LINEAR AUTOMORPHISMS THAT ARE SYMPLECTOMORPHISMS

STANISŁAW JANECZKO & ZBIGNIEW JELONEK

Abstract. Let K be a field of real or complex numbers. Let (X ≅ K^{2n}, ω) be a symplectic space and take 0 < k < n, N = ( \begin{matrix}2n \\ 2k \end{matrix} ). Let L_{1}, ..., L_{N} ⊂ X be 2k−dimensional linear subspaces, which are in a sufficiently general position. We show that if F : X → X is a linear automorphism, which preserves the form ∧^{k}ω on all subspaces L_{1}, ..., L_{N}, then F is an ϵ_{k}−symplectomorphism (i.e. F^{∗}ω = cω, where c^{k} = 1). In particular if K = ℝ and k is odd then F must be a symplectomorphism. We observe also that the set A_{l,2r} of all l-dimensional linear subspaces on which the form ω has rank ≤ 2r is linear in the Grassmannian G(l, 2n), i.e., there is a linear subspace L such that A_{l,2r} = L ∩ G(l, 2n). In particular the set A_{l,2r} can be computed effectively.

1. Introduction.

Let K be a field such that K = ℝ or K = ℂ. Let (X, ω) be a symplectic space, i.e., X ≅ K^{2n} is vector space and ω is a bilinear, non-degenerate skew-symmetric form. The symplectic component of a linear subspace L ⊂ X is defined as the subspace

L^{ω} = {x ∈ X : ω(x, y) = 0∀_{y∈L}}.

This space may not be transversal to L. A subspace L ⊂ X is called: isotropic if L ⊂ L^{ω}, coisotropic if L^{ω} ⊂ L, symplectic if L ∩ L^{ω} = {0} and Lagrangian if L^{ω} = L. L is symplectic if and only if ω |_{L} is non degenerate form. For any subspace L we have dimL + dimL^{ω} = dimX and (L^{ω})^{ω} = L. For (X, ω) there exists a basis, called the Darboux basis, u_{1}, …, u_{n}, v_{1}, …, v_{n}, such that

ω(u_{i}, u_{j}) = ω(v_{i}, v_{j}) = 0, ω(u_{i}, v_{j}) = δ_{ij}.

If L ⊂ X is a subspace, then there is a basis u_{1}, …, u_{k}, v_{1}, …, v_{k}, w_{1}, … w_{l} of L such that ω |_{L} (u_{j}, v_{k}) = δ_{jk} and all other pairings ω |_{L} (•, •) vanish. This basis extends to a Darboux basis for (X, ω) and the integer 2k is the rank of ω |_{L} .

We say that a linear automorphism F : X → X is a symplectomorphism (or symplectic on X) if F^{∗}ω = ω, i.e. , if ω(x, y) = ω(F (x), F (y)) for every x, y ∈ X. If L ⊂ X is a linear subspace, then we say that F is symplectic on L if ω(x, y) = ω(F (x), F (y)) for every x, y ∈ L. The group of authomorphisms of (X, ω) is called the symplectic group and is denoted by Sp(X, ω) or in Darboux basis we write Sp(2n, ℝ) as real 2n × 2n matrices A, which satisfy A^{T} J_{0}A = J_{0}, where J_{0} is 2n×2n matrix of ω in Darboux form. If we identify ℝ^{2n} with ℂ^{n} in the usual way ℝ^{2n} ∋ (x, y) ⟶ x + iy, then multiplication by J_{0} in ℝ^{2n} corresponds to multiplication by i in ℂ^{n}. And with this identification the complex linear group GL(n, ℂ) is a subgroup of GL(2n, ℝ) and the unitary group U(n) is a maximal, compact subgroup of Sp(2n, ℝ); U(n) = Sp(2n, ℝ) ∩ O(2n, ℝ).

Date: October 15, 2002.

1991 Mathematics Subject Classification. 51N10, 51N20, 15A04.

Key words and phrases. linear mappings, symplectic space, symplectomorphism.

The second author was partially supported by the KBN grant number 2PO3A 01722.

If L is a subspace of (X, ω), then obviously the form ω |_{L} and its exterior powers (volume form) are symplectic invariants (cf. [1]). The natural questions we may formulate now are following:

1. To how extend such linear subspaces data preserved, determine the group of interesting transformations of symplectic space?

2. Is this possible to collect such partial ”information” from the family of subspaces and construct the complete system of invariants for symplectic group?

In this paper, using the methods and ideas introduced in [4], we establish the structural data on subspaces and find the groups of automorphisms defined by them. The criteria for symplectomorphisms are confirmed by the geometrical intuition coming from the standard symplectic geometry. While symplectic structures naturally arise in diverse contexts such as Hamiltonian mechanics, field theory, differential and algebraic geometry we focus on the motivation to extend the considerations including the time reversible (including antisymplectomorhisms) Hamiltonian systems (cf. [6]) and composite systems with weakened phase space structure.

Let N = ( \begin{matrix}2n \\ l \end{matrix} ) and G = G(l, 2n) ⊂ ℙ^{N−1} denote a Grassmannian, which parame-

terizes all l−dimensional linear subspaces of X. Let {L_{1}, ..., L_{m}} ⊂ G(l, 2n). We say that subspaces L_{1}, ..., L_{m} are co-planar, if there is a hyperplane Λ ⊂ ℙ^{N−1} which contains all of them.

If subspaces L_{1}, ..., L_{m} are not co-planar, then we can always choose a sub-family L_{1}, ..., L_{k} of not co-planar subspaces with k = N and conversely it is easy to construct a collection L_{1}, ..., L_{N} which is not co-planar. Hence we can always assume that m = N. We prove:

Theorem. Let (X, ω) be a symplectic space of dimension 2n and let F : X → X be a linear automorphism. Let 0 < k < n be a natural number. Assume, that F preserves the form ∧^{k}ω on a collection L_{1}, ..., L_{N} of 2k−dimensional subspaces, which are not co-planar. Then F is an ϵ_{k}−symplectomorphism. (i.e., F^{∗}ω = ϵω, where ϵ^{k} = 1). In particular if K = ℝ and the number k is odd, then F is a symplectomorphism.

Corollary. Let (X, ω) be a symplectic space of dimension 2n and let F : X → X be a linear automorphism. Let 0 < k < l ≤ n be natural numbers such that (k, l) = 1. Assume, that F preserves forms ∧^{k}ω and ∧^{l}ω. Then F is a symplectomorphism.

We observe also, that the following is true:

Theorem. Let (X, ω) be a 2n-dimensional symplectic space and let 0 < l < 2n be a natural number. The set A_{l} of all l-dimensional linear subspaces on which the form ω degenerates (i.e., has not a maximal rank) is co-planar in the Grassmannian G(l, 2n) ⊂ ℙ^{N−1}. More precisely, if A_{l,2r} denote the set of all l-dimensional linear subspaces on which the form ω has rank ≤ 2r, then the set A_{l,2r−2} is co-planar in the set A_{l,2r}, i.e., there is a linear projective subspace L ⊂ ℙ^{N−1} such that A_{l,2r} ∩ L = A_{l,2r−2}. Moreover, the set A_{l,2r} is an irreducible algebraic variety and we can find effectively equations of this set.

At the end we introduce a 2k−dimensional symplectic volume as svlol_{2k}(v_{1}, ..., v_{2k}) = det[ω(v_{i}, v_{j})] and we show the following:

Theorem Let (X, ω) be a symplectic space of dimension 2n and let F : X → X be a

linear automorphism. Let 0 < k < n and M = 2 ( \begin{matrix}2n \\ 2k \end{matrix} ) . Assume, that F preserves 2k

dimensional symplectic volume, on a collection L_{1}, ..., L_{M} of 2k−dimensional subspaces, which are not two-co-planar. Then F is an ϵ_{2k}−symplectomorphism.

2. The case of k = 1.

Let (X, ω) be a symplectic space, i.e., X ≅ K^{2n} is vector space and ω is a bilinear, non-degenerate skew-symmetric form.

Let L ⊂ X be a 2−dimensional linear subspace. If vectors v_{1}, v_{2} form a basis of L, then the line K(v_{1} ∧ v_{2}) is uniquely determined by L and it does not depend on the basis v_{1}, v_{2}. This line determines a unique point Ψ(L) in the Grassmannian G(2, 2n) ⊂ ℙ^{N−1}. Let us recall the notion of co-planar spaces: Definition 2.1. Let {L_{1}, ..., L_{N}} be affine 2−dimensional subspaces of X. We say that subspaces {L_{1}, ..., L_{N}} are co-planar, if points {Ψ(L_{1}), ..., Ψ(L_{N})} ⊂ G(2, 2n) are co-planar, i.e., if there is a hyperplane Λ ⊂ ℙ^{N−1} containing all points {Ψ(L_{1}), ..., Ψ(L_{N})}. Theorem 2.1. Let X be a symplectic space and let F : X → X be a linear automorphism. Assume, that F is symplectic on a collection L_{1}, ..., L_{N} of 2−dimensional subspaces, which are not co-planar. Then F is a symplectomorphism.

Proof. It is well-known (see e.g., [1]), that we can always choose a suitable vector basis e_{1}, ..., e_{2n} in X, (Darboux basis), such that ω((∑ v_{i}e_{i}), (∑ w_{i}e_{i})) = ∑_{0<i≤n}(v_{i}w_{i+n} − v_{i+n}w_{i}). ω = ∑_{i=1}^{n} e_{i}^{∗} ∧ e_{i+n}^{∗} in the conjugate basis e_{1}^{∗}, ..., e_{2n}^{∗}.

Let us denote by \overline{G(2, 2n)} ⊂ ⋀^{2} X := Y the set of all vectors (v ∧ w), where v, w ∈ X. It is well-known that the set \overline{G(2, 2n)} is not contained in any linear hyperplane in Y.

Let (v_{1}^{i}, v_{2}^{i}) be basis of a linear subspace L_{i}, i = 1, ..., N. It is easy to see that the linear subspaces {L_{1}, ..., L_{N}} are co-planar if vectors {(v_{1}^{i} ∧ v_{2}^{i})}_{i=1,...,N} ⊂ ⋀^{2} X := Y are co-planar in Y.

Let (v_{1}^{i}, v_{2}^{i}) be a basis of a linear space L_{i}, i = 1, ..., N and take u_{i} = v_{1}^{i} ∧ v_{2}^{i}. Now, let us consider a mapping R := ⋀^{2} F : Y → Y. In the space Y we have the basis (e_{ij} = e_{i} ∧ e_{j}, 0 < i < j ≤ 2n). For a vector y = ∑ y_{ij}e_{ij} let η(y) = ∑_{i=1}^{n} y_{ii+n}. Of course η is a linear form on Y.

Observe that η(v ∧w) = ω(v, w). Consequently, the mapping F is symplectic if and only if for every v, w ∈ X we have η(v ∧ w) = η(R(v ∧ w)). However the form η(y) − η(R(y)) is linear on Y and by the assumption it does vanish on the set of vectors u_{i}, i = 1, ..., N. Since the latter set is not co-planar in Y, we have that the form η(y) − η(R(y)) vanishes identically on Y. This means that ω(v ∧ w) = ω(F (v) ∧ F (w)) for every v, w ∈ X, i.e., the mapping F is a symplectomorphism. □

From the proof we have the following: Corollary 2.1. Let (X, ω) be a 2n-dimensional symplectic space. Then the set of all 2-dimensional linear subspaces on which the form ω degenerates is co-planar in the Grassmannian G(2, 2n).

Proof. Indeed, this set is given by the equation {y ∈ G(2, 2n) : η(y) = 0}. □

Remark 2.1. The Lagrangian Grassmann manifold Λ_{2} is isomorphic to a non-singular quadric in ℙ^{4}. Moreover, if K = ℝ, then this quadric has the signature (+ + + − −).

Indeed, the Grassmannian G(2, 4) is given by the equation y_{12}y_{34} − y_{13}y_{24} + y_{14}y_{23} = 0 (see e.g., [2], p.211). The manifold Λ_{2} is a section of the Grassmannian G(2, 4) by the hyperplane y_{13} +y_{24} = 0 (cf. the proof of Theorem 2.1). Consequently Λ_{2} in homogeneous coordinates y_{12}, y_{34}, y_{24}, y_{14}, y_{23} has the equation y_{12}y_{34} + y_{24}^{2} + y_{14}y_{23} = 0.

We apply the above results to the case of more general symplectic transformations. Let (X, ω), (Y, μ) be two symplectic spaces, X ≅ 𝕂^{2n}, Y ≅ 𝕂^{2m}, m ≥ n.

Definition 2.2. Let F be a linear subspace of X ⊕ Y, dimF = m + n. We say that F is an ϵ - symplectic relation if F is a Lagrangian subspace of the product symplectic space (X ⊕ Y, μ ⊖ ϵω), i.e. μ ⊖ ϵω |_{F} = 0, where ϵ = ±1, μ ⊖ ϵω = π_{Y}^{∗} μ − π_{X}^{∗}ϵω and π_{X}(π_{Y}) : X ⊕ Y ⟶ X(Y) are the canonical projections.

Remark 2.2. If F is an ϵ - symplectic relation of (X ⊕Y, μ⊖ϵω), m = n, ϵ = +1(−1), and π_{X} |_{F}: F ⟶ X is onto, then F is graph of a symplectomorphism (anti-symplectomorphism). If m > n, ϵ = +1(−1), and the projection π_{X} |_{F} is onto, then ϵ - symplectic relation F is graph of a linear symplectic reduction (anti-reduction). In this case W = π_{Y} (F) is a coisotropic subspace of (Y, μ), i.e. the μ - complementary space W^{μ} is a subspace of W, and f is graph of a linear projection of W onto X along W^{μ}.

Let F be a linear subspace of X ⊕ Y. Let S be an affine subspace of X. We define the image F (S) of S by F, F (S) = {y ∈ Y : ∃_{x∈S}(x, y) ∈ F }.

Theorem 2.2. Let (X, ω), (Y, μ) be two symplectic spaces dimX = 2n, dimY = 2m, m ≥ n. Let F be a linear subspace of the product symplectic space (X ⊕ Y, μ ⊖ ϵω), dimF = m+n. Assume that π_{X} |_{F} is onto and the form μ⊖ϵω is isotropic on a collection of subspaces (L_{1} ⊕ F (L_{1})), …, (L_{N} ⊕ F (L_{N})) ⊂ X ⊕ Y, where L_{1}, …, L_{N} ⊂ X are not co-planar, 2-dimensional subspaces of X. Then

1. F is graph of an ϵ - symplectomorphism if m = n,

2. F is ϵ - symplectic reduction if m > n.

Proof. First we consider the case m = n. By the assumption F is a graph of a linear map ϕ : X ⟶ Y. Let μ and ω be in Darboux form. We see that the corresponding linear forms w̄ on ⋀^{2} X; ω̄(∑ x_{ij}e_{ij}) = ∑_{i=1}^{n} x_{ii+n}, and μ̄ on ⋀^{2} Y, μ̄(∑ y_{ij}h_{ij}) = ∑_{i=1}^{m} y_{ii+m}, where e_{ij} = e_{i} ∧ e_{j}, h_{ij} = h_{i} ∧ h_{j} and {e_{i}}, {h_{i}} are the corresponding Darboux bases on X and Y, define the linear form on ⋀^{2} X,

μ̄(Φ(x)) − ϵω̄(x), (2.1)

where Φ = ⋀^{2} ϕ : ⋀^{2} X ⟶ ⋀^{2} Y, ω̄(v ∧ w) = ω(v, w), μ̄(s ∧ u) = μ(s, u). As in the proof of Theorem 2.1, let (v_{1}^{i}, v_{2}^{i}) be basis of a linear subspace L_{i}, i = 1, …, N, u_{i} = v_{1}^{i} ∧ v_{2}^{i}, By assumption the form μ̄(Φ(x)) − ϵω̄(x) vanishes on the set of vectors u_{i}, i = 1, …, N. Since {u_{i}}_{i=1}^{N} is not co-planar in ⋀^{2} X, we know that the form (2.1) vanishes identically on ⋀^{2} X, i.e. F is Lagrangian so we see that Kerϕ = {0}, otherwise ω has to be degenerated. Obviously ϕ is an ϵ - symplectomorphism. Indeed μ(ϕ(v), ϕ(w)) − ϵω(v, w) = 0 for every v, w ∈ X.

In the case m > n we choose a linear section ψ : X ⟶ Y of the canonical projection π_{X} |_{F}: F ⟶ X and get the vanishing of the form

μ(ψ(v), ψ(w)) − ϵω(v, w) = 0

for all sections ψ and vectors v, w ∈ X. From this, using the previous arguments, we deduce that F is Lagrangian and so an ϵ - reduction consequently. □

3. The general case

Let L ⊂ X be a 2k−dimensional linear subspace. If vectors v_{1}, ..., v_{2k} form a basis of L, then the line K(v_{1} ∧ ... ∧ v_{2k}) is uniquely determined by L and it does not depend on the basis v_{1}, ..., v_{2k}. This line determine a unique point Ψ(L) in the Grassmannian G(2k, 2n) ⊂ ℙ^{N−1}.

Let us recall the notion of co-planar spaces:

Definition 3.1. Let {L_{1}, ..., L_{N}} be linear 2k−dimensional subspaces of X. We say that subspaces {L_{1}, ..., L_{N}} are co-planar, if the points {Ψ(L_{1}), ..., Ψ(L_{N})} ⊂ G(2k, 2n) are coplanar, i.e., if there is a hyperplane Λ ⊂ ℙ^{N−1} containing all points {Ψ(L_{1}), ..., Ψ(L_{N})}.

Definition 3.2. Let (X, ω) be a symplectic space and let F : X → X be a linear automorphism. We say that F is an ϵ_{k}−symplectomorphism if F^{∗}ω = ϵω, where ϵ^{k} = 1. Additionally we say that F is an anti-symplectomorphism if F^{∗}ω = −ω.

Remark 3.1. We can treat an ϵ_{k}−symplectomorphism as a root from symplectomorphism. Indeed, if F is an ϵ_{k}−symplectomorphism, then F^{k} is a symplectomorphism.

Theorem 3.1. Let (X, ω) be a symplectic space of dimension 2n and let F : X → X be a linear automorphism. Let 0 < k < n. Assume, that F preserves the form ∧^{k}ω on a collection L_{1}, ..., L_{N} of 2k−dimensional subspaces, which are not co-planar. Then F is an ϵ_{k}−symplectomorphism. In particular if K = ℝ and the number k is odd, then F is a symplectomorphism.

Proof. Let us denote by \overline{G(2k, 2n)} ⊂ ⋀^{2k} X := Y the set of all vectors (v_{1} ∧ ... ∧ v_{2k}), where v_{1}, ..., v_{2k} ∈ X. It is well-known, that the set \overline{G(2k, 2n)} is not contained in any linear hyperplane in Y.

Let (v_{1}^{i}, ..., v_{2k}^{i}) be a basis of a linear subspace L_{i}, i = 1, ..., N. It is easy to see that linear subspaces {L_{1}, ..., L_{N}} are co-planar if vectors {u_{i} := (v_{1}^{i} ∧...∧v_{2k}^{i})}_{i=1,...,N} ⊂ ⋀^{2k} X := Y are co-planar in Y.

In the space Y we have the basis (e_{i_{1}...i_{2k}} = e_{i_{1}} ∧ ... ∧ e_{i_{2k}}, 0 < i_{1} < i_{2}... < i_{2k} ≤ 2n). For a vector y = x_{1} ∧ ... ∧ x_{2k} ∈ \overline{G(2k, 2n)} let y = ∑ y_{i_{1}...i_{2k}}e_{i_{1}...i_{2k}}. It is easy to see that mappings y_{i_{1}...i_{2k}} treated as functions of vectors x_{1}..., x_{2k} form a basis of the space of all 2k-linear skew-symmetric forms on X. In particular there are numbers a_{i_{1}...i_{2k}} such that ∧^{k}ω(y) = ∑ a_{i_{1}...i_{2k}}y_{i_{1}...i_{2k}}. Take η(x_{1} ∧ ... ∧ x_{2k}) = ∧^{k}ω(x_{1}, ..., x_{2k}). We have η(y) = ∑ a_{i_{1}...i_{2k}}y_{i_{1}...i_{2k}} and consequently we can treat η as a linear form on the whole of Y.

Now, let us consider a mapping R := ⋀^{2k} F : Y → Y. Consequently, the mapping F preserves the form ∧^{k}ω on L_{i} if and only if η(u_{i}) = η(R(u_{i})). However the form η(y) − η(R(y)) is linear on Y and by the assumption it does vanish on the set of vectors u_{i}, i = 1, ..., N. Since the latter set is not co-planar in Y, we have that the form η(y) − η(R(y)) vanishes identically on Y. This means that ∧^{k}ω(v_{1}, ..., v_{2k}) = ∧^{k}ω(F (v_{1}), ..., F (v_{2k})) for every v_{1}, ..., v_{2k} ∈ X, i.e., the mapping F preserves the form ∧^{k}ω. Now we prove the following

Lemma 3.1. Let (X, ω) be a symplectic space. Let W ⊂ X be a 2k−dimensional symplectic subspace of X, i.e. (W, ω |_{W}) is a symplectic space. If

∧^{k}ω(v, w_{1}, …, w_{2k−1}) = 0

for every {w_{1}, …, w_{2k−1}}, w_{i} ∈ W, then v is complementary to W with respect to ω.

Proof. Because W is symplectic we can choose the basis of X, {e_{i}}_{i=1}^{2n}, such that ω is in Darboux form on W, i.e., ω |_{W} = ∑_{i=1}^{k} e_{i}^{∗}∧e_{i+k}^{∗} and W is generated by vectors e_{i}, i ≤ 2k. Moreover, we can assume that the subspace V, complementary to W is generated by vectors e_{i}, i > 2k. Now let v = ∑_{i=1}^{2n} a_{i}e_{i}. Then taking the (2k − 1)−elements e_{1}, …, ”j”, …, e_{2k} and applying to ∧^{k}ω, we get

∧^{k}ω(v, e_{1}, …, ”j”, …, e_{2k}) = ±a_{j} ∧^{k} ω(e_{1}, e_{2}, …, e_{2k}) = ±a_{j}k!,

so from this and by the assumption we have a_{j} = 0, j = 1, ..., 2k. This means that v ∈ V. □

Now consider a general vector subspace V ⊂ X of dimension 2k + 2. Take a general subspace L_{V} ⊂ V of dimension 2k and let K_{V} ⊂ V be a subspace, which is complementary to L_{V} with respect to ω. Since the form ∧^{k+1}ω is proportional to the usual volume form Ω on V and any linear mapping changes the volume form multiplying it by a suitable constant, we have that there is a non-zero constant c such that

Ω(F (v_{1}), ..., F (v_{2k+2})) = cΩ(v_{1}, ..., v_{2k+2}) (3.1)

for every v_{1}, ..., v_{2k+2} ∈ V. Now let us take in (3.1) v_{1}, ..., v_{2k} ∈ L_{V} and v = v_{2k+1}, w = v_{2k+2} ∈ K_{V} .

Computing directly we have (note that v, w are complementary to v_{1}, ..., v_{2k}!):

Ω(v_{1}, ..., v_{2k}, v, w) = aω(v, w) ∧^{k} ω(v_{1}, ..., v_{2k}), (3.2)

where a is a suitable non-zero constant. Since F preserves ∧^{k}ω, using the Lemma 3.1, we can deduce that the subspace F (V_{L}) is complementary to the subspace F (K_{L}) (in the space F (V)). In fact ∧^{k}ω(F (v), F (v_{1}), …, F (v_{2k−1}) = 0 for all v_{1}, …, v_{2k−1} ∈ L_{V} , so the Lemma works.

Hence similarly as before we have

Ω(F (v_{1}), ..., F (v_{2k}), F (v), F (w)) = aω(F (v), F (w)) ∧^{k} ω(F (v_{1}), ..., F (v_{2k}). (3.3)

Now from (3.2) and (3.3) we conclude that

ω(F (v), F (w)) = cω(v, w).

Since spaces K_{V} are not all co-planar in G(2, V) we deduce (similarly as in the proof of Theorem 2.1) that F^{∗}ω = cω on V. Finally, since F^{∗} ∧^{k} ω = ∧^{k}ω = c^{k} ∧^{k} ω, we have c^{k} = 1. Thus there is only finite number of possibilities on c. Let ϵ_{1}, ..., ϵ_{k} be all roots from 1 of order k (if K = ℝ, then we consider 1 and -1 only). Now in the Grassmannian G(2, X) consider subsets S_{i} := {y : ϵ_{i}η(y) = η(∧^{2}F (y))}. Since G(2, X) = ⋃_{i=1}^{k} S_{i} we have that there is an index i such that the set S_{i} is not co-planar in G(2, X). Consequently, it must be ϵ_{i}η(y) = η(∧^{2}F (y)) identically, i.e., F^{∗}ω = ϵ_{i}ω holds on the whole of X. Moreover, if k is odd and K = ℝ, we have ϵ_{i} = 1. □

Corollary 3.1. Let (X, ω) be a symplectic space and let F : X → X be a linear automorphism. Let 0 < l < k ≤ n be natural numbers such that (k, l) = 1. Assume, that F preserves the form ∧^{k}ω and the form ∧^{l}ω. Then F is a symplectomorphism. Remark 3.2. The simplest example of an ϵ_{k}−symplectomorphism F : X → X is the following mapping:

J_{k}(∑_{i=1}^{n}(v_{i}e_{i} + v_{i+n}e_{i+n})) = ∑_{i=1}^{n}(v_{i}e_{i} + ϵ_{k}v_{i+n}e_{i+n}),

in Darboux coordinates, where ϵ_{k} is a primitive root from 1 of order k. Thus if the linear mapping F preserves the form ∧^{k}ω then it is of the form F = J_{l}S, where S is a symplectomorphism and l is a number, which divides k.

From the proof we have also the following: Corollary 3.2. Let (X, ω) be a 2n-dimensional symplectic space. Then the set of all 2kdimensional linear subspaces on which the form ω degenerates (i.e., has not a maximal rank) is co-planar in the Grassmannian G(2k, 2n).

4. Geometry of the set A_{l,2r}.

Let A_{l,2r} ⊂ G(l, 2n) denote the set of all l−dimensional linear subspaces of X on which the form ω has rank ≤ 2r. Of course A_{l,2r} ⊂ A_{l,2r+2} if 2r + 2 ≤ l. We have the following: Theorem 4.1. Let (X, ω) be a symplectic space of dimension 2n. Then the set A_{2k,2k−2} ⊂ G(2k, 2n) ⊂ ℙ^{N−1} is an irreducible algebraic subset of G(2k, 2n) and it linearly span a hyperplane in ℙ^{N−1}. More generally, for r < k the set A_{2k,2r} is also irreducible and linear in the G(2k, 2n), i.e., there is a linear projective subspace L ⊂ ℙ^{N−1} such that A_{2k,2r} = G(2k, 2n) ∩ L. Moreover the set A_{2k,2r} can be computed effectively.

Proof. First assume that K = ℂ. Let A = A_{2k,2k−2} denote the set of all 2k dimensional subspaces of rank < 2k.

By the Andreotti-Salmon Theorem (see e.g., [5]) the embedded Grassmanian G(2k, 2n) ⊂ ℙ^{N−1} is projectively factorial. In particular the Picard group of G(2k, 2n) is generated by the hyperplane section. It can be easily deduced from this that for every hyperplane H ⊂ ℙ^{N−1} the set H ∩ G(2k, 2n) is an irreducible variety which is not contained in a proper linear subspace of H (cf. [3], Lemma 3.18). Let H ⊂ ℙ^{N−1} be a hyperplane, such that A = H ∩ G(2k, 2n) (cf. Corollary 3.2). Thus, by the above, we have that A linearly span the hyperplane H.

Now let us consider the set A_{2k,2r}. It is irreducible since it contains an orbit of the symplectic group as a dense subset U = A_{2k,2r} \ A_{2k,2r−2}. We show that there is a linear subspace L, such that A_{2k,2r} = L ∩ G(2k, 2n). Let us take a sequence of vectors (x_{1}, ..., x_{2k}) ∈ X^{2k}. Let 𝔸 be the matrix which has (coordinates of) vectors x_{i_{1}}, ..., x_{i_{2k−2r}} as rows. Let δ_{s_{1},...,s_{2k−2r}}(x_{i_{1}}, ..., x_{i_{2k−2r}}) denote the main minor of the matrix 𝔸 determined by columns indexed by s_{1}, ..., s_{2k−2r}. Consider all possible skew-symmetric forms of the type

∧^{2r}ω(x_{j_{1}}, ..., x_{j_{2r}})δ_{s_{1},...,s_{2k−2r}}(x_{i_{1}}, ..., x_{i_{2k−2r}}),

where {j_{1}, ..., j_{2r}} ∪ {i_{1}, ..., i_{2k−2r}} = {1, ..., 2k}. It is not difficult to check that they simultaneously vanish only on vectors x_{1} ∧ ... ∧ x_{2k} which belongs to A_{2k,2r}. On the other hand these skew-symmetric functions can be treated as linear forms on ℙ^{N−1} (cf. the proof of Theorem 3.1). Of course we can find these linear forms effectively (as functions of variables y_{i_{1},...,i_{2k}}, cf. the proof of Theorem 3.1). Since we know the equations of the Grassmannian G(2k, 2n) (see [2], p. 211), we can computed the set A_{2k,2r} effectively. This finishes the proof in the case K = ℂ.

Now we sketch the proof for the case K = ℝ. As before define the set A ⊂ G(2k, 2n) as the set of all linear subspaces of X of dimension 2k and of rank < 2k. This set has a stratification on smooth subsets A_{r} = {W ∈ A : rankW = 2r}, where r = 0, 1, ..., k − 1 and A_{i} ⊂ cl(A_{i+1}). Moreover, every such subset is homogeneous with respect to the induced action of the group Spin(2n, K). Let us take a real subspace L ∈ A_{k−1}. Let H ⊂ Sp(2n, ℝ) be a stabilizer of L in the group Sp(2n, ℝ). Thus dim A =dim A_{k−1}= dim Sp(2n, ℝ)−dim H. Now let us complexify X. Let H′ ⊂ Sp(2n, ℂ) be a stabilizer of the subspace L ⨂ ℂ in the group Sp(2n, ℂ).

Let A′ ⊂ G(2k, 2n, ℂ) be the set of all complex 2k−dimensional linear subspaces of X ⨂ ℂ of dimension 2k and of rank < 2k. Then A′ has the same (complex) dimension as the orbit of the subspace L ⨂ ℂ and this dimension is equal to dim Sp(2n, ℂ)−dim H′. But H′ contains a complexification of the subgroup H, thus dim_{ℂ}H ≥ dim_{ℝ}H and consequently dim_{ℂ}A′ ≤ dim_{ℝ}A. But in the complex case we have dim A′ =dim G(2k, 2n, ℂ) − 1. From this we have immediately that dim A =dim G(2k, 2n, ℝ) − 1. This means that the complexification of the set A is the set A′. Thus the set A spans linearly a (real) hyperplane if and only if the set the set A′ spans a (complex) hyperplane. Now we can finish the proof as above. □

From the proof we see that the part of the Theorem 4.1 can be generalized:

Corollary 4.1. Let (X, ω) be a symplectic space of dimension 2n. Let l, r be integral numbers such that l ≤ 2n and 2r < l. Then there is a proper linear subspace L ⊂ ℙ^{N−1} such that A_{l,2r} = L ∩ G(l, 2n). Moreover, we can computed equations of A_{l,2r} effectively. In particular it is true for the Grassmann Lagrangian manifold Λ_{n} = A_{n,0}.

Definition 4.1. Let {L_{1}, ..., L_{N−1}} ⊂ A_{2k,2k−2} be linear 2k−dimensional subspaces of X (of rank < 2k). We say that subspaces {L_{1}, ..., L_{N−1}} are in general position, if they linearly span a hyperplane in ℙ^{N−1}.

Remark 4.1. By the Theorem above every sufficiently general subset {L_{1}, ..., L_{N−1}} ⊂ A_{2k,2k−2} is in general position.Moreover, we can find such subspaces L_{i} with the rank L_{i} = 2k − 2.

A slightly more general version of Theorem 3.1 is:

Theorem 4.2. Let (X, ω) be a symplectic space of dimension 2n and let F : X → X be a linear automorphism. Let 0 < k < n. Let {L_{1}, ..., L_{N−1}} be linear 2k−dimensional subspaces of X of rank < 2k, which are in general position. Assume, that F transforms subspaces L_{1}, ..., L_{N−1} onto subspaces which also have rank < 2k. Then there is a non-zero constant c, such that F^{∗}ω = cω.

Proof. First assume that K = ℂ. Let L be a 2k dimensional subspace of rank 2k. There is a constant a, such that F^{∗}∧^{k}ω = a∧^{k}ω on L. As in the proof of Theorem 3.1, let us denote by \overline{G(2k, 2n)} ⊂ ⋀^{2k} X := Y the set of all vectors (v_{1} ∧ ... ∧ v_{2k}), where v_{1}, ..., v_{2k} ∈ X. Let A = {L_{1}, ..., L_{N−1}}.

Let H ⊂ ℙ^{N} be a hyperplane, such that A_{2k,2k−2} = H ∩ G(2k, 2n). Thus, by the assumption, we have that A linearly span the hyperplane H. Since L ∉ H we can easily deduced that the set B := {L} ∪ A is not co-planar in Y. By the assumption we have for W ∈ B:

aη(u_{W}) = η(R(u_{W})),

where u_{W} ∈ \overline{G(2k, 2n)} ⊂ ⋀^{2k} X := Y is a vector determined by W.

This implies that a linear form aη(y)−η(R(y)) does vanish on the set of vectors u_{W} , W ∈ B. Since the set B is not co-planar in Y, we have that the form aη(y) − η(R(y)) vanishes identically on Y. This means that a ∧^{k} ω(v_{1}, ..., v_{2k}) = ∧^{k}ω(F (v_{1}), ..., F (v_{2k})) for every v_{1}, ..., v_{2k} ∈ X, i.e., F^{∗} ∧^{k} ω = a ∧^{k} ω.

Now we can repeat word by word the proof of Theorem 3.1 to see that there is a constant c, such that F^{∗}(ω) = cω and c^{k} = a. Since F is a linear automorphism, we have c ≠ 0. □

Corollary 4.2. Let (X, ω) be a symplectic space of dimension 2n and let F : X → X be a linear automorphism. Let 0 < k < n. Assume, that F transforms 2k dimensional subspaces of rank 2k − 2 onto subspaces which also have rank < 2k. Then there is a non-zero constant c, such that F^{∗}ω = cω. Corollary 4.3. Let (X, ω) be a symplectic space of dimension 2n and let F : X → X be a linear automorphism. Let 0 < l < 2n and let 2r + 2 ≤ l. Assume, that F transforms the set A_{l,2r} into the same set. Then there is a non-zero constant c, such that F^{∗}ω = cω.

Proof. Let B_{2r+2,2r} denote the set of 2r + 2 dimensional subspaces of rank 2r. Since every subspace from B_{2r+2,2r} is contained in some subspaces from A_{l,2r}, it is easy to see that F transforms the set B_{2r+2,2r} into the same set. Hence we are done by Corollary 4.2. □

Corollary 4.4. Let (X, ω) be a symplectic space of dimension 2n and let F : X → X be a linear automorphism. Let 2 ≤ l ≤ n and assume, that F transforms l−dimensional isotropic (e.g. Lagrangian) subspaces onto subspaces of the same type. Then there is a non-zero constant c, such that F^{∗}ω = cω.

5. Symplectic volume

In this section we introduce and study the symplectic volume. Definition 5.1. Let (X, ω) be a 2n-dimensional symplectic space. Let v_{1}, ..., v_{2k} ∈ X be vectors. By a symplectic 2k-volume of a a collection v_{1}, ..., v_{2k} we mean the number svol_{2k}(v_{1}, ..., v_{2k}) := det([ω(v_{i}, v_{j})]_{1≤i,j≤2k}).

The symplectic volume has the following nice property: Proposition 5.1. Let (X, ω) be a symplectic space and let v_{1}, ..., v_{2k} ∈ X be vectors. Then svol_{2k}(v_{1}, ..., v_{2k}) = 0 if and only if vectors v_{1}, ..., v_{2k} are either linearly dependent or the space (W =< v_{1}, ..., v_{2k} >, ω |_{W}) is not symplectic. Moreover, if vectors v_{1}, ..., v_{2k} are linearly independent then rank[ω(v_{i}, v_{j})]_{1≤i,j≤2k} = rank ω |_{W} .

Proof. If vectors v_{1}, ..., v_{2k} are either linearly dependent, then we can easily see (e.g., by Cramer rules) that det([ω(v_{i}, v_{j})]_{1≤i,j≤2k}) = 0.

Assume that vectors v_{1}, ..., v_{2k} are linearly independent. There is a basis e_{1}, ..., e_{2k} of the subspace (W =< v_{1}, ..., v_{2k} > such that the form ω′ = ω |_{W} has in this basis a canonical form. In particular rank ω′ = rank [ω(e_{i}, e_{j})]_{1≤i,j≤2k}. But there is a non-singular 2k × 2k matrix 𝔸 = [a_{ij}], such that v_{i} = ∑_{j=1}^{2k} a_{ij}e_{j}. Using bilinearity of ω, we have by the direct computation

[ω(v_{i}, v_{j})] = 𝔸[ω(e_{i}, e_{j})]𝔸^{T} .

This finishes the proof. □

Theorem 5.1. Let (X, ω) be a symplectic space and let v_{1}, ..., v_{2k} ∈ X be vectors. Then (∧^{k}ω(v_{1}, ..., v_{2k}))^{2} = (k!)^{2}svol_{2k}(v_{1}, ..., v_{2k}).

Proof. We can assume that K = ℂ. Let us choose the basis {e_{1}, ..., e_{2n}} on X such that ω = ∑ e_{i}^{∗} ∧ e_{i+n}^{∗}. Now for vectors v_{1}, ..., v_{2k} ∈ X let us consider the determinant svol_{2k}(v_{1}, ..., v_{2k}) := det([ω(v_{i}, v_{j})]_{1≤i,j≤2k}). We can treat this determinant as a polynomial function on X^{2k} of degree 2k. Since the matrix [ω(v_{i}, v_{j})] is skew-symmetric we have in fact svol_{2k}(v_{1}, ..., v_{2k}) = P f(v_{1}, ..., v_{2k})^{2}, where P f(v_{1}, ..., v_{2k}) is also a polynomial (the Pfaffian of [ω(v_{i}, v_{j})]).

Let us note that P f(v_{1}, ..., v_{2k})^{2} = svol_{2k}(v_{1}, ..., v_{2k}) = 0 if and only if vectors v_{1}, ..., v_{2k} are either linearly dependent or the space (W =< v_{1}, ..., v_{2k} >, ω |_{W}) is not symplectic (cf. Proposition 5.1). The same property has the polynomial function ∧^{k}ω(v_{1}, ..., v_{2k}). Moreover it is not difficult to check that the latter polynomial is irreducible and of degree k. This means that two polynomials: P f(v_{1}, ..., v_{2k}) and ∧^{k}ω(v_{1}, ..., v_{2k}) have the same degree and the same zero-sets. Since the second polynomial is irreducible we have by the Hilbert Nullstellensatz that there is a constant C such that P f(v_{1}, ..., v_{2k}) = C ∧^{k} ω(v_{1}, ..., v_{2k}). In particular we have C^{2}(∧^{k}ω(v_{1}, ..., v_{2k}))^{2} = svol_{2k}(v_{1}, ..., v_{2k}). Moreover, by the direct computations we have (∧^{k}ω(e_{1}, ..., e_{2k}))^{2} = (k!)^{2} and svol_{2k}(e_{1}, ..., e_{2k}) = 1. Thus C^{2} = (k!)^{−2} and (∧^{k}ω(v_{1}, ..., v_{2k}))^{2} = (k!)^{2}svol_{2k}(v_{1}, ..., v_{2k}). □

To state the next result we need additional notion. Let us recall that if L ⊂ X is a 2k−dimensional linear subspace. and vectors v_{1}, ..., v_{2k} form a basis of L, then the line K(v_{1} ∧...∧v_{2k}) is uniquely determined by L and it does not depend on the basis v_{1}, ..., v_{2k}. This line determine a unique point Ψ(L) in the Grassmannian G(2k, 2n) ⊂ ℙ^{N−1}, where

N = ( \begin{matrix}2n \\ 2k \end{matrix} ).

Definition 5.2. Let {L_{1}, ..., L_{M}} be linear 2k−dimensional subspaces of X. We say that subspaces {L_{1}, ..., L_{M}} are two-co-planar, if the points {Ψ(L_{1}), ..., Ψ(L_{N})} ⊂ G(2k, 2n) are two-co-planar, i.e., if there are hyperplanes Λ_{1}, Λ_{2} ⊂ ℙ^{N−1} such that the union Λ_{1} ∪Λ_{2} contains all points {Ψ(L_{1}), ..., Ψ(L_{M})}.

Remark 5.1. Let us take M = ( \begin{matrix}N \\ 2 \end{matrix} ). Let \{L_{1}, ..., L_{M}\} be linear 2k−dimensional

subspaces of X. It is not difficult to check that if this subspaces are sufficiently general, then they are not two-co-planar.

Theorem 5.2. Let (X, ω) be a symplectic space of dimension 2n and let F : X → X be a linear automorphism. Let 0 < k < n. Assume, that F preserves 2k dimensional symplectic volume, on a collection L_{1}, ..., L_{M} of 2k−dimensional subspaces, which are not two-co-planar. Then F is an ϵ_{2k}−symplectomorphism.

Proof. By Theorem above on every L_{i} we have either F^{∗}(∧^{k}ω) = ∧^{k}ω or F^{∗}(∧^{k}ω) = −∧^{k}ω.

Let us denote by \overline{G(2k, 2n)} ⊂ ⋀^{2k} X := Y the set of all vectors (v_{1} ∧ ... ∧ v_{2k}), where v_{1}, ..., v_{2k} ∈ X.

Let (v_{1}^{i}, ..., v_{2k}^{i}) be a basis of a linear subspace L_{i}, i = 1, ..., N. It is easy to see that linear subspaces {L_{1}, ..., L_{N}} are co-planar if vectors {u_{i} := (v_{1}^{i} ∧...∧v_{2k}^{i})}_{i=1,...,N} ⊂ ⋀^{2k} X := Y are co-planar in Y.

In the space Y we have the basis (e_{i_{1}...i_{2k}} = e_{i_{1}} ∧ ... ∧ e_{i_{2k}}, 0 < i_{1} < i_{2}... < i_{2k} ≤ 2n). For a vector y = x_{1} ∧ ... ∧ x_{2k} ∈ \overline{G(2k, 2n)} let y = ∑ y_{i_{1}...i_{2k}}e_{i_{1}...i_{2k}}. It is easy to see that mappings y_{i_{1}...i_{2k}} treated as functions of vectors x_{1}..., x_{2k} form a basis of the space of all 2k-linear skew-symmetric forms on X. In particular there are numbers a_{i_{1}...i_{2k}} such that ∧^{k}ω(y) = ∑ a_{i_{1}...i_{2k}}y_{i_{1}...i_{2k}}. Take η(x_{1} ∧ ... ∧ x_{2k}) = ∧^{k}ω(x_{1}, ..., x_{2k}). We have η(y) = ∑ a_{i_{1}...i_{2k}}y_{i_{1}...i_{2k}} and consequently we can treat η as a linear form on the whole of Y.

Now, let us consider a mapping R := ⋀^{2k} F : Y → Y. Consequently, F^{∗} ∧^{k} ω = ± ∧^{k} ω on L_{i} if and only if ±η(u_{i}) = η(R(u_{i})). However forms η(y) ± η(R(y)) are linear on Y and by the assumption vectors u_{i}, i = 1, ..., M are contained in the union of kernels of these forms. Since vectors u_{i}, i = 1, ..., M are not two-co-planar in Y, we have that either the form η(y) − η(R(y)) or the form η(y) + η(R(y)) vanishes identically on Y. This means that either ∧^{k}ω(v_{1}, ..., v_{2k}) = ∧^{k}ω(F (v_{1}), ..., F (v_{2k})) for every v_{1}, ..., v_{2k} ∈ X or −∧^{k} ω(v_{1}, ..., v_{2k}) = ∧^{k}ω(F (v_{1}), ..., F (v_{2k})) for every v_{1}, ..., v_{2k} ∈ X , i.e., F^{∗}∧^{k} ω = ±∧^{k} ω.

Now we can repeat word by word the proof of Theorem 3.1 to see that there is a constant c, such that F^{∗}(ω) = cω. Moreover, we have c^{k} = 1 or c^{k} = −1. In both cases c^{2k} = 1, hence F is an ϵ_{2k}−symplectomorphism. □

Corollary 5.1. Let X be a symplectic space of dimension 2n and let F : X → X be a linear automorphism. Let 0 < k < n. Assume, that F preserves 2k dimensional symplectic volume. Then F is an ϵ_{2k}−symplectomorphism. Corollary 5.2. Let X be a real symplectic space of dimension 2n and let F : X → X be a linear automorphism. Let 0 < k < n. Assume, that F preserves 2k dimensional symplectic volume. Then F is either a symplectomorphism or it is an anti-symplectomorphism.

References

[1] V.I. Arnold, A.B. Givental, Symplectic Geometry, in Dynamical Systems IV , Springer Verlag, 1990.

[2] P. Griffiths, J. Harris, Principles of Algebraic Geometry, A Wiley Interscience Publication, New York 1978.

[3] Z. Jelonek, Affine varieties with finite group of automorphisms, Math. Z. 216, 575-591, (1994).

[4] Z. Jelonek, Affine automorphisms that are isometries, Linear Algebra & its Applications, (to appear).

[5] S. Łojasiewicz, Introduction to complex analytic geometry, Birkhäuser Verlag, Berlin 1991.

[6] J. Montaldi, Caustics in time reversible hamiltonian systems, Lecture Notes in Math. 1463, Springer-Verlag 1991, p.266-277.

[7] A. Weinstein, Lectures on Symplectic Manifolds, C.B.M.S. Conference Series 29 (American Mathematical Society, 1977)

(Z. Jelonek) Instytut Matematyczny PAN, Polska Akademia Nauk, Św. Tomasza 30, 31-027 Kraków, Poland, and Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany

  E-mail address: najelone@cyf-kr.edu.pl

(S. Janeczko) Instytut Matematyczny PAN, ul. Śniadeckich 8, 00-950 Warszawa, Poland, Wydział Matematyki i Nauk Informacyjnych, PW, Pl. Politechniki 1, 00-661, Warszawa, Poland

  E-mail address: janeczko@ise.pw.edu.pl