ ) ∈ mZ}. This gives a strong constraint on possible subsets J .
-To realise that ϑ is not N-regular, one can use Theorem 7.2(i). That is, if k[g 1 ] G 0 has a free generator of degree that does not belong to {d j | j ∈ P m−1 }, then ϑ cannot be N-regular.
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