Isospectrality of Margulis-Smilga spacetimes for irreducible representations of real split semisimple Lie groups
IISOSPECTRALITY OF MARGULIS-SMILGA SPACETIMESFOR IRREDUCIBLE REPRESENTATIONS OF REAL SPLITSEMISIMPLE LIE GROUPS
SOURAV GHOSH
Abstract.
In this article we show that, under certain conditions, equal-ity of the Margulis-Smilga invariant spectra of two Margulis-Smilgaspacetimes induce an automorphism of the ambient affine Lie group.In particular, we show that equality of the Margulis-Smilga invariantspectra of two Margulis-Smilga spacetimes, coming from the adjointrepresentation of a real split simple algebraic Lie group G with trivialcenter and Lie algebra g , induce an automorphism of the affine group G (cid:110) g . Contents
Introduction 2Acknowledgements 51. Jordan Decomposition 52. Weights and eigenspaces 63. Characteristic Polynomial 94. Unit eigenspace projection 105. Margulis-Smilga invariant 126. Isospectrality: fixed linear part 147. Isospectrality of norm: zero spectrum 158. Isospectrality of norm: fixed linear part 179. Isospectrality of norm: general case 1910. Isospectrality: general case 22Appendix A. Normal subgroups 23References 26
Date : September 29, 2020.The author acknowledge support from the following grant: OPEN/16/11405402. a r X i v : . [ m a t h . G T ] S e p SOURAV GHOSH
Introduction
Let G be a noncompact real semisimple Lie group, let V be a finite di-mensional vector space and let R : G → GL ( V ) be a faithful irreduciblerepresentation. We denote the affine group obtained from this representa-tion by G (cid:110) R V . In the main body of this article, when there is no confusion,we will omit the subscript R from G (cid:110) R V . Moreover, for a hyperbolic groupΓ, let ρ : Γ → G (cid:110) R V be a representation such that the projection of ρ ( γ )in G is loxodromic for all non identity element γ ∈ Γ. In a series of works[Smi16a, Smi18, Smi16b] Smilga shows that if the faithful irreducible repre-sentation R satisfy a few conditions (which are in particular satisfied by theadjoint representation), then for the nonabelian free group Γ with finitelymany generators, there exist ρ : Γ → G (cid:110) R V such that ρ (Γ) act properly on V . In such a situation, we call the quotient space ρ (Γ) \ V a Margulis-Smilgaspacetime . More precisely,
Definition 0.1.
Let G be a real semisimple Lie group of noncompact typewith trivial center, let V be a finite dimensional vector space and let R : G → GL ( V ) be a faithful irreducible representation. Moreover, let ρ : Γ → G (cid:110) R V be a representation with its linear part L ρ : Γ → G be such that L ρ ( γ ) isloxodromic for all non identity element γ ∈ Γ and L ρ (Γ) is Zariski denseinside G . Then ρ (Γ) \ V is called a Margulis-Smilga spacetime if and only if ρ (Γ) act properly on V .Interest in these spaces started from counter intuitive examples intro-duced by Margulis [Mar83, Mar84], to answer a question of Milnor [Mil77]regarding the Auslander Conjecture [Aus64]. The examples constructed byMargulis are examples of Margulis-Smilga spacetimes when G ∼ = SL ( R ) and R is the adjoint representation. Note that the adjoint action of SL ( R ) canalso be seen as the linear action of SO (2 ,
1) on R . Later on, similar ex-amples were constructed for the linear action of SO (2 n, n −
1) on R n − by Abels–Margulis–Soifer [AMS02]. Recently, similar examples were con-structed for adjoint representaions of any noncompact semisimple Lie groupin [Smi16a] and for any general representation satisfying certain special cri-teria in [Smi18, Smi16b]. These criteria are technical in nature but theycan be roughly translated to mean the following: the unit eigenspace of R g for every loxodromic element g ∈ G contains a nontrivial subspace V g suchthat any two subspaces V g and V h can be canonically identified with eachother via a map π g,h : V g → V h with π g,g − never being the identity map (formore details please see the Main Theorem at page 4 of [Smi16b]). In the casewhere G is split, the spaces V g for g ∈ G are precisely the unit eigenspaces of R g and the criteria on R boils down to the existence of nontrivial zero weightspaces and the action of the Weyl group on the zero weight space being non-trivial. A complete classification of such representations has recently beenobtained, in the split case by LeFloch–Smilga [LFS18] and in the generalcase by Smilga [Smi20].In the original construction of Margulis [Mar83, Mar84] a certain realvalued invariant played a central role in the detection of proper affine ac-tions. These invariants are called Margulis invariants . Later, similar realvalued invariants were introduced in [AMS02] to detect proper affine actions
ARGULIS-SMILGA SPACETIMES 3 of representations into SO (2 n, n − (cid:110) R n − . Recently, these invariantswere generalized by Smilga in [Smi16a, Smi18, Smi16b] into vector valuedinvariants to detect proper affine actions into G (cid:110) R V . We call these gen-eralized vector valued invariants introduced by Smilga as Margulis-Smilga invariants. The definition of these invariants are also technical in nature butroughly they can be thought of as follows: for any ( g, X ) ∈ G (cid:110) R V we denotethe projection of X onto V g , with respect to some canonical decompositionof eigenspaces of R g , by X g , then the Margulis-Smilga invariant M ( g, X ) isthe class [ π g,h ( X g ) | h ∈ G ] (for a more precise definition in the split casesee Definition 5.3 and for the general case see Definition 7.19 of [Smi16b]).The marked spectrum of Margulis-Smilga invariants of a Margulis-Smilgaspacetime closely resemble the marked length spectrum of a hyperbolic sur-face. In fact, it is not very difficult to show that the marked Margulis-Smilgainvariant spectrum of a Margulis-Smilga spacetime ρ satisfy the following: M ( ρ ( γ )) (cid:54) = 0 for all non identity element γ ∈ Γ. Also, the additivity ofMargulis-Smilga invariants play a crucial role in constructing examples ofproper affine actions. In the case when G ∼ = SL ( R ) and R is the adjoint rep-resentation, Drumm–Goldman [DG01] showed that the Margulis-Smilga in-variant spectrum of two Margulis-Smilga spacetimes ρ and (cid:37) are same if andonly if there exists an isomorphism σ : SL ( R ) (cid:110) Ad sl ( R ) → SL ( R ) (cid:110) Ad sl ( R )such that ρ = σ ◦ (cid:37) . Later, Kim [Kim05] generalized this result for examplesconstructed by Abels–Margulis–Soifer. In the case when G ∼ = SO (2 n, n − R is the inclusion map into GL ( R n − ), he showed that the Margulis-Smilga invariant spectrum of two Margulis-Smilga spacetimes ρ and (cid:37) aresame if and only if there exists an isomorphism σ : SO (2 n, n − (cid:110)R n − → SO (2 n, n − (cid:110) R n − such that ρ = σ ◦ (cid:37) . Recently, the author proved[Gho19] that an infinitesimal version of Kim’s result is also true. In thisarticle we generalize the isospectrality results from [DG01] and [Kim05] toshow that for a large class of other interesting cases too the Margulis-Smilgainvariant spectrum of a Margulis-Smilga spacetime do indeed determine theMargulis-Smilga spacetime.Now we mention the criteria we use in this article more precisely andstate the Theorems we prove. Convention 0.2.
Throughout this article unless otherwise stated we followthe following convention for the tuple ( G , V , R , Γ , ρ, (cid:37) ) :1. G denotes a real split connected semisimple algebraic Lie group withtrivial center,2. V denotes a finite dimensional vector space with dim V > R : G → GL ( V ) denotes a faithful irreducible algebraic representationwhich admits zero as a weight,4. G (cid:110) R V denote an affine group whose multiplication law is defined asfollows: for all g, h ∈ G and X, Y ∈ V ,( g, X )( h, Y ) := ( gh, X + R ( g ) Y ) , L : G (cid:110) R V → G denotes the map for which L ( g, X ) = g for all g ∈ G and X ∈ V ,6. T : G (cid:110) R V → V denotes the map for which T ( g, X ) = X for all g ∈ G and X ∈ V , SOURAV GHOSH
7. Γ denote a word hyperbolic group with finitely many generators,8. ρ : Γ → G (cid:110) R V (respectively (cid:37) ) denote injective homomorphismssuch that L ◦ ρ (Γ) (respectively L ◦ (cid:37) (Γ)) is Zariski dense inside G and L ◦ ρ ( γ ) (respectively L ◦ (cid:37) ( γ )) are loxodromic for all non identityelement γ ∈ Γ.We use the above convention and show the following:
Theorem 0.3.
Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2, let L ◦ ρ = L ◦ (cid:37) and let M ( ρ ( γ )) = M ( (cid:37) ( γ )) for all γ ∈ Γ. Then there exists an innerautomorphism σ of G (cid:110) R V such that σ ◦ ρ = (cid:37) .Moreover, we prove two stronger results for a special class of represen-tations R . Let R be absolutely irreducible and let R be conjugate to itsdual ( R t ) − . We call representations which are conjugate to their dual as self-contragredient (for more details see Section 3.11 of [Sam90]). Then byLemma 1.3 of [Gro71] the representation R admits an invariant symmetricbilinear form B R . We denote the norm coming from this bilinear form by (cid:107) · (cid:107) R and prove the following results: Theorem 0.4.
Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2, let R be an ab-solutely irreducible self-contragredient representation, let L ◦ ρ = L ◦ (cid:37) and let (cid:107) M ( ρ ( γ )) (cid:107) R = (cid:107) M ( (cid:37) ( γ )) (cid:107) R for all γ ∈ Γ. Then there exists an automorphism σ : G (cid:110) R V → G (cid:110) R V such that σ ◦ ρ = (cid:37) . Moreover, σ is conjugation by anelement ( A , Y ) ∈ O ( B R ) (cid:110) V such that A centralizes R ( G ). Theorem 0.5.
Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2, let R an ab-solutely irreducible self-contragredient representation and let (cid:107) M ( ρ ( γ )) (cid:107) R = (cid:107) M ( (cid:37) ( γ )) (cid:107) R for all γ ∈ Γ. Then either of the following holds:1. both ρ (Γ) and (cid:37) (Γ) are Zariski dense inside some conjugates of G ,2. there exists an automorphism σ : G (cid:110) R V → G (cid:110) R V such that σ ◦ ρ = (cid:37) and σ is conjugation by an element ( A , Y ) ∈ O ( B R ) (cid:110) V such that A normalizes R ( G ).In fact, we also prove the following characterization: Theorem 0.6.
Let ( G , V , R , Γ , ρ ) be as in Convention 0.2 and let R an abso-lutely irreducible self-contragredient representation. Then the following areequivalent:1. (cid:107) M ( ρ ( γ )) (cid:107) R = 0 for all γ ∈ Γ,2. M ( ρ ( γ )) = 0 for all γ ∈ Γ,3. is conjugate to L ◦ ρ (Γ) = ( e, Y ) ρ (Γ)( e, Y ) − for some Y ∈ V .Hence we obtain the following result: Theorem 0.7.
Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2 and let ρ and (cid:37) be two Margulis-Smilga spacetimes. Then the following holds:1. If ρ and (cid:37) are conjugate via some inner automorphism of G (cid:110) R V , thenthey have the same Margulis-Smilga invariant spectrum.2. If ρ , (cid:37) have the same marked Margulis-Smilga invariant spectrum and L ◦ ρ = L ◦ (cid:37) , then there exists σ , an inner isomorphism of G (cid:110) R V ,such that ρ = σ ◦ (cid:37) . ARGULIS-SMILGA SPACETIMES 5
3. If ρ , (cid:37) have the same marked Margulis-Smilga invariant spectrumand R is absolutely irreducible self-contragredient, then R preserves asymmetric bilinear form B R and there exists ( A , Y ) ∈ O ( B R ) (cid:110) V suchthat ρ = ( A , Y ) (cid:37) ( A , Y ) − .We note that the adjoint representations of connected real split sim-ple algebraic Lie groups with trivial center are absolutely irreducible self-contragredient representations. Hence, along with all the known results fromthe literature, our result also covers in its full generality, adjoint represen-tations of real split simple algebraic Lie groups with trivial center. Further-more, we note that the techniques used in this article to prove Theorem 0.7can be used to prove a more general result in the split case which mightinclude some more representations R but we do not include it in this articlebecause the conditions on R does not look natural enough in that generality. Acknowledgements
I would like to thank Ilia Smilga for explaining me his work in [Smi16b]and Arghya Mondal for helpful discussions.1.
Jordan Decomposition
In this section we recall certain basic results about the structure theory ofreal semisimple algebraic Lie groups of noncompact type with trivial centerand their Jordan decomposition. These results will be used later in thearticle to obtain our main result.Let G be a real semisimple algebraic Lie group of noncompact type withtrivial center and let g be its Lie algebra. We denote the identity elementof G by e . Let C g be the conjugation map on G i.e. for any g, h ∈ G we have C g ( h ) = ghg − and let Ad g be the differential of this at identity.Hence we obtain a homomorphism Ad : G → SL ( g ). Moreover, let ad bethe differential of Ad at the identity element. We fix a Cartan involution θ : g → g and consider the corresponding decomposition g = k ⊕ p where k (respectively p ) is the eigenspace of eigenvalue 1 (respectively -1). Let a bethe maximal abelian subspace of p . We denote the space of linear forms on a by a ∗ and for all α ∈ a ∗ we define g α := { X ∈ g | ad H ( X ) = α ( H ) X for all H ∈ a } . We call α ∈ a ∗ a restricted root if and only if both α (cid:54) = 0 and g α (cid:54) = 0. LetΣ ⊂ a ∗ be the set of all restricted roots. As g is finite dimensional, it followsthat Σ is finite. Moreover, we note that g = g ⊕ (cid:77) α ∈ Σ g α . We choose a ++ , a connected component of a \ ∪ α ∈ Σ ker( α ) and denote itsclosure by a + . Let K ⊂ G (respectively A ⊂ G ) be the connected subgroupwhose Lie algebra is k (respectively a ) and let A + := exp ( a + ). We note that K is a maximal compact subgroup of G .Let B be the Killing form on g i.e. for any X, Y ∈ g we have B ( X, Y ) := tr ( ad X ◦ ad Y ) and denote B ( X, X ) by (cid:107) X (cid:107) B . We define Σ + ⊂ Σ to be
SOURAV GHOSH the set of restricted roots which take positive values on a + and note thatΣ = Σ + (cid:116) − Σ + . We consider the following nilpotent subalgebras: n ± := (cid:77) ± α ∈ Σ + g α . Let K , A , N be the Lie subgroups of G generated respectively by k , a , n + . Let g ∈ G . Then1. g is called elliptic if and only if some conjugate of g lies in K ,2. g is called hyperbolic if and only if some conjugate of g lies in A ,3. g is called unipotent if and only if some conjugate of g lies in N . Theorem 1.1 (Jordan decomposition) . Let G be a connected real semisim-ple algebraic Lie group of noncompact type with trivial center. Then forany g ∈ G , there exist unique g e , g h , g u ∈ G such that the following hold:1. g = g e g h g u ,2. g e is elliptic, g h is hyperbolic and g u is unipotent,3. the elements g e , g h , g u commute with each other. Definition 1.2.
Let G be a connected real semisimple algebraic Lie groupof noncompact type with trivial center and let g ∈ G . Then the Jordanprojection of g , denoted by Jd g , is the unique element in a + such that g h isa conjugate of exp ( Jd g ). Remark 1.3.
We note that Jd is continuous. Indeed, we use Lemmas 6.32and 6.33 (ii) of [BQ16] and Appendix V.4 of [Whi72] to deduce it (see also[Tit71]). Definition 1.4.
Let G be a connected real semisimple algebraic Lie groupof noncompact type with trivial center and let g ∈ G . Then g is called loxodromic if and only if Jd g ∈ a ++ .Moreover, let M be the centralizer of a inside K and m be the Lie subalgebraof g coming from M . We note that g = n + ⊕ g ⊕ n − and g = a ⊕ m . Remark 1.5. If G is split then m is trvial and M is a finite group (seeTheorem 7.53 of [Kna02]). Proposition 1.6 (Dang, see Proposition 2.31 of [Dan19]) . Let G be a con-nected real semisimple algebraic Lie group of noncompact type with trivialcenter and let g ∈ G be loxodromic. Then the following holds:1. g u is trivial,2. for h g ∈ G with g h = h g exp ( Jd g ) h − g , we have m g := h − g g e h g ∈ M ,3. for ( h g , m g ) ∈ G × M as above, we have g = h g m g exp ( Jd g ) h − g ,4. if ( h, m ) ∈ G × M satisfy g = hm exp ( Jd g ) h − , then there exists aunique c ∈ MA such that h = h g c and m = c − m g c .We observe that the Jordan projection are invariant under conjugation,i.e. for all g, h ∈ G we have Jd hgh − = Jd g .2. Weights and eigenspaces
In this section we recall some basic results about the structure theory offinite dimensional faithful irreducible representations of real semisimple Lie
ARGULIS-SMILGA SPACETIMES 7 groups of noncompact type with trivial center. These results will be usedlater in the article to obtain our main result.Let G be a real semisimple Lie group of noncompact type with trivialcenter, V be a finite dimensional vector space and let R : G → GL ( V ) be afaithful irreducible representation. Hence, we obtain a Lie algebra represen-tation ˙ R : g → gl ( V ), by taking the differential ˙ R of the representation R atthe identity. We recall that a ∗ denotes the space of all linear forms on a andfor all λ ∈ a ∗ we define V λ := { X ∈ V | ˙ R H ( X ) = λ ( H ) X for all H ∈ a } . We call λ ∈ a ∗ a restricted weight of the representation R if and only if both λ (cid:54) = 0 and V λ (cid:54) = 0. Let Ω ⊂ a ∗ be the set of all restricted weights. As V isfinite dimensional, it follows that Ω is finite. Moreover, we note that V = V ⊕ (cid:77) λ ∈ Ω V λ . Remark 2.1.
Henceforth, we will only consider representations R such that V is nontrivial and we will denote (cid:76) λ ∈ Ω V λ by V (cid:54) =0 . Notation 2.2.
Henceforth we will also use the expression R g to denote R ( g )for any g ∈ G . Lemma 2.3.
Let R : G → GL ( V ) and ˙ R : g → gl ( V ) be as above. Then forany X ∈ a and t ∈ R we have R (exp( tX )) = exp( t ˙ R ( X )). Proof.
We observe that both { R (exp( tX )) } t ∈ R and { exp( t ˙ R ( X )) } t ∈ R are oneparameter subgroups of GL ( V ) passing through identity element with thesame tangent vector i.e. R (exp(0)) = exp(0) and ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 R (exp( tX )) = ˙ R ( X ) = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( t ˙ R ( X ) . Hence for all t ∈ R we have R (exp( tX )) = exp( t ˙ R ( X )). (cid:3) Lemma 2.4.
Let g ∈ A and let X g ∈ a be such that g = exp( X g ). Thenfor any X ∈ V λ we have R g X = exp( λ ( X g )) X .In particular, we obtain that V λ = { X ∈ V | R g ( X ) = exp( λ ( X g )) X for all g ∈ A } . Proof. As A is abelian we note that the exponential map from a to A issurjective. Moreover, we observe thatexp( t ˙ R ( X g )) (cid:12)(cid:12) t =0 X = X = exp( tλ ( X g )) | t =0 X,∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( t ˙ R ( X g )) X = ˙ R ( X g ) X = λ ( X g ) X = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( tλ ( X g )) X. Hence, both exp( t ˙ R ( X g )) X and exp( tλ ( X g )) X have the same value at t = 0and their derivatives at t = 0 are also equal. It follows that exp( t ˙ R ( X g )) X =exp( tλ ( X g )) X for all t ∈ R . We take t = 1 and use Lemma 2.3 to concludethat R g X = exp( λ ( X g )) X . Therefore, we obtain V λ ⊂ { X ∈ V | R g ( X ) = exp( λ ( X g )) X for all g ∈ A } . SOURAV GHOSH
Furthermore, let X ∈ V be such that R g ( X ) = exp( λ ( X g )) X for all g ∈ A .Hence, for any Y ∈ a we observe that˙ R ( Y ) X = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( ˙ R ( tY )) X = ∂∂t (cid:12)(cid:12)(cid:12)(cid:12) t =0 exp( λ ( tY )) X = λ ( Y ) X. Therefore, it follows from the definition of V λ that V λ ⊃ { X ∈ V | R g ( X ) = exp( λ ( X g )) X for all g ∈ A } , and we conclude our result. (cid:3) Lemma 2.5.
Let G be as in Convention 0.2. Then for any g ∈ MA we have R g V λ = V λ . Moreover, for λ = 0 we have V = { X ∈ V | R g ( X ) = X for all g ∈ MA } . Proof. As G is split we have m = 0. Hence M is a discrete group. It followsthat the connected component of M containing identity is a singleton. Nowwe use Theorem 7.53 of [Kna02] and observe that M ⊂ exp( i a ). Hence forany m ∈ M there exists X m ∈ a such that m = exp( iX m ). It follows thatfor any X ∈ V λ we have(2.1) R m X = exp( i ˙ R ( X m )) X = exp( iλ ( X m )) X. Also, as X and R m X lie inside the real part we obtain that exp( iλ ( X m )) ∈ R .Therefore, R m X ∈ V λ if and only if X ∈ V λ .Finally, using the definition of V we obtain that V ⊃ { X ∈ V | R g ( X ) = X for all g ∈ MA } . Also, using Equation 2.1, for X ∈ V we get R m X = exp(0) X = X andusing Lemma 2.4 we obtain that R g X = X for all g ∈ MA . Hence, V ⊂ { X ∈ V | R g ( X ) = X for all g ∈ MA } and we conclude our result. (cid:3) Lemma 2.6.
Let G be as in Convention 0.2 and let g ∈ G be a loxodromicelement. Then the dimension of the unit eigenspace of R g is atleast dim V .Moreover, the set of loxodromic elements h ∈ G such that the dimensionof the unit eigenspace of R h is exactly dim V , is a non-empty open densesubset of G . Proof.
We use Proposition 1.6 and Lemma 2.5 to deduce that the uniteigenspace of R g for a loxodromic element g is atleast dim V .Suppose Y ∈ a is such that α ( Y ) (cid:54) = 0 for all α ∈ Σ and λ ( Y ) (cid:54) = 0 forall λ ∈ Ω, then by Lemma 2.4 we get that R (exp( Y )) X = exp( λ ( Y )) X . Weagain use Theorem 7.53 of [Kna02] and the fact that G is split to concludethat for any m ∈ M there exists X m ∈ a such that m = exp( iX m ) and R m X = exp( iλ ( X m )) X with exp( iλ ( X m )) ∈ R . Hence, exp( iλ ( X m )) = ± X ∈ V λ with λ (cid:54) = 0 we have R ( m exp( Y )) X = exp( λ ( Y )) R m X = ± exp( λ ( Y )) X (cid:54) = X. ARGULIS-SMILGA SPACETIMES 9
Also, using Lemma 2.5 we obtain that R g ( X ) = X for all g ∈ MA and forall X ∈ V . Therefore, our result follows by using Remark 1.3, Proposition1.6 and observing that the set a \ (cid:32) (cid:91) α ∈ Σ ker( α ) ∪ (cid:91) λ ∈ Ω ker( λ ) (cid:33) is a non-empty open dense subset of a as the sets Σ and Ω are finite. (cid:3) Characteristic Polynomial
In this section we recall the definition of the minimal polynomial and thecharacteristic polynomial of a linear transformation. We also prove somepreliminary results which will play a central role in the proof of our maintheorem.
Definition 3.1.
Let A ∈ gl ( V ) and let I ∈ gl ( V ) be the diagonal matrixwith all its diagonal entries equal to 1. Then the characteristic polynomialof A in the indeterminate x is defined by the following expression:det( x I − A ) . Notation 3.2.
Let ( G , V , R ) be as in Convention 0.2 and let g ∈ G . Wealternately denote R ( g ) by R g . Then R g ∈ GL ( V ) ⊂ gl ( V ). Hence ( R e − R g ) ∈ gl ( V ). Also we observe that R e = I . In order to simplify our notations, inthis article we will denote the characteristic polynomial of ( R e − R g ) in theindeterminate x by CP g , i.e. CP g ( x ) = det( x R e − ( R e − R g )) = det(( x − R e + R g ) . Theorem 3.3 (Cayley–Hamilton, see [Fro77]) . Let g ∈ G and let CP g bethe characteristic polynomial of ( R e − R g ). Then CP g ( R e − R g ) = 0. Remark 3.4.
One can use the Cayley–Hamilton Theorem 3.3 to deducethat the characteristic polynomial of ( R e − R g ) has the following expression: CP g ( x ) = dim V (cid:88) k =0 ( − dim V − k tr ( ∧ dim V − k ( R e − R g )) x k . Hence the coefficients of the characteristic polynomial are also algebraic.Let R [ x ] be set of all polynomials in the indeterminate x and note that R [ x ] is a principal ideal domain i.e. any ideal is generated by a singlepolynomial, which is unique up to units in R [ x ]. Now for A ∈ gl ( V ) weconsider: I A := { p ( x ) ∈ R [ x ] | p ( A ) = 0 } and observe that I A is a proper ideal of R [ x ]. Definition 3.5.
The minimal polynomial of A ∈ gl ( V ) is the unique monicpolynomial which generates I A . It is the monic polynomial of least degreeinside I A . Notation 3.6.
Let g ∈ G . In order to simplify our notations, in this articlewe will denote the minimal polynomial of ( R e − R g ) in the indeterminate x by MP g ( x ). Remark 3.7.
We observe that by definition CP g ( x ) ∈ I ( R e − R g ) and hence MP g ( x ) divides CP g ( x ). In fact, one can deduce that CP g ( x ) and MP g ( x ) havethe same irreducible factors in R [ x ]. Moreover, when g is loxodromic, usingProposition 14 in Chapter 7.5.8 of [Bou03] we get that MP g ( x ) has no multiplefactors. Lemma 3.8.
Let g ∈ G be a loxodromic element. Then R g is diagonalizableover C . Proof.
We use Theorem 2.4.8 (ii) of [Spr09] to conclude our result. (cid:3)
Proposition 3.9.
Let g ∈ G be loxodromic and let CP g ( x ) be the character-istic polynomial of ( R e − R g ) with variable x . Then P g ( x ) := CP g ( x ) /x dim V is a polynomial.Moreover, let P g ( x ) = (cid:80) dim( V (cid:54) =0 ) k =0 a k ( g ) x k . Then the coefficient a k ( g ), forany k ∈ { , , .., dim V } , is algebraic in g . Proof.
Using Lemma 2.6 we obtain that R g has eigenvalue 1 with multiplicityatleast dim V . Hence, ( R e − R g ) has atleast dim V many 0 as eigenvalues.Moreover, by Lemma 3.8 we know that R g is diagonalizable over C . Hence CP g ( x ) is divisible by x dim V and it follows that P g ( x ) is a polynomial.As CP g ( x ) is the characteristic polynomial of ( R e − R g ), we have CP g ( x ) = dim V (cid:88) k =0 ( − dim V − k tr ( ∧ dim V − k ( R e − R g )) x k . We denote dim V by n and dim V by n . Moreover, as x n divides CP g ( x )we have tr ( ∧ n − k ( R e − R g )) = 0 for all 0 ≤ k ≤ n − P g ( x ) = n (cid:88) k = n ( − n − k tr ( ∧ n − k ( R e − R g )) x k − n = n − n (cid:88) k =0 ( − n − n − k tr ( ∧ n − n − k ( R e − R g )) x k . As R is algebraic, we conclude by observing that a k ( g ) = ( − n − n − k tr ( ∧ n − n − k ( R e − R g ))is algebraic in g for all k . (cid:3) Lemma 3.10.
Let g ∈ G be a loxodromic element such that the dimensionof the unit eigenspace of R g is exactly dim V . Then P g (0) (cid:54) = 0. Proof.
We use Lemma 3.8 and the fact that V = V ⊕ V (cid:54) =0 to conclude ourresult. (cid:3) Unit eigenspace projection
In this section we deduce a formula for the projection operator ontoeigenspace of unit eigenvalues with respect to the eigenspace decomposi-tion of a linear operator. Moreover, for ( G , V , R ) as in Convention 0.2 and g ∈ G , we relate the unit eigenspace projections of R g with the projectiononto the zero weight space of R . ARGULIS-SMILGA SPACETIMES 11
Let π be the projection onto the V component with respect to thedecomposition: V = V ⊕ V (cid:54) =0 . Lemma 4.1.
Let g ∈ G be a loxodromic element and let P g be as in Propo-sition 3.9. Then ( R e − R g ) P g ( R e − R g ) = 0 . Proof.
Let g h be the hyperbolic part of g with respect to the Jordan decom-position. Let h ∈ G be such that g h = h (exp λ g ) h − . We use Proposition1.6 to obtain that c := h − gh ∈ MA . As g is loxodromic, using Lemma 3.8we obtain that R c is diagonalizable over C . It follows that the minimal poly-nomial MP c ( x ) of ( R e − R c ) is a product of distinct monic linear factors andhence is divisible by x but not by x (See Remark 3.7). Also MP c ( x ) divides CP c ( x ) and hence MP c ( x ) divides x P c ( x ). We also know that MP c ( R e − R c ) = 0and it follows that ( R e − R c ) P c ( R e − R c ) = 0.As g = hch − we have R e − R g = R h ( R e − R c ) R − h . Hence P g = P c and P g ( R e − R g ) = P c ( R e − R g ) = R h P c ( R e − R c ) R − h . Therefore, we conclude that( R e − R g ) P g ( R e − R g ) = R h ( R e − R c ) P c ( R e − R c ) R − h = 0 . (cid:3) Proposition 4.2.
Let c ∈ MA and let P c be as in Proposition 3.9. Then forany X ∈ V : P c ( R e − R c ) X = P c (0) π ( X ) . Proof. As R c is diagonalizable over complex numbers, R e − R c is also diago-nalizable over complex numbers. Moreover, as R c Z = Z for all Z ∈ V wehave V ⊂ ker( R e − R c ). We will prove our result in two separate cases:1. V (cid:54) = ker( R e − R c ): In this case P c ( x ) is divisible by x . Hence P c ( x ) isdivisible by the minimal polynomial MP c ( x ). Moreover, MP c ( R e − R c )vanishes and we obtain that P c ( R e − R c ) = 0. Also, as P c ( x ) is divisibleby x we have P c (0) = 0. Therefore, P c ( R e − R c ) X = 0 = P c (0) π ( X ) . V = ker( R e − R c ): In this case, given any Y ∈ V (cid:54) =0 there exists an Y (cid:48) ∈ V (cid:54) =0 such that ( R e − R c ) Y (cid:48) = Y . Indeed, as R c V (cid:54) =0 ⊂ V (cid:54) =0 weobtain that ( R e − R c ) : V (cid:54) =0 → V (cid:54) =0 is a linear map with kernel V ∩ V (cid:54) =0 = { } and hence ( R e − R c ) isinvertible on V (cid:54) =0 . Therefore, for any X ∈ V there exists Y ∈ V (cid:54) =0 such that ( X − π ( X )) = ( R e − R c ) Y . It follows that P c ( R e − R c )( X − π ( X )) = P c ( R e − R c )( R e − R c ) Y = 0and hence for any X ∈ V we have P c ( R e − R c ) X = P c ( R e − R c ) π ( X ).Moreover, as ( R e − R c ) π ( X ) = 0, we conclude by observing that P c ( R e − R c ) π ( X ) = dim( V (cid:54) =0 ) (cid:88) k =0 a k ( c )( R e − R c ) k π ( X )= a ( c ) π ( X ) = P c (0) π ( X ) . Our result is complete. (cid:3)
Proposition 4.3.
Let g ∈ G be a loxodromic element such that the dimen-sion of the unit eigenspace of R g is exactly dim V . Then the map P g (0) − P g ( R e − R g ) : V → V is the projection onto the unit eigenspace of R g with respect to the eigenspacedecomposition of R g . Proof.
We use Lemma 3.10 and observe that P g (0) (cid:54) = 0. Hence the map P g (0) − P g ( R e − R g ) : V → V is a well defined linear map. Moreover, as g isloxodromic, we use Proposition 1.6 and obtain that there exists h ∈ G suchthat c := h − gh ∈ MA . Now we use Proposition 4.2 and obtain that P c (0) − P c ( R e − R c ) X = π ( X )for all X ∈ V . Hence, P c (0) − P c ( R e − R c ) = π is a projection operatorprojecting onto V . Moreover, as c = h − gh , we deduce that P g (0) − P g ( R e − R g ) = R h ◦ π ◦ R − h . It follows that P g (0) − P g ( R e − R g ) is a projection onto the space R h V . There-fore, we will be done once we show that R h V is the unit eigenspace of R g .Finally, we observe that R c X = X if and only if R g R h X = R h X and concludeour result using Lemmas 2.5 and 2.6. (cid:3) Margulis-Smilga invariant
In this section we define the Margulis-Smilga invariants correspondingto a faithful irreducible representation of a real split connected semisimpleLie group with trivial center. We also relate these invariants with the uniteigenspace projections introduced in the previous section.Let ( G , V , R ) be as in Convention 0.2. We consider the group G (cid:110) V asfollows: for any g, h ∈ G and X, Y ∈ V we have ( g, X ) , ( h, Y ) ∈ G (cid:110) V and( g, X )( h, Y ) := ( gh, X + R g Y ). Moreover, we denote the affine action of G (cid:110) V on V by Af i.e. for any ( g, X ) ∈ G (cid:110) V and Y ∈ V we have: Af ( g,X ) Y := R g Y + X. Let L : G (cid:110) V → G be the map such that L ( g, X ) = g and let T : G (cid:110) V → V be the map such that T ( g, X ) = X for all g ∈ G and x ∈ V . Image under L of ( g, X ) ∈ G (cid:110) V is called the linear part of ( g, X ) and the image under T is called the translation part of ( g, X ). Lemma 5.1.
Let ( g, X ) ∈ G (cid:110) V be such that g is loxodromic and g h beits hyperbolic part with respect to the Jordan decomposition. Let h , h besuch that h exp ( Jd g ) h − = g h = h exp ( Jd g ) h − . Then π ( R − h X ) = π ( R − h X ). Proof.
We recall that by Lemma 2.5, for any c ∈ MA we have R c V (cid:54) =0 = V (cid:54) =0 and R c X = X for any X ∈ V .Also by Proposition 1.6 there exist some c ∈ MA such that h = h c .For i ∈ { , } we denote the component of R − h i X inside V (cid:54) =0 by Y i and ARGULIS-SMILGA SPACETIMES 13 the component of R − h i X inside V by Z i . As h = h c we deduce that( Y + Z ) = R c ( Y + Z ) and hence R c Y − Y = Z − R c Z = Z − Z . We notice that ( R c Y − Y ) ∈ V (cid:54) =0 , ( Z − Z ) ∈ V and V ∩ V (cid:54) =0 = { } .Therefore, Z = Z and we conclude that π ( R − h X ) = π ( R − h X ). (cid:3) Lemma 5.2.
Let ( g, X ) ∈ G (cid:110) V be such that g is loxodromic and g h beits hyperbolic part with respect to the Jordan decomposition. Let h ∈ G besuch that g h = h exp ( Jd g ) h − . Then for any Y ∈ V we have π ( R − h ( Af ( g,X ) Y − Y )) = π ( R − h X ) . Proof. As Af ( g,X ) Z = R g Z + X and π is linear, proving this lemma isequivalent to showing that π ( R − h ( R g Z − Z )) = 0 for all Z ∈ V . We denote Y := R − h Z and observe that R − h ( R g Z − Z ) = R h − gh Y − Y. We recall from Proposition 1.6 that l := h − gh ∈ MA . Hence using Lemma2.5 we deduce that R l ( Y − π ( Y )) ∈ V (cid:54) =0 and it follows that π ( R l ( Y − π ( Y ))) = 0 . Also, as l ∈ MA , using Lemma 2.5 we have R l π ( Y ) = π ( Y ). Therefore, weconclude by observing that π ( R l ( Y − π ( Y ))) = π ( R l Y − R l π ( Y )) = π ( R l Y − π ( Y ))= π ( R l Y ) − π ( π ( Y )) = π ( R l Y ) − π ( Y )= π ( R l Y − Y )and hence π ( R − h ( R g Z − Z )) = π ( R l Y − Y ) = 0. (cid:3) Definition 5.3.
Let ( g, X ) ∈ G (cid:110) V be such that g is loxodromic and g h beits hyperbolic part with respect to the Jordan decomposition. Let h ∈ G besuch that g h = h exp ( Jd g ) h − . Then the Margulis-Smilga invariant of ( g, X )denoted by M ( g, X ) is defined as follows: M ( g, X ) := π ( R − h X ) . Remark 5.4.
Note that by Definition 6.2 of [Smi16b], Proposition 7.8 of[Smi16b] and Lemma 2.5, the definition of a Margulis-Smilga invariant givenhere is the same as the the definition of a Margulis invariant given in Def-inition 7.19 of [Smi16b]. Smilga was the first to modify real valued Mar-gulis invariants into vector valued invariants and he used these invariantsin [Smi16a, Smi18, Smi16b] to construct proper affine actions of Schottkygroups.
Proposition 5.5.
Let g ∈ G be a loxodromic element and let h ∈ G be suchthat g h = h exp ( Jd g ) h − . Then for any Y ∈ V we have P g ( R e − R g ) Y = P g (0) R h M ( g, Y ) . Proof.
Let c := h − gh . Then by Proposition 1.6 we have c ∈ MA . Now usingProposition 4.2 we obtain that P c ( R e − R c ) X = P c (0) π ( X ) for all X ∈ V .Also, we have P c ( x ) = P g ( x ). Hence, we deduce that P g ( R e − R g ) Y = R h P c ( R e − R c ) R − h Y = P c (0) R h π ( R − h Y ) = P g (0) R h M ( g, Y )and our result follows. (cid:3) Isospectrality: fixed linear part
In this section we show that the Margulis-Smilga invariant spectrum oftwo faithful irreducible algebraic representations with fixed linear parts ofa split connected semisimple Lie group with trivial center are completelydetermined by the isomorphism class.Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2. We denote L ( ρ ( γ )) by L ρ ( γ ), T ( ρ ( γ )) by T ρ ( γ ), M ( ρ ( γ )) by M ρ ( γ ). Definition 6.1.
The map M ρ : Γ → V (respectively M (cid:37) ) is called the marked Margulis-Smilga invariant spectrum of the representation ρ (respectively (cid:37) ). Proposition 6.2.
Let ( G , V , R , Γ , ρ ) be as in Convention 0.2. Then either ρ (Γ) is Zariski dense inside G (cid:110) V or ρ (Γ) is conjugate to L ρ (Γ) under theaction of some element of { e } (cid:110) V . Proof.
Let X be the Zariski closure of ρ (Γ) inside G (cid:110) V and let γ ∈ Γ.As ρ ( γ ) ρ (Γ) ρ ( γ ) − = ρ (Γ), we obtain that ρ ( γ ) X ρ ( γ ) − = X . Also, L is ahomomorphism. Hence, we have L ρ ( γ ) L ( X ) L ρ ( γ ) − = L ( X ) for all γ ∈ Γ. As X normalizes { e } (cid:110) V using the Corollary to Proposition A at page 54 of[Hum75] we obtain that X ( { e } (cid:110) V ) is a Zariski closed subgroup of G (cid:110) V .Also, as L ρ (Γ) is Zariski dense inside G , we obtain that g L ( X ) g − = L ( X )for all g ∈ G . It follows that L ( X ) is normal inside G . Moreover, as L ρ (Γ) isZariski dense inside G and L ρ (Γ) ⊂ L ( X ), we deduce that L ( X ) = G .Now we consider the map L | X : X → G and note thatker( L | X ) = ( { e } (cid:110) V ) ∩ X . We will prove our result in two parts as follows: (cid:5)
Let ker( L | X ) be trivial, then L | X is an isomorphism. Hence, for all g ∈ G there exists X g ∈ V such that X gh = X g + R g X h and X = { ( g, X g ) | g ∈ G } .As G is connected, we use Whitehead’s Lemma (see end of section 1.3.1 inpage 13 of [Rag07]) and deduce that there exists X ∈ V such that X g = X − R g X . Therefore, we have T ρ ( γ ) = X − R L ρ ( γ ) X for all γ ∈ Γ. Hence ρ ( γ ) = ( e, X )( L ρ ( γ ) , e, X ) − for all γ ∈ Γ. (cid:5) Let ker( L | X ) be non trivial. Then there exist X ∈ V with X (cid:54) = 0 suchthat ( e, X ) ∈ ker( L | X ). As ker( L | X ) is normal inside X we obtain that forany ( h, Y ) ∈ X we have ( h, Y )( e, X )( h, Y ) − ∈ ker( L | X ). We also notice thatfor all ( h, Y ) ∈ G (cid:110) V we have( h, Y )( e, X )( h, Y ) − = ( h, e, X )( h, − = ( e, R h X ) . Moreover, as L ( X ) = G we deduce that ( e, R h X ) ∈ ker( L | X ) for all h ∈ G . As R is irreducible, we use Lemma A.1 and obtain that ker( L | X ) = ( { e } (cid:110) V ).Furthermore, as L ( X ) = G , we conclude that X = G (cid:110) V .Therefore, either ρ (Γ) is Zariski dense inside G (cid:110) V or ρ (Γ) is conjugate to L ρ (Γ) under the action of some element of { e } (cid:110) V and our result follows. (cid:3) ARGULIS-SMILGA SPACETIMES 15
Theorem 6.3.
Let ( G , V , R , Γ , ρ ) be as in Convention 0.2 and M ρ ( γ ) = 0 forall γ ∈ Γ. Then ρ (Γ) is conjugate to L ρ (Γ) under the action of some elementof { e } (cid:110) V . Proof. As M ρ ( γ ) = 0 for all γ ∈ Γ, using Proposition 5.5 we obtain that P L ρ ( γ ) ( R e − R L ρ ( γ ) ) T ρ ( γ ) = 0 for all γ ∈ Γ. We consider the following map: (cid:107) : G (cid:110) V → R ( g, X ) (cid:55)→ P g ( R e − R g ) X and observe that it is algebraic. We denote the zero set of (cid:107) by Z (cid:107) i.e. Z (cid:107) := { ( g, X ) ∈ G (cid:110) V | (cid:107) ( g, X ) = 0 } . We choose a X (cid:54) = 0 inside V and a loxodromic element g ∈ G such thatthe dimension of the unit eigenspace of R g is exactly dim V . Moreover, let h ∈ G be such that hgh − ∈ MA . Then using Lemma 3.10 and Proposition5.5 we obtain that (cid:107) ( g, R h X ) = P g ( R e − R g ) R h X = P g (0) R h π ( X ) = P g (0) R h X (cid:54) = 0 . Hence Z (cid:107) (cid:40) G (cid:110) V and it follows that X , the Zariski closure of ρ (Γ) inside G (cid:110) V , is a proper subvariety of G (cid:110) V i.e. X ⊂ Z (cid:107) (cid:40) G (cid:110) V . Finally, weconclude our result by using Proposition 6.2. (cid:3) Theorem 6.4.
Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2, let L ρ = L (cid:37) andlet M ρ ( γ ) = M (cid:37) ( γ ) for all γ ∈ Γ. Then there exists an inner automorphism σ of G (cid:110) V such that σ ◦ ρ = (cid:37) . Proof.
Let η := ( L ρ , T ρ − T (cid:37) ). We observe that for all γ ∈ Γ we have M η ( γ ) = M ρ ( γ ) − M (cid:37) ( γ ) = 0 . Therefore, using Theorem 6.3 we obtain that there exists Y ∈ V such that η ( γ ) = ( e, Y )( L ρ ( γ ) , e, Y ) − for all γ ∈ Γ. Hence, for all γ ∈ Γ it followsthat T ρ ( γ ) − T (cid:37) ( γ ) = Y − R L ρ ( γ ) Y and we conclude by observing that for all γ ∈ Γ, the following hold : ρ ( γ ) = ( e, Y ) (cid:37) ( γ )( e, Y ) − . (cid:3) Isospectrality of norm: zero spectrum
In this section we restrict the space of representations we are working withand only consider those representations which are absolutely irreducible andself-contragredient. We do this in order to introduce an invariant norm onthe vector space. Moreover, we characterize faithful irreducible algebraicrepresentations of a real split connected semisimple algebraic Lie group withtrivial center whose normed Margulis-Smilga invariant spectrum is zero. Wenote that the results in this section doesn’t follow directly from results inthe previous section as the invariant norm in question might not be positivedefinite.Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2 and also let R be an absolutelyirreducible self-contragredient representation. We note that by Lemma 1.3of [Gro71] the representation R admits an invariant symmetric bilinear form.We denote this bilinear form by B R and the norm coming from this invariantform by (cid:107) · (cid:107) R . Theorem 7.1.
Let ( G , V , R , Γ , ρ ) be as in Convention 0.2, let R be an abso-lutely irreducible self-contragredient representation and (cid:107) M ρ ( γ ) (cid:107) R = 0 for all γ ∈ Γ. Then ρ (Γ) is conjugate to L ρ (Γ) under the action of some element of { e } (cid:110) V . Proof. As (cid:107) M ρ ( γ ) (cid:107) R = 0 for all γ ∈ Γ, using Proposition 5.5 we obtain that (cid:107) P L ρ ( γ ) ( R e − R L ρ ( γ ) ) T ρ ( γ ) (cid:107) R = 0 for all γ ∈ Γ. We consider the following map: (cid:105) : G (cid:110) V → R ( g, X ) (cid:55)→ (cid:107) P g ( R e − R g ) X (cid:107) R and observe that it is algebraic. We denote the zero set of (cid:105) by Z (cid:105) i.e. Z (cid:105) := { ( g, X ) ∈ G (cid:110) V | (cid:105) ( g, X ) = 0 } . We use Lemma 2.6 and Lemma 3.10 to obtain that there exists c ∈ MA suchthat P c (0) (cid:54) = 0. As B R , the invariant form of R , is orthogonal, we use Lemma1.1 of [Gro71] to conclude that the restriction of B R on V is a non-degenerateorthogonal form. Hence, V admits vectors which are not self-orthogonal.Let X ∈ V be such that (cid:107) X (cid:107) R (cid:54) = 0. We use Proposition 4.2 and obtain (cid:105) ( c, X ) = (cid:107) P c ( R e − R c ) X (cid:107) R = P c (0) (cid:107) π ( X ) (cid:107) R = P c (0) (cid:107) X (cid:107) R (cid:54) = 0 . Hence Z (cid:105) (cid:40) G (cid:110) V and it follows that X , the Zariski closure of ρ (Γ) inside G (cid:110) V , is a proper subvariety of G (cid:110) V i.e. X ⊂ Z (cid:105) (cid:40) G (cid:110) V . Finally, weconclude our result by using Proposition 6.2. (cid:3) Corollary 7.2.
Let ( G , V , R , Γ , ρ ) be as in Convention 0.2. Then the follow-ing are equivalent:1. M ρ ( γ ) = 0 for all γ ∈ Γ,2. ρ (Γ) is conjugate to L ρ (Γ) under the action of some element of { e } (cid:110) V .Moreover, if R is an absolutely irreducible self-contragredient representation.Then the following are equivalent:3. (cid:107) M ρ ( γ ) (cid:107) R = 0 for all γ ∈ Γ,4. M ρ ( γ ) = 0 for all γ ∈ Γ. Proof.
We use theorem 6.3 to obtain that (1) = ⇒ (2). Now we show that(2) = ⇒ (1). Let ρ = ( e, X ) L ρ ( e, X ) − for some X ∈ V and let γ ∈ Γ and let h ∈ G be such that L ρ ( γ ) h = h exp ( Jd L ρ ( γ ) ) h − . Then h − L ρ ( γ ) h =: c ∈ MA .Therefore, we deduce that M ρ ( γ ) = π ( R − h T ρ ( γ )) = π ( R − h ( X − L ρ ( γ ) X )) = π ( R − h X − R c R − h X ) = 0 . Now let R be an absolutely irreducible self-contragredient representation.As M ρ ( γ ) = 0 implies (cid:107) M ρ ( γ ) (cid:107) R = 0 for any γ ∈ Γ we have (4) = ⇒ (3) andusing Theorem 7.1 we get that (3) = ⇒ (2). Also we have (2) = ⇒ (1).Hence (3) = ⇒ (4). (cid:3) Corollary 7.3.
Let ( G , V , R , Γ , ρ ) be as in Convention 0.2 and there existsa γ ∈ Γ such that M ρ ( γ ) (cid:54) = 0. Then ρ (Γ) is Zariski dense inside G (cid:110) V . Proof.
We use Proposition 6.2 to obtain that either ρ (Γ) is Zariski denseinside G (cid:110) V or ρ (Γ) is conjugate to L ρ (Γ) under the action of some elementof { e } (cid:110) V . We observe that if ρ (Γ) is not Zariski dense inside G (cid:110) V , then ρ (Γ) is conjugate to L ρ (Γ) under the action of some element of { e } (cid:110) V andwe obtain a contradiction using Corollary 7.2. (cid:3) ARGULIS-SMILGA SPACETIMES 17 Isospectrality of norm: fixed linear part
In this section we consider faithful absolutely irreducible algebraic self-contragredient representations of a real split connected semisimple algebraicLie group with trivial center, which are conjugate to their dual and we showthat the normed Margulis-Smilga invariant spectra of two such representa-tions with fixed linear parts are completely determined by the isomorphismclass.Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2 and let R be an absolutely ir-reducible self-contragredient representation. Hence, R preserves an invariantsymmetric bilinear form, which we denote by B R . Moreover, we denote thenorm associated to B R by (cid:107) · (cid:107) R . Lemma 8.1.
Let ( g, X ) , ( h, Y ) ∈ G (cid:110) V be such that their linear parts areloxodromic and (cid:107) M ( g, X ) (cid:107) R = (cid:107) M ( h, Y ) (cid:107) R . Then (cid:107) P g (0) P h ( R e − R h ) Y (cid:107) R = (cid:107) P h (0) P g ( R e − R g ) X (cid:107) R . Proof.
We use Proposition 5.5 and observe that (cid:107) P g (0) P h ( R e − R h ) Y (cid:107) R = (cid:107) P g (0) P h (0) M ( h, Y ) (cid:107) R = (cid:107) P g (0) P h (0) M ( g, X ) (cid:107) R = (cid:107) P h (0) P g ( R e − R g ) X (cid:107) R . Our result follows. (cid:3)
Lemma 8.2.
Let ג : G (cid:110) ( V ⊕ V ) → R be such that for all ( g, X, Y ) ∈ G (cid:110) ( V ⊕ V ) we have: ג ( g, X, Y ) := P g (0) ( (cid:107) P g ( R e − R g ) X (cid:107) R − (cid:107) P g ( R e − R g ) Y (cid:107) R ) , and let Z ג := { ( g, X, Y )) | ג ( g, X, Y ) = 0 } ⊂ G (cid:110) ( V ⊕ V ). Then Z ג (cid:54) = G (cid:110) ( V ⊕ V ) . Proof.
We use Lemma 2.6 and Lemma 3.10 to obtain that there exists c ∈ MA such that P c (0) (cid:54) = 0. As B R , the invariant form of R , is orthogonal, weuse Lemmas 1.1 and 1.3 of [Gro71] to conclude that the restriction of B R on V is a non-degenerate orthogonal form. Hence, V admits vectors whichare not self-orthogonal. Let X ∈ V be such that (cid:107) X (cid:107) R (cid:54) = 0. We choose Y = 0 and using Proposition 4.2 we observe that ג ( c, X, Y ) = ג ( c, X,
0) = P c (0) ( (cid:107) P c ( R e − R c ) X (cid:107) R − (cid:107) P c ( R e − R c )0 (cid:107) R )= P c (0) (cid:107) P c (0) π ( X ) (cid:107) R = P c (0) (cid:107) X (cid:107) R (cid:54) = 0 . Hence, Z ג is a proper subvariety of ( G (cid:110) ( V ⊕ V )), concluding our result. (cid:3) Remark 8.3.
We denote the projections onto the left and right coordinatesof G (cid:110) ( V ⊕ V ) by π ρ and π (cid:37) respectively i.e. π ρ , π (cid:37) : G (cid:110) ( V ⊕ V ) → G (cid:110) V be such that for all ( g, X, Y ) ∈ G (cid:110) ( V ⊕ V ) we have π ρ ( g, X, Y ) = ( g, X )and π (cid:37) ( g, X, Y ) = ( g, Y ). Proposition 8.4.
Let η : Γ → G (cid:110) ( V ⊕ V ) be a representation whose Zariskiclosure inside G (cid:110) ( V ⊕ V ) is a proper subvariety. Moreover, let ρ := π ρ ◦ η , (cid:37) := π (cid:37) ◦ η and both ρ , (cid:37) are Zariski dense inside G (cid:110) V . Then there existsa continuous automorphism σ : G (cid:110) V → G (cid:110) V such that σ ◦ ρ = (cid:37) . Proof.
Let us denote the Zariski closure of η (Γ) inside G (cid:110) ( V ⊕ V ) by X . As η ( γ ) η (Γ) η ( γ ) − = η (Γ) for all γ ∈ Γ, we obtain that η ( γ ) X η ( γ ) − = X . Wenote that both π ρ and π (cid:37) are homomorphisms. Hence ρ ( γ ) π ρ ( X ) ρ ( γ ) − = π ρ ( X ) and (cid:37) ( γ ) π (cid:37) ( X ) (cid:37) ( γ ) − = π (cid:37) ( X ). As both ρ (Γ) and (cid:37) (Γ) are Zariskidense inside G (cid:110) V , we obtain that both π ρ ( X ) and π (cid:37) ( X ) are normal inside G (cid:110) V (using the Corollary at page 54 of [Hum75]). Moreover, as π ρ ( X ) ⊃ ρ (Γ), π (cid:37) ( X ) ⊃ (cid:37) (Γ), we use Proposition A.2 and obtain that π ρ ( X ) = G ρ (cid:110) V and π (cid:37) ( X ) = G (cid:37) (cid:110) V for some normal subgroups G ρ , G (cid:37) of G . Also, as ρ (Γ)and (cid:37) (Γ) are Zariski dense inside G (cid:110) V we obtain that G ρ = G = G (cid:37) .We denote ker( π ρ | X ) by N ρ and ker( π (cid:37) | X ) by N (cid:37) . Hence,dim N ρ = dim X − dim G (cid:110) V = dim N (cid:37) . Moreover, we have dim X (cid:12) dim G (cid:110) ( V ⊕ V ) and it follows that dim N ρ =dim N (cid:37) (cid:12) dim V . Therefore, N ρ = X ∩ ( { e } (cid:110) ( { } ⊕ V )) (cid:40) { e } (cid:110) ( { } ⊕ V )and N (cid:37) = X ∩ ( { e } (cid:110) ( V ⊕ { } )) (cid:40) { e } (cid:110) ( V ⊕ { } ). Moreover, as N ρ is normalinside X , we obtain that for all ( g, X, Y ) ∈ X :( g, X, Y ) N ρ ( g, X, Y ) − = N ρ . But for all Z ∈ V we have ( g, X, Y )( e, , Z )( g, X, Y ) − = ( e, , R g Z ). As R isirreducible, for Z (cid:54) = 0 the group generated by { ( e, , R g Z ) | g ∈ G } is equal to { e } (cid:110) ( { }⊕ V ) (for more details see Lemma A.1). Therefore, if ( e, , Z ) ∈ N ρ for Z (cid:54) = 0 then { e } (cid:110) ( { } ⊕ V ) = N ρ (cid:40) { e } (cid:110) ( { } ⊕ V ), a contradiction. Itfollows that N ρ is trivial. Using similar arguments we also obtain that N (cid:37) istrivial. Hence both π ρ | X and π (cid:37) | X are isomorphisms. Now we conclude byobserving that σ := π (cid:37) | X ◦ π ρ | − X is a continuous automorphism of G (cid:110) V and σ ◦ ρ = π (cid:37) | X ◦ π ρ | − X ◦ ρ = π (cid:37) | X ◦ η = (cid:37). (cid:3) Proposition 8.5.
Let σ : G (cid:110) V → G (cid:110) V be a continuous automorphism.Then there exists ( A , Y ) ∈ GL ( V ) such that the action of σ is conjugationby ( A , Y ). Proof.
We observe that σ induces a continuous additive map ˜ σ : V → V .As continuous additive maps between vector spaces are linear and σ is anisomorphism, ˜ σ is an invertible linear map. Hence, there exists A ∈ GL ( V )such that σ ( e, X ) = ( e, A X ) for all X ∈ V . Moreover, for g σ ∈ G and Y g σ ∈ V ,let σ ( g,
0) = ( g σ , Y g σ ). Then Y g σ h σ = Y g σ + R g σ Y h σ for all g σ , h σ ∈ G . As G is connected, we use Whitehead’s Lemma (see end of section 1.3.1 in page13 of [Rag07]) to deduce that there exists Y ∈ V such that Y g σ = Y − R g σ Y .We also note that for all g ∈ G we have AR g = R g σ A . Indeed, for any X ∈ V :( g σ , Y g σ + A X ) = σ ( e, X ) σ ( g,
0) = σ ( g, σ ( e, R − g X ) = ( g σ , Y g σ + R g σ AR − g X ) , and it follows that σ ( g, X ) = ( A , Y )( R g , X )( A , Y ) − . (cid:3) Theorem 8.6.
Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2, let R be anabsolutely irreducible self-contragredient representation, let L ρ = L (cid:37) and let (cid:107) M ρ ( γ ) (cid:107) R = (cid:107) M (cid:37) ( γ ) (cid:107) R for all γ ∈ Γ. Then there exists an automorphism σ : G (cid:110) V → G (cid:110) V such that σ ◦ ρ = (cid:37) . Moreover, σ is conjugation by anelement ( A , Y ) ∈ O ( B R ) (cid:110) V such that A centralizes R ( G ). ARGULIS-SMILGA SPACETIMES 19
Proof.
We will prove this result in three parts. (cid:5)
Let (cid:107) M ρ ( γ ) (cid:107) R = 0 for all γ ∈ Γ. Hence (cid:107) M (cid:37) ( γ ) (cid:107) R = 0 for all γ ∈ Γ. We use Corollary 7.2 and obtain that there exists
X, Y ∈ V such that( e, X ) ρ ( e, X ) − = L ρ = ( e, Y ) (cid:37) ( e, Y ) − . Hence, ( e, X − Y ) ρ ( e, X − Y ) − = (cid:37) . (cid:5) Let there exists γ ∈ Γ such that (cid:107) M ρ ( γ ) (cid:107) R (cid:54) = 0. Hence (cid:107) M (cid:37) ( γ ) (cid:107) R (cid:54) = 0and using Corollary 7.3 we obtain that both ρ (Γ) and (cid:37) (Γ) are Zariski denseinside G (cid:110) V .Let η : Γ → G (cid:110) ( V ⊕ V ) be such that for all γ ∈ Γ we have η ( γ ) = ( L ρ ( γ ) , T ρ ( γ ) , T (cid:37) ( γ )) . Let ג and Z ג be as in Lemma 8.2. We use Lemma 8.1 and obtain that η (Γ) ⊂ Z ג . Hence X , the Zariski closure of η (Γ) inside G (cid:110) ( V ⊕ V ) is a subvarietyof Z ג . It follows that X is a proper subvariety of G (cid:110) ( V ⊕ V ). Now weuse Proposition 8.4 and obtain that there exists a continuous automorphism σ : G (cid:110) V → G (cid:110) V such that σ ◦ ρ = (cid:37) . (cid:5) We use Proposition 8.5 and obtain that there exists ( A , Y ) ∈ G (cid:110) V such that σ ( g, X ) = ( A , Y )( R g , X )( A , Y ) − . Also, as L ρ = L (cid:37) we obtain that AR g = R g A for all g ∈ G . Moreover, as X ⊂ Z ג , we use Lemma 4.1 and for all( g, X ) ∈ G (cid:110) V we obtain that P g (0) (cid:107) AP g ( R e − R g ) X (cid:107) R = P g (0) (cid:107) P g ( R e − R g ) X (cid:107) R . Hence, for c ∈ MA with P c (0) (cid:54) = 0, X ∈ V and g = hch − we get that (cid:107) AR h X (cid:107) R = (cid:107) R h X (cid:107) R . As R is irreducible we deduce that (cid:107) A Y (cid:107) R = (cid:107) Y (cid:107) R forall Y ∈ V . It follows that A ∈ O ( B R ). Hence, A is in the centralizer of R ( G )inside O ( B R ) and our result follows. (cid:3) Isospectrality of norm: general case
In this section we consider faithful absolutely irreducible algebraic self-contragredient representations of a real split connected semisimple algebraicLie group with trivial center and we show that the normed Margulis-Smilgainvariant spectrum of two such representations are completely determinedby the isomorphism class.Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2 and let R be an absolutely ir-reducible self-contragredient representation. Hence, R preserves an invariantsymmetric bilinear form, which we denote by B R . Moreover, we denote thenorm associated to B R by (cid:107) · (cid:107) R . Remark 9.1.
Let N r , N l be two nontrivial proper normal subgroups of G (cid:110) V such that ι : ( G (cid:110) V ) / N r → ( G (cid:110) V ) / N l is a continuous isomor-phism. We denote the set of all ( g ι , X g ι , g (cid:48) ι , Y g ι ) ∈ ( G (cid:110) V × G (cid:110) V ) suchthat ( g (cid:48) ι , Y g ι ) N l = ι (( g ι , X g ι ) N r ) by D ι i.e. D ι := { ( g ι , X g ι , g (cid:48) ι , Y g ι ) | ( g (cid:48) ι , Y g ι ) N l = ι (( g ι , X g ι ) N r ) } . Lemma 9.2.
Let ℵ : G (cid:110) V × G (cid:110) V → R be such that for all ( g, X, h, Y ) ∈ G (cid:110) V × G (cid:110) V we have: ℵ ( g, X, h, Y ) := (cid:107) P g (0) P h ( R e − R h ) Y (cid:107) R − (cid:107) P h (0) P g ( R e − R g ) X (cid:107) R , and let Z ℵ := { ( g, X, h, Y ) ∈ G (cid:110) V × G (cid:110) V | ℵ ( g, X, h, Y ) = 0 } . Then for all ι as mentioned in Remark 9.1 we have D ι \ Z ℵ (cid:54) = ∅ . In particular, we have Z ℵ (cid:40) G (cid:110) V × G (cid:110) V . Proof.
Let N r , N l be any two nontrivial proper normal subgroups of G (cid:110) V such that ι : ( G (cid:110) V ) / N r → ( G (cid:110) V ) / N l is a continuous isomorphism. Weuse Proposition A.2 and observe that N r = G r (cid:110) V and N l = G l (cid:110) V , forsome nontrivial proper normal subgroup G r , G l of G . Now using the thirdisomorphism Theorem of groups we obtain that ( G (cid:110) V ) / N r is isomorphicto G / G r and ( G (cid:110) V ) / N l is isomorphic to G / G l . Therefore, ι : ( G (cid:110) V ) / N r → ( G (cid:110) V ) / N l gives rise to an isomorphism ι : G / G r → G / G l . Now we useLemmas 2.6 and 3.10 to observe that the set S := { g ∈ G | P g (0) (cid:54) = 0 } isan open dense subset of G . Moreover, as G / G r and G / G l are the quotientsof G by some group action, the projection maps π r : G → G / G r and π l : G → G / G l are open. Hence π r ( S ) and π l ( S ) are open dense subsets of G / G r and G / G l respectively. It follows that ι ◦ π r ( S ) is an open dense subsetof G / G l and hence ι ◦ π r ( S ) ∩ π l ( S ) is an open dense subset of G / G l . Let p ∈ ι ◦ π r ( S ) ∩ π l ( S ). Then there exists g ι , g (cid:48) ι ∈ S such that p = π l ( g (cid:48) ι ) = ι ◦ π r ( g ι ) i.e. g (cid:48) ι G l = ι ( g ι G r ). It follows that P g ι (0) (cid:54) = 0, P g (cid:48) ι (0) (cid:54) = 0 and( g (cid:48) ι , Y ) N l = ι (( g ι , X ) N r ) for all X, Y ∈ V .As B R , the invariant form of R , is orthogonal, we use Lemma 1.1 of [Gro71]to conclude that the restriction of B R on V is a non-degenerate orthogonalform. Hence, V admits vectors which are not self-orthogonal. Let V ∈ V be such that (cid:107) V (cid:107) R (cid:54) = 0. Moreover, let h ∈ G be such that hg ι h − ∈ MA . Wechoose Y g ι = 0, X g ι = R h V and using Proposition 4.2 we observe that ℵ ( g ι , X g ι , g (cid:48) ι , Y g ι ) = ℵ ( g ι , X g ι , g (cid:48) ι , (cid:107) P g ι (0) P g (cid:48) ι ( R e − R g (cid:48) ι )0 (cid:107) R − (cid:107) P g (cid:48) ι (0) P g ι ( R e − R g ι ) X g ι (cid:107) R = −(cid:107) P g (cid:48) ι (0) P g ι (0) R h ◦ π ◦ R − h ( X g ι ) (cid:107) R = −(cid:107) P g (cid:48) ι (0) P g ι (0) V (cid:107) R (cid:54) = 0 . Hence, the set ( G (cid:110) V × G (cid:110) V ) \ Z ℵ is non empty and in particular it contains( g ι , X g ι , g (cid:48) ι , Y g ι ) with ( g (cid:48) ι , Y g ι ) N l = ι (( g ι , X g ι ) N r ), concluding our result. (cid:3) Remark 9.3.
We denote the projections onto the left and right componentsof G (cid:110) V × G (cid:110) V by π l and π r respectively, i.e. π l , π r : G (cid:110) V × G (cid:110) V → G (cid:110) V besuch that for all ( g, X, h, Y ) ∈ G (cid:110) V × G (cid:110) V we have π l ( g, X, h, Y ) = ( g, X )and π r ( g, X, h, Y ) = ( h, Y ). Proposition 9.4.
Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2, let ρ (Γ) and (cid:37) (Γ) both be Zariski dense inside G (cid:110) V and X , the Zariski closure of ( ρ, η )(Γ)inside G (cid:110) V × G (cid:110) V , satisfy D ι \ X (cid:54) = ∅ for all ι mentioned in Remark 9.1.Then there exists a continuous automorphism σ : G (cid:110) V → G (cid:110) V such that σ ◦ ρ = (cid:37) . Proof.
As ( (cid:37), ρ )( γ )( (cid:37), ρ )(Γ)( (cid:37), ρ )( γ ) − = ( (cid:37), ρ )(Γ) for all γ ∈ Γ, we obtainthat ( (cid:37), ρ )( γ ) X ( (cid:37), ρ )( γ ) − = X . We recall the projection maps π l and π r from Remark 9.3 and observe that they are homomorphisms. Hence, itfollows that (cid:37) ( γ ) π l ( X ) (cid:37) ( γ ) − = π l ( X ) and ρ ( γ ) π r ( X ) ρ ( γ ) − = π r ( X ) for all γ ∈ Γ. As both ρ (Γ) and (cid:37) (Γ) are Zariski dense inside G (cid:110) V , we deduce thatboth π l ( X ) and π r ( X ) are normal inside G (cid:110) V (using the Corollary at page54 of [Hum75]). We observe that π l ( X ) ⊃ (cid:37) (Γ), π r ( X ) ⊃ ρ (Γ) and we useProposition A.2 to obtain that π l ( X ) = G l (cid:110) V and π r ( X ) = G r (cid:110) V for some ARGULIS-SMILGA SPACETIMES 21 normal subgroups G l , G r of G . Moreover, both ρ (Γ) and (cid:37) (Γ) are Zariskidense inside G (cid:110) V and it follows that G l = G r = G .Now we consider the following two normal subgroups of X : N l := ker( π l | X )and N r := ker( π r | X ). As ( (cid:37), ρ )(Γ) ⊂ X and N l is normal in X , for all γ ∈ Γwe have ( (cid:37) ( γ ) , ρ ( γ )) N l ( (cid:37) ( γ ) , ρ ( γ )) − ⊂ N l . Moreover, as N l = ker( π l ) ∩ X ,we obtain that N l ⊂ { o } × G (cid:110) V , where o := ( e, N l is of the form ( o, n ) and we obtain that( (cid:37) ( γ ) , ρ ( γ ))( o, n )( (cid:37) ( γ ) , ρ ( γ )) − = ( o, ρ ( γ ))( o, n )( o, ρ ( γ )) − for all γ ∈ Γ. As ρ (Γ) is Zariski dense inside G (cid:110) V , we obtain that N l isnormal inside { o } × G (cid:110) V . Similarly, we obtain that N r is normal inside G (cid:110) V × { o } . Moreover, as π l ( X ) = G (cid:110) V = π r ( X ), we obtain thatdim( N l ) = dim( X ) − dim( G (cid:110) V ) = dim( N r ) . Now using Proposition A.2 we deduce that either of the following holds:1. N l = { o } × G (cid:110) V and N r = G (cid:110) V × { o } ,2. N l = { o } × G l (cid:110) V and N r = G r (cid:110) V × { o } for some nontrivial propernormal subgroups G l , G r of G ,3. both are trivial.We consider these three cases separately below: (cid:5) If N l = { o }× G (cid:110) V and N r = G (cid:110) V ×{ o } , then we obtain a contradiction.Indeed, we have G (cid:110) V × G (cid:110) V = N r N l ⊂ X (cid:40) G (cid:110) V × G (cid:110) V . (cid:5) Suppose N l = { o } × G l (cid:110) V and N r = G r (cid:110) V × { o } . Then by Goursat’slemma [Gou89] we get that the image of X inside ( G (cid:110) V ) / N r × ( G (cid:110) V ) / N l is given by the graph of an isomorphism σ : ( G (cid:110) V ) / N r → ( G (cid:110) V ) / N l .Now we want to show that σ is continuous. We consider the projections p r : G (cid:110) V × { o } → ( G (cid:110) V × { o } ) / ( G r (cid:110) V × { o } ) ,p l : { o } × G (cid:110) V → ( { o } × G (cid:110) V ) / ( { o } × G l (cid:110) V ) , and let π (cid:48) l : X / ( N r N l ) → ( G (cid:110) V ) / N r and π (cid:48) r : X / ( N r N l ) → ( G (cid:110) V ) / N l respectively be the quotient maps induced by p r ◦ ( π l | X ) and p l ◦ ( π r | X ). Wenote that σ = ( π (cid:48) l ) − ◦ π (cid:48) r . Hence σ is a continuous isomorphism. It followsthat, for all g, g (cid:48) ∈ G , and X, X (cid:48) ∈ V with σ (( g, X ) N r ) = ( g (cid:48) , X (cid:48) ) N l we have( g, X, g (cid:48) , X (cid:48) ) ∈ X i.e. D σ ⊂ X . Hence, ∅ = D σ \ X (cid:54) = ∅ , a contradiction. (cid:5) Suppose both N l and N r are trivial then by Goursat’s lemma [Gou89]we get that X inside G (cid:110) V × G (cid:110) V is the graph of an automorphism σ of G (cid:110) V . We can choose σ to be ( π l | X ) − ◦ ( π r | X ) and conclude by observingthat it is continuous. (cid:3) Theorem 9.5.
Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2, let R be anabsolutely irreducible self-contragredient representation and let (cid:107) M ρ ( γ ) (cid:107) R = (cid:107) M (cid:37) ( γ ) (cid:107) R for all γ ∈ Γ. Then either of the following holds:1. both ρ (Γ) and (cid:37) (Γ) are Zariski dense inside some conjugates of G ,2. there exists ( A , Y ) ∈ O ( B R ) (cid:110) V such that A normalizes R ( G ) and ρ isconjugate to (cid:37) by ( A , Y ). Proof.
We will prove this result in three parts. (cid:5)
Let (cid:107) M ρ ( γ ) (cid:107) R = 0 for all γ ∈ Γ. It follows that (cid:107) M (cid:37) ( γ ) (cid:107) R = 0 for all γ ∈ Γ. We use Corollary 7.2 and obtain that there exists
X, Y ∈ V such that ρ = ( e, X ) L ρ ( e, X ) − and (cid:37) = ( e, Y ) L (cid:37) ( e, Y ) − . It follows that both ρ (Γ) and (cid:37) (Γ) are Zariski dense inside some conjugates of G . (cid:5) Let there exist γ ∈ Γ such that (cid:107) M ρ ( γ ) (cid:107) R (cid:54) = 0. Hence (cid:107) M (cid:37) ( γ ) (cid:107) R (cid:54) = 0 andusing Corollary 7.3 we obtain that both ρ (Γ) and (cid:37) (Γ) are Zariski denseinside G (cid:110) V .We use Lemma 8.1 and observe that ℵ ( (cid:37) ( γ ) , ρ ( γ )) = 0 for all γ ∈ Γ.Hence X , the Zariski closure of ( (cid:37), ρ )(Γ) inside G (cid:110) V × G (cid:110) V , is a subvarietyof Z ℵ and using Lemma 9.2 we obtain that D ι \ X (cid:54) = ∅ for all ι as mentionedin Remark 9.1. Now using Proposition 9.4 we deduce that there exists acontinuous automorphism σ : G (cid:110) V → G (cid:110) V such that σ ◦ ρ = (cid:37) . (cid:5) Let σ : G (cid:110) V → G (cid:110) V be as above. We use Proposition 8.5 and obtainthat there exists ( A , Y ) ∈ G (cid:110) V such that σ ( g, X ) = ( A , Y )( g, X )( A , Y ) − .Now we recall Notation 3.2 and Proposition 3.9 to obtain that P g = P g σ Moreover, as X ⊂ Z ℵ , we use Lemma 4.1 and for all ( g, X ) ∈ G (cid:110) V weobtain that P g (0) (cid:107) AP g ( R e − R g ) X (cid:107) R = P g (0) (cid:107) P g ( R e − R g ) X (cid:107) R . Hence, for c ∈ MA with P c (0) (cid:54) = 0, X ∈ V and g = hch − we get that (cid:107) AR h X (cid:107) R = (cid:107) R h X (cid:107) R . As R is irreducible we deduce that (cid:107) A Y (cid:107) R = (cid:107) Y (cid:107) R forall Y ∈ V . It follows that A ∈ O ( B R ). Hence, A is in the normalizer of R ( G )inside O ( B R ) and our result follows. (cid:3) Isospectrality: general case
In this section, we show that the Margulis-Smilga invariant spectra oftwo faithful absolutely irreducible algebraic self-contragredient representa-tions of a connected real split semisimple algebraic Lie group with trivialcenter are completely determined by the isomorphism class of Margulis-Smilga spacetimes. Finally, as a corollary we prove the main result and oneapplication.
Theorem 10.1.
Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2, let R be an ab-solutely irreducible self-contragredient representation and let M ρ ( γ ) = M (cid:37) ( γ )for all γ ∈ Γ. Then either of the following holds:1. both ρ (Γ) and (cid:37) (Γ) are Zariski dense inside some conjugates of G ,2. there exists a continuous automorphism σ : G (cid:110) V → G (cid:110) V such that σρ = (cid:37) and σ is conjugation by an element ( A , Y ) ∈ O ( B R ) (cid:110) V suchthat A normalizes R ( G ). Proof. As M ρ ( γ ) = M (cid:37) ( γ ) for all γ ∈ Γ, we obtain that (cid:107) M ρ ( γ ) (cid:107) R = (cid:107) M (cid:37) ( γ ) (cid:107) R for all γ ∈ Γ. Hence, using Theorem 9.5 we obtain our result. (cid:3)
Theorem 10.2.
Let ( G , V , R , Γ , ρ, (cid:37) ) be as in Convention 0.2 and let ρ and (cid:37) be two Margulis-Smilga spacetimes. Then the following holds:1. If ρ and (cid:37) are conjugate via some inner automorphism of G (cid:110) V , thenthey have the same Margulis-Smilga invariant spectrum.2. If ρ , (cid:37) have the same Margulis-Smilga invariant spectrum and L ρ = L (cid:37) ,then there exists σ , an inner isomorphism of G (cid:110) V , such that ρ = σ ◦ (cid:37) .3. If ρ , (cid:37) have the same Margulis-Smilga invariant spectrum and R is anabsolutely irreducible self-contragredient representation, then thereexists ( A , Y ) ∈ O ( B R ) (cid:110) V such that ρ = ( A , Y ) (cid:37) ( A , Y ) − . ARGULIS-SMILGA SPACETIMES 23
Proof.
We will prove this result in three parts. (cid:5)
Let ( g, Y ) ∈ G (cid:110) V be such that ( g, Y ) ρ ( g, Y ) − = (cid:37) and for all γ ∈ Γlet h γ ∈ G be such that L ρ ( γ ) = h γ exp( Jd L ρ ( γ ) ) h − γ . Then for all γ ∈ Γ wehave Jd L ρ ( γ ) = Jd L (cid:37) ( γ ) and L (cid:37) ( γ ) = gh γ exp( Jd L ρ ( γ ) )( gh γ ) − . Hence, for all γ ∈ Γ, we deduce that M (cid:37) ( γ ) = π ( R − gh γ T (cid:37) ( γ )) = π ( R − gh γ ( Y + R g T ρ ( γ ) − R g R L ρ ( γ ) R − g Y ))= π ( R − h γ ( R − g Y + T ρ ( γ ) − R L ρ ( γ ) R − g Y ))= π ( R − h γ T ρ ( γ )) + π ( R − h γ (( R e − R L ρ ( γ ) ) R − g Y ))= M ρ ( γ ) + π (( R e − R exp( Jd L ρ ( γ ) ) ) R − gh γ Y ) = M ρ ( γ ) . (cid:5) Let ρ , (cid:37) be two Margulis-Smilga spacetimes with L ρ = L (cid:37) and the sameMargulis-Smilga invariant spectrum. We use Theorem 6.4 to obtain thatthere exists σ , an inner isomorphism of G (cid:110) V , such that ρ = σ ◦ (cid:37) . (cid:5) Let ρ , (cid:37) be two Margulis-Smilga spacetimes with the same Margulis-Smilga invariant spectrum and let R be an absolutely irreducible self contra-gredient representation. Hence ρ and (cid:37) , act properly on V . It follows thatfor all γ ∈ Γ \ { e } we have M ρ ( γ ) (cid:54) = 0 and M (cid:37) ( γ ) (cid:54) = 0. Now we use Corollary7.3 and Theorem 10.1 to obtain our result. (cid:3) Theorem 10.3.
Let G be a real split connected simple algebraic Lie groupwith trivial center, let g be its Lie algebra with Killing form B , let Ad : G → GL ( g ) be the adjoint representation and let ρ and (cid:37) be two Margulis-Smilgaspacetimes. Then the following holds:1. If ρ and (cid:37) are conjugate via some inner automorphism of G (cid:110) Ad g , thenthey have the same Margulis-Smilga invariant spectrum.2. If ρ , (cid:37) have the same Margulis-Smilga invariant spectrum and L ρ = L (cid:37) , then there exists σ , an inner isomorphism of G (cid:110) Ad g , such that ρ = σ ◦ (cid:37) .3. If ρ , (cid:37) have the same Margulis-Smilga invariant spectrum then thereexists ( A , Y ) ∈ SO ( B ) (cid:110) g such that ρ = ( A , Y ) (cid:37) ( A , Y ) − . Proof.
We observe that g is finite dimensional, the Killing form B is a non-degenerate symmetric bilinear form. As G is connected and with trivialcenter we obtain that the adjoint representation is faithful. Also, as G issimple we obtain that Ad is irreducible. It follows that the complexificationof Ad is also irreducible and hence Ad is absolutely irreducible. Moreover, byProposition 4.4.5 of [Spr09] we obtain that Ad is algebraic. We observe that a is a zero weight space of Ad and hence Ad admits zero as a weight. Also, asthe Killing form is not degenerate we obtain that Ad is self-contragredient.Hence our result follows from Theorem 10.2. (cid:3) Appendix A. Normal subgroups
In this section we prove some results about the normal subgroups of affinegroups of the form G (cid:110) V , where G is a connected real split semisimplealgebraic Lie group with trivial center acting on a vector space V via afaithful irreducible algebraic representation R : G → GL ( V ). We expect that these results are known in the community but we could not find anappropriate reference in the literature. Lemma A.1.
Let G be a connected real split semisimple Lie group, let V be a finite dimensional vector space with dim V >
1, let R : G → GL ( V ) bean irreducible algebraic representation and let X ∈ V with X (cid:54) = 0. Thenthe additive group generated by { R g X | g ∈ G } ⊂ V is V . Proof.
If possible let us assume that R g X = X for all g ∈ G . Then R ( G ) fixesthe line R X and R X (cid:40) V , a contradiction to the fact that the representation R is irreducible.Hence we can assume that there exists a g ∈ G such that R g X (cid:54) = X . Weuse Lemma 2.6 and the continuity of the action of G to deduce that we canchoose g such that g is loxodromic and the dimension of the unit eigenspaceof R g is exactly dim V . Let m ∈ M , Z ∈ a ++ and h ∈ G be such that g = hm exp( Z ) h − and for all λ ∈ Ω ∪ { } let Y λ ∈ V λ be such that Y := R − h X = Y + (cid:88) λ ∈ Ω Y λ . Moreover, as Ω is finite, we can slightly perturb g and make sure that for all λ, ν ∈ Ω we have λ ( Z ) (cid:54) = ν ( Z ) whenever λ (cid:54) = ν . As gX (cid:54) = X we obtain that Y (cid:54) = Y and there exists µ ∈ Ω such that Y µ (cid:54) = 0. We observe that λ ( Z ) (cid:54) = 0for λ ∈ Ω and choose t λ := (cid:18) λ ( Z ) (cid:19) . It follows that R exp( t λ Z ) Y λ = 2 R exp( Z ) Y λ . Indeed, R exp( t λ Z ) Y λ = exp( λ ( t λ Z )) Y λ = exp(log 2 + λ ( Z )) Y λ = 2 exp( λ ( Z )) Y λ = 2 R exp( Z ) Y λ . Therefore, for all λ ∈ Ω we have (2 R exp( Z ) − R exp( t λ Z ) ) Y λ = 0. It follows thatfor all λ ∈ (Ω ∪ { } ) \ { µ } and R µ := (cid:0) R exp( Z ) − R e (cid:1) (cid:89) ν ∈ Ω \{ µ } (cid:0) R exp( Z ) − R exp( t ν Z ) (cid:1) , we have R µ Y λ = 0. Therefore, we obtain that R µ Y ∈ V µ and R µ Y = R µ Y µ .Moreover, we observe that R µ Y µ = (cid:0) R exp( Z ) − R e (cid:1) (cid:89) ν ∈ Ω \{ µ } (cid:0) R exp( Z ) − R exp( t ν Z ) (cid:1) Y µ = (exp( µ ( Z )) − (cid:89) ν ∈ Ω \{ µ } (2 exp( µ ( Z )) − exp( t ν µ ( Z ))) Y µ . and (exp( µ ( Z )) − (cid:81) ν ∈ Ω \{ µ } (2 exp( µ ( Z )) − exp( t ν µ ( Z ))) (cid:54) = 0. Hence (cid:8)(cid:0) R exp( tZ ) − R exp( sZ ) (cid:1) R µ Y | t, s ∈ R (cid:9) = R Y µ . Let S be the additive group generated by { R g X | g ∈ G } ⊂ V and hence R − h X ∈ S . We observe that R µ is inside the additive group generated by theset { R g | g ∈ G } ⊂ gl ( V ). It follows that R Y µ ⊂ S . Also, we observe thatthe additive group generated by R ( G ) R Y µ is the same as the vector spacegenerated by R ( G ) R Y µ . Moreover, the vector space generated by R ( G ) R Y µ ARGULIS-SMILGA SPACETIMES 25 is invariant under the action of R ( G ) and using the irreducibility of the rep-resentation R we obtain that R ( G ) R Y µ generates V . Therefore, we concludethat S = V . (cid:3) Proposition A.2.
Let G be a connected real split semisimple algebraic Liegroup with trivial center, let V be a finite dimensional vector space withdim V >
1, let R : G → GL ( V ) be a faithful irreducible algebraic represen-tation and let N be a normal subgroup of G (cid:110) V . Then N is either of thefollowing subgroups:1. the trivial group,2. G i (cid:110) V , where G i is a normal subgroup of G . Proof.
Let N be a nontrivial normal subgroup of G (cid:110) V . Then there exists( g, X ) ∈ N with ( g, X ) (cid:54) = ( e, h, Y ) ∈ G (cid:110) V we observethat ( h, Y ) − = ( h − , − R − h Y ) and hence( h, Y )( g, X )( h, Y ) − = ( hgh − , Y + R h X − R hgh − Y ) . It follows that ( g, X ) ∈ N if and only if ( hgh − , Y + R h X − R hgh − Y ) ∈ N for all h ∈ G and Y ∈ V .Now we consider the linear projection map L : G (cid:110) V → G and observe thatfor all h ∈ G , h L ( N ) h − ⊂ L ( N ). It follows that L ( N ) is a normal subgroupof G . We prove our result in the following two parts:1. L ( N ) is trivial: As N is nontrivial, in this case we see that there exists X (cid:54) = 0 such that ( e, X ) ∈ N . Hence for all h ∈ G we have( h, e, X )( h, − = ( e, R h X ) ∈ N . As the representation R is irreducible using Lemma A.1 we obtain that( e, Y ) ∈ N for all Y ∈ V . Therefore, we deduce that N = { e } (cid:110) V .2. L ( N ) is a nontrivial normal subgroup of G : In this case also we seethat there exists X (cid:54) = 0 such that ( e, X ) ∈ N . Indeed, if not then N ∩ ( { e } (cid:110) V ) = { ( e, } and hence L | N : N → G is an isomorphism onto L ( N ). It follows that for all g ∈ L ( N ) thereexist X g ∈ V such that X gh = X g + R g X h and N = { ( g, X g ) | g ∈ L ( N ) } . Since N is normal inside G (cid:110) V , for all Y ∈ V we have( e, Y )( g, X g )( e, Y ) − = ( g, Y + X g − R g Y ) ∈ N . Hence Y + X g − R g Y = X g for all Y ∈ V , a contradiction. Therefore,there exists X (cid:54) = 0 such that ( e, X ) ∈ N . Hence for all h ∈ G we have( h, e, X )( h, − = ( e, R h X ) ∈ N . As the representation R is irreducible, using Lemma A.1 we obtainthat ( e, Y ) ∈ N for all Y ∈ V and we deduce that L ( N ) (cid:110) V ⊂ N . Also, N ⊂ L − ( L ( N )) = L ( N ) (cid:110) V . It follows that L ( N ) (cid:110) V = N Therefore, the only nontrivial normal subgroups of G (cid:110) V are of the form G i (cid:110) V where G i is a normal subgroup of G . (cid:3) References [AMS02] H. Abels, G. A. Margulis, and G. A. Soifer. On the Zariski closure of the linearpart of a properly discontinuous group of affine transformations.
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