The Kontsevich--Zorich cocycle over Veech--McMullen family of symmetric translation surfaces

We describe the Kontsevich--Zorich cocycle over an affine invariant orbifold coming from a (cyclic) covering construction inspired by works of Veech and McMullen. In particular, using the terminology in a recent paper of Filip, we show that all cases of Kontsevich--Zorich monodromies of $SU(p,q)$ type are realized by appropriate covering constructions.

Let k be a positive integer, and let be an integer at least equal to 3. We denote by R the rotation of C centered at 0 of angle 2π/ , by S the symmetry z → −z w.r.t. the imaginary axis. We write Z m for the standard cyclic group with m elements andZ m for the Z mhomogeneous space of pairs of consecutive elements of Z m .
Let Q even ⊂ C be the closed regular polygon whose vertices are the roots of unity of order . Let Q odd := S(Q even ) = −Q even .
Consider k copies of Q even , indexed by the even elements of Z 2k , and k copies of Q odd , indexed by the odd elements of Z 2k .
The vertices of Q even and Q odd are indexed by A(even, j) = R j (1), A(odd, j) = −A(even, j), ∀j ∈ Z .
For j ∈ Z and i ∈ Z 2k , with parity , we denote by A(i, j), M (i, (j, j + 1)), ℵ ± (i, j) the copies in Q i of A( , j), M ( , (j, j + 1), ℵ ± ( , j). Definition 1.1. The translation surface M k, is obtained from the disjoint union of the Q i , i ∈ Z 2k by identifying through the appropriate translation, for each i ∈ Z 2k , j ∈ Z , the segment ℵ + (i, j) with the segment ℵ − (i + 1, j + 1). We denote by ℵ((i, i + 1), (j, j + 1)) the image of these segments in M k, .
1.2. Basic properties and symmetries of the translation surface M k, . We start by computing the ramification at the singular set, associated to the M (i, (j, j +1)) and A(i, j).
For each j ∈Z , the points M (i, j ), i ∈ Z 2k are identified into a single point M (j ) on M k, where the total angle is 2πk.
On the other hand, when rotating counterclockwise around A(i, j), a sector of angle π( −2) in Q i is followed by a sector of the same angle at A(i − 1, j − 1) in Q i−1 . The points of M k, corresponding to the A(i, j) are therefore naturally indexed by the orbits of the transformation (i, j) → (i − 1, j − 1) on Z 2k × Z . We denote by A(∆) the point of M k, associated to an orbit ∆. The number of such orbits is the greatest common divisor of 2k and . The total angle at such a point A(∆) is 2πk( −2) . We denote by Σ the set of marked points M (j ), A(∆) of M k, , of cardinality + . The genus of M k, is thus given by Observe that and have the same parity.
Remark 1.2. The translation surface M 1, has been first studied by Veech [7]. He shows that it is a Veech surface and that the image of the Veech group in P SL(2, R) is the lattice generated by R and the parabolic element It follows easily that the the subset Σ M ⊂ M 1, consisting of the points M j , j ∈Z is invariant under the group of affine homeomorphisms of M 1, . The same is true of the the subset Σ A consisting of the one (if is odd) or two (if is even) points A(∆). . , x ∈ C and a constant c ∈ C * : see Veech [7]. More generally, M k, is the covering of M 1, given by the Riemann surface z 2k = (x − x 1 ) . . . (x − x ) and the holomorphic one-form cdx/y k : see McMullen [5]. For this reason, we define the Veech-McMullen family F k, of translation surfaces the Riemman surfaces y 2k = (x − x 1 ) . . . (x − x ) equipped with cdx/y k for arbitrary choices of distinct points x 1 , . . . , x ∈ C and constants c ∈ C * . The Veech-McMullen family F 1, for odd, resp. even, is the hyperelliptic component of the stratum H(( − 2 − )/ , . . . , ( − 2 − )/ ) of translation surfaces with conical singularities with total angle 2πk( −2) : see [4]. In general, the Veech-McMullen family F k, is an affine suborbifold of the stratum H(k − 1, . . . , k − 1 , k( − 2 − )/ , . . . , k( − 2 − )/ ) given by a covering construction. Therefore, the Kontsevich-Zorich cocycle over the Teichmüller geodesic flow on F k, is coded by the hyperelliptic Rauzy diagrams with arrows decorated by certain matrices (describing actions on the homology of canonical translation surfaces in F k,l ).
In particular, following the discussion in our previous paper [1], one can associate a Rauzy-Veech group RV (k, ) to F k, . By definition, the matrices in RV (k, ) preserve the natural symplectic intersection form on the absolute homology of the translation surfaces in F k, . Remark 1.4. In this notation, the main result from our previous paper [1] asserts that RV (1, k) is naturally isomorphic to an explicit finite-index subgroup of the integral symplectic group Sp(2g, Z) (where g is the genus of M k, ).
1.4. Statement of the main result. In this paper, we study the structure of the Rauzy-Veech groups of F k, .
This result provides explicit examples showing that all cases of SU (p, q) Kontsevich-Zorich monodromies discussed in Filip's paper [2] actually occur.
1.5. Organization of the paper. In Section 2, we study a decomposition H 1 ⊕ · · · ⊕ H k of the first absolute homology group of M k, . In Section 3, we describe the restrictions RV (k, )| Hr of the Rauzy-Veech group R(k, ) to the summands of the decomposition H 1 (M k, , R) = H 1 ⊕ · · · ⊕ H k in terms of complex matrices on the vector space C equipped with adequate hermitian forms. In Section 3, we reduce the proof of Theorem 1.5 to the investigation of certain groups of complex matrices. Finally, we analyse in Section 4 the relevant groups of matrices in order to establish Theorem 1.5. Remark 1.7. We hope that the arguments in this paper might be useful to study the question of non-continuity of the central Oseledets subspaces of the Kontsevich-Zorich cocycle.
2. HOMOLOGY GROUPS 2.1. Subgroups of affine diffeomorphisms. We now define a finite subgroup G of order 4k of the group of affine diffeomorphisms of M k, .
This two actions commute and they combine to define an action of the product group Z 2k × Z on M k, .
We now define an affine involution σ of M k, whose derivative is S.
The involution σ conjugates the action of every element (r, s) ∈ Z 2k × Z to the action of (−r, −s). Combining the action of σ and the action of Z 2k × Z defines an action of a group G of order 4k . It is the usual dihedral group of order 4k when = 1. The group G is the group of permutations ofZ 2k × Z of the form (i , j) → (r, s) ± (i , j), for r ∈ Z 2k , s ∈ Z .
2.2. Conjugacy classes in G. Let 1 2k , 1 be the standard generators of the corresponding cyclic groups.
• Assume first that is odd. There are k + 1 conjugacy classes of G contained in Z 2k × Z , more precisely 2 elements of order 2 and k − 1 pairs of distinct elements inverse to each other. The other 2 conjugacy classes are those of σ and σ1 2k and have size k . • Assume now that is even. There are k + 2 conjugacy classes of G contained in Z 2k × Z , more precisely 4 elements of order 2 and k − 2 pairs of distinct elements inverse to each other. The other 4 conjugacy classes are those of σ, σ1 2k , σ1 , σ1 2k 1 and have size 1 2 k . 2.3. Irreducible representations of G over R or C. The irreducible representations of G over C are all defined over R and have dimension 1 or 2.
For (r, s) ∈ Z 2k × Z , the representation π r,s sends σ to S, 1 2k to a rotation of angle π r k and 1 to a rotation of angle 2π s . The character χ r,s of π r,s vanishes on G − (Z 2k × Z ) and its values on Z 2k × Z are given by When (r, s) has order > 2, the representation π r,s is irreductible and is conjugated by S to π −r,−s .

2.4.
Irreducible representations of G over Q. Let Π := 2k / be the least common multiple of 2k and , which is also the least common multiple of the elements of G. By a general result of Brauer (see [6, Theorem 24, Section 12.3]), which is easily checked in the case of G, all irreducible representations of G over C are actually defined over the cyclotomic field Q(Π). To see how to group together the π r,s in order to obtain the irreducible representations of G over Q, consider the action of the Galois group (Z Π ) * of Q(Π) over Q on R(2k, ) defined by t.(r, s) = (tr, ts).
The 1-dimensional representations of G are all defined over Q. Similarly, each of the points in R(2k, ) \ R * (2k, ) is fixed by (Z Π ) * . By [6, Theorem 29, section 13.1], we conclude that the irreducible representations of G over Q of dimension > 1 are parametrized by the orbits of the action of (Z Π ) * on R * (2k, ): for an orbit O, the associated representation is π O := ⊕ (r,s)∈O π r,s . Remark 2.1. To compute the orbits of the action of (Z Π ) * on R(2k, ), it is sufficient to consider the action over Z 2k × Z , as −1 ∈ (Z Π ) * . Then one uses the Chinese remainder theorem to split Π into prime powers. Write Π = p p Cp , 2k = p p Ap , = p p Bp , with C p = max(A p , B p ) for each prime p. The action of (Z Π ) * on Z 2k × Z is the product of the actions of ( For any (r , s ) in the order of (r, s), the order of r (resp. s ) is also p a (resp. p b ). When min(a, b) = 0, there is exactly one orbit with these orders. When min(a, b) > 0, the stabilizer of (r, s) as above has cardinality equal to p C−max(a,b) , hence the corresponding orbit has cardinality equal to p max(a,b)−1 (p − 1). As the number of pairs (r, s) with orders (p a , p b ) is equal to p a+b−2 (p − 1) 2 , the number of orbits associated to these values of a, b is equal to p min(a,b)−1 (p − 1).

2.5.
Decomposition of the first relative homology group. The classes of the oriented segments ℵ(i , j ) ( i ∈Z 2k , j ∈Z ) obviously span the first relative homology group Going around the boundary of Q i gives the relation providing 2k − 1 independent relations between the ℵ(i , j ). As these are the only relations between the ℵ(i , j ).
Denote by (e i ), (E i ,j ) the canonical bases of Q Z 2k , QZ 2k ×Z respectively. We equip Q, Q Z 2k , QZ 2k ×Z with structures of G-module by defining We have then an exact sequence of G-modules.
The maps in this exact sequence are as follows. The map from Q to Q Z 2k sends 1 to i e i . The map from Q Z 2k to QZ 2k ×Z sends e i to j (E {i,i+1},j − E {i−1,i},j ). The map from QZ 2k ×Z to H 1 (M k, , Σ, Q) sends E i ,j to the class of ℵ(i , j ).
2.6. Decomposition of the first absolute homology group. The exact sequence of Gmodules for relative homology reads This gives When is odd, it is equal to 1 on all of G − (Z 2k × Z ). This gives in this case When is even, χ Σ M takes the value 0 on the conjugacy classes of σ and σ1 2k , and the value 2 on the conjugacy classes of σ1 and σ1 2k 1 . This gives in this case When is odd, it is equal to 1 on all of G − (Z 2k × Z ). This gives When is even, χ σ A takes the value 0 on the conjugacy classes of σ and σ1 2k 1 , and the value 2 on the conjugacy classes of σ1 and σ1 2k . This gives In the formula χ ab = χ rel − χ Σ M − χ Σ A + 1 we have now computed all the terms in the right-hand side. We obtain χ ab = (r,s)∈R(2k, ),r =0,s =0, r 2k + s / ∈Z χ r,s .
From this character formula, we get
is invariant under the action of (Z Π ) * , which has the same orbits in Z 2k that the action of (Z 2k ) * . But the only integers n > 1 such that (Z n ) * = {±1} are 2, 3, 4, 6. This proves the proposition.
In general, the subspaces H r are given by the eigenspaces of the generator of the group of deck transformations of M k, → M 1, . In particular, the decompostion H 1 (M k, , R) = ⊕ 0<r k H r can be extended to all translation surfaces of the Veech-McMullen family F k, .

2.7.
Relation with the symplectic intersection form. Let χ σ be the 1-dimensional character on G with kernel Z 2k × Z . The intersection form ω on H 1 (M k, , Q) satisfies, for Proposition 2.5. The 2-dimensional summands in the decomposition of the G-module H 1 (M k, , Q) in the last subsection are mutually ω-orthogonal.
Proof. Let E, F be two such distinct summands.The intersection form defines an homomorphism u from E to the dual F * of F . We have to prove that u = 0. If v ∈ E belongs to the kernel of u, then the same is true for g.v, for any g ∈ G. As E is irreducible, this proves that u is either invertible or equal to 0. If u is invertible, the formula above shows that it is an isomorphism of G-modules from E to the tensor product of the contragredient representation of F by χ σ . But this tensor product is isomorphic as G-module to F itself. As E, F are distinct, we conclude that u cannot be invertible, hence u = 0.
For 0 < r k, let H r be the sum of the summands in the decomposition of H 1 (M k, , Q) with characters χ r,s , s varying according to the prescription above. Assume now that 0 < r < k. We equip H r with the complex structure such that defines an hermitian 1 form on H r whose imaginary part is ω.
The signature of this hermitian form is calculated in Subsection 3.6.
3. HYPERELLIPTIC RAUZY DIAGRAMS AND RAUZY-VEECH GROUPS RV (k, ) 3.1. Review of the description of hyperelliptic Rauzy diagrams. In this subsection, we recall the content of Subsection 2.1 of our previous paper [1].
The hyperelliptic Rauzy class R d over A d and the associated Rauzy diagram D d are inductively defined as follows.
The Rauzy class R d contains the central vertex π * = π * (d): The arrows of D d are given by the following one-to-one maps R t , R b from R d to itself: . The elements of R d correspond bijectively to the words in {t, b} of length < d − 1 via the following map W d : let W d (π * (d)) be the empty word, W d (j t (π)) is the word tW d−1 (π) and W d (j b (π)) is the word bW d−1 (π).
One recovers from W d (π) the winners of the arrows starting from π as follows: the winner of the arrow of top type starting from π is the letter d ; similarly, the winner of the arrow of bottom type starting from π is the letter As it turns out, all non-trivial simple loops in R d are elementary: they consist of arrows of the same type. Any such loop γ contains a unique vertex π such that γ passes through π but γ * (π) does not contain any arrow of γ, and, furthermore, π is the vertex of γ such that |W d (π)| = d − 1 − |γ| is minimal. In the sequel, we denote by γ the non-oriented loop at π * (d) defined by γ := γ * (π) * γ * (γ * (π)) −1 .

3.2.
Mapping classes attached to the arrows of D d . In this subsection, we essentially review the content of Section 4 of our previous paper [1]. Given π ∈ R d , denote by M π the canonical translation surface with combinatorial data π whose length data λ can and suspension data τ can are: We obtain M π by identifying parallel sides of an appropriate polygon P π . The set of marked points of M π is denoted by Σ π , and the middle points of the sides of P π together with a base point O π in the interior of P π form a subset Σ * π of M π . The Rauzy-Veech operation associated to each arrow γ : π → π of D d is encoded by the isotopy class of a homeomorphism H γ : (M π , Σ π ∪ Σ * π ) → (M π , Σ π ∪ Σ * π ) constructed in Subsection 4.1 of [1].
In this context, given γ a simple loop in R d , the action of γ as a isotopy class on M π * was computed in Proposition 4.5 of [1]: it is a Dehn twist about the straight line joining the midpoints of the sides of P π * indexed by the letter of A d winning in the loop γ.
Note that the elements of Mod(π * ) can be viewed also as mapping classes on the translation surface M 1, where = d + 1, and, a fortiori, they can be lifted to M k, via the natural projection M k, → M 1, .
The Rauzy-Veech group RV (1, ) was computed in our previous paper [1]: it is isomorphic to a (explicit) finite-index subgroup of Sp(2g, Z) (where g is the genus of M 1, ).
Remark 3.2. In general, the natural projection M k, → M 1, takes H k to H 1 (M 1, , R) in such a way that RV (k, )| H k is isomorphic to RV (1, ), so that RV (k, )| H k is the explicit finite-index subgroup described in Theorem 2.9 of our previous paper [1].

3.3.
Lifting the action of the loops in R d : top case. Let γ, π, γ be as in the previous subsection. Let k be an integer 2. Let M k,d+1 be the surface considered in the first section. We have a canonical projection M k,d+1 → M 1,d+1 . The action of γ , as an isotopy class of the translation surface M 1,d+1 with marked points at the M (j) and the A(δ), was already discussed in the previous subsection. We describe now the lift of this action to M k,d+1 .
Assume first that γ is of top type. Let w : This gives Let 0 < r < k, 0 < s < := d + 1 with r 2k + s = 1; write We have Then we have We also compute the image of V cos (p , r) and V sin (p , r) for p = p. We write p = d − 1 − 2w . Assume first that p < p (i.e., w > w). One has .
When p > p (i.e., w < w), one obtains r)). In summary, for each 0 < r < k, we deduce that • The subspace H r is fixed by the lift of the action of γ (which commutes with the action of Z 2k ); therefore, the Rauzy-Veech group RV (k, ) gives rise to welldefined groups RV (k, )| Hr (obtained by restriction to H r ); • H r is the direct sum of the 2-dimensional subspace generated by V cos (p, r) and V sin (p, r) on which γ acts by a rotation of −π(1 + r k ), and a subspace of codimension 2 on which γ acts by the identity.

3.4.
Lifting the action of the loops in R d : bottom case. In the same setting that in the last subsection, we now assume that γ is of bottom type.
We also compute the image of V cos (p , r) and V sin (p , r) for p = p. We write p = −d + 1 + 2w . Assume first that p > p (i.e., w > w). One has r)), and similarly L b p (V sin (p , r)) = V sin (p , r) + L b p (V sin (p, r)). Thus we see that L b p is the inverse of L t p ! In summary, the group RV (k, )| Hr is generated by the operators L t p | Hr .

3.5.
Formulas for L t p as a complex operator. Let ρ := exp(iπ r k ). The complex structure on H r is given by 1 2k .v = ρv. We have thus For p > p, L t p (V cos (p , r)) = V cos (p , r) − ρ −1 V cos (p, r), and for p < p L t p (V cos (p , r)) = V cos (p , r) − V cos (p, r).
For p > p , we have Proof. Observe first that, although V i (p) is only a relative homology class with nonzero boundary equal to is an absolute homology class so the intersection form ω is well-defined in the formulas of the lemma.
In the second case, it is easy to represent the cycles V i (p) (resp. V i (p )) by paths from M (w + 1, w + 2) (resp. M (w + 1, w + 2)) to M (0, 1) in Q i so that the intersection takes place only at M (0, 1). A direct inspection at this point gives the formula of the lemma.
In the first case, we choose two distinct representations for each V i (p) as paths from M (w + 1, w + 2) to M (0, 1) so that the intersection takes place only at M (w + 1, w + 2) and M (0, 1). Again a direct inspection at these points gives the formula of the lemma.
Using the first part of the lemma, we get (as i x i = 0) We have thus Z(p), Z(p) := −k 1 + cos π r k sin(π r k ) .
We now compute for p > p . We have (using the second part of Lemma 3.3) This gives .
We finally get, for p > p , It is probably nicer to scale the hermitian form by the factor −k 1+cos π r k sin(π r k ) in order to have Z(p), Z(p) = 1 for all p. One has then, for p > p Z(p), Z(p ) = 1 2 (1 − i tan πr 2k ).
Let u := tan πr 2k . Consider the hermitian form on C A d defined by and Observe that A 1 is positive, so we can diagonalize simultaneously A 1 and A 2 . Let indeed ξ := exp 2iπ (recall that = d + 1). Define, for 0 < s < , We have then, for z 1 , . . . , z d ∈ C For s = s , one has and (z s z s ξ s−s + z s z s ξ s −s ) = 0.
On the other hand, for 0 < s < , we have We have therefore In particular, this shows that the hermitian form on H r has the signature described in Theorem 1.5 (as expected from McMullen's paper [5]).

3.7.
Matricial description of RV (k, )| Hr . Our discussion so far is summarized as follows. The group RV (k, )| Hr is generated by the operators L t p , p ∈ A d , given by These operators preserve the hermitian form At this point, we reduced the proof of Theorem 1.5 to the analysis of the group generated by the matrices above.

MATRICES
In this section, we complete the proof of Theorem 1.5 by studying the groups of matrices. Since we do not need anymore to make reference to the homology groups translation surfaces M k, , we are going to rewrite below the formulas from Subsection 3.7 using a slightly more abstract notation.
We denote by A d the set of values of p. • (e p ) is the canonical basis of C A d .

The operators. For
Observe that the inverse L −1 p = L b p is given by i.e. the same formula, changing ζ to ρ and inverting the order on A d .

The invariant sesquilinar form. Let Q α be the sesquilinear form on C
has infinite order if ρ is a root of unity but ρ 4 = 1, Proof. The second assertion is clear. For the first, if ρ is a root of unity but ρ 4 = 1, ρ 6 = 1, there exists a Galois-conjugate ρ of ρ such that ρ + ρ −1 < −1. Then the corresponding Galois-conjugate of the matrix of L b −1 •L t 1 has infinite order. The same is true of the matrix of L b −1 • L t 1 .

4.4.2.
The cases α = 1 6 , 1 4 . In these cases, the sesquilinear form has signature (2, 0). The coefficients of the matrices of the group generated by L 1 and L −1 belong to the ring of integers of the quadratic field Q(ρ) and are bounded, hence the group generated by L 1 and L −1 is finite. Observe that L b −1 • L t 1 is parabolic, i.e satisfies µ = 1, ω = 0.
The operator L b −1 • L t 1 belongs to SU (Q α ) and has infinite order hence the closed (for the usual topology) subgroup generated by L b −1 • L t 1 is a one-parameter group isomorphic to a circle, consisting of those transformations of SU (Q α ) having the same eigenvectors An infinitesimal generator of this one-parameter group is the element X of the Lie algebra su(Q α ) which satisfies X.v ± = ±iv ± , where v ± are the eigenvectors of The coefficients of X are given by For n ∈ Z, write ad(L n −1 )X =: X(n), which is the infinitesimal generator of the previous one-parameter group conjugated by L n −1 . Setting also L n −1 v ± =: v ± (n) =: a ± (n)e −1 + e 1 , we have We now show that the vector fields X = X(0), X(1), X(2) are linearly independent (over C). The sequences s(n) := a + (n) + a − (n) and p(n) := a + (n)a − (n) satisfy the recurrence relations s(n + 1) = −ζs(n) − 2, p(n + 1) = ζ 2 p(n) + ζs(n) + 1 with initial conditions s(0) = −1 − ρ + ρ 2 , p(0) = ρ. We have thus the vector fields X = X(0), X(1), X(2) are indeed linearly independent. The Lie algebra su(Q α ) has dimension 3, hence is generated by X(0), X(1), X (2) .
We conclude that the intersection of SU (Q α ) with the group generated by L 1 and L −1 is dense (for the usual topology) in SU (Q α ).
The operator L b −1 • L t 1 belongs to SU (Q α ) and has eigenvalues λ + > 1 and λ − = λ −1 + . Let v ± be the associated eigenvectors.The Zariski closure of the group generated by L b −1 • L t 1 is the one-parameter group having for infinitesimal generator a vectorfield X satisfying X.v ± = ±v ± . The same calculation than in the previous case shows that X, ad(L −1 )X and ad(L 2 −1 )X span su(Q α ). We conclude that the intersection of SU (Q α ) with the group generated by L 1 and L −1 is dense (for the Zariski topology) in SU (Q α ).

4.5.
The induction step in the generic case.
We also denote by ι p the embedding of C A d−1 into C A d such that ι p (e q ) = e ιp(q) , and by H p the hyperplane of C A d which is the image of this embedding.
From the defining formulas, for all p ∈ A d , q ∈ A d−1 , the hyperplane H p is invariant under L ιp(q) and we have In order to avoid confusion, we denote by Q α the sesquilinear form on C A d−1 denoted by Q α previously, and keep the notation Q α for the sesquilinear form on C A d . Proof. This is clear from subsection 4.2 When ρ d+1 = 1, we denote by H p the 1-dimensional subspace which is the Q α -orthogonal of H p . As Q α is non degenerate and p∈A d H p = {0}, C A d is the direct sum of the H p .
When moreover ρ d = 1, the restriction of Q α to each H p is non degenerate by the lemma. Therefore C A d is the direct sum of H p and H p . From the formulas for L q , q = p, we see that the line H p is point wise fixed under all L q , q = p.

A result on stabilizers.
Proposition 4.3. Let d ≥ 3, Q a non degenerate sesquilinear form on C d , u 1 , u 2 ∈ C d linearly independent vectors such that Q(u 1 ) = Q(u 2 ) = 0. Let SU (Q) be the special unitary group of Q, and , for j = 1, 2, let G j be the stabilizer of u j in SU (Q). Then the smallest subgroup containing G 1 ∪ G 2 is SU (Q).
Proof. Let G be the smallest subgroup of SU (Q) containing G 1 ∪G 2 . Let c be the common value of the Q(u j ). It is sufficient to prove that G acts transitively on {Q(u) = c}. Indeed, assume this is true, and let h be an element of SU (Q); there exists g ∈ G such that g(u 1 ) = h(u 1 ); then g −1 h ∈ G 1 and h ∈ G. To prove that G acts transitively on {Q(u) = c}, we observe that, as {Q(u) = c} is connected, it is sufficient to show that the orbits of G have non empty interior (and so are open) in {Q(u) = c}.
Let u 0 be a vector such that Q(u 0 ) = c. Proof. Recall the following Fact: for p + q ≥ 2 and b = 0, SU (p, q) acts transitively on Assume for instance that < Gu 0 , u 1 > has non empty interior in C. Let W be a non-empty open set which is contained in < Gu 0 , u 1 > and is disjoint from the circle {|z| = Q(u 1 )}. For any w ∈ W , the intersection of {Q(u) = Q(u 1 )} with {< u, u 1 >= w} consists of vectors of the form u = αu 1 + v, with α = w Q(u1) , < v, u 1 >= 0, Q(v) = (1 − |α| 2 )Q(u 1 ). As |α| = 1, it follows from the fact recalled above that the intersection of {Q(u) = Q(u 1 )} with {< u, u 1 >= w} is contained in Gu 0 . Proof. We may assume for instance that From the fact recalled above, the set G 1 v 0 consists of the vectors v orthogonal to u 1 satisfying Q(v) = (1 − |α| 2 )Q(u 1 ). Any linear projection of this set on C has nonempty interior, which proves the assertion of the lemma.
We can now end the proof of the proposition. In view of the two lemmas above, we know that Gu 0 has non empty interior in {Q(u) = c} except perhaps if both < Gu 0 , u 1 > and < Gu 0 , u 2 > are contained in the circle {|z| = c}. We will now prove that this exceptional case is impossible. Write as before u 0 = αu 1 + v 0 , u 2 = βu 1 + v 2 , with < v 0 , u 1 >=< v 2 , u 1 >= 0, v 2 = 0. Exchanging u 1 , u 2 if necessary, we may assume that v 0 = 0. We may also assume that |α| = 1, Q(v 0 ) = 0. Choose a 2-dimensional subspace E of the hyperplane H orthogonal to u 1 with the following properties: • the subspace E contains v 0 .
• The restriction of Q to E is non degenerate.
• The orthogonal projection of v 2 on E is = 0.
Such a choice is possible because the last two conditions are open and dense amongst 2-dimensional subspaces of H containing v 0 . Choose a basis e, f of E such that v 0 = e+f and Q(xe + yf ) = |x| 2 − |y| 2 . For any a, b ∈ C with |a| 2 − |b| 2 = 1, we can find g ∈ G 1 such that g.e = ae + bf, g.f =be +āf.
Therefore any vector of the form ze +zf , with z ∈ C * belongs to G 1 v 0 . Let se + tf = 0 be the orthogonal projection of v 2 on E. Then zs −zt belongs to < G 1 v 0 , v 2 > for any z ∈ C * . As the set {zs −zt, z ∈ C * } contains a straight segment in C, the set < G 1 u 0 , u 2 >= αβc+ < G 1 v 0 , v 2 > is not contained in the circle {|z| = c}.
This concludes the proof of the proposition.

Application.
Proposition 4.6. Assume that d ≥ 3, ρ d , ρ d+1 = 1. Assume also that the intersection of the group generated by the operators L q , q ∈ A d−1 , on C A d−1 with the special unitary group SU (Q α ) is dense (resp. Zariski dense) in SU (Q α ) . Then the intersection of the group generated by the operators L p , p ∈ A d , on C A d with the special unitary group SU (Q α ) is dense (resp. Zariski dense) in SU (Q α ) .
Proof. Let p 1 , p 2 be two distinct elements of A d . Denote by u 1 , u 2 generators of H p1 , H p2 respectively, satisfying Q α (u 1 ) = Q α (u 2 ) = 0. Such a choice is possible because the restrictions of Q α to H p1 , H p2 have the same signature by lemma 4.2. By the assumption of the proposition, for i ∈ {1, 2}, the intersection of the group generated by the operators L p , p ∈ A d , p = p i , with the special unitary group SU (Q α ) is dense (resp. Zariski dense) in the stabilizer G i of u i in SU (Q α ). By Proposition 4.3 , the smallest group containing We thus obtain the conclusion of the proposition .
4.6. The induction step in the non exceptional degenerate case.
4.6.1. The setting. We assume in this subsection that d 4 and that (d + 1)α is an integer. Therefore the sesquilinear form Q α on C A d is degenerate. As 0 < α < 1 2 , dα is not an integer and the sesquilinear form Q α on C A d−1 is non-degenerate. As the restriction of Q α to each hyperplane H p is isomorphic to Q α (Lemma 4.2), the kernel of Q α has dimension 1.
Remark 4.7. In the case d = 3, we must have α = 1 4 ; then the induction hypothesis (see below) is not satisfied.
As Q α is invariant under each L p the kernel of Q α is invariant under the L p . But the eigenvalues of L p are 1 with multiplicity (d − 1) and −ζ with multiplicity 1, and the eigenvector associated to the eigenvalue −ζ is e p , which is not an eigenvector of L q for q = p. We conclude that the kernel of Q α is pointwise fixed by each L p .
We make the following induction hypothesis: on C A d−1 , the intersection of the subgroup generated by the L p , p ∈ A d−1 with the special unitary group SU (Q α ) is dense for the ordinary topology (resp. Zariski dense) in SU (Q α ). 4.6.2. The induction step. As the restriction of Q α to each H p is non-degenerate, the kernel Ce of Q α is not contained in any H p .
Let us denote by SU * (Q α ) the subgroup of GL(C A d ) formed by linear automorphisms which preserve Q α , fix e (and not simply the line Ce) and have determinant 1. If one writes these automorphisms in the basis (e, e 3−d , . . . , e d−1 ) (using that C A d = Ce ⊕ H 1−d ), the matrix takes a block triangular form with g ∈ SU (Q α ). Let D be the subgroup of SU * (Q α ) formed of automorphisms whose matrix in the selected basis satisfies v = 0.
Proof. Let D be such a subgroup . We identify Clearly V (D ) is an additive subgroup of (C A d−1 ) * . Moreover, as  Proof. We first observe that , as N 3, the orbit SU (Q).v 0 of a vector v 0 is equal to Indeed, Witt's theorem implies that the orbit U (Q).v 0 is as stated. As N 3, there exists a vector v 1 in this orbit with at least one coordinate vanishing (in an orthogonal basis for Q) . But then the image of the stabilizer of v 1 (in U (Q)) by the determinant map is the full unit circle. This means that the orbits U (Q).v 1 and SU (Q).v 1 are equal.
Let V be an additive subgroup of C N which is also SU (Q)-invariant. It is sufficient to show that , if V contains a non-zero vector, then V has non-empty interior.
is not constant in a neighborhood of v 0 in V 0 , hence its image contains a non-trivial interval. This implies that V has non-empty interior.
Corollary 4.10. Under the induction hypothesis stated above, the intersection of the subgroup generated by the L p with SU (Q α ) is dense (resp. Zariski dense) in SU * (Q α ).
Proof. Let G be the closure (resp. the Zariski closure) of the intersection of the subgroup generated by the L p , 1 − d p d − 1 with SL(n, C). We have seen earlier that G is contained in SU * (Q α ).
Let G be the closure (resp. the Zariski closure) of the intersection of the subgroup generated by the L p , 3 − d p d − 1 with SL(n, C. We have seen earlier that G ⊂ D. It follows from the induction hypothesis that G = D. As L 1−d • L −1 d−1 has determinant 1 but does not preserve H 1−d , G is not equal to D. It follows from the proposition that G = SU * (Q α ).

4.7.
From the degenerate case to the non-degenerate case. 4.7.1. The setting. We assume in this subsection that d 5 and that dα is an integer. Therefore the sesquilinear form Q α is non degenerate , but the restrictions of Q α to each hyperplane H p , which are isomorphic to Q α , are degenerate. It means that the Q α -orthogonal of H p is a line Cw p contained in H p .
We make the following induction hypothesis: The intersection of the subgroup generated by the L p , p ∈ A d−1 with SL(C A d−1 ) is dense for the ordinary topology (resp. Zariski dense) in the subgroup SU * (Q α ) defined in the previous subsection. 4.7.2. Stabilizers. Let Q be a non-degenerate non-definite sesquilinear form on C N . Let w be a nonzero vector such that Q(w) = 0. Let H be the hyperplane Q-orthogonal to Cw. It contains Cw. Denote by Q the restriction of Q to H, which is degenerate. Choose a vector w such that < w, w >= 1. Denote by H the orthogonal of the plane Cw ⊕ Cw , and by Q the restriction of Q to H , which is non-degenerate. We have direct sums In the next proposition, the group SU * (Q ) was defined in the last subsection.
Proposition 4.11. We have an exact sequence The homomorphism θ from Stab(w) to SU * (Q ) is induced by restriction to H. The exact sequence is not split.
Proof. If a matrix of the form (4.1) is unitary, we must have ω = 1 because the scalar product < w, w > is preserved.
As ω ≡ 1, the homomorphism θ takes values in SU * (Q ). It is onto by Witt's theorem. An elementary computation shows that the kernel of θ is equal to K (see also below).
It remains to show that θ has no section. Assume by contradiction that such a section σ exists. Consider This determines h as a semi-linear isomorphism from (H ) * to H . On the other hand, as σ is a homomorphism, we have is a symmetric function of v, v . As h is antilinear, we should have v(h(v )) ≡ 0. This is not true since h is an isomorphism.
We now obtain in our particular setting: Corollary 4.12. Let p ∈ A d . The intersection with SU (Q α ) of the subgroup generated by the L q , q = p, is Zariski dense in the stabilizer Stab(w p ).
Remark 4.13. Probably one doesn't get the density in the usual topology, even if we started from this form of the induction hypothesis.
Proof. We already know that L q (w p ) = w p for q = p. Therefore the Zariski closure G p of the intersection with SU (Q α ) of the subgroup generated by the L q , q = p is a Zariski closed subgroup contained in Stab(w p ). On the other hand, the induction hypothesis (applied to the restriction of Q α to H p , which is isomorphic to Q α ) implies that restriction to H p induces an homomorphism of G p onto SU * (Q α ).
The kernel of the homomorphism from G p onto SU * (Q α ) is a Zariski closed subgroup of K hence it is either equal to K (in which case G p = Stab(w p )) or to {1}.(If this subgroup is only closed for the usual topology, it could be an infinite discrete subgroup of K). But the second case is impossible since the exact sequence of the proposition is not split.  Proof. Let u ∈ N (w, c). It is sufficient to see that the determinant of an element g ∈ SU (Q) which stabilizes w and u can have any determinant of modulus one. This is clear since the restriction of Q to the orthogonal of < u, w > is non-degenerate and the restriction of g to this subspace is any unitary matrix.
Lemma 4.15. Let F be a nontrivial linear subspace of C N . Assume that the restriction of Q to F is non-degenerate.
(1) If the restriction of Q to F is indefinite, any translate u + F intersects {Q = 0}.
(2) One has now Q(u + f ) = Q(u) + Q(f ) Q(u); the conclusion follows. Proof. Indeed, one has Q(w + zw ) = |z| 2 Q(w ) + 2 (z < w , w >) with < w , w > = 0 as Q is non degenerate and Q(w) = 0. Proof. Let G be the smallest subgroup containing G 1 , . . . , G N . It is sufficient to show that G acts transitively on Q := {Q(u) = 0, u = 0}. As this last set is connected, it is sufficient to show that any orbit of G in Q has non empty interior.
Lemma 4.18. Let u 0 ∈ Q. There exists an index 1 i N such that the image of G.u 0 by the map u →< u, w i > has non empty interior in C.
Let F be the codimension 2 subspace of C N orthogonal to w i , w j . As < w i , w j > = 0, the restriction of Q to F is nondegenerate.
• If the restriction of Q to F is indefinite, for any c i close to c i there exist u 2 ∈ Q such that < u 2 , w i >= c i , < u 2 , w j >= c j . By Lemma 4.14, one has u 2 ∈ G j .u 1 . This proves the assertion of the lemma in this case. • Assume that the restriction of Q to F is positive definite (the negative case is symmetric). Then the restriction of Q to F ⊥ is indefinite. Identify F ⊥ to C 2 through u → (< u, w i >, < u, w j >). We have Q(c i , c j ) 0 by Lemma 4.15. If Q(c i , c j ) < 0, we proceed as in the first case to get the conclusion of the lemma (using again Lemma 4.15). If Q(c i , c j ) = 0, by Lemma 4.16 there exists c j close to c j such that Q(c i , c j ) < 0. Then there exists u 1 ∈ Q such that < u 1 , w i >= c i , < u 1 , w j >= c j . One has u 1 ∈ G i .u 1 from Lemma 4.14. Then the end of the argument is the same than for Q(c i , c j ) < 0.
From the lemma, we may assume c i :=< u 0 , w i > is different from 0 and that a small neighborhood of c i in C is contained in the image of G.u 0 by the map u →< u, w i >. Let u ∈ Q be close to u 0 . There exists u 1 ∈ G.u 0 such that < u 1 , w i >=< u, w i > ( = 0). By Lemma 4.14, one has u ∈ G i .u 1 ⊂ G.u 0 .
Putting together Proposition 4.17 and Corollary 4.12, we obtain Corollary 4.19. The intersection with SU (Q α ) of the subgroup generated by the L p is Zariski dense in SU (Q α ).
Proof. We apply Proposition 4.17, taking for w p (p ∈ A d ) a generator of the orthogonal H p of H p . We have Q α (w p ) = 0. If we had < w p , w q >= 0 for some distint p, q ∈ A d , the vector w p would belong to H p ∩ H q , and two of its coordinates in the canonical basis would vanish. But we know from the diagonalisation formulas of subsection 4.3 that it is not so. From Corollary 4.12, the Zariski closure of the intersection with SU (Q α ) of the subgroup generated by the L p contains the stabilizer of each w p . Therefore it is equal to SU (Q α ).
4.8. Exceptional case I: α = 1/3. At this stage, we can conclude that the Zariski closure of the intersection with SU (Q α ) of the subgroup generated by the L p is equal to SU ((Q α ) when α = 1 6 , 1 4 , 1 3 and (d + 1)α is not an integer ( so that Q α is non-degenerate. We may even replace Zariski closure by closure for the usual topology when the form is definite, i.e (d + 1)α < 1. Indeed it is sufficient to proceed by induction on the dimension d, starting with the results of Subsection 4.4, and applying successively either Proposition 4.6, Corollary 4.10, or Corollary 4.19.
In the two exceptional cases α = 1 6 , 1 4 , the group generated by L 1 , L −1 in dimension 2 is a finite group so it is not a basis for a successful induction. These two cases will be dealt with in later subsections. However, the case α = 1 3 does not need any supplemental work. Remember that Q 1 3 is degenerate for d = 2. The kernel of Q α is generated by e := e −1 + je 1 and fixed by L −1 and L 1 . observed in Subsection 4.4 that the intersection of the group generated by L −1 and L 1 with SL(2, C) contains a parabolic matrix. The determinant of L −1 and L 1 is −ζ, a sixth root of unity. This is sufficient to show that the Zariski closure G of the intersection of the group generated by L −1 and L 1 with SL(2, C) is the stabilizer of e in SL(2, C). Indeed, G is contained in this stabilizer. There are only three types of Zariski closed subgroups (over R!) of this stabilizer: the two trivial subgroups, and (given any vector f independent of e) the subgroup G f of the stabilizer formed of elements g such that g.f − f is a real multiple of e (There is a one-parameter family, parametrized by the 1-dimensional real projective space, of such subgroups). Here , the existence of a parabolic element guarantees that G is not reduced to the identity. It cannot be of the intermediate form , because conjugating by powers of L 1 an element g such that g.f − f = e, we get elements g such that g .f − f = ωe for any sixth root of unity. Therefore G is equal to the full stabilizer.
Thus the argument from Subsubsection 4.7 can be used to conclude the desired result (even though the dimension d is not as high as assumed there). 4.9. Exceptional case II: α = 1/4. We assume in this subsection that α = 1 4 . Then, we have seen in Subsection 4.4 that the group generated by L −1 and L 1 is finite. For d = 3, the form Q 1 4 is degenerate, but the Zariski closure G of the intersection with SL(3, C) of the group generated by L −2 , L 0 and L 2 is strictly smaller than the group SU * (Q 1 4 ) described in Subsection 4.6 (see below). It is only from dimension 4 that we get a "big" group generated by the L p .
Consider first the case d = 2. It can be checked that the group Γ generated by L −1 and L 1 has order 96. The property of Γ that will be useful in the sequel is Lemma 4.20. The representation of Γ on C 2 R 4 induced by the inclusion Γ ⊂ U (Q 1 4 ), is irreducible over R.
with γ ∈ Γ. We restrict to the subgroup G −3 of finite index such that γ = 1 Γ . As in Subsection 4.7, writing that the form Q 1 4 is preserved gives (when γ = 1 Γ ) Lemma 4.22. Conversely, any matrix of the prescribed form satisfying these relations belongs to G −3 .
Proof. Essentially the same than in Proposition 4.11 and Corollary 4.12.
The 5-dimensional Lie algebra g −3 of G −3 is therefore the set of matrices of the form with v −1 , v 1 ∈ C and s ∈ R.
Similarly, from the action of L −3 , L −1 , L 1 , one obtains a Zariski closed group G 3 whose Lie algebra g 3 is the set of matrices of the form with u −1 , u 1 ∈ C and r ∈ R.
Lemma 4.23. The smallest Lie algebra containing g −3 and g 3 is the Lie algebra su(Q 14 ).
Proof. With A, B as above, we have After adding appropriate elements A , B of g −3 , g 3 respectively, the matrix AB − BA + A + B is equal to These five matrices, together with g −3 and g 3 , span a 15-dimensional vector space. As su(Q 1 4 ), being isomorphic to su(3, 1), has also dimension 15, the lemma is proved. ). This provides an appropriate starting point for the induction for α = 1/4. The results in higher dimension follow. 4.10. Exceptional case III: α = 1/6. The proof is identical to the previous case α = 1/4 due to the following fact, an improvement (in generality) on Lemma 4.20 Proposition 4.25. For any α ∈ (0, 1/2), any d 2, such that (d + 1)α is not an integer, there is no non trivial R-subspace of C A d which is invariant under every L p , p ∈ A d .
Proof. The kernel of L p − id is the hyperplane The other eigenvalue of L p is equal to −ζ and is simple. The associated eigenspace is Ce p . We claim that the intersection p∈A d H p is trivial (if ζ d+1 = 1). Indeed, the intersection H p ∩H p+2 is contained in {x p = ζx p+2 } for p < d−1, and the intersection H d−1 ∩H 1−d is contained in {ζx 1−d = ζ −1 x d−1 }.
Let p ∈ A d , and let W be a R-subspace of C A d which is invariant under L p . If W is not contained in H p , it contains Ce p : indeed, if w ∈ W has the form h + te p with h ∈ H p and t = 0, then w − L p (w) = (1 + ζ)te p belongs to W and Ce p ⊂ W as ζ is not real.