Maximal representations of complex hyperbolic lattices in SU(m,n)
aa r X i v : . [ m a t h . G R ] M a y MAXIMAL REPRESENTATIONS OF COMPLEXHYPERBOLIC LATTICES INTO
SU( m, n ) M. B. POZZETTI
Abstract.
Let Γ denote a lattice in SU(1 , p ) , with p greater than 1.We show that there exists no Zariski dense maximal representation withtarget SU( m, n ) if n > m > . The proof is geometric and is basedon the study of the rigidity properties of the geometry whose pointsare isotropic m -subspaces of a complex vector space V endowed with aHermitian metric h of signature ( m, n ) and whose lines correspond tothe m dimensional subspaces of V on which the restriction of h hassignature ( m, m ) . Introduction
Let Γ be a finitely generated group and G be a connected semisimple Liegroup. It is an interesting problem to select and study some connected com-ponents of the representation variety Hom(Γ , G ) that consist of homomor-phisms ρ : Γ → G that are well behaved and, ideally, reflect some interestinggeometric properties of the group Γ . The best example of this frameworkis the case in which Γ is the fundamental group Γ g of a closed surface ofgenus g ≥ and G is PSL ( R ) . In this case the Teichmüller space arises asa component of Hom(Γ g , PSL ( R )) // PSL ( R ) that can be selected by meansof a cohomological invariant [Gol80].In the more general setting in which G is any Hermitian Lie group, theso-called maximal representations form a well studied union of connectedcomponents of the character variety Hom(Γ g , G ) //G which generalize theTeichmüller component [BGPG06, BIW10]. In analogy with holonomy repre-sentations of hyperbolizations, maximal representations can be characterizedas those representations that maximize an invariant, the Toledo invariant ,that can be defined in terms of bounded cohomology. Such representationsare discrete and faithful, and give rise to interesting geometric structures[BIW10, GW12]. Maximal representations in Hermitian Lie groups were
Date : July 30, 2018.
Key words and phrases.
Complex hyperbolic space, Shilov boundary, maximal repre-sentation, tight embedding, tube-type subdomain.I want to thank Marc Burger and Alessandra Iozzi for suggesting the topic of this article,for sharing with me their ideas and for many insightful conversation about the content ofthe paper. This work was partially supported by the Swiss National Science Foundationproject 200020-144373. I am grateful for the hospitality of Princeton University andUniversità di Pisa where part of this work was completed. first studied by Toledo in [Tol89] where he proves that a maximal repre-sentation ρ : Γ g → SU(1 , q ) fixes a complex geodesic, and by Hernandez[Her91] who studied maximal representations ρ : Γ g → SU(2 , q ) and showedthat the image must stabilize a symmetric space associated to the group SU(2 , . In general any maximal representation stabilizes a tube-type sub-domain [BIW10]. Despite this, a remarkable flexibility result holds for max-imal representations ρ of fundamental groups of surfaces: if the image of ρ is a Hermitian Lie group of tube type, then ρ admits a one parameter familyof deformations consisting of Zariski dense representations [BIW10, KP09].An analogue of the Toledo invariant was defined by Burger and Iozzi in[BI00] for representations of a lattice Γ in SU(1 , p ) with values in a Hermit-ian Lie group G . This allows to select a union of connected components of Hom(Γ , G ) consisting of maximal representations . These generalize maximalrepresentations of fundamental groups of surfaces: the fundamental group ofa surface is a lattice in PU(1 ,
1) = PSL ( R ) . However, if p is greater than one,a different behavior is expected: Goldman and Milson proved local rigidityfor the standard embedding of Γ in SU(1 , q ) [GM87], and Corlette provedthat maximal representations of uniform complex hyperbolic lattices withvalues in SU(1 , q ) all come from the standard construction [Cor88]. Thepicture for rank one targets was completed independently by Koziarz andMaubon [KM08a] and by Burger and Iozzi [BI08]: any maximal represen-tation of a lattice in SU(1 , p ) with values in SU(1 , q ) admits an equivarianttotally geodesic holomorphic embedding H p C → H q C . Koziarz and Maubongeneralized this result to the situation in which the target group is classicalof rank 2 and the lattice is cocompact [KM08b] . It is conjectured that everymaximal representation of a complex hyperbolic lattice with target a Her-mitian Lie group is superrigid, namely it extends, up to a representation of Γ in the compact centralizer of the image, to a representation of the ambientgroup SU(1 , p ) .In this article we show that the conjecture holds for Zariski dense repre-sentations in SU( m, n ) , with m different from n : Theorem 1.1.
Let Γ be a lattice in SU(1 , p ) with p > . If m is differentfrom n , then every Zariski dense maximal representation of Γ into PU( m, n ) is the restriction of a representation of SU(1 , p ) . This immediately implies the following:
Corollary 1.2.
Let Γ be a lattice in SU(1 , p ) with p > . There are noZariski dense maximal representations of Γ into SU( m, n ) , if < m < n . Exploiting results of [BIW09], and the classification of maximal repre-sentations between Hermitian Lie groups [Ham11, Ham12, HP14], we areable to use our main theorem to give a structure theorem for all maximalrepresentations ρ : Γ → SU( m, n ) . In his recent preprint Spinaci studies maximal representations of cocompact Kählergroups admitting an holomorphic equivariant map [Spi14]
AXIMAL REPRESENTATIONS INTO
SU( m, n ) Theorem 1.3.
Let ρ : Γ → SU( m, n ) be a maximal representation. Thenthe Zariski closure L = ρ (Γ) Z splits as the product SU(1 , p ) × L t × K where L t is a Hermitian Lie group of tube type without irreducible factors that arevirtually isomorphic to SU(1 , , and K is a compact subgroup of SU( m, n ) .Moreover there exists an integer k such that the inclusion of L in SU( m, n ) can be realized as ∆ × i × Id : L → SU(1 , p ) m − k × SU( k, k ) × K <
SU( m, n ) where ∆ : SU(1 , p ) → SU(1 , p ) m − k is the diagonal embedding, i : L t → SU( k, k ) is a tight holomorphic embedding and K is contained in the compactcentralizer of ∆ × i ( L ) . It is possible to show that there are no tube-type factors in the Zariskiclosure of the image of ρ by imposing some non-degeneracy hypothesis onthe associated linear representation of Γ into GL( C m + n ) : Corollary 1.4.
Let Γ be a lattice in SU(1 , p ) , with p > and let ρ bea maximal representation of Γ into SU( m, n ) . Assume that the associatedlinear representation of Γ on C n + m has no invariant subspace on which therestriction of the Hermitian form has signature ( k, k ) for some k . Then(1) n ≥ pm ,(2) ρ is conjugate to ρ × χ ρ where ρ is the restriction to Γ of the di-agonal embedding of m copies of SU(1 , p ) in SU( m, n ) and χ ρ is arepresentation χ ρ : Γ → K , where K is a compact group. Recently Klingler proved that all the representations of uniform complexhyperbolic lattices that satisfy a technical algebraic condition are locallyrigid [Kli11]. As a particular case his main theorem implies that if Γ is acocompact lattice in SU(1 , p ) and ρ : Γ → SU( m, n ) is obtained by restrictingto Γ the diagonal inclusion of SU(1 , p ) in SU( m, n ) , then ρ is locally rigid.Since the invariant defining the maximality of a representation is constanton connected components of the representation variety, we get a new proofof Klingler’s result in our specific case, and the generalization of this latterresult to non-uniform lattices: Corollary 1.5.
Let Γ be a lattice in SU(1 , p ) , with p > , and let ρ bethe restriction to Γ of the diagonal embedding of m copies of SU(1 , p ) in SU( m, n ) . Then ρ is locally rigid. Our proof of Theorem 1.1 is inspired by Margulis’ beautiful proof of su-perrigidity for higher rank lattices: in order to show that a representation ρ : Γ → G extends to the group H in which Γ sits as a lattice, it is enoughto exhibit a ρ -equivariant algebraic map φ : H/P → G/L for some par-abolic subgroups P of H and L of G . The existence of measurable ρ -equivariant boundary maps φ : H/P → G/L where
P < H is a minimalparabolic subgroup and G is a linear algebraic group is by now well under-stood [BI04, Fur81, BF14], and the crucial part in the proof of superrigidity M. B. POZZETTI for our representations is to show that such a measurable equivariant bound-ary map must indeed be algebraic. In general not every representation ofa complex hyperbolic lattice is superrigid: for example Livne constructedin his PhD dissertation a lattice in
SU(1 , that surjects onto a free group(cfr. [DM93, Chapter 16]), moreover Mostow constructed examples of lat-tices Γ , Γ in SU(1 , admitting a surjection Γ ։ Γ with infinite kernel(cfr. [Mos80, Tol03]). These examples show that many representations ofcomplex hyperbolic lattice do not extend to SU(1 , and hence some addi-tional information on the boundary map φ is needed in order to deduce itsalgebraicity.We restrict our interest to maximal representations precisely to be ableto gather some information on a measurable boundary map φ . The maxi-mality of a representation ρ can be rephrased as a property of the inducedpullback map ρ ∗ : H ( G, R ) → H (Γ , R ) in bounded cohomology. Oneof the advantages of bounded cohomology with respect to ordinary coho-mology is that it can be isometrically computed from the complex of L ∞ functions on some suitable boundary of the group [BM02] and, in all geo-metric cases known so far [BI02], the pullback map in bounded cohomologycan be implemented using boundary maps. In particular we exploit resultsof [BI09] and we show that the fact that the representation ρ is maximalimplies that a ρ -equivariant measurable boundary map must preserve someincidence structure on the boundary (this was proven in [BI08] in the casein which the image is of rank one).To describe more precisely this incidence structure, recall that one of thekey features of the complex hyperbolic space is the existence of complexgeodesics tangent to any vector in T H p C : these are precisely the totally ge-odesic holomorphic embeddings of the Poincaré disc in H p C . The boundariesof these subspaces produce a family of circles in ∂ H p C , the socalled chains ,that form an incidence structure that was first studied by Cartan in [Car32].Under many respects, the natural generalization to higher rank of the visualboundary of the complex hyperbolic space is the Shilov boundary of a Her-mitian symmetric space and the generalization of a complex geodesic, whenmaximal representations are involved, is a maximal tube-type subdomain.All these objects have an explicit linear description: it is well known thatthe boundary of the complex hyperbolic space can be identified with theset of isotropic lines in C p +1 , and it is easy to check that a triple of lines x, y, z is contained in a chain if and only if dim h x, y, z i = 2 . Similarly theShilov boundary S m,n of SU( m, n ) can be described as the set of maximalisotropic subspaces of C m + n and, again, a triple of transverse isotropic sub-spaces x, y, z in S m,n is contained in the boundary of a tube-type subdomainprecisely when dim h x, y, z i = 2 m . In such case we will say that x, y, z arecontained in an m -chain .As it turns out, if ρ : Γ → SU( m, n ) is a maximal representation and φ : ∂ H p C → S m,n is a measurable ρ -equivariant boundary map, then φ inducesa map from the chain geometry of ∂ H p C to the geometry whose space is S m,n AXIMAL REPRESENTATIONS INTO
SU( m, n ) and whose lines are the m -chains. Therefore most of this paper is devoted tothe study of these geometries. We generalize some results of Cartan [Car32]and Goldman [Gol99] and this allows us to prove a strong rigidity result formeasurable maps that preserve this geometry, that is a higher rank analogueof the main theorem of [Car32]: Theorem 1.6.
Let p > , < m < n and let φ : ∂ H p C → S m,n be ameasurable map whose essential image is Zariski dense. Assume that, foralmost every triple with dim h x, y, z i = 2 , it holds dim h φ ( x ) , φ ( y ) , φ ( z ) i =2 m . Then φ coincides almost everywhere with a rational map. Outline of the paper.
In Section 2, after recalling the relevant conceptsabout Hermitian symmetric spaces and continuous bounded cohomology, weprove that a measurable boundary map associated with a maximal represen-tation induces a map between chain geometries. Sections 3 to 5 are devotedto prove Theorem 1.6: in Section 3 we study the chain geometry of S m,n andprove some properties of the incidence structure of chains; in Section 4 weshow that the restriction to almost every chain of a measurable boundarymap associated with a maximal representation is rational; in Section 5 weshow that this information is already enough to conclude. We finish thearticle with Section 6, where we prove all the remaining results announcedin this introduction. 2. Preliminaries
Hermitian Symmetric spaces.
Let G be a connected semisimple Liegroup of noncompact type with finite center and let K be a maximal com-pact subgroup. We will denote by X = G/K the associated symmetric space.Throughout this article we will be only interested in
Hermitian symmetricspaces, that is in those symmetric spaces that admit a G -invariant complexstructure J . It is a classical fact [Kor00, Theorem III.2.6] that these sym-metric spaces admit a bounded domain realization, that means that theyare biholomorphic to a bounded convex subspace of C n on which G acts viabiholomorphisms. An Hermitian symmetric space is said to be of tube-type if it is also biholomorphic to a domain of the form V + i Ω where V is a realvector space and Ω ⊂ V is a proper convex open cone. Hermitian symmetricspaces were classified by Cartan [Car35], and are the symmetric spaces asso-ciated to the exceptional Lie groups E ( − and E ( − together with 4families of classical domains: the ones associated to SU( p, q ) , of type I p,q inthe standard terminology , the ones associated to SO ∗ (2 p ) , of type II p , thesymmetric spaces, III p , of the groups Sp(2 p, R ) , and the symmetric spacesof the group IV p associated to SO (2 , p ) . It is well known that the onlyspaces that are not of tube type are the symmetric space of E ( − andthe families I p,q with q = p and II p with p odd. It follows from the clas-sification that any Hermitian symmetric space contains maximal tube-type In Cartan’s original terminology [Car35] the families
III p and IV p are exchanged M. B. POZZETTI subdomains, and those are all conjugate under the G -action, are isometri-cally and holomorphically embedded and have the same rank as the ambientsymmetric space.The G -action via biholomorphism on the bounded domain realization of X extends continuously on the topological boundary ∂ X . If the real rankof G is greater than or equal to two, ∂ X is not an homogeneous G -space,but contains a unique closed G -orbit, the Shilov boundary S G of X . If X is irreducible, the stabilizer of any point s of S G is a maximal parabolicsubgroup of G . In the reducible case, if X = X × . . . × X n is the de Rhamdecomposition in irreducible factors whose isometry group is G i , then S G splits as the product S G × . . . × S G n as well. Moreover when Y is a maximaltube-type subdomain of X , the Shilov boundary of Y embeds in the Shilovboundary of X .The diagonal action of G on the pairs of points ( s , s ) ∈ S G has a uniqueopen orbit corresponding to pairs of opposite parabolic subgroups. Twopoints in S G are transverse if they belong to this open orbit. Whenever a pair ( s , s ) of transverse points of S G is fixed, there exists a unique maximal tube-type subdomain Y = G T /K T of X such that s i belongs to S G T . In particularthis implies that the Shilov boundaries of maximal tube-type subdomainsdefine a rich incidence structure in S G .Given three points in S G there won’t, in general, exist a maximal tube-type subdomain Y of X whose Shilov boundary contains all the three points.However it is possible to determine when this happens with the aid of theKähler form. Recall that, since X is a Hermitian symmetric space, it ispossible to define a differential two form via the formula ω ( X, Y ) = g ( X, J Y ) where g denotes the G -invariant Riemannian metric normalized so that itsminimal holomorphic sectional curvature is − , and J is the complex struc-ture of X . Since ω is G -invariant, it is closed: this is true for every G -invariant differential form on a symmetric space. This implies that X is aKähler manifold and ω is its Kähler form. Let X (3) denote the triples ofpairwise distinct points in X and let us consider the function β X : X (3) → R ( x, y, z ) → π R ∆( x,y,z ) ω where we denote by ∆( x, y, z ) any smooth geodesic triangle having ( x, y, z ) as vertices. Since ω is closed, Stokes theorem implies that β X is a welldefined continuous G -invariant cocycle and it is proven in [CØ03] that itextends continuously to the triples of pairwise transverse points in the Shilovboundary. If a triple ( s , s , s ) ∈ S doesn’t consist of pairwise transversepoints, the limit of β X ( x i , x i , x i ) as x ij approaches s j is not well defined, butClerc proved that, restricting only to some preferred sequences (the one thatconverge radially to s j ), it is possible to get a measurable extension of β X tothe whole Shilov boundary. The obtained extension β S : S G → R is called AXIMAL REPRESENTATIONS INTO
SU( m, n ) the Bergmann cocycle and it is a measurable strict cocycle. The maximalityof the Bergmann cocycle detects when a triple of points is contained in theShilov boundary of a tube-type subdomain: Proposition 2.1. (1) β S is a strict alternating G -invariant cocycle withvalues in [ − rk X , rk X ] ,(2) If β S ( s , s , s ) = rk X then the triple ( s , s , s ) is contained in theShilov boundary of a tube-type subdomain.(3) The Bergmann cocycle is a complete invariant for the G action ontriples of pairwise transverse points contained in a tube type subdo-main.Proof. The first fact was proven in [Cle07], the second can be found in[BIW09, Proposition 5.6], the third follows from the transitivity of the G -action on maximal tube type subdomains of S G and [CN06, Theorem 5.2]. (cid:3) We will call a triple ( s , s , s ) in S G satisfying | β S ( s , s , s ) | = rk( X ) a maximal triple.In the case where G is SU(1 , p ) , that is a finite cover of the connected com-ponent of the identity in Isom ( H p C ) , the maximal tube-type subdomains arecomplex geodesics of H p C and the Bergmann cocycle coincides with Cartan’sangular invariant c p [Gol99, Section 7.1.4]. Following Cartan’s notation wewill denote by chains the boundaries of the complex geodesics.2.2. Continuous (bounded) cohomology and maximal representa-tions.
We introduce now the concepts we will need about continuous andcontinuous bounded cohomology, standard references are respectively [BW00]and [Mon01]. A quick introduction to the relevant aspects of continuousbounded cohomology can also be found in [BI09].Throughout the section G will be a locally compact second countablegroup, every finitely generated group fits in this class when endowed withthe discrete topology. The continuous cohomology of G with real coefficients, H n c ( G, R ) is the cohomology of the complex (C n c ( G, R ) G , d) where C n c ( G, R ) = { f : G n +1 → R | f is a continuous function } , the invariants are taken with respect to the diagonal action, and the differ-ential d n : C n c ( G, R ) → C n +1c ( G, R ) is defined by the expression d n f ( g , . . . , g n +1 ) = n +1 X i =0 ( − i f (( g , . . . , ˆ g i , . . . , g n +1 ) . Similarly the continuous bounded cohomology H n cb ( G, R ) of G is the coho-mology of the subcomplex (C n cb ( G, R ) G , d) of (C n c ( G, R ) G , d) consisting ofbounded functions. The inclusion i : C n cb ( G, R ) G → C n c ( G, R ) G induces, in We choose the normalization of [Cle07], the normalization chosen in [DT87] is suchthat β DT = π · β S , the one of [BIW10] is such that β BIW = β S M. B. POZZETTI cohomology, the so-called comparison map c : H n cb ( G, R ) → H n c ( G, R ) . TheBanach norm on the cochain modules C n cb ( G, R ) defined by k f k ∞ = sup ( g ,...,g n ) ∈ G n +1 | f ( g , . . . , g n ) | induces a seminorm on H n cb ( G, R ) that is usually referred to as the canonicalseminorm or Gromov’s norm .Most of the results about continuous and continuous bounded cohomologyare based on the functorial approach to the study of these cohomological the-ories that is classical in the case of continuous cohomology and was developedby Burger and Monod [BM99] in the setting of continuous bounded cohomol-ogy. This allows to show that the cohomology of many different complexesrealizes canonically the given cohomological theory. Since we will only needapplications of this machinery that are already present in the literature wewill not describe it any further here and we refer instead to [BW00, Mon01]for details on this nice subject.A first notable application of this approach to continuous cohomology isvan Est Theorem [vE53, Dup76] that realizes the continuous cohomologyof a semisimple Lie group in terms of G -invariant differential forms on theassociated symmetric space: Theorem 2.2 (van Est) . Let G be a semisimple Lie group without compactfactors, then Ω n ( X , R ) G ∼ = H n c ( G, R ) . Under this isomorphism the differential form ω corresponds to the class ofthe cocycle c ω defined by the formula c ω ( g , . . . , g n ) = 1 π Z ∆( g x,...g n x ) ω for any fixed basepoint x in X . Let us now focus more specifically on the second bounded cohomology ofa Hermitian Lie group G . By van Est isomorphism the module H ( G, R ) is isomorphic to the vector space of the G -invariant differential 2-forms on X which are generated, as a real vector space, by the Kähler classes of theirreducible factors of the symmetric space X . The class corresponding viavan Est isomorphism to the Kähler class ω of X is represented by the cocycle c ω ( g , g , g ) = β X ( g x, g x, g x ) where x ∈ X is any fixed point.It was proven in [DT87] for the irreducible classical domains and in [CØ03]in the general case that the absolute value of the cocycle c ω is boundedby rk ( X ) , hence the class [ c ω ] is in the image of the comparison map c :H ( G, R ) → H ( G, R ) . Moreover, if G is a connected semisimple Lie groupwith finite center and without compact factors, the comparison map c isinjective (hence an isomorphism) in degree 2 [BM99]. We will denote by κ bG the bounded Kähler class , that is the class in H cb ( G, R ) satisfying c ( κ bG ) =[ c ω ] . The Gromov norm of κ bG can be computed explicitly: AXIMAL REPRESENTATIONS INTO
SU( m, n ) Theorem 2.3 ([DT87, CØ03, BIW09]) . Let G be a Hermitian Lie groupwith associated symmetric space X and let κ bG be its bounded Kähler class.If k · k denotes the Gromov norm, then k κ bG k = rk ( X ) . Let now M be a locally compact second countable topological group, G a Lie group of Hermitian type, ρ : M → G a continuous homomorphism.The precomposition with ρ at the cochain level induces a pullback map inbounded cohomology ρ ∗ b : H ( G, R ) → H ( M, R ) that is norm non in-creasing. Tight homomorphisms were first defined in [BIW09], these arehomomorphisms ρ for which the pullback map is norm preserving, namely k ρ ∗ ( κ bM ) k = k κ bM k . In the same paper the following structure theorem isproven: Theorem 2.4 ([BIW09, Theorem 7.1]) . Let L be a locally compact secondcountable group, G a connected algebraic group defined over R such that G = G ( R ) ◦ is of Hermitian type. Suppose that ρ : L → G is a continuoustight homomorphism. Then(1) The Zariski closure H = ρ ( L ) Z is reductive.(2) The group H = H ( R ) ◦ almost splits as a product H nc H c where H c iscompact and H nc is of Hermitian type.(3) If Y is the symmetric space associated to H nc , then the inclusion of Y in X is totally geodesic and the Shilov boundary S H nc sits as asubspace of S G . In the case when also L is an Hermitian Lie group, tight homomorphismscan be completely classified: in fact it is possible to prove that, if L has nosimple factor locally isomorphic to SU(1 , , then the map ρ is equivariantwith an holomorphic map (see [Ham12] for the case when L is simple, and[HP14] for the general case) and in particular the classification of [Ham11]applies. In our setting this implies the following; Theorem 2.5.
Let i : L → SU( m, n ) be a tight homomorphism, assume thatno factor of L is locally isomorphic to SU(1 , . Then each simple factor of L is either isomorphic to SU( s, t ) or is of tube type. Moreover if L = L t × L nt where L t is the product of all the irreducible factors of tube type, then thereexists an orthogonal decomposition C m,n = C k,k ⊕ C m − k,n − k such that L t isincluded in SU( C k,k ) and L nt is included in SU( C m − k,n − k ) .Proof. This can be found in [HP14]. (cid:3)
A key feature of bounded cohomology is that, whenever Γ is a lattice in G , it is possible to construct a left inverse T • b : H • b (Γ) → H • cb ( G ) of therestriction map. Indeed the bounded cohomology of Γ can be computedfrom the complex (C • cb ( G, R ) Γ , d) and the transfer map T • b can be definedon the cochain level by the formula T k b ( c ( g , . . . , g k )) = Z Γ \ G c ( gg , . . . , gg k ) d µ ( g ) where µ is the measure on Γ \ G induced by the Haar measure of G providedit is normalized to have total mass one. It is worth remarking that whenwe consider instead continuous cohomology (without boundedness assump-tions), a transfer map can be defined with the very same formula only forcocompact lattices, but the restriction map is in general not injective if thelattice is not cocompact.Let us fix a representation ρ : Γ → G . Since H (SU(1 , p ) , R ) = R κ b SU(1 ,p ) ,the class T ∗ b ρ ∗ ( κ bG ) is a scalar multiple of the Kähler class κ b SU(1 ,p ) . The generalized Toledo invariant of the representation ρ is the number i ρ suchthat T ∗ b ρ ∗ ( κ bG ) = i ρ κ b SU(1 ,p ) . A consequence of Theorem 2.3, and the factthat the transfer map is norm non-increasing, is that | i ρ | ≤ rk ( X ) . Definition 2.6.
A representation ρ is maximal if | i ρ | = rk ( X ) . Clearlymaximal representation are in particular tight representations.The following lemma will be useful at the very end of the article, in theproof of Corollary 1.5: Lemma 2.7.
The generalized Toledo invariant is constant on connected com-ponents of the representation variety.Proof.
This is proven in [BI08, Page 4]. (cid:3)
Boundary maps and maximal representations.
The existence ofmeasurable boundary maps for Zariski dense homomorphisms in algebraicgroups is by now classical:
Proposition 2.8. [BI04, Proposition 7.2]
Let Γ be a lattice in SU(1 , p ) , G a Lie group of Hermitian type and let ρ : Γ → G be a Zariski denserepresentation. Then there exists a ρ -equivariant measurable map φ : ∂ H p C →S G such that, for almost every pair of points x, y in ∂ H p C , φ ( x ) and φ ( y ) aretransverse. Let us now fix a measurable map φ : ∂ H p C → S G , and define the essentialZariski closure of φ to be the minimal Zariski closed subset V of S G suchthat µ ( φ − ( V )) = 1 . Such a set exists since the intersection of finitely manyclosed subset of full measure has full measure and S G is an algebraic variety,in particular it is Noetherian. We will say that a measurable boundary map φ is Zariski dense if its essential Zariski closure is the whole S G . Proposition 2.9.
Let ρ be a Zariski dense representation, then φ is Zariskidense.Proof. Indeed let us assume by contradiction that the essential Zariski clo-sure of φ ( ∂ H p C ) is a proper Zariski closed subset V of S G . The set V is The original definition of the generalized Toledo invariant given in [BI00] used con-tinuous cohomology only. However it is proven in [BI07, Lemma 5.3] that the invariantthat was originally defined in [BI00], i ρ in the notation of that article, and the invariantwe defined here, that there was denoted by t b ( ρ ) , coincide. AXIMAL REPRESENTATIONS INTO
SU( m, n ) ρ (Γ) -invariant: indeed for every element γ in Γ , we get µ ( φ − ( ρ ( γ ) V )) = µ ( γφ − ( V )) = 1 , hence, in particular, ρ ( γ ) V = V by minimality of V .Let us now recall that the Shilov boundary S G is an homogeneous spacefor G , and let us fix the preimage W of V under the projection map G → G/Q = S G . W is a proper Zariski closed subset of G , moreover if g is anyelement in W , the Zariski dense subgroup ρ (Γ) of G is contained in W g − and this gives a contradiction. (cid:3) One of the advantages of bounded cohomology when proving rigiditystatements is that the bounded cohomology of a group can be computedfrom a suitable boundary of the group itself, for example when Γ is a lat-tice in SU(1 , p ) , the complex (L ∞ alt (( ∂ H p C ) • , R ) Γ , d ) realizes isometrically thebounded cohomology of Γ [BM02]. Moreover, exploiting functoriality prop-erties of bounded cohomology, one can implement the pullback via a measur-able boundary map provided by Proposition 2.8 thus getting the followingresult: Proposition 2.10 ([BI09, Theorem 2.41] ) . Let Γ be a lattice in SU(1 , p ) and let G be a Hermitian Lie group. Let ρ : Γ → G be a representation, β S : ( S G ) → R the Bergmann cocycle and φ : ∂ H p C → G/Q be a measurable ρ -equivariant boundary map. For almost every triple ( x, y, z ) in ∂ H p C , theformula i ρ c p ( x, y, z ) = Z Γ \ SU(1 ,p ) β S ( φ ( gx ) , φ ( gy ) , φ ( gz ))d µ ( g ) holds. We will now show that, since β S is a strict G -invariant cocycle and SU(1 , p ) acts transitively on pairs of distinct points of ∂ H p C , the equality holds forevery triple of pairwise distinct points (this is an adaptation in our contextof an argument due to Bucher: cfr. the proof of [BBI13, Proposition 3] incase n = 3 ). Lemma 2.11.
The equality in Proposition 2.10 holds for every triple ( x, y, z ) of pairwise distinct points.Proof. The formula of Proposition 2.10 is an equality between
SU(1 , p ) -invariant strict cocycles: clearly this is true for the left-hand side, moreoverthe expression on the right-hand side is a strict cocycle since β S is, and is SU(1 , p ) invariant since φ is ρ -equivariant and β S is G -invariant.Let us now fix a SU(1 , p ) -invariant full measure set O ⊆ ( ∂ H p C ) on whichthe equality holds. Since O is of full measure, an application of Fubini’sTheorem is that for almost every pair ( y , y ) ∈ ( ∂ H p C ) the set of points z ∈ ∂ H p C such that ( y , y , z ) ∈ O is of full measure. Let us fix a pair ( y , y ) for which this holds and denote by W the set of points z such that ( y , y , z ) ∈ O .Let us now fix a triple ( x , x , x ) of points in ∂ H p C . Since the SU(1 , p ) action on ∂ H p C is transitive on pairs of distinct points, for every i there exist an element g i such that ( x i , x i +1 ) = ( g i y , g i y ) . Let us now fix a point x in the full measure set g W ∩ g W ∩ g W . Since x is in g i W , we get that g − i x ∈ W , and hence ( x i , x i +1 , x ) = g i ( y , y , g − i x ) ∈ O . In particular,computing the cocycle identity on the 4tuple ( x , x , x , x ) we get that theidentity of Proposition 2.10 holds for the triple ( x , x , x ) . (cid:3) Corollary 2.12.
Let ρ : Γ → G be a maximal representation and let φ : ∂ H p C → S G be a ρ -equivariant measurable boundary map. Then for al-most every maximal triple ( x, y, z ) ∈ ( ∂ H p C ) , the triple ( φ ( x ) , φ ( y ) , φ ( z )) iscontained in the Shilov boundary of a tube-type subdomain and is a maximaltriple.Proof. Let us fix a positively oriented triple ( x, y, z ) of points on a chain.We know from Lemma 2.11 that the equality Z SU(1 ,p ) / Γ β S ( φ ( gx ) , φ ( gy ) , φ ( gz )) dg = rk ( X ) holds: since ρ is maximal, then i ρ = rk( X ) , and since ( x, y, z ) are on a chainthen c p ( x, y, z ) = 1 .Since k β S k ∞ = rk( X ) , it follows that β S ( φ ( gx ) , φ ( gy ) , φ ( gz )) = rk ( X ) for almost every g in SU(1 , p ) . By Proposition 2.1, this implies that foralmost every g ∈ SU(1 , p ) , the triple ( φ ( gx ) , φ ( gy ) , φ ( gz )) is contained inthe boundary of a tube-type subdomain. Since maximal triples in ∂ H p C forman SU(1 , p ) -orbit, the fact that the result holds for almost every element g implies that the result holds for almost every triple of positively orientedpoints in a chain. The same argument applies for negatively oriented triples. (cid:3) Definition 2.13.
A measurable map φ preserves the chain geometry if, foralmost every pair x, y in ∂ H p C , the images φ ( x ) , φ ( y ) are transverse sub-spaces and, for almost every maximal triple ( x, y, z ) ∈ ( ∂ H p C ) , the triple ( φ ( x ) , φ ( y ) , φ ( z )) is maximal.This amounts to saying that the map φ induces an almost everywheredefined morphism ( φ, ˆ φ ) from the geometry ∂ H p C × C whose points are pointsin ∂ H p C and whose lines are the chains, to the geometry S G × T whosepoints are points in S G and whose lines are the Shilov boundaries of maximaltube-type subdomains of S G . The morphism ( φ, ˆ φ ) has the property that itpreserves the incidence structure almost everywhere.Purpose of the next sections is to show that a measurable Zariski densemap φ : ∂ H p C → S SU( m,n ) that preserves the chain geometry coincides almosteverywhere with an algebraic map. AXIMAL REPRESENTATIONS INTO
SU( m, n ) The Chain geometry of S m,n For the rest of the paper we will restrict our attention to the HermitianLie group
SU( m, n ) consisting of complex matrices that preserve a non-degenerate Hermitian form of signature ( m, n ) with m ≤ n , and denote,for the sake of brevity, by S m,n the Shilov boundary of SU( m, n ) that waspreviously denoted by S SU( m,n ) . The purpose of this section is to understandsome features of the incidence structure of the subsets of S m,n that arise asShilov boundaries of the maximal tube-type subdomains of the symmetricspace associated to SU( m, n ) . For reasons that will be explained later wewill call m - chains such subsets. The main tool that we will introduce in ourinvestigations is a projection map π x , depending on the choice of a point x ∈ S m,n . The map π x associates to a point y that is transverse to x theuniquely determined m -chain that contains both x and y . The central resultsof the section are Proposition 3.11 and Proposition 3.13.3.1. A model for S m,n . Throughout the paper we will realize
SU( m, n ) asthe subgroup of SL( m + n, C ) that preserves the Hermitian form h representedwith respect to the standard basis by the matrix h = (cid:20) m − Id n − m m (cid:21) . Wewill denote by Gr m ( C m,n ) the Grassmannian of m -dimensional subspaces of C m + n . It is well known that the Shilov boundary S m,n can be realized asthe subset of Gr m ( C m,n ) consisting of subspaces that are isotropic for theform h . Both S m,n and the action of SU( m, n ) on it are real algebraic .It is classical and easy to verify that the unique open SU( m, n ) -orbit on S m,n consists of pairs of points whose underlying vector spaces are transverse .In particular we will often identify a point x ∈ S m,n with its underlyingvectorspace, and we will use the notation x ⋔ y , that would be more suited forthe linear setting, with the meaning that the pair ( x, y ) is a pair of transversepoints. We will use the notation S (2) m,n for the set of pairs of transverse points,and we will denote the set of points in S m,n that are transverse to a givenpoint x by S xm,n := { y ∈ S m,n | y ⋔ x } . If x and y are transverse point, the linear span h x, y i is a m dimensionalsubspace V of C m + n on which h has signature ( m, m ) . The Shilov boundaryof the maximal tube type subdomain containing x and y is the set S V = { z ∈ Gr m ( V ) | h | z = 0 } = S m,n ∩ Gr m ( V ) . Definition 3.1. An m - chain in S m,n is a subspaces of the form S V for somelinear subspace V of C m,n on which h restricts to an Hermitian form ofsignature ( m, m ) . More details can be found in [Poz14] and in [BI04] where it is also possible to find thedescription of a complex variety S m , n such that S m,n = S m , n ( R ) Clearly any pair of transverse points x, y in S m,n uniquely determines a m -chain S h x,y i with the property that both x and y belong to S h x,y i . We willdenote by T x,y such a chain.In the case m = 1 , that is X m,n = H n C , the 1-chains are boundaries ofcomplex geodesics or chains in Cartan’s terminology. This is the reason whywe chose to call the Shilov boundaries of maximal tube-type subdomains m -chains . To be more consistent with Cartan’s notation, we omit the 1, andsimply call chains the -chains.3.2. The Heisenberg model H m,n ( x ) . We now want to give a model forthe Zariski open subset of S m,n consisting of points transverse to a givenpoint x . The model we are introducing is sometimes referred to as (theboundary of) a Siegel domain of genus two and was studied, for example, byKoranyi and Wolf in [KW65]. In the case m = 1 this model is described in[Gol99, Chapter 4] but our conventions here will be slightly different.In the rest of the paper, for each complex matrix X we will denote by X T the transpose of X , by X ∗ = X T the transpose conjugate of X . If, moreover, X is invertible we will denote by X − the inverse of X and by X −∗ theinverse of X ∗ . Moreover we will indicate a point V in the Grassmannian Gr m ( C m + n ) with a ( n + m ) · m dimensional matrix: we will understand sucha matrix as an ordered basis of the subspace V . Clearly two matrices X, Y represent the same element in Gr m ( C m + n ) if and only if there exists a matrix G ∈ GL m ( C ) such that X = Y G . A direct computation gives that a point x ∈ Gr m ( C m + n ) represented by the matrix (cid:20) X X X (cid:21) belongs to S m,n if and onlyif X ∗ X + X ∗ X − X ∗ X = 0 where X and X have m rows and X has n − m rows.Let us focus on the maximal isotropic subspace v ∞ = h e i | ≤ i ≤ m i = h Id m i ∈ S m,n . The set of points S v ∞ m,n that are transverse to v ∞ admit a basis of the form h XY Id m i with X ∗ + X − Y ∗ Y = 0 . We will identify such a set with the linearspace M (( n − m ) × m, C ) × u ( m ) where u ( m ) is the set of antiHermitianmatrices. We use the symbol H m,n ( v ∞ ) for such a linear space, that will beunderstood as parametrizing S v ∞ m,n via the map H m,n ( v ∞ ) = M (( n − m ) × m, C ) × u ( m ) → S v ∞ m,n ( X, Y ) (cid:20) Y + X ∗ X/ X Id m (cid:21) . We refer to H m,n ( v ∞ ) as the Heisenberg model. This is because, as we willnow see, H m,n ( v ∞ ) identifies with the generalized Heisenberg group that isthe nilpotent radical of the stabilizer of v ∞ . AXIMAL REPRESENTATIONS INTO
SU( m, n ) Let us denote by Q the maximal parabolic subgroup of SU( C m + n , h ) thatis the stabilizer of v ∞ . It is easy to verify that Q = A B E C F A −∗ mn − mm (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ∈ GL m ( C ) , C ∈ U( n − m ) ,A − B − F ∗ C = 0 E ∗ A −∗ + A − E − F ∗ F = 0det C det A det A −∗ = 1 . The group Q can be written as L ⋉ N where L = A C
00 0 A −∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A ∈ GL m ( C ) ,C ∈ U( n − m )det C det A det A −∗ = 1 is reductive and N = Id E ∗ F E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F ∗ + F − E ∗ E = 0 is nilpotent.The group L is the stabilizer of the two transverse points v ∞ and v = h e n +1 , . . . , e n + m i of S m,n . If we denote by a the determinant of the matrix A , an explicit isomorphism between GL m ( C ) × SU( n − m ) and L is given by: GL m ( C ) × SU ( n − m ) → L ( A, B ) (cid:20) A aa − B
00 0 A −∗ (cid:21) . Similarly the 2 step nilpotent group N can be identified with M ( n × m, C ) ⋉ u ( m ) : M (( n − m ) × m, C ) ⋉ u ( m ) → N ( E, F ) (cid:20) Id E ∗ F + EE ∗ /
20 Id E (cid:21) . It is particularly easy to describe the action of L and N on H m,n ( v ∞ ) :the group N acts by left multiplication according to the group structure on M ( n × m, C ) ⋉ u ( m )( E, F ) · ( X, Y ) = (cid:18) E + X, F + Y + E ∗ X − X ∗ E (cid:19) and L acts via right-left matrix multiplication on the first factor and conju-gation on the second: ( A, B ) · ( X, Y ) = ( aa − BXA ∗ , AY A ∗ ) . The projection π x . We consider the projection on the first factor π v ∞ : H m,n ( v ∞ ) → M (( n − m ) × m, C ) . Under the natural identification H m,n ( v ∞ ) ∼ = N , this projection corresponds to the group homomorphismwhose kernel is the center u ( m ) of N . Purpose of this section is to givea geometric interpretation of the quotient space M (( n − m ) × m, C ) : it corresponds to a parametrization of the space of m -chains through the point v ∞ . In order to make this statement more precise let us consider the set W v ∞ = { V ∈ Gr m ( C m + n ) | v ∞ < V, h | V has signature ( m, m ) } . The following lemma gives an explicit identification of W v ∞ with the quotientspace M (( n − m ) × m, C ) = N/ u ( m ) : Lemma 3.2.
There exists a bijection between M (( n − m ) × m, C ) and W v ∞ defined by the formula i : M (( n − m ) × m, C ) → W v ∞ A A ∗ Id0 ⊥ . Proof.
Let V be a point in W v ∞ . Then V ⊥ is a ( n − m ) dimensional subspaceof C m + n that is contained in v ⊥∞ . This implies that V ⊥ admits a basis of theform (cid:2) A B (cid:3) T where A has m rows and n − m columns and B is a square n − m dimensional matrix. Since the restriction of h on V has signature ( m, m ) , the restriction of h to V ⊥ is negative definite, in particular thematrix B must be invertible. This implies that, up to changing the basis of V ⊥ , we can assume that B = Id n − m . This gives the desired bijection. (cid:3) W v ∞ parametrizes the m -chains containing the point v ∞ . We will callthem vertical chains: the intersection T v ∞ of a vertical chain T with theHeisenberg model H m,n ( v ∞ ) consists precisely of a fiber of the projection onthe first factor in the Heisenberg model: Lemma 3.3.
Let T ⊂ S m,n be a vertical chain and V be its associated linearsubspace. If we denote by p T in M (( n − m ) × m, C ) , the point p T = i − ( V ) ,we have:(1) for every x in T v ∞ , then π v ∞ ( x ) = p T ,(2) T v ∞ = π − v ∞ ( p T ) ,(3) the center M of N acts simply transitively on T v ∞ .Proof. (1) An element w of H m,n ( v ∞ ) with basis (cid:2) X Y Id m (cid:3) T belongsto the chain T if and only if Y = p T : indeed the requirement that V ⊥ iscontained in w ⊥ restates as (cid:2) X ∗ Y ∗ Id m (cid:3) − Id 0Id 0 0 p ∗ T Id0 = p ∗ T − Y ∗ . This implies that for every w ∈ T , we have π v ∞ ( w ) = p T .Viceversa if π v ∞ ( w ) = p T , then w is contained in V and this proves (2).(3) The fact that M acts simply transitively on T v ∞ is now obvious:indeed M acts on the Heisenberg model by vertical translation stabilizingevery vertical chain. (cid:3) AXIMAL REPRESENTATIONS INTO
SU( m, n ) The stabilizer Q of v ∞ naturally acts on the space W v ∞ and it is easy todeduce explicit formulae for this action from the formulae given in Section3.2. In the sequel, when this will not cause confusion, we will identify W v ∞ with M (( n − m ) × m, C ) considering implicit the map i − . It is worthremarking that everything we did so far doesn’t really depend on the choiceof the point v ∞ , and a map π x : S xm,n → W x can be defined for every point x ∈ S m,n . We decided to stick to the point v ∞ , since the formulae in theexplicit expressions are easier.3.4. Projections of chains.
We now want to understand what are the pos-sible images under the map π v ∞ of other chains. We define the intersectionindex of an m -chain T with a point x ∈ S m,n by i x ( T ) = dim( x ∩ V T ) where V T is the m dimensional linear subspace of C m + n associated to T .Clearly ≤ i v ∞ ( T ) ≤ m , and i v ∞ ( T ) = m if and only if the chain T isvertical. In general we will call k -vertical a chain whose intersection indexis k : with this notation vertical chains are m -vertical. Sometimes we willcall horizontal the chains that are 0-vertical (in particular each point in thechain is transverse to v ∞ ).In our investigations it will be precious to be able to relate different situ-ations via the action of the group G = SU( C m + n , h ) , under this respect thefollowing lemma will be fundamental: Lemma 3.4.
For every k ∈ { , . . . m } the group G acts transitively on(1) the pairs ( x, T ) where x ∈ S m,n is a point and T is an m -chain with i x ( T ) = k ,(2) the triples ( x, y, T ) where x ⋔ y , y ∈ T and i x ( T ) = k .In particular the intersection index is a complete invariant of m -chains upto the SU( m, n ) -action.Proof. We will prove directly the second statement. By transitivity of the G -action on the set of transverse pairs we can assume that x = v ∞ , y = v ,in particular this reduces the proof to showing that L acts transitively on theset of chains through v . It is not hard to show that the orthogonal to sucha chain T has a basis of the form (cid:2) n − m Z (cid:3) T for some matrix Z : anyvector contained in the orthogonal to v has vanishing components in v ∞ ,moreover, since the orthogonal to a chain is positive definite we can assumethat the central block is the identity up to changing the basis. Moreover itholds that m − rk( Z ) = i v ∞ ( T ) . The statement is now obvious. (cid:3) As explained at the beginning of the section we want to give a parametriza-tion of a generic chain T and study the restriction of π v ∞ to T . In view ofLemma 3.4, it is enough to understand, for every k , the parametrization andthe projection of a single k -vertical chain. The k -vertical chain we will dealwith is the chain with associated linear subspace V k = h e i , e j + e m + j − k + e n + j , v | ≤ i ≤ k < j ≤ m i . Lemma 3.5. T k is the linear subspace associated to a k -vertical chain T k .Proof. V k is a m -dimensional subspace containing v . Moreover V k splitsas the orthogonal direct sum V k = V k ⊥ ⊕ V k == h e i , e n + i | ≤ i ≤ k i ⊥ ⊕ h e j + e m + j − k + e n + j , e n + j | k + 1 ≤ j ≤ m i == h v ∞ ∩ V k , e n + i | ≤ i ≤ k i ⊥ ⊕ h e j + e m + j − k + e n + j , e n + j | k + 1 ≤ j ≤ m i . Since v ∞ ∩ V k = h e , . . . , e k i , we get that i v ∞ ( T k ) is k . Since h | V k hassignature ( k, k ) and h | V k has signature ( m − k, m − k ) , we get that therestriction of h on V k has signature ( m, m ) and this concludes the proof. (cid:3) Lemma 3.6. H m,n ( v ∞ ) ∩ T k consists precisely of those subspaces of C m + n that admit a basis of the form E ∗ E + C E ∗ XE Id + XE Id + X k
00 Id m − k with E ∈ M (( m − k ) × k, C ) X ∈ U( m − k ) C ∈ u ( k ) . The projection of T k is contained in an affine subspace of M (( n − m ) × m, C ) of dimension m − km , and consists of the points of M (( n − m ) × m, C ) thathave expression (cid:2) E Id+ X (cid:3) with E ∈ M (( m − k ) × k, C ) and X ∈ U( m − k ) .Proof. It is enough to check that the orthogonal to V k is V ⊥ k = h e m + j + e n + j + k , e m + l | ≤ j ≤ m − k < l ≤ n − m i . This implies that any m -dimensional subspace z of V k , that is transverse to v ∞ , has a basis of the form z = Z Z Z Z Z Z km − km − kn − m + kkm − k and therestrictionof h to z is zeroif and only if Z ∗ + Z = Z ∗ Z (11) Z ∗ + Z = Z ∗ Z (12) Z ∗ + Z = Z ∗ Z (21) Z ∗ + Z = Z ∗ Z (22) . Equation (22) restates as Z = Id + X for some X ∈ U( m − k ) : indeeda square matrix Z satisfies the equation Z ∗ + Z = Z ∗ Z , if and only if theequation ( Z − Id) ∗ ( Z − Id) = Z ∗ Z − Z − Z ∗ + Id = Id holds, which meansthat Z − Id belongs to U( m − k ) .This concludes the proof of the first part of the lemma: the ( m − k ) × k matrix Z can be chosen arbitrarily, Equation (12) uniquely determines Z in function of Z and Z , and Equation (11) determines the Hermitian partof Z in function of Z , but is satisfied independently on the antiHermitianpart of Z . This proves the first part of the lemma. AXIMAL REPRESENTATIONS INTO
SU( m, n ) The second part is a direct consequence of the parametrization of T v ∞ k wejust gave, together with the identification of W v ∞ and M (( n − m ) × m, C ) given in Lemma 3.2. (cid:3) Definition 3.7.
We will call a subset of W x that is the projection of a k -vertical chain a ( m, k ) -circle .The reason for the name circle is due to the fact that, in the case ( m, n ) =(1 , the projections of horizontal chains are Euclidean circles in C . Thisfact was first observed and used by Cartan in [Car32]. In fact every Eu-clidean circle C ⊆ C p − is a circle in our generalized definition, namely isthe projection of some 1-chain of ∂ H p C . Indeed we know from Lemma 3.6that the Euclidean circle (1 + e it , , . . . , ∈ C p − is the projection of thechain associated to the linear subspace h e + e , e p +1 i of C p +1 . Moreover theset of Euclidean circles is a homogeneous space under the group of Euclideansimilarities of C p − and the group Q = stab( v ∞ ) acts on C p − as the fullgroup of Euclidean similarities.In the general case it is important to record both the dimension of the m -chain that is projected and the dimension of the U( m − k ) factor in theprojection. This explains the notation. We will call generalized circle anysubset of M (( n − m ) × m, C ) arising as a projection of an m -chain. Inparticular a generalized circle is an ( m, k ) -circle for some k .The ultimate goal of this section is to understand the possible lifts of agiven ( m, k ) -circle. We begin by analyzing the stabilizers in SU( C m + n , h ) ofsome configurations: Lemma 3.8.
The stabilizer in
SU( C m + n , h ) of the triple ( v ∞ , v , T k ) is thesubgroup S of L ∼ = GL m ( C ) × SU( n − m ) consisting of pairs of the form (cid:18)(cid:20) Y X yy − C (cid:21) , (cid:20) C C (cid:21)(cid:19) with C ∈ U( m − k ) C ∈ U( n − m + k ) X ∈ M ( k × ( m − k ) , C ) Y ∈ GL k ( C ) , y = det( Y ) . Proof.
We determined in Section 3.2 that the stabilizer L in SU( C m + n , h ) ofthe pair ( v ∞ , v ) is isomorphic to GL m ( C ) × SU( n − m ) . The stabilizer ofthe triple ( v ∞ , v , T k ) is clearly contained in L and consists precisely of theelements of L stabilizing V ⊥ k .In the proof of Lemma 3.6 we saw that the subspace V ⊥ k has a basis ofthe form h Id n − m X i where X denotes the m × ( n − m ) matrix (cid:2) Id m − k (cid:3) .From the explicit expression of elements in L we get A aa − C A −∗ X = aa − CA −∗ X ∼ = aa − A −∗ XC − . In turn the requirement that aa − A −∗ XC − = X , that is X = aa − A ∗ XC ,implies, in the suitable block decomposition for the matrices, that aa − (cid:20) A ∗ A ∗ A ∗ A ∗ (cid:21) (cid:20) m − k (cid:21) (cid:20) C C C C (cid:21) = aa − (cid:20) A ∗ A ∗ (cid:21) (cid:20) C C C C (cid:21) = aa − (cid:20) A ∗ C A ∗ C A ∗ C A ∗ C (cid:21) . This implies that A ∗ = aa − C − and C = A = 0 . Moreover since C is unitary, also C must be 0, and both C and C must be unitary. Thisconcludes the proof. (cid:3) Let us now denote by o the point o = π v ∞ ( v ) = 0 in W v ∞ and by C k the ( m, k ) -circle that is the projection of T k . We will denote by S the stabilizerin Q of the pair ( o, C k ) . Lemma 3.9.
The stabilizer of the pair ( o, C k ) is the group S = Stab Q ( o, C k ) = M ⋊ S where, as above, we denote by M the center of the nilpotent radical N of Q and by S the stabilizer in Q of the pair ( v , T k ) .Proof. Recall that any element in Q can be uniquely written as a product nl where n is in N , and l belongs to L , the Levi component of Q . Sinceany element in S fixes, by assumption, the point o = π v ∞ ( v ) and since anyelement in L fixes o , if nl is in S then n ( o ) = o that, in turn, implies that n belongs to M . Hence S is of the form M ⋊ S for some subgroup S of L .Let now X be a point in W v ∞ = M (( n − m ) × m, C ) . The action of ( A, C ) ∈ L on W v ∞ is X aa − CXA ∗ . We want to show that if C k ispreserved then ( A, C ) must belong to S . We have proven in Lemma 3.6that any point z ∈ π v ∞ ( T k ) can be written as (cid:2) E Id+ Z (cid:3) for some matrices E ∈ M (( m − k ) × k, C ) and Z ∈ U( m − k ) . Explicit computations give that (cid:20) E Id + Z (cid:21) = aa − (cid:20) C C C C (cid:21) (cid:20) E Id + Z (cid:21) (cid:20) A ∗ A ∗ A ∗ A ∗ (cid:21) == aa − (cid:20) C E C (Id + Z ) C E C (Id + Z ) (cid:21) (cid:20) A ∗ A ∗ A ∗ A ∗ (cid:21) . Since A is invertible, E is arbitrary and both Id and − Id are in U( m − k ) ,the matrix C must be zero. Hence C must have the same block form of agenuine element of S . In particular C is invertible. Since aa − C ( EA ∗ +(Id + Z ) A ∗ ) must be an element of Id + U ( m − k ) for every E , we get that A ∗ must be zero.The result now follows from Claim 3.10 below. (cid:3) Claim 3.10.
Let C ∈ U( l ) and A ∈ GL l ( C ) be matrices and let U denotethe set U = { Id + X | X ∈ U( l ) } ⊂ M ( l × l, C ) . If C U A ∗ = U then A = C . AXIMAL REPRESENTATIONS INTO
SU( m, n ) Proof.
Let us consider the birational map i : M ( l × l, C ) → M ( l × l, C ) X X − that is defined on a Zariski open subset O of M ( l × l, C ) .The image, under the involution i , of U is the set L = { W | Id − W ∗ − W = 0 } = 12 Id + u ( l ) . Moreover i ( CXA ∗ ) = A −∗ i ( X ) C − , hence in order to show that the sub-group preserving U consists precisely of the pairs ( A, A ) , it is enough tocheck that the subgroup of U( l ) × GL l ( C ) preserving L consists precisely ofthe pairs ( A, A ) with A ∈ U( l ) .This last statement amounts to show that the only matrix B ∈ GL l ( C ) such that Id − W ∗ B ∗ − BW = 0 for all W ∈ L is the identity itself. Choosing W to be Id we get that B ∗ + B = 2Id hence in particular B = Id + Z with Z ∈ u ( l ) . Since moreover L = { Id + M | M ∈ u ( l ) } we have to show that if ZM + M ∗ Z ∗ = ZM + M Z = 0 for all M in u ( l ) then Z must be zero, andthis can be easily seen, for example by choosing M to be the matrix that iszero everywhere apart from the l -th diagonal entry where it is equal to i . (cid:3) We now have all the ingredients we need to prove the first crucial resultof the section. Recall from Section 3.1 that every pair of transverse points x, y in S m,n uniquely determines an m -chain T x,y that is the unique chainthat contains both x and y . Proposition 3.11.
Let x ∈ S m,n be a point, T be a chain with i x ( T ) = k , t ∈ T be a point, y = π x ( t ) ∈ W x . Then(1) T is the unique lift of the ( m, k ) -circle π x ( T ) through the point t ,(2) for any point t in T x,t = π − x ( y ) there exists a unique m chainthrough t which lifts π x ( T ) .Proof. (1) As a consequence of Lemma 3.4, in order to prove the statements,we can assume that the triple ( x, t, T ) is the triple ( v ∞ , v , T k ) . Let T ′ be another m -chain containing the point t that lifts the ( m, k ) -circle C k , aconsequence of Lemma 3.4 is that there exists an element g ∈ L such that ( v ∞ , v , T k ) = g ( v ∞ , v , T ′ ) . Moreover, since π v ∞ ( T ′ ) = C k , we get that g ∈ S . But we know that S ∩ L = S and this proves that T ′ = T k .(2) This is a consequence of the first part, together with the observation that M acts transitively on the vertical chain T v ,v ∞ . (cid:3) We conclude the section by determining what are the lifts of a point y that are contained in an m -chain T . For every m -chain T we consider thesubgroup M T = Stab M ( T ) . Clearly if t is a lift of a point y ∈ W v ∞ that is contained in T , then all theorbit M T · t consists of lifts of y that are contained in T . We want to show that also the other containment holds, namely that the lifts are precisely the M T orbit of any point. Lemma 3.12.
For the chain T k we have M T k = i ( E k ) where E k = { X ∈ u ( m ) | X ij = 0 if i > k or j > k } = (cid:26)(cid:20) X
00 0 (cid:21) (cid:12)(cid:12)(cid:12) X ∈ u ( k ) (cid:27) , and i : u ( m ) → N is the inclusion of the center of the group.Proof. We already observed that the orthogonal to V k is V ⊥ k = h e m + j + e n + j + k , e l + m | ≤ j ≤ m − k < l ≤ n − m i . Moreover an element of M stabilizes T k if and only if it stabilizes V ⊥ k . If now m = h Id 0 E i is an element of M , then the image m · ( e m + j + e n + j + k ) = P E ij e j + k + e m + j + e n + j + k that belongs to V ⊥ k if and only if the ( j + k ) -th column of the matrix E is zero. This implies that the subgroup of M that fixes V ⊥ k is contained in i ( E k ) . Viceversa it is easy to check that i ( E k ) belongs to SU( V k ) , in particular it preserves T k . (cid:3) We denote by Z T the intersection of the linear subspace V T underlying T with v ∞ : Z T = v ∞ ∩ V T . In the standard case in which T = T k we will denote by Z k the subspace Z T k which equals to the span of the first k vectors of the standard basis of C m . Proposition 3.13.
Let T be a k -vertical chain, then(1) If g ∈ Q is such that gT = T k , then M T = g − M T k g .(2) For any point x ∈ T , we have π − v ∞ ( π v ∞ ( x )) ∩ T = M T x .(3) If n ∈ N , then M nT = M T .(4) If a ∈ GL( m ) is such that a ( Z T ) = Z k , then M T = i ( a − E k a −∗ ) .Proof. (1) This follows from the definition of M T k and M T and the fact that M is normal in Q .(2) Let us first consider the case T = T k . In this case the statement is aneasy consequence of the explicit parametrization of the chain T k we gave inLemma 3.6: any two points in T k that have the same projection are in thesame M T k orbit. The general case is a consequence of the transitivity of Q on k -vertical chains: let g ∈ Q be such that gT = T k and let us denote by y the point gx . Then we know that M T k y = π − v ∞ ( π v ∞ ( y )) ∩ T k . This impliesthat M T x = g − M T k gx = g − ( M T k y ) = g − ( π − v ∞ ( π v ∞ ( y )) ∩ T k ) == g − π − v ∞ ( π v ∞ ( y )) ∩ g − T k == π − v ∞ ( π v ∞ ( x )) ∩ T. Where in the last equality we used that the Q action on H m,n ( v ∞ ) inducesan action of Q on W v ∞ so that the projection π v ∞ is Q equivariant.(3) This is a consequence of the fact that M is in the center of N : M nT = AXIMAL REPRESENTATIONS INTO
SU( m, n ) nM T n − = M T .(4) By (3) we can assume that T is a chain through the point v : indeedthere exists always an element n ∈ N such that nT contains v , moreoverboth M nT = M T and Z nT = Z T (the second assertion follows from the factthat any element in N acts trivially on v ∞ ).Since v ∈ T and we proved in Lemma 3.4 that L is transitive on k -verticalchains through v , we get that there exists a pair ( C, A ) ∈ U( n − m ) × GL m ( C ) such that, denoting by g the corresponding element in L , we have gT = T k .It follows from (1) that M T = g − M T k g , in particular we have A − C −
00 0 A ∗ Id 0 E A C
00 0 A −∗ = Id 0 A − EA −∗ and hence the subgroup M T is the group i ( A − E k A −∗ ) . Moreover, since gT = T k we have in particular that gZ T = Z k and hence A ( Z T ) = Z k if weconsider Z T as a subspace of v ∞ .In order to conclude the proof it is enough to check that for every a ∈ GL m ( C ) with a ( Z T ) = Z k the subgroups a − E k a −∗ coincide. Indeed it isenough to check that for every element a ∈ GL m ( C ) such that a ( Z k ) = Z k then a − E k a −∗ = E k . But if a satisfies this hypothesis, the matrix a −∗ hasthe form h A A A i . In particular we can compute: a − Xa −∗ = (cid:20) A ∗ A ∗ A ∗ (cid:21) (cid:20) X
00 0 (cid:21) (cid:20) A A A (cid:21) = (cid:20) A ∗ X A
00 0 (cid:21) and the latter matrix still belongs to E k . (cid:3) The restriction to a chain is rational
In this section we prove that the chain geometry defined in Section 3 isrigid in the following sense:
Theorem 4.1.
Let φ : ∂ H p C → S m,n be a measurable, chain geometry pre-serving, Zariski dense map. Then for almost every chain C in ∂ H p C therestriction φ | C coincides almost everywhere with a rational map. Let us recall that, whenever a point x ∈ S m,n is fixed, the center M x ofthe nilpotent radical N x of the stabilizer Q x of x in SU( C m + n , h ) acts onthe Heisenberg model H m,n ( x ) . Moreover, for every m -chain T containingthe point x , the M x action is simply transitive on the Zariski open subset T x of T . The picture above is true for both ∂ H p C ∼ = S ,p and S m,n where, if x ∈ ∂ H p C , the group M x can be identified with u (1) , and, if φ : ∂ H p C → S m,n is the boundary map, M φ ( x ) ∼ = u ( m ) .The idea of the proof is to show that, for almost every point x ∈ ∂ H p C , theboundary map is equivariant with respect to a measurable homomorphism h : M x → M φ ( x ) . Since such homomorphism must be algebraic, we get thatthe restriction of φ to almost every chain through x must be algebraic. In order to define the homomorphism h we will prove first that a map φ satisfying our assumptions induces a measurable map φ x : W x → W φ ( x ) .Here W x can be identified with C p − and W φ ( x ) can be identified with M (( n − m ) × m, C ) , both these identifications are non canonical but we fix themonce and forall. We will then use the map φ x to define a cocycle α : M x × ( ∂ H p C ) x → M φ ( x ) with respect to which φ is equivariant. We will then showthat α is independent on the point x and hence coincides almost everywherewith the desired homomorphism.4.1. First properties of chain preserving maps.
Recall from Section2 that a map φ is Zariski dense if the essential Zariski closure of φ ( ∂ H p C ) is the whole S m,n , or, equivalently if the preimage under φ of any properZariski closed subset of S m,n is not of full measure. Moreover, by definition,a measurable boundary map preserves the chain geometry if the image under φ of almost every pair of distinct points is a pair of transverse points, andthe image of almost every maximal triple ( x , x , x ) in ( ∂ H p C ) , is containedin an m -chain.We will denote by T the set of chains in ∂ H p C , and by T m the set of m -chains of S m,n . The set T is a smooth manifold, indeed an open subsetof the Grassmannian Gr ( C p +1 ) , and we will endow T with its Lebesguemeasure class.The following lemma, an application in this context of Fubini’s theorem,gives the first property of a chain geometry preserving map: Lemma 4.2.
Let φ : ∂ H p C → S m,n be a chain geometry preserving map. Foralmost every chain C ∈ T there exists an m -chain ˆ φ ( C ) ∈ T m such that, foralmost every point x in C , φ ( x ) ∈ ˆ φ ( C ) .Proof. There is a bijection between the set ( ∂ H p C ) { } consisting of triples ofdistinct points on a chain and the set T { } = { ( C, x, y, z ) | C ∈ T and ( x, y, z ) ∈ C (3) } . In turn the projection onto the first factor endows the manifold T { } withthe structure of a smooth bundle over T . In particular Fubini’s theoremimplies that, for almost every chain C ∈ T and for almost every triple ( x, y, z ) ∈ C , the triple ( φ ( x ) , φ ( y ) , φ ( z )) belongs to the same m -chain ˆ φ ( C ) .Moreover ˆ φ ( C ) has the desired properties again as a consequence of Fubinitheorem. (cid:3) We can now use the fact that each pair of transverse points a, b in S m,n uniquely determines a chain T a,b to reformulate Lemma 4.2 in the followingway: Corollary 4.3.
Let φ : ∂ H p C → S m,n be a measurable chain preserving map.Then there exists a measurable map ˆ φ : T → T m such that, for almost everypair ( x, T ) ∈ ∂ H p C × T with x ∈ T , then φ ( x ) ∈ ˆ φ ( T ) . AXIMAL REPRESENTATIONS INTO
SU( m, n ) Proof.
The only thing that we have to check is that the map ˆ φ is measurable,but this follows from the fact that the map associating to a pair ( x, y ) ∈ S (2) m,n the m -chain T x,y is algebraic. (cid:3) Recall that, if x is a point in ∂ H p C , we denote by W x the set of chainsthrough x . We use the identification of W x as subvariety of Gr ( C ,p ) toendow the space W x with its Lebesgue measure class. Corollary 4.4.
For almost every x ∈ ∂ H p C , almost every chain in W x satis-fies Lemma 4.2.Proof. It is again an application of Fubini’s theorem. Let us indeed considerthe manifold T { } = { ( C, x ) | C ∈ T , x ∈ C } the projection on the first twofactor T { } → T { } realizes the first manifold as a smooth bundle over thesecond with fiber ( R × R ) \ ∆ . In particular for almost every pair in T { } thechain satisfies the assumption of Corollary 4.3. Since T { } is a bundle over ∂ H p C with fiber W x over x , the statement follows applying Fubini again. (cid:3) We will call a point x that satisfies the hypotheses of Corollary 4.4 generic for the map φ . Let us now fix, for the rest of the section, a point x that isgeneric for the map φ and consider the diagram H ,p ( x ) φ / / π x (cid:15) (cid:15) H m,n ( φ ( x )) π φ ( x ) (cid:15) (cid:15) W x φ x / / W φ ( x ) . Lemma 4.5. If x is generic for φ , there exists a measurable map φ x such thatthe diagram commutes almost everywhere. Moreover φ x induces a measurablemap ˆ φ x from the set of circles of W x to the set of generalized circles of W φ ( x ) such that, for almost every chain T , we have that ˆ φ ( T ) is a lift of ˆ φ x ( π x ( T )) .Proof. The fact that a map φ x exists making the diagram commutative ona full measure set is a direct application of Corollary 4.4.Since the set of horizontal chains in ∂ H p C is a smooth bundle over the setof Euclidean circles in W x ∼ = C p − , we have that, for almost every Euclideancircle C , the map ˆ φ is defined on almost every chain T with π x ( T ) = C .Moreover a Fubini-type argument implies that, for almost every circle C ,the diagram commutes when restricted to the preimage of C .This implies that the projections ˆ φ x ( C ) := π φ ( x ) ( ˆ φ ( T i )) coincide for almostevery lift T i of C if C satisfies the hypotheses of the previous paragraph, andthis concludes the proof. (cid:3) A measurable cocycle.
Recall that if
H, K are topological groupsand Y is a Borel H -space, then a map α : H × Y → K is a Borel cocycle ifit is a measurable map such that, for every h , h in H and for almost every y ∈ Y , it holds α ( h h , y ) = α ( h , h · y ) α ( h , y ) . Proposition 4.6.
Let φ be a measurable, chain preserving map φ : ∂ H p C →S m,n . For almost every point x in ∂ H p C there exists a measurable cocycle α : M x × ( ∂ H p C ) x → M φ ( x ) such that φ is α -equivariant.Proof. Let us fix a point x generic for the map φ . For almost every pair ( e, y ) where e ∈ M x and y ∈ H ,p ( x ) , we have that the points φ ( y ) and φ ( ey ) are onthe same vertical chain in H m,n ( φ ( x )) . In particular there exists an element α ( e, y ) ∈ M φ ( x ) such that α ( e, y ) φ ( y ) = φ ( ey ) . We extend α by definingit to be 0 on pairs that do not satisfy this assumption. The function α ismeasurable since φ is measurable.We now have to show that the map α we just defined is actually a cocycle.In order to do this let us fix the set O of points z for which ˆ φ ( T x,z ) is m -vertical and φ ( z ) ∈ ˆ φ ( T x,z ) . O has full measure as a consequence of Lemma4.2. Let us now fix two elements e , e ∈ M x . For every element z in thefull measure set O ∩ e − O ∩ e e − O , the three points φ ( z ) , φ ( e z ) , φ ( e e z ) belong to the same vertical m -chain, moreover, by definition of α , we have α ( e e , z ) φ ( z ) = φ ( e ( e z ))= α ( e , e z ) φ ( e z )= α ( e , e z ) α ( e , z ) φ ( z ) . The conclusion follows from the fact that the action of M φ ( x ) on H m,n ( φ ( x )) is simply transitive. (cid:3) Proposition 4.7.
Let us fix a point x . Assume that there exists a measurablefunction β : M x × W x → M φ ( x ) such that for every e ∈ M x , for almost every T in W x and for almost every z in T , the equality α ( e, z ) = β ( e, T ) holds.Then the restriction of the boundary map φ to almost every chain throughthe point x is rational.Proof. We are assuming that for every e in M x for almost every T in W x and for almost every z in T , the equality α ( e, z ) = β ( e, T ) holds. Fubini’sTheorem then implies that for every T in a full measure subset F of W x , foralmost every e in M x and almost every z in T the equality α ( e, z ) = β ( e, T ) holds. In particular for every vertical chain T in F and almost every pair ( e , e ) in M x we have β ( e , T ) β ( e , T ) = β ( e e , T ) : it is in fact enough tochose e and e so that the equality of α ( e i , z ) and β ( e i , T ) holds for almostevery z and compute the cocycle identity for α in a point z that works bothfor e and e .It is classical that if π : G → J satisfies π ( xy ) = π ( x ) π ( y ) for almost everypair ( x, y ) in G , then π coincides almost everywhere with an actual Borelhomomorphism (cfr. [Zim84, Theorem B.2]). In particular for every T in F , we can assume (up to modifying β | T on a zero measure subset) that therestriction of β to T is a measurable homomorphism β T : M x → M φ ( x ) andhence coincides almost everywhere with an algebraic map. Since the actionof M x and M φ ( x ) on each vertical chain is algebraic and simply transitive,we get that for almost every vertical chain T the restriction of φ to T isalgebraic. (cid:3) AXIMAL REPRESENTATIONS INTO
SU( m, n ) The fact that we let β depend on the vertical chain T might be surprising,and it is probably possible to prove that the cocycle α coincides almosteverywhere with an homomorphism that doesn’t depend on the vertical chain T . However since it suffices to prove that the restriction of α to almost everyvertical chain coincides almost everywhere with an homomorphism, and sincethis reduces the technicalities involved, we will restrict to this version.The rest of the section is devoted to prove, using the chain geometry of S m,n , that the hypothesis of Proposition 4.7 is satisfied whenever φ is aZariski dense, chain geometry preserving map and m < n . In the followingproposition we deal with a preliminary easy case, in which the geometricpicture behind the general proof should be clear. Proposition 4.8.
Let φ : ∂ H p C → S m,n be a measurable, Zariski dense,chain geometry preserving map, and let n ≥ m . Then the restriction of φ to almost every chain coincides almost everywhere with a rational map.Proof. We want to apply Proposition 4.7 and show that the cocycle α : M x × ∂ H p C → M φ ( x ) only depends on the vertical chain a point belongs to.We consider the set F ⊆ W x of chains F such that(1) ˆ φ ( F ) is an m -vertical chain, hence in particu-lar φ x ( F ) is defined,(2) for almost every circle C containing the point F ∈ W x , for almost every chain T lifting C the diagram of Lemma 4.5 commutes almosteverywhere. F Tz
It follows from the proof of Lemma 4.5 that the set F is of full measure,moreover, we get, applying Fubini, that if F is an element in F , for almostevery point z in F and almost every chain T through z the diagram of Lemma4.5 commutes almost everywhere when restricted to T . In particular, usingFubini again, this implies that for almost every point w in ∂ H p C the diagramof Lemma 4.5 commutes almost everywhere when restricted to the chain T z,w .Let us now fix a chain F ∈ F and denoteby O the full measure set of points in F for which that holds. For every element e ∈ M x we also consider the full measureset O e = O ∩ e − O . We claim that giventwo points z , z ∈ O e the cocycle α ( e, z i ) has the same value β ( e, F ) . In fact let usconsider the set A z ,z ,e ⊆ ∂ H p C consistingof points w such that(1) φ ( w ) ∈ ˆ φ ( T w,z ) ∩ ˆ φ ( T w,z ) (2) φ ( ew ) ∈ ˆ φ ( T ew,ez ) ∩ ˆ φ ( T ew,ez ) (3) dim h φ ( z ) , φ ( z ) , φ ( w ) i = 3 m . F T z ,w z T z ,w wz T ez ,ew = eT z ,w ez T ez ,ew = eT z ,w ewez We claim that the set A z ,z ,e is not empty. Indeed, by definition of O e , the set of points w satisfying the first two assumption is of full mea-sure. Moreover, since n ≥ m , the set C of points in S m,n such that dim h φ ( z ) , φ ( z ) , φ ( w ) i < m is a proper Zariski closed subset of S m,n . Sincethe map φ is Zariski dense, the preimage of C cannot have full measure, andthis implies that A z ,z ,e has positive measure, in particular it contains atleast one point. The third assumption on the point w implies that the m -chain containing φ ( w ) and φ ( z i ) is horizontal for i = 1 , .Let us fix a point w ∈ A z ,z ,e and consider the m -chain ˆ φ ( eT w,z i ) for i =1 , . The m -chain ˆ φ ( eT w,z i ) is a lift of the ( m, -circle C i = π φ ( x ) ( ˆ φ ( T w,z i )) that contains both the points φ ( ez i ) and φ ( ew ) . In particular, since α ( e, z i ) ˆ φ ( T w,z i ) is a lift of C i containing φ ( ez i ) we get that α ( e, z i ) ˆ φ ( T w,z i ) = ˆ φ ( eT w,z i ) . Simi-larly we get that α ( e, w ) ˆ φ ( T w,z i ) = ˆ φ ( eT w,z i ) . This gives that α ( e, z i ) − α ( e, w ) ∈ M ˆ φ ( T w,zi ) , but the latter group is the trivial group since we know thatthe chain ˆ φ ( T w,z i ) is 0-vertical. This implies that α ( e, z ) = α ( e, w ) = α ( e, z ) . (cid:3) Proof of Theorem 4.1.
Let us now go back to the setting of Theorem4.1: we fix a measurable, chain geometry preserving, Zariski dense map φ : ∂ H p C → S m,n , a generic point x ∈ ∂ H p C such that for almost every chain t ∈ W x , for almost every point y ∈ t , φ ( y ) ∈ ˆ φ ( t ) . We want to show that themeasurable cocycle α : ∂ H p C \{ x } × u (1) → u ( m ) coincides on almost everyvertical chain with a measurable homomorphism. In particular it is enoughto show that for almost every pair z , z on a vertical chain in ∂ H p C , the values α ( e, z ) and α ( e, z ) coincide. For a generic pair ( z , z ) we have that thetriple ( φ ( x ) , φ ( z ) , φ ( z )) is contained in a tube type subdomain and hencewe can compose the map φ with an element of the group SU( m, n ) so that φ ( x ) = v ∞ , φ ( z ) = v and φ ( z ) = v d , here and in the following we denoteby v d the subspace with basis (cid:2) Id 0 d (cid:3) T for some diagonal matrix d withall entries equal to ± i . In fact it is proven in [CN06, Theorem 5.2] that theBergmann cocycle is a complete invariant for the SU( m, n ) action on triplesof pairwise transverse points in an m -chain, and varying the matrix d onegets that β S ( v , v ∞ , v d ) achieves all possible values (cfr. Proposition 2.1).For the rest of the section we restrict to the case n < m , since theotherwise Theorem 4.1 follows from Proposition 4.8 and denote by l theinteger l = n − m and k = 2 m − n . In analogy with the proof of Proposition4.8 we denote by A z ,z ,e to be the full measure subset of ∂ H p C consisting ofpoints with(1) φ ( w ) ∈ ˆ φ ( T w,z ) ∩ ˆ φ ( T w,z ) , (2) φ ( ew ) ∈ ˆ φ ( T ew,ez ) ∩ ˆ φ ( T ew,ez ) , (3) dim h v , v d , φ ( w ) i = m + n. We will consider the subset D v ,v d of S m,n defined by D v ,v d = { w ∈ S m,n | w is transverse to v , v d and h v , v d i} . AXIMAL REPRESENTATIONS INTO
SU( m, n ) It is easy to verify that D v ,v d consists of points w such that both chains T v ,w and T v d ,w are well defined and k -vertical: indeed in our assumptions w is transverse to h v , v d i = h v ∞ , v i . In particular dim h w, v , v ∞ i = n + m and we get n + m = dim h w, v i + dim v ∞ − dim( v ∞ ∩ h v , w i ) which implies that i v ∞ ( T v ,w ) = 3 m − ( n + m ) = k .Our next goal is to associate to any point w in D v ,v d subgroups E ( w ) , I ( w ) which, when we consider a point w that is image of a point z in A v ,v d ,e , rep-resent, respectively, the error allowed by the point z for the cocycle α ( e, v ) and some information on the difference α ( e, v ) − α ( e, v d ) obtained applyingthe strategy of Proposition 4.8 to the point z .In order to define the subgroups properly, we use the identification M v ∞ = u ( m ) provided in Section 3 and use the linear structure on u ( m ) . Moreoverwe will denote by H the positive definite bilinear form on u ( m ) given by H ( A, B ) = tr A ∗ B . We also denote by β v the map β v : D v ,v d ⊆ S m,n → Gr k ( v ∞ ) w → h v , w i ∩ v ∞ similarly we define β v d so that β v d ( w ) = h v d , w i ∩ v ∞ .We want to understand the subspaces on which the possible defect ofthe cocycle α to be an homomorphism are confined. Whenever two k -dimensional subspaces Z i of C m are fixed we denote by S ( Z , Z ) the sub-space: S ( Z , Z ) = h z z ∗ − z z ∗ | z i ∈ Z i i < u ( m ) . The map S is useful to define the error subgroup E ( w ) associated to apoint w in D v ,v d : E ( w ) = S ( β v ( w ) , β v ( w )) + S ( β v d ( w ) β v d ( w )) . For each point z in the set A z ,z ,e the error group E ( φ ( z )) bounds theerror of the cocycle α : Lemma 4.9.
For every point z in A z ,z ,e we get α ( e, z ) − α ( e, z ) ∈ E ( φ ( z )) . Proof.
By the assumption on z we have that ˆ φ ( T ez,ez ) and ˆ φ ( T z,z ) projectto the same ( m, k ) -circle, and in particular we get that α ( e, z ) − α ( e, z ) ∈ M ˆ φ ( T z,z ) . In the same way one gets that α ( e, z ) − α ( e, z ) ∈ M ˆ φ ( T z,z ) . It follows from Proposition 3.13 that if g i ∈ GL( m ) is such that g β v ( φ ( w )) = h e , . . . , e k i (resp. g β v d ( φ ( w )) = h e , . . . , e k i ) we have that M ˆ φ ( T w,zi ) = i ( g − i E k g −∗ i ) , and this proves our first claim since it is easy to check, fromthe definition of the set E ( φ ( w )) that E ( φ ( w )) = g − E k g −∗ + g − E k g −∗ . (cid:3) In particular it is enough to show that, for almost every pair z , z on avertical chain, the intersection T z ∈A z ,z ,e E ( φ ( z )) = { } . In fact this wouldimply that the restriction of α to almost every chain essentially doesn’tdepend on the choice of the point, hence coincides with a measurable homo-morphism. In order to do this we define another subgroup of M v ∞ associatedto a point w ∈ D v ,v d . The information associated to w will be I ( w ) = S (cid:16) ( β v ( w ) ⊥ , β v d ( w ) ⊥ (cid:17) Here the orthogonals are considered with respect to the standard Hermitianform on v ∞ = C m . It is easy to verify that I ( w ) is contained in the orthogonalto E ( w ) with respect to the orthogonal form on u ( m ) given by H ( A, B ) = tr ( A ∗ B ) .We postpone the proof of the following technical lemma to the next sec-tion: Lemma 4.10.
For every proper subspace L of u ( m ) the set C ( L ) = { z ∈D v ∞ v ,v d | I ( z ) ⊆ L } is a proper Zariski closed subset of D v ∞ v ,v d ⊆ S m,n . We now conclude the proof of Theorem 4.1 assuming Lemma 4.10.
Proof of Theorem 4.1 .
We choose m points w , . . . , w m in ∂ H p C such that h I ( φ ( w )) i = u ( m ) : we work by induction and assume that there exist j points w , . . . , w j with dim L j = dim h I ( φ ( w i )) | i ≤ j i ≥ j . If the set L j isequal to the whole u ( m ) we are done. Otherwise it follows from Lemma 4.10that the subset C ( L j ) of D v ∞ v ,v d is a proper Zariski closed subset of D v ∞ v ,v d .In particular, since φ is Zariski dense, its essential image cannot be con-tained in C ( L j ) ∪ ( S m,n \D v ∞ v ,v d ) that is a Zariski closed subset of S m,n . Hencewe can find a point w j +1 in the full measure set A z ,z ,e such that I ( φ ( w j +1 )) is not contained in L j , and this implies that L j +1 = h I ( φ ( w i )) | i ≤ j + 1 i strictly contains L j , hence has dimension strictly bigger than j . This com-pletes the proof of Theorem 4.1. (cid:3) Possible errors.
A crucial step in the proof of Lemma 4.10 is to showthat the map β = β v × β v d : D v ,v d ⊆ S m,n → Gr k ( v ∞ ) is surjective (cfr.Proposition 4.12). As a preparation for this result we give a parametrizationof the image of D v ,v d under the map π v × π v d : D v ,v d → W v × W v d . It iseasy to check that explicit parametrizations of W v and W v d are given by M ( l × m, C ) → W v A → l A ∗ ⊥ M ( l × m, C ) → W v d A → A ∗ Id l dA ∗ ⊥ AXIMAL REPRESENTATIONS INTO
SU( m, n ) Moreover a point A i in W v i correspond to a k -vertical chain if rk( A i ) = m − k .This allows us to give an explicit description of the image: Lemma 4.11.
Under the parametrizations above, the image of the map π v × π v d : D v ∞ v ,v d → W v × W v d is the closed subset C v ,v d of M ( l × m, C ) × M ( l × m, C ) defined by C v ,v d = (cid:26) ( A , A ) (cid:12)(cid:12)(cid:12)(cid:12) A A ∗ ∈ Id + U( m ) A , A have maximal rank (cid:27) . Proof.
We already observed that each point w in D v ,v d uniquely determinestwo k -vertical chains T w,v , T w,v d . In particular for each pair ( A , A ) in theimage of π v × π v d we have that A i has maximal rank. We will now show that A A ∗ ∈ Id + U( m ) if an only if the intersection of linear subspaces associatedto the two m -chains contains a maximal isotropic subspace.Two m -chains T , T intersect in S m,n if and only if the intersection V ∩ V of their underlying vector spaces V , V contains a maximal isotropicsubspace. In turn this is equivalent to the requirement that ( V ∩ V ) ⊥ has signature (0 , l ) . Indeed, since V ⊥ has signature (0 , l ) and is containedin ( V ∩ V ) ⊥ , we get that the signature of any subspace of ( V ∩ V ) ⊥ is ( k , l + k ) for some k , k . On the other hand if V ∩ V contains a maximalisotropic subspace z , then ( V ∩ V ) ⊥ ⊆ z ⊥ and the latter space has signature (0 , l ) . In particular the signature of ( V ∩ V ) ⊥ would be (0 , l ) , and clearlythe orthogonal of a subspace of signature (0 , l ) contains a maximal isotropicsubspace.Since ( V ∩ V ) ⊥ = h V ⊥ , V ⊥ i , we are left to check that the requirementthat signature of this latter subspace is (0 , l ) is equivalent to the requirementthat A A ∗ belongs to Id + U ( l ) . If now we pick a pair ( A , A ) ∈ M ( l × m, C ) × M ( l × m, C ) representing a pair of subspaces ( V , V ) ∈ W v × W v d wehave that the subspace ( V ∩ V ) ⊥ is spanned by the columns of the matrix A ∗ Id Id A ∗ dA ∗ . It is easy to compute the restriction of h to the given generating system of ( V ∩ V ) ⊥ : (cid:20) A A Id − A d (cid:21) Id − IdId A ∗ Id Id A ∗ dA ∗ = (cid:20) A A Id − A d (cid:21) A ∗ dA ∗ − Id − Id0 A ∗ = (cid:20) − Id A A ∗ − Id A A ∗ − Id − Id (cid:21) The latter matrix is negative semidefinite and has rank l if and only if(4.1) A A ∗ A A ∗ − A A ∗ − A A ∗ = ( A A ∗ − Id)( A A ∗ − Id) − Id = 0 . In this case the restriction of h to ( V ∩ V ) ⊥ has signature (0 , l ) . The intersection V ∩ V contains maximal isotropic subspaces that aretransverse to v and v d if and only if the radical of V ∩ V , which coincideswith the radical of ( V ∩ V ) ⊥ , is transverse to both subspaces. It is easy toverify that this is always the case if A and A have maximal rank. (cid:3) We now turn to the analysis of the map β : D v ∞ v ,v d ⊆ S m,n → Gr k ( v ∞ ) z → ( h v , z i ∩ v ∞ , h v d , z i ∩ v ∞ ) . Proposition 4.12.
The map β is surjective.Proof. If we denote by ζ the uniquely defined map with the property that β = ζ ◦ ( π v × π v d ) , then it is easy to check that the map ζ has the followingexpression, with respect to the coordinates described above: ζ : W v × W v d → Gr k ( v ∞ ) ( A , A ) (ker( A ) , ker( A )) . In order to conclude the proof it is enough to show that any pair ( V , V ) of k -dimensional subspaces of v ∞ can be realized as the kernels of a pair ofmatrices satisfying Equation 4.1.We first consider the case in which the subspaces V , V intersect trivially,of course this can only happen if k ≤ l . In this case there exists an element g ∈ U( m ) such that gV = e V = h e . . . , e k i and that e V = gV is spanned bythe columns of the matrix h B Id k i where B is a matrix in M ( k × k, C ) . Clearly V i is the kernel of A i if and only if e V i is the kernel of e A i = A i g − and A i g − satisfies the equation 4.1 if and only if A i does. In particular it is enough toexhibit matrices e A i whose kernel is e V i .Let us first notice that, for any matrix B ∈ M ( k × k, R ) there exists amatrix X ∈ GL k ( C ) such that XB is a diagonal matrix D whose elementsare only 0 or 1. Let us now consider the matrices e A ∗ = (cid:20) (cid:21) kl − k e A ∗ = (cid:20) X D
00 0 Id (cid:21) kl − k By construction e V is the kernel of e A and e V is the kernel of e A , moreoverwe have that e A e A ∗ satisfies Equation 4.1: e A ∗ e A =2 (cid:20) (cid:21) X ∗ D ∗
00 Id = (cid:20) D ∗
00 2Id (cid:21) ∈ Id + U ( m ) . This implies that there is a point z ∈ S m,n with β ( z ) = ( e V , e V ) .The general case, in which the intersection of V i is not trivial, is analogous:we can assume, up to the U( m ) action that V ∩ V = h e , . . . , e s i and we canrestrict to the orthogonal to V ∩ V with respect to the standard Hermitianform. (cid:3) We can now prove Lemma 4.10.
AXIMAL REPRESENTATIONS INTO
SU( m, n ) Proof of Lemma 4.10.
The subspace C ( L ) is Zariski closed since the sub-spaces of u ( m ) that are contained in L form a Zariski closed subset of theGrassmanian Gr( u ( m )) of the vector subspaces of u ( m ) , moreover the sub-space I ( z ) is obtained as the composition I ( z ) = S ◦ β of two regular maps.In order to verify that C ( L ) is a proper subset, unless L = u ( m ) , it isenough to verify that the subspaces of the form S ( Z ⊥ , Z ⊥ ) with Z i transversesubspaces span the whole u ( m ) . Once this is proven, the result follows fromthe the surjectivity of β : since β is surjective, the preimage of a properZariski closed subset is a proper Zariski closed subset. The fact that thespan h z z ∗ − z z ∗ | z , z ∈ C m linearly independent i is the whole u ( m ) follows from the fact that every matrix of the form iz z ∗ is in the span, since such a matrix can be obtained as the difference z ( z − iz ) ∗ − ( z − iz ) z ∗ − ( z z ∗ − z z ∗ ) . (cid:3) The boundary map is rational
In this section we will show that a Zariski dense, chain geometry preservingmap φ : ∂ H p C → S m,n whose restriction to almost every chain is rational,coincides almost everywhere with a rational map.Assume that the chain geometry preserving map φ is rational and let usfix a point x . Since the projection π φ ( x ) : S φ ( x ) m,n → W φ ( x ) is regular we getthat the map φ x : W x → W φ ( x ) induced by φ is rational as well. The firstresult of the section is that the converse holds, namely that if there existenough many generic points s , . . . , s l in ∂ H p C such that φ s i is rational, thenthe original map φ had to be rational as well.In what follows we will denote by l the smallest integer bigger than m/ ( n − m ) . Lemma 5.1.
There exists a Zariski open subset
O ⊂ S lm,n , such that for any ( x , . . . , x l ) in O , there exists a Zariski open subset D x ,...,x l ⊂ S m,n such thatfor every z ∈ D x ,...,x l we have l \ i =1 h z, x i i = z. Proof.
Let us consider the set F of ( l + 1) -tuples ( x , . . . , x l , z ) in S l +1 m,n withthe property that T li =1 h z, x i i = z . This is a Zariski open subset of S l +1 m,n :indeed, since z is clearly contained in the intersection, the set F is definedby the equation dim T li =1 h z, x i i ≤ m . In order to conclude the proof it isenough to show that for each z ∈ S m,n the set of tuples ( x , . . . , x l ) withthe property that ( x , . . . , x l , z ) ∈ F is non empty: this implies that the set F is a non empty Zariski open subset, and in particular there must exist aZariski open subset of S lm,n consisting of l -tuples ( x , . . . , x l ) satisfying thehypothesis of the lemma. Let us then fix a point z ∈ S m,n . We denote by A kz the set of k -tuples x = ( x , . . . , x k ) in S zm,n such that dim T ki =1 h z, x i i = max { m − ( n − m )( k − , m } . In order to conclude the proof it is enough to exhibit, for every k -tupla x in A kz a non empty subset B of S m,n such that ( x, b ) ∈ A k +1 z foreach b in B . If we denote by V k the subspace T ki =1 h z, x i i , that has, by ourassumption on the tuple x , dimension m − ( n − m )( k − , we can take B to be the Zariski open subset B = { x k +1 ∈ S m,n | x k +1 ⋔ V k , x k +1 ⋔ z } . The set B is not empty since both transversality conditions are non-empty,Zariski open conditions, and for this choice we get dim T k +1 i =1 h z, x i i = dim( V k ∩ h z, x k +1 i )= dim V k + dim h z, x k +1 i − dim h V k , x k +1 i == 2 m − ( n − m )( k −
1) + 2 m − min { m + n, m − ( n − m )( k −
1) + m } == max { m − ( n − m ) k, m } . (cid:3) It is worth remarking that, if n ≥ m , the set S (2) m,n is contained in O andfor x , x transverse the set D x ,x consists of the points z that are transverseto x , x and h x , x i . This is consistent with the notation in Section 4.4. Ingeneral we will assume (up to restricting D x ,...,x l to a smaller Zariski opensubset) that each z in D x ,...,x l is transverse to x i for each i . Lemma 5.2.
Let ( x , . . . , x l ) be an l -tuple of pairwise transverse points inthe set O defined in Lemma 5.1. There exist a quasiprojective subset C x ,...,x l of W x × . . . × W x l such that the map β x ,...,x l = π x × . . . × π x l : D x ,...,x l → W x × . . . × W x l gives a birational isomorphism.Proof. We consider the set C ′ x ,...,x l consisting of tuples ( t , . . . , t l ) with theproperty that the associated linear subspaces intersect in an m -dimensionalisotropic subspace, and that t j − π j ( x i ) has maximal rank for every i, j . Withthis choice C ′ x ,...,x l is quasiprojective since the condition that the intersectionhas dimension at least m and that the restriction of h to the intersection isdegenerate are closed condition (defined by polynomial), the condition thatthe intersection has dimension at most m is an open condition. The set C x ,...,x l is the subset of C ′ x ,...,x l that is the image of β x ,...,x l .The fact that the map β x ,...,x l gives a birational isomorphism follows fromthe fact that a regular inverse to β x ,...,x l is given by the algebraic map thatassociates to an l -tuple of points their unique intersection. (cid:3) We now have all the ingredients we need to prove the following
Proposition 5.3.
Let us assume that for almost every point x ∈ ∂ H p C themap φ x coincides almost everywhere with a rational map. The same is truefor φ . AXIMAL REPRESENTATIONS INTO
SU( m, n ) Proof.
Let us fix l points t , . . . , t l which are generic in the sense of Lemma4.2, which satisfy that φ t i coincides almost everywhere with a rational map,and with the additional property that the l -tupla ( φ ( t ) , . . . , φ ( t l )) belongsto the set O . We can find such points since the map φ is Zariski dense andthe set O is Zariski open. Let us now consider the diagram ∂ H p C \ { t , . . . , t l } φ / / π t × ... × π tl (cid:15) (cid:15) D φ ( t ) ,...,φ ( t l ) ⊆ S m,nβ φ ( t ,...,φ ( tl ) (cid:15) (cid:15) W t × . . . × W t n φ t × ... × φ tn / / C φ ( t ) ,...,φ ( t l ) . A consequence of Lemma 4.5 and of the definition of the isomorphisms β φ ( t ) ,...,φ ( t l ) is that the diagram commutes almost everywhere. In partic-ular, since the isomorphism β φ ( t ) ,...φ ( t l ) is birational, and π t × . . . × π t l isrational, we get that φ coincides almost everywhere with a rational map. (cid:3) Let us now fix a point x in ∂ H p C , and identify the space W x with C p − .We want to study the map φ x : C p − → W φ ( x ) . It follows from Lemma 3.6restricted to the case m = 1 that the projections of chains in ∂ H p C to C p − are Euclidean circles C ⊂ C p − (possibly collapsed to points). Lemma 5.4. If x is generic in the sense of Lemma 4.2, the restriction of φ x to almost every Euclidean circle C of C p − is rational.Proof. It follows from the explicit parametrization of a chain given in Lemma3.6 that, whenever a point t in π − x ( C ) is fixed, the lift map l : C → T isalgebraic, where T is the unique lift of C containing t .In particular, if T is a chain such that the restriction of φ to T coincidesalmost everywhere with a rational map, the restriction of φ x to C = π x ( T ) coincides almost everywhere with a rational map. We can now use a Fubinibased argument to get that, for almost every circle C , the restriction of φ x to C is rational: for almost every chain T , the restriction to T coincidesalmost everywhere with a rational map, the conclusion follows from the factthat the space of chains that do not contain x is a full measure subset ofthe space of chains in ∂ H p C that forms a smooth bundle over the space ofEuclidean circles in C p − . (cid:3) An usual Fubini type argument implies now the following
Corollary 5.5.
For almost every complex affine line
L ⊆ C m , for almostevery Euclidean circle C contained in L , the restriction of φ x to C is alge-braic. Moreover the same is true for almost every point p in L and almostevery circle C containing p . In order to conclude the proof we will apply many times the following wellknown lemma that allows to deduce that a map is rational provided thatthe restriction to sufficiently many subvarieties is rational. Given a map φ : A × B → C and given a point a ∈ A we denote by a φ : B → C the map a φ ( b ) = φ ( a, b ) in the same way, if b is a point in B , b φ will the note the map b φ ( a ) = φ ( a, b ) Lemma 5.6 ([Zim84, Theorem 3.4.4]) . Let φ : R n + m → R be a measurablefunction. Let us consider the splitting R n + m = R n × R m . Assume that foralmost every a ∈ R n the function a φ : R m → R coincides almost everywherewith a rational function and for almost every b ∈ R m the function b φ : R n → R coincides almost everywhere with a rational function, then φ coincidesalmost everywhere with a rational function. This easily gives that the restriction of φ x to any complex affine line L in C p − coincides almost everywhere with a rational map: Lemma 5.7.
For almost every affine complex line
L ⊂ C p − , the restriction φ x | L coincides almost everywhere with a rational map.Proof. Let us fix a line L satisfying the hypothesis of Corollary 5.5 anddenote by φ L : C → W φ ( x ) the restriction of φ x to L , composed with a linearidentification of L with the complex plane C . By the second assertion ofCorollary 5.5, we can find a point p ∈ C such that for almost every Euclideancircle C through p the restriction of φ L to C coincides almost everywherewith a rational map. Let us now consider the birational map i p : C → C defined by i p ( z ) = ( z − p ) − , and let us denote by ψ L the composition ψ L = φ L ◦ i − p . Since the image under i p of Euclidean circles through thepoint p are precisely the affine real lines of C that do not contain , weget that the restriction of ψ L to almost every affine line coincides almosteverywhere with a rational map. In particular a consequence of Lemma 5.6is that the map ψ L itself coincides almost everywhere with a rational map.Since φ L coincides almost everywhere with ψ L ◦ i p we get that the same istrue for the map φ L and this concludes the proof. (cid:3) Applying Lemma 5.6 again we deduce the following proposition:
Proposition 5.8.
Let φ : ∂ H p C → S m,n be a map with the property that foralmost every chain C the restriction of φ to C coincides almost everywherewith a rational map. Then for almost every point x ∈ ∂ H p C the map φ x coincides almost everywhere with a rational map. In turn this was the last missing ingredient to prove Theorem 1.66.
Conclusion
The last step of Margulis’ original proof of superrigidity involves showingthat if a Zariski dense representation ρ : Γ → H of a lattice Γ in the algebraicgroup G admits an algebraic boundary map, then it extends to a represen-tation of G (cfr. [Mar91] and [Zim84, Lemma 5.1.3]). The same argumentapplies here to deduce our main theorem: Proof of Theorem 1.1.
Let ρ : Γ → PU( m, n ) be a Zariski dense maximalrepresentation and let ψ : ∂ H p C → S m,n be a measurable ρ -equivariant AXIMAL REPRESENTATIONS INTO
SU( m, n ) boundary map, that exists as a consequence of Proposition 2.8 (the dif-ference between SU( m, n ) and PU( m, n ) plays no role here, since the actionof SU( m, n ) on S m,n factors through the projection to the adjoint form ofthe latter group). The essential image of ψ is a Zariski dense subset of S m,n as a consequence of Proposition 2.9, moreover Corollary 2.12 implies that ψ preserves the chain geometry.Since we proved that any measurable, Zariski dense, chain preservingboundary map ψ coincides almost everywhere with a rational map (cfr. The-orem 1.6), we get that there exists a ρ -equivariant rational map φ : ∂ H p C →S m,n . The ρ -equivariance follows from the fact that φ coincides almost ev-erywhere with ψ that is ρ -equivariant. In particular, for every γ in Γ the seton which φ ( γx ) = ρ ( γ ) φ ( x ) is a Zariski closed, full measure set, and henceis the whole ∂ H p C .Since φ is ρ -equivariant and rational it is actually regular: indeed the setof regular points for φ is a non-empty, Zariski open, Γ -equivariant subset of ∂ H p C . Since, by Borel density [Zim84, Theorem 3.2.5], the lattice Γ is Zariskidense in SU(1 , p ) and ∂ H p C is an homogeneous algebraic SU(1 , p ) space, theonly Γ -invariant proper Zariski closed subset of ∂ H p C is the empty set, andthis implies that the set of regular points of φ is the whole ∂ H p C .In the sequel it will be useful to deal with complex algebraic groups andcomplex varieties in order to exploit algebraic results based on Nullstellen-satz. This is easily achieved by considering the complexification. We will de-note by G the algebraic group SL( p +1 , C ) and by H the group PSL( m + n, C ) endowed with the appropriate real structures so that SU(1 , p ) = G ( R ) and PU( m, n ) = H ( R ) . Since ∂ H p C and S m,n are homogeneous spaces that areprojective varieties, there exist parabolic subgroups P <
SL( p + 1 , C ) and Q <
PSL( m + n, C ) such that ∂ H p C = ( G/P )( R ) = G ( R ) / ( P ∩ G ( R )) and S m,n = ( H/Q )( R ) .The algebraic ρ -equivariant map φ : ∂ H p C → S m,n lifts to a map φ : G ( R ) → S m,n and we can extend the latter map uniquely to an algebraicmap T : G → H/Q using the fact that G ( R ) is Zariski dense in G . Theextended map T is ρ -equivariant since G ( R ) is Zariski dense in G : wheneveran element γ ∈ Γ is fixed, the set { g ∈ G | T ( γg ) = ρ ( γ ) T ( g ) } is Zariskiclosed and contains G ( R ) .Let us now focus on the graph of the representation ρ : Γ → H as asubset Gr( ρ ) of the group G × H . Since ρ is an homomorphism, Gr( ρ ) is asubgroup of G × H , hence its Zariski closure Gr( ρ ) Z is an algebraic subgroup.The image under the first projection π of Gr( ρ ) Z is a closed subgroup of G : indeed the image of a rational morphism (over an algebraically closedfield) contains an open subset of its closure, since in our case π is a grouphomomorphism, its image is an open subgroup that is hence also closed.Moreover π (Gr( ρ ) Z ) contains Γ that is Zariski dense in G by Borel density,hence equals G . We now want to use the existence of the algebraic map T and the factthat ρ (Γ) is Zariski dense in H to show that Gr( ρ ) Z is the graph of anhomomorphism. In fact it is enough to show that Gr( ρ ) Z ∩ ( { id } × H ) =(id , id) . Let (id , f ) be an element in Gr( ρ ) Z ∩ ( { id } × H ) . Since H isabsolutely simple being an adjoint form of a simple Lie group, and N = T h ∈ H hQh − is a normal subgroup of H , it is enough to show that f ∈ N or, equivalently, that f fixes pointwise H/Q .But, since T is a regular map, and the actions of G on itself and of H on H/Q are algebraic, we get that the stabilizer of the map T under the G × H -action, Stab G × H ( T ) = { ( g, h ) | (( g, h ) · T )( x ) = h − T ( gx ) = T ( x ) , ∀ x ∈ G } , is a Zariski closed subgroup of G × H . Moreover Stab G × H ( T ) contains Gr( ρ ) and hence also Gr( ρ ) Z . In particular (id , f ) belongs to the stabilizer of T ,hence the element f of H fixes the image of T pointwise. Since the image of T is ρ (Γ) -invariant, ρ (Γ) is Zariski dense and the set of points in H/Q thatare fixed by f is a closed subset, f acts trivially on H/Q . (cid:3) We can now prove Theorem 1.3:
Proof of Theorem 1.3.
Let ρ : Γ → SU( m, n ) be a maximal representa-tion and let L be the Zariski closure of ρ (Γ) in SL( m + n, C ) . Here, asabove, SU( m, n ) = H ( R ) with respect to a suitable real structure on H =SL( m + n, C ) . Since the representation ρ is tight, we get, as a consequenceof Theorem 2.4, that L ( R ) almost splits a product L nc × L c where L nc is asemisimple Hermitian Lie group tightly embedded in SU( m, n ) and K = L c is a compact subgroup of SU( m, n ) .Let us consider L , . . . , L k the simple factors of L nc , namely L nc , beingsemisimple, almost splits as the product L × . . . × L k where L k are simpleHermitian Lie groups. The first observation is that none of the groups L i can be virtually isomorphic to SU(1 , . In that case the composition ofthe representation ρ with the projection L nc → L i would be a maximalrepresentation of a complex hyperbolic lattice with values in a group thatis virtually isomorphic to SU(1 , and this is ruled out by [BI08]: indeedBurger and Iozzi prove, as the last step in their proof of [BI08, Theorem 2],that there are no maximal representations of complex hyperbolic lattices in PU(1 , .This implies that the inclusion i : L nc → SU( m, n ) fulfills the hypothesesof Theorem 2.5. In particular it is enough to prove that each factor L i whichis not of tube type is isomorphic to SU(1 , p ) and the composition of ρ withthe projection to L i is conjugate to the inclusion. Since, by Corollary 1.2,there is no Zariski dense representation of Γ in SU( m i , n i ) if < m i < n i , weget that m i = 1 . Moreover, since the only Zariski dense tight representationof SU(1 , p ) in SU(1 , q ) is the identity map, we get that n i = p and the AXIMAL REPRESENTATIONS INTO
SU( m, n ) composition of ρ with the projection to L i is conjugate to the inclusion.This concludes the proof. (cid:3) Proof of Corollary 1.4.
We know that the Zariski closure of the represen-tation ρ is contained in a subgroup of SU( m, n ) isomorphic to SU(1 , p ) t × SU( m − t, m − t ) × K . The product M = SU(1 , p ) t × SU( m − t, m − t ) corresponds to a splitting C m,n = V ⊕ . . . ⊕ V t ⊕ W ⊕ Z where the restrictionof h to V i is non-degenerate and has signature (1 , p ) and the restriction of h to W is non-degenerate and has signature ( m − t, m − t ) . The subspace W is left invariant by M hence also by K (since K commutes with M andall the invariant subspaces for M have different signature). In particular thelinear representation of Γ associated with ρ leaves invariant a subspace onwhich h has signature ( k, k ) for some k greater than 1 unless there are nofactors of tube-type in the decomposition of L . This latter case correspondsto standard embeddings. (cid:3) Proof of Corollary 1.5.
Let us denote by ρ : Γ → SU( m, n ) the standardrepresentation. Since by Lemma 2.7 the generalized Toledo invariant is con-stant on components of the representation variety, we get that any otherrepresentation ρ in the component of ρ is maximal. By Theorem 1.3 thisimplies that ρ (Γ) Z almost splits as a product K × L t × SU(1 , p ) , and is con-tained in a subgroup of SU( m, n ) of the form SU(1 , p ) t × SU( m − t, m − t ) × K .If the group L t is trivial then ρ is a standard embedding, and is hence con-jugate to ρ up to a character in the compact centralizer of the image of ρ .In particular this would imply that ρ is locally rigid.Let us then assume by contradiction that there are representations ρ i arbitrarily close to ρ and with the property that the tube-type factor of theZariski closure of ρ i (Γ) is non trivial. Up to modifying the representations ρ i we can assume that the compact factor K in the Zariski closure of ρ i istrivial.By Theorem 1.3 this implies that ρ i (Γ) is contained in a subgroup of SU( m, n ) isomorphic to SU( m, pm − , moreover we can assume, up toconjugate the representations ρ i in SU( m, n ) , that the Zariski closure of ρ i is contained in the same subgroup SU( m, pm − for every i . Since therepresentations whose image is contained in the subgroup SU( m, pm − isa closed subspace of Hom(Γ , G ) /G , we get that the image of ρ is containedin SU( m, pm − and this is a contradiction, since the image of the diagonalembedding doesn’t leave invariant any subspace on which the restriction of h has signature ( m, pm − . (cid:3) References [BBI13] M. Bucher, M. Burger, and A. Iozzi. A dual interpretation of the Gromov-Thurston proof of Mostow rigidity and volume rigidity for representations ofhyperbolic lattices. In
Trends in harmonic analysis , volume 3 of
Springer IN-dAM Ser. , pages 47–76. Springer, Milan, 2013. [BF14] U. Bader and A. Furman. Boundaries, rigidity of representations, and Lyapunovexponents.
ArXiv e-prints , April 2014.[BGPG06] S. B. Bradlow, O. García-Prada, and P. B. Gothen. Maximal surface group rep-resentations in isometry groups of classical Hermitian symmetric spaces.
Geom.Dedicata , 122:185–213, 2006.[BI00] M. Burger and A. Iozzi. Bounded cohomology and representation varieties oflattices in
PU(1 , n ) . Preprint announcement , 2000.[BI02] M. Burger and A. Iozzi. Boundary maps in bounded cohomology. Appendix to:“Continuous bounded cohomology and applications to rigidity theory” [Geom.Funct. Anal. (2002), no. 2, 219–280; MR1911660 (2003d:53065a)] by Burgerand N. Monod. Geom. Funct. Anal. , 12(2):281–292, 2002.[BI04] M. Burger and A. Iozzi. Bounded Kähler class rigidity of actions on Hermitiansymmetric spaces.
Ann. Sci. École Norm. Sup. (4) , 37(1):77–103, 2004.[BI07] M. Burger and A. Iozzi. Bounded differential forms, generalized Milnor-Woodinequality and an application to deformation rigidity.
Geom. Dedicata , 125:1–23,2007.[BI08] M. Burger and A. Iozzi. A measurable Cartan theorem and applications to de-formation rigidity in complex hyperbolic geometry.
Pure Appl. Math. Q. , 4(1,Special Issue: In honor of Grigory Margulis. Part 2):181–202, 2008.[BI09] M. Burger and A. Iozzi. A useful formula from bounded cohomology. In
Géométries à courbure négative ou nulle, groupes discrets et rigidités , volume 18of
Sémin. Congr. , pages 243–292. Soc. Math. France, Paris, 2009.[BIW09] M. Burger, A. Iozzi, and A. Wienhard. Tight homomorphisms and Hermitiansymmetric spaces.
Geom. Funct. Anal. , 19(3):678–721, 2009.[BIW10] M. Burger, A. Iozzi, and A. Wienhard. Surface group representations with max-imal Toledo invariant.
Ann. of Math. (2) , 172(1):517–566, 2010.[BM99] M. Burger and N. Monod. Bounded cohomology of lattices in higher rank Liegroups.
J. Eur. Math. Soc. (JEMS) , 1(2):199–235, 1999.[BM02] M. Burger and N. Monod. Continuous bounded cohomology and applications torigidity theory.
Geom. Funct. Anal. , 12(2):219–280, 2002.[BW00] A. Borel and N. Wallach.
Continuous cohomology, discrete subgroups, and rep-resentations of reductive groups , volume 67 of
Mathematical Surveys and Mono-graphs . American Mathematical Society, Providence, RI, second edition, 2000.[Car32] E. Cartan. Sur le groupe de la géométrie hypersphérique.
Comment. Math. Helv. ,4(1):158–171, 1932.[Car35] E. Cartan. Sur les domaines bornés homogènes de l’espace den variables com-plexes.
Abh. Math. Sem. Univ. Hamburg , 11(1):116–162, 1935.[Cle07] J. L. Clerc. An invariant for triples in the Shilov boundary of a bounded sym-metric domain.
Comm. Anal. Geom. , 15(1):147–173, 2007.[CN06] J.-L. Clerc and K.-H. Neeb. Orbits of triples in the Shilov boundary of a boundedsymmetric domain.
Transform. Groups , 11(3):387–426, 2006.[CØ03] J. L. Clerc and B. Ørsted. The Gromov norm of the Kaehler class and the Maslovindex.
Asian J. Math. , 7(2):269–295, 2003.[Cor88] K. Corlette. Flat G -bundles with canonical metrics. J. Differential Geom. ,28(3):361–382, 1988.[DM93] P. Deligne and G. D. Mostow.
Commensurabilities among lattices in
PU(1 , n ) ,volume 132 of Annals of Mathematics Studies . Princeton University Press,Princeton, NJ, 1993.[DT87] A. Domic and D. Toledo. The Gromov norm of the Kaehler class of symmetricdomains.
Math. Ann. , 276(3):425–432, 1987.[Dup76] J. Dupont. Simplicial de Rham cohomology and characteristic classes of flatbundles.
Topology , 15(3):233–245, 1976.
AXIMAL REPRESENTATIONS INTO
SU( m, n ) [Fur81] H. Furstenberg. Rigidity and cocycles for ergodic actions of semisimple Lie groups(after G. A. Margulis and R. Zimmer). In Bourbaki Seminar, Vol. 1979/80 ,volume 842 of
Lecture Notes in Math. , pages 273–292. Springer, Berlin-NewYork, 1981.[GM87] W. M. Goldman and J. J. Millson. Local rigidity of discrete groups acting oncomplex hyperbolic space.
Invent. Math. , 88(3):495–520, 1987.[Gol80] W. M. Goldman.
Discontinuous groups and the Euler class . ProQuest LLC, AnnArbor, MI, 1980. Thesis (Ph.D.)–University of California, Berkeley.[Gol99] W. M. Goldman.
Complex hyperbolic geometry . Oxford Mathematical Mono-graphs. The Clarendon Press Oxford University Press, New York, 1999. OxfordScience Publications.[GW12] O. Guichard and A. Wienhard. Anosov representations: domains of discontinuityand applications.
Invent. Math. , 190(2):357–438, 2012.[Ham11] O. Hamlet. Tight holomorphic maps, a classification.
ArXiv e-prints , October2011.[Ham12] O. Hamlet. Tight maps, a classification.
ArXiv e-prints , June 2012.[Her91] L. Hernández. Maximal representations of surface groups in bounded symmetricdomains.
Trans. Amer. Math. Soc. , 324(1):405–420, 1991.[HP14] O. Hamlet and B. Pozzetti. Classification of tight homomorphisms.
ArXiv e-prints , November 2014.[Kli11] B. Klingler. Local rigidity for complex hyperbolic lattices and Hodge theory.
Invent. Math. , 184(3):455–498, 2011.[KM08a] V. Koziarz and J. Maubon. Harmonic maps and representations of non-uniformlattices of
PU( m, . Ann. Inst. Fourier (Grenoble) , 58(2):507–558, 2008.[KM08b] V. Koziarz and J. Maubon. Representations of complex hyperbolic lattices intorank 2 classical Lie groups of Hermitian type.
Geom. Dedicata , 137:85–111, 2008.[Kor00] A. Korányi. Function spaces on bounded symmetric domains. In
Analysis andgeometry on complex homogeneous domains , volume 185 of
Progress in Mathe-matics , pages 185–281. Birkhäuser Boston Inc., Boston, MA, 2000.[KP09] I. Kim and P. Pansu. Local rigidity in quaternionic hyperbolic space.
J. Eur.Math. Soc. (JEMS) , 11(6):1141–1164, 2009.[KW65] A. Korányi and J. A. Wolf. Realization of hermitian symmetric spaces as gener-alized half-planes.
Ann. of Math. (2) , 81:265–288, 1965.[Mar91] G. A. Margulis.
Discrete subgroups of semisimple Lie groups , volume 17 of
Ergeb-nisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics andRelated Areas (3)] . Springer-Verlag, Berlin, 1991.[Mon01] N. Monod.
Continuous bounded cohomology of locally compact groups , volume1758 of
Lecture Notes in Mathematics . Springer-Verlag, Berlin, 2001.[Mos80] G. D. Mostow. On a remarkable class of polyhedra in complex hyperbolic space.
Pacific J. Math. , 86(1):171–276, 1980.[Poz14] B. Pozzetti. PhD Thesis. 2014.[Spi14] M. Spinaci. Rigidity of maximal holomorphic representations of Kähler groups .
ArXiv e-prints , September 2014.[Tol89] D. Toledo. Representations of surface groups in complex hyperbolic space.
J.Differential Geom. , 29(1):125–133, 1989.[Tol03] D. Toledo. Maps between complex hyperbolic surfaces.
Geom. Dedicata , 97:115–128, 2003. Special volume dedicated to the memory of Hanna Miriam Sandler(1960–1999).[vE53] W. T. van Est. Group cohomology and Lie algebra cohomology in Lie groups. I,II.
Nederl. Akad. Wetensch. Proc. Ser. A. = Indagationes Math. , 15:484–492,493–504, 1953.[Zim84] R. J. Zimmer. Ergodic theory and semisimple groups , volume 81 of
Monographsin Mathematics . Birkhäuser Verlag, Basel, 1984.
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