Jean-Philippe Anker

AND BARTOSZ Bertrand Rémy

CONSTRUCTION OF BROOKS-LINDENSTRAUSS KERNELS ON AFFINE BUILDINGS OF ARBITRARY REDUCED TYPE, WITH APPLICATIONS

This article deals with harmonic analysis on affine buildings. Its main goal is to construct suitable kernels associated to a discrete multitemporal wave equations on the latter spaces, the long-standing motivation being to contribute to progress in arithmetic quantum unique ergodicity (AQUE) on certain Riemannian manifolds.

I

Quantum unique ergodicity (QUE) deals with certain limit measures on compact (or finite-volume) negatively curved manifolds or locally symmetric manifolds of non-compact type. More precisely, on a negatively curved compact manifold we consider a sequence ( : ∈ N) of 2 -normalized eigenfunctions of the positive Laplacian Δ corresponding to eigenvalues → ∞. In this context, Z. Rudnick and P. Sarnak [START_REF] Rudnick | The behaviour of eigenstates of arithmetic hyperbolic manifolds[END_REF] conjectured that the only weak-★ cluster value of the sequence of probability measures |( ) | 2 dvol is dvol itself. When is a locally symmetric space whose universal cover = / is an irreducible Riemannian symmetric space of non-compact type, the analogous conjecture has a more group-theoretic formulation since then the identity component = Isom( ) • is a simple Lie group and is a maximal compact subgroup (they are all conjugated). The conjectured statement on limit measures is the same but one makes the assumption that the functions ( : ∈ N) are joint eigenfunctions of the algebra D of all -invariant differential operators on descended to = Γ\ .

Though it is possible to attack these conjectures from a purely analytic perspective (see e.g. [START_REF] Anantharaman | Quantum ergodicity and delocalization of Schrödinger eigenfunctions[END_REF]), in the locally symmetric case some specific choices of eigenfunctions ( : ∈ N) can be made, relying on groupand representation-theoretic considerations. Indeed, if one assumes that Γ the fundamental group of is an arithmetic lattice in , then an additional commutative algebra of difference operators becomes available. More precisely, since can be seen as the Archimedean points of an algebraic group G over a number field , there is a Hecke algebra ℋ associated with an adelic realization of the homogeneous space Γ\ whose elements commute with D. Thus it makes sense to work with joint eigenfunctions of both D and ℋ which is the context of arithmetic quantum unique ergodicity (AQUE). These additional arithmetic invariance conditions may be considered as a strong restriction, but they are automatically fulfilled if the D-spectrum is multiplicity free, which is actually conjectured under suitable assumptions. The most striking and complete results in AQUE are those obtained by E. Lindenstrauss for compact hyperbolic surfaces [START_REF] Lindenstrauss | Invariant measures and arithmetic quantum unique ergodicity[END_REF] together with the finite volume case proved by K. Soundararajan [START_REF] Soundararajan | Quantum unique ergodicity for SL 2 (Z)[END_REF], but more and more cases became available in recent years.

Still in the arithmetic situation, there is an intermediate context in which the additional arithmetic invariance assumption only involves a single place of . In this case, the corresponding subalgebra ℋ of ℋ is a convolution algebra of functions on the non-Archimedean Lie group G( ) where denotes the completion of with respect to . Equivalently, ℋ can be seen as some commutative algebra 0 of averaging operators acting on functions defined on the vertices of the Bruhat-Tits building attached to G( ). This roughly explains why the main technique of the present paper is the purely geometric (building-theoretic) counterpart to harmonic analysis of bi--invariant functions on G( ) where is a suitably chosen maximal compact subgroup of G( ), as developed by I. Satake and I.G. Macdonald [START_REF] Macdonald | Spherical functions on a group of -adic type[END_REF].

We need to introduce additional concepts in order to state our results whose main outcome is the construction of a wave kernel on an affine building. Let us recall that an affine building [START_REF] Tits | Reductive groups over local fields, Automorphic forms, representations and -functions[END_REF] is a simplicial complex covered by subcomplexes called apartments which are all isomorphic to a given Euclidean tiling. The (discrete) symmetry group of the model tiling is called the Weyl group of the building, and apartments in the building are requested to satisfy the following incidence conditions:

-any two simplices must be contained in an apartment; -given any two apartments, there must be an isomorphism between them fixing their intersection. These axioms allow one to define a real-valued distance between arbitrary points making an affine building a non-positively curved contractible space. They also lead to a vectorial distance between vertices (i.e. 0-simplices) of particular type and called special vertices: A vertex is called special if its stabilizer in the Weyl group (of any apartment containing it) is the full linear part of the Weyl group; we denote by the set of special vertices.

Given an affine building a (purely geometric) harmonic analysis can be developed and leads to a Gelfand-Fourier transform with no group involved (see Subsection 1.4 and [START_REF]Spherical harmonic analysis on affine buildings[END_REF] for more details). First of all, the geometry of an apartment embedded in defines a root system Φ with Weyl group . Let us denote by the Euclidean space containing it; this space contains the co-weight lattice in which the vectorial distance :

× → + takes its values, see Section 1.3 for details. For the purpose of applications, we consider a commutative Banach algebra 1 densely containing 0 whose spectrum Σ is contained in the complexification C of , so that the Gelfand-Fourier transform ℱ maps 1 into the algebra C (Σ) of continuous functions on Σ, which is defined as [START_REF] Borel | Existence of discrete cocompact subgroups of reductive groups over local fields[END_REF].

Theorem A. Assume that the root system Φ of the affine building is reduced. Set = =1 where 1 , . . . , is the canonical co-weight basis in . Let 0 = 0 + 0 be an element of Σ. Then there exist ( 0 ), ( 0 ) ∈ N such that for all integers ( 0 ) and ( 0 ), there is a function : × → C with the following properties:

(i) for all , ∈ the value of ( , ) depends only on the vectorial distance ( , ); moreover, ( , ) vanishes when ( , )

+1

where > 0 is a constant depending only on the root system Φ; (ii) there are > 0 and > 0 such that

sup , ∈ | ( , )| - ;
(iii) for all ∈ and ∈ Σ, we have

F ( , •) ( ) -1 while F ( , •) ( 0 )
.

Such kernels were first constructed by Sh. Brooks and E. Lindenstrauss in [START_REF] Sh | Graph eigenfunctions and quantum unique ergodicity[END_REF][START_REF]Non-localisation of eigenfunctions on large regular graphs[END_REF][START_REF]Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF] for homogeneous trees and next by Z. Shem-Tov in [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF] for affine buildings of type ˜ with 2. Our construction works for any affine building of reduced type and our kernel differs from the one in [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF].

Let us briefly comment on the proof of Theorem A. To construct the kernel we seek for building blocks satisfying (i) and (ii). For this purpose in the case of homogeneous trees, Sh. Brooks and E. Lindenstrauss used a fundamental solution to the discrete wave equation. Notice that, on real hyperbolic spaces, fundamental solutions to the shifted wave equation are known to possess similar properties (see for instance [START_REF] Tataru | Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation[END_REF]). Therefore we introduce a discrete multitemporal wave equation on affine buildings which is the analog of the system studied by M.A. Semenov-Tian-Shansky [START_REF] Semenov-Tjan-Šanskiȋ | Harmonic analysis on Riemannian symmetric spaces of negative curvature, and scattering theory[END_REF] on non-compact symmetric spaces. We consider a particular non-trivial solution to this equation and show that it satisfies finite propagation speed and uniform exponential space-time decay (see Theorem 2.3). Inspired by Sh. Brooks and E. Lindenstrauss [START_REF] Sh | Graph eigenfunctions and quantum unique ergodicity[END_REF][START_REF]Non-localisation of eigenfunctions on large regular graphs[END_REF][START_REF]Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF], we define a kernel satisfying (i), (ii), and (iii). The last property is proved by a careful study of its Gelfand-Fourier transform on the spectrum Σ. Lastly let us note again that the statement of Theorem A and its proof make no use of any group action.

Let us explain now how the above result fits in a general strategy elaborated to prove some cases of AQUE. The general scheme of proof introduced by E. Lindenstrauss [START_REF] Lindenstrauss | Invariant measures and arithmetic quantum unique ergodicity[END_REF] consists of three steps which are quite independent from one another (see [START_REF]Entropy bounds and quantum unique ergodicity for Hecke eigenfunctions on division algebras, Probabilistic methods in geometry, topology and spectral theory[END_REF]Introduction] for more details).

• Microlocal lift. This step consists in lifting the problem from the locally symmetric space = Γ\ / to a geometric bundle over (the unit tangent bundle in rank one) and eventually to the homogeneous space Γ\ . However, even if initially it was proved geometrically, there is now a representation-theoretic approach which works in fairly general setup [START_REF] Silberman | On quantum unique ergodicity for locally symmetric spaces[END_REF]. The outcome is a reformulation of the problem where the main objects of study are limits = lim →∞ | | 2 d where d is the normalized Haar measure on Γ\ , and where are 2 -normalized eigenfunctions of both ( ) and ℋ (or part of it). Here denotes the Lie algebra of , and ( ) the center of its universal enveloping algebra.

• Positive entropy. A consequence of the previous step is that any cluster value of

(| | 2 d : ∈ N)
is invariant under a maximal split torus of , denoted by in the sequel. In this step, the goal is to prove that for all regular elements ∈ (or at least for some specific ones), almost every ergodic component of a limit measure has positive entropy. This is the place where our construction may play a rôle. In this general scheme, the use of a kernel as in Theorem A occurs in the second step, following ideas initially due to Sh. Brooks and E. Lindenstrauss [START_REF] Sh | Graph eigenfunctions and quantum unique ergodicity[END_REF][START_REF]Non-localisation of eigenfunctions on large regular graphs[END_REF][START_REF]Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF] and enables one to deal with suitable "explicit" partitions of Γ\ . This leads to the following positive entropy result whose proof consists of checking that the lemmas used by Z. Shem-Tov in [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF]Theorem 1.1] to deal with the case = SL (R) can be used verbatim once his specific kernel for SL is replaced by ours.

Theorem B. Let be a simple real Lie group which is R-split, and let Γ be an arithmetic subgroup of . Let be an -invariant probability measure on the homogeneous space = Γ\ . Assume that = lim →∞ | | 2 d is a weak-★ limit where > 0 and are some 2 -normalized joint eigenfunctions of both ( ) and of the Hecke algebra at some fixed prime . Then any regular element ∈ acts with positive entropy on each ergodic component of .

As already mentioned, modulo some progresses on the classification of invariant measures on homogeneous spaces, Theorem B may contribute to provide some additional cases of AQUE involving a single place. In particular, it may be useful in the following situation (see Subsection 4.2 for more details): Let G be an absolutely almost simple group over Q and let be a prime number. We set ∞ = G(R) and = G(Q ), we choose a maximal compact subgroup in . Let us denote by Γ[ 1 ] (resp. by Γ) the Z[ 1 ]-points (resp. the Z-points) of G, for instance defined by means of some embedding G < GL . In good cases (e.g. when G is simply connected), we have

= Γ[ 1 ] and Γ[ 1 ] ∩ = Γ, hence a bijection Γ ∞ ↦ → Γ[ 1 ] ( ∞ , 1) between Γ\ ∞ and Γ[ 1 ]\( ∞ ×
)/ . As recalled in Subsection 4.1, since we are eventually interested in ∞ only, we can adjust the choice of G so that is split over Q , hence the Bruhat-Tits building of has a reduced root system and thus admits kernels as in our Theorem A. This will lead to a non-compact but finite volume quotient space Γ\ ∞ . To deal with compact spaces Γ\ ∞ one has to assume that G is anisotropic over Q. This can be achieved either by choosing G so that

′ is compact at some other non-Archimedean place ′ (implying strong restrictions on the absolute type of G) or by assuming that the rational points of G at some Archimedean place are compact (with no restriction on the absolute root system but requiring to pass from Q to a number field for the ground field of G).

The structure of the paper goes as follows: In Section 1, we introduce the basics about affine buildings and harmonic analysis thereon (again, we do it without calling on Bruhat-Tits theory, in fact without using any group action). In Section 2, we define a discrete multitemporal wave equation and use Fourier analysis to study a particular solution for which we prove finite propagation speed and uniform space-time estimates.

In Section 3, we give a detailed proof of Theorem A. In Section 4, we describe the general context in which constructed kernels could be used and prove Theorem B.

We conclude the introduction with some notation. In the whole paper, N (resp. N * ) is the set of integers 0 (resp. > 0). is the maximal dimension of its simplices. A face of a simplex is a non-empty subset ⊆ . For a simplex we denote by St( ) the collection of simplices containing ; in particular, St( ) is a simplicial complex. Two abstract simplicial complexes and ′ are isomorphic if there is a bijection : ( ) → ( ′ ) such that for all = { 1 , . . . , } ∈ we have ( ) = { ( 1 ), . . . , ( )} ∈ ′ . With every abstract simplicial complex one can associate its geometric realization | | in the vector space of functions → R with finite support, see e.g. [22, §2].

A

A set equipped with a collection of equivalence relations {∼ : ∈ } where = {0, . . . , }, is called a chamber system and the elements of are called chambers. A gallery of type = 1 . . . in is a sequence of chambers ( 1 , . . . , ) such that for all ∈ {1, 2, . . . }, we have -1 ∼

, and -1 ≠ . If ⊆ , a residue of type is a subset of such that any two chambers can be joined by a gallery of type = 1 . . . with 1 , . . . , ∈ . From a chamber system we can construct an abstract simplicial complex where each residue of type corresponds to a simplex of dimension -# . Then, for a given vertex , we denote by C ( ) the set of chambers containing .

A Coxeter group is a group given by a presentation

: ( ) , = 1, for all , ∈
where = ( , ) × is a symmetric matrix with entries in Z ∪ {∞} such that for all , ∈ ,

, = 2 if ≠ , 1 if = .
For a word = 1 • • • in the free monoid we denote by an element of of the form

= 1 • • • . The length of ∈ , denoted ℓ( ),
is the smallest integer such that there is a word = 1 • • • and = . We say that is reduced if ℓ( ) = . A Coxeter group may be turned into a chamber system by introducing in the following collection of equivalence relations:

∼ ′ if and only if = ′ or = ′ . The corresponding simplicial complex Σ is called Coxeter complex. A simplicial complex
is called a building of type Σ if it contains a family of subcomplexes called apartments such that (Bi) each apartment is isomorphic to Σ, (Bii) any two simplices of lie in a common apartment, (Biii) for any two apartments and ′ having a chamber in common there is an isomorphism : → ′ fixing ∩ ′ pointwise. The rank of the building is the cardinality of the set . We always assume that is irreducible. A simplex is a chamber in if it is a chamber in any of its apartments. By ( ) we denote the set of chambers in . Using the building axioms we see that ( ) has a chamber system structure. However, it is not unique. A geometric realization of the building is its geometric realization as an abstract simplicial complex. In this article we assume that the system of apartments in is complete meaning that any subcomplex of isomorphic to Σ is an apartment. We denote by Aut( ) the group of automorphisms of the building .

Affine Coxeter complexes.

In this section we recall basic facts about root systems and Coxeter groups. A general reference is [START_REF]Chapters 4-6, Elements of Mathematics[END_REF], which deals with Coxeter systems attached to reduced root systems. Since at the beginning the root system may be non-reduced we will also refer to [START_REF] Mantero | Eigenvalues of the vertex set Hecke algebra of an affine building[END_REF][START_REF] Parkinson | Buildings and Hecke algebras[END_REF].

Let Φ be an irreducible, but not necessarily reduced, finite root system in a Euclidean space . The Z-span of Φ is the root lattice . Let { 1 , . . . , } be a base of Φ, and let Φ + denote the corresponding subset of positive roots. We denote + = ∈ : , > 0 for every ∈ Φ + the positive Weyl chamber.

For a positive root ∈ Φ + , its height is defined as

ht( ) = =1 wherever = =1 with ∈ N. Since Φ is irreducible, there is a unique highest root 0 = =1 with ∈ N. We set 0 = {1, . . . , } and = {0} ∪ { ∈ 0 : = 1}. Let Φ ∨ = { ∨ = 2
, : ∈ Φ} be the dual root system. The Z-span of Φ ∨ is the co-root lattice . Let + = ∈Φ + N ∨ . The dual basis { 1 , . . . , } to { 1 , . . . , } consists of fundamental co-weights and its Z-span is the co-weight lattice . Let + = =1 N be the cone of dominant co-weights, and let =1 N * be the subcone of strongly dominant co-weights. Note that for a given , the family - ; is increasing in while the family + ; is decreasing. To each wall we associate ; the orthogonal reflection in defined by

; ( ) = - , - ∨ .
Set 0 = 0 ;1 , and = ;0 for each ∈ 0 .

The finite Weyl group is the subgroup of GL( ) generated by { : ∈ 0 }. Let us denote by 0 the longest element in . The fundamental sector in defined as

0 = + = ∈ 0 R + = ∈ 0 + ;0
is the fundamental domain for the action of on .

The affine Weyl group is the subgroup of Aff ( ) generated by { : ∈ }. Observe that is a Coxeter group. The hyperplanes H give the geometric realization of its Coxeter complex Σ Φ . To see this, let (Σ Φ ) be the family of closures of the connected components of \ ∈ H . By 0 we denote the fundamental chamber (or fundamental alcove), i.e.

0 = ∈ : , 0 1 and , 0 for all ∈ 0 = ∈ 0 + ;0 ∩ - 0 ;1
which is the fundamental domain for the action of on . Moreover, the group acts simply transitively on (Σ Φ ). This allows us to introduce a chamber system in (Σ Φ ): For two chambers and ′ and ∈ , we set ∼ ′ if and only if = ′ or there is ∈ such that = . 0 and ′ = . 0 . The vertices of 0 are {0, 1 / 1 , . . . , / }. Let us denote the set of vertices of all ∈ (Σ Φ ) by (Σ Φ ). Under the action of , the set (Σ Φ ) is made up of + 1 orbits .0 and .( / ) for all ∈ 0 . Thus setting Σ Φ (0) = 0, and Σ Φ ( / ) = for ∈ 0 , we obtain the unique labeling Σ Φ : (Σ Φ ) → such that any chamber ∈ (Σ Φ ) has one vertex with each label.

For each simplicial automorphism : Σ Φ → Σ Φ there is a permutation of the set such that for all chambers and ′ , we have ∼ ′ if and only if ( ) ∼ ( ) ( ′ ), and

Σ Φ ( ( )) = ( Σ Φ ( )) for all ∈ (Σ Φ ).
A vertex is called special if for each ∈ Φ + there is such that belongs to

; . The set of all special vertices is denoted by (Σ Φ ).

Given ∈ and ∈ , the set = + . 0 is called a sector in Σ Φ with a base vertex . Moreover, by [1, Corollary 3.20], an affine Coxeter complex Σ Φ uniquely determines the affine Weyl group but not a finite root system Φ. In fact, the root systems C and BC have the same affine Weyl group.

Affine buildings.

A building of type Σ is called an affine building if Σ is a Coxeter complex corresponding to an affine Weyl group. Select a chamber 0 in ( ) and an apartment 0 containing 0 . Using an isomorphism 0 : 0 → Σ such that 0 ( 0 ) = 0 , we define the labeling in 0 by 0 ( ) = Σ ( 0 ( )) for all ∈ ( 0 ). Now, thanks to the building axioms the labeling can be uniquely extended to : ( ) → . To turn ( ) into a chamber system over we declare that two chambers and ′ are -adjacent if they share all vertices except the one of type (equivalently, they intersect along an -panel). For each ∈ ( ) and ∈ , we define ( ) = # ′ ∈ ( ) : ′ ∼ -1. In all the paper, we assume that ( ) only depends on , i.e. that ( ) is independent of , and therefore the building is regular; we henceforth write instead of ( ). We also assume that 1 < ( ) < ∞ and therefore the building is thick and locally finite. A vertex of is special if it is special in any of its apartments. The set of special vertices is denoted by . We choose the finite root system Φ in such a way that Σ is its Coxeter complex. In all cases except when the affine group has type C or BC , the choice is unique. In the remaining cases we select C if 0 = , otherwise we take BC . This guarantees that ( ) = ( + ′ ) for all , ′ ∈ , see the discussion in [START_REF] Mantero | Eigenvalues of the vertex set Hecke algebra of an affine building[END_REF]Section 2.13]. In this article all buildings have reduced type.

Given two special vertices , ∈ , let be an apartment containing and and let : → Σ be a type-rotating isomorphism such that ( ) = 0 and ( ) ∈ 0 , see [START_REF] Parkinson | Buildings and Hecke algebras[END_REF]Definition 4.1.1]. We set ( , ) = ( ) ∈ + . For ∈ + and ∈ , we denote by ( ) the set of all special vertices ∈ such that ( , ) = . The building axioms entail that the cardinality of ( ) depends only on , see [START_REF]Spherical harmonic analysis on affine buildings[END_REF]Proposition 1.5]. Let be the common value. We fix once and for all an origin which is a special vertex of the chamber 0 . Let us define a multiplicative function on by

0 ( ) = ∈Φ +
, with ∈ where = whenever ∈ . for ∈ 0 . For ∈ having the reduced expression

= 1 • • • , we set = 1 • • • . Then (1) = ( -1 ) ( -1 ) 0 ( )
where = ∈ : . = , and for any subset ⊆ we have set

( -1 ) = ∈ -1 .
1.4. Spherical harmonic analysis. In this subsection we summarize spherical harmonic analysis on affine buildings (see [START_REF] Macdonald | Spherical functions on a group of -adic type[END_REF][START_REF]Spherical harmonic analysis on affine buildings[END_REF]). We consider the averaging operators

( ) = 1 ∈ ( ) ( ) for all ∈ +
acting on functions : → C. Let 0 = C-span{ : ∈ + } be a commutative unital involutive algebra ( ) whose characters can be expressed in terms of Macdonald functions (see [START_REF]Spherical harmonic analysis on affine buildings[END_REF])

(2) ( ) = -1 2 0 ( ) ( -1 ) ∈ c( . ) . , with ∈ + and ∈ C where c( ) = ∈Φ + 1 --1 -, ∨ 1 --, ∨ is the so-called c-function; the values of
where the denominator of the c-function vanishes can be obtained by taking appropriate limits. Specifically, every multiplicative functional on 0 is given by the evaluation ℎ ( ) = ( ) for all ∈ + , at certain ∈ C . Moreover, ℎ = ℎ ′ if and only if . + 2 = . ′ + 2 (see [19, Theorem 3.3.12(ii)]). Recall that Macdonald functions as given by the formula (2) are linear combinations for all ∈ + .

Conversely, every is a linear combination of 's with . Let 2 be the closure of 0 in the operator norm on ℓ 2 ( ). Then 2 is a commutative unital ★ -algebra whose spectrum 2 equals modulo ⋉ 2 ( ), and the Gelfand-Fourier transform F ( ) = ℎ ( ) with ∈ identifies 2 with C ( 2 ) the ★ -algebra of (bounded) continuous functions on 2 . Finally, the following inversion formula holds: for every ∈ 0 and , ∈ ;

(4) ( ) ( ) = 1 2
( -1 ) | | ∫ F ( ) ( ) d |c( )| 2
where denotes the Dirac measure at , = ( , ) and

(5)

= ∈ : , ∨ 2 for all ∈ Φ ,
The involution is induced by ★ = ★ where ★ = -0 . . Specifically, ℎ extends to a bounded character on 2 if and only if ∈ .

is a fundamental domain for the lattice 2 in , see [START_REF]Spherical harmonic analysis on affine buildings[END_REF]. Actually we need another completion of 0 : As in [12, Section 6], we consider the following commutative unital involutive Banach algebra:

1 = = ∈ + : 1 = ∈ + | | < ∞ .
Notice that on the one hand the character ℎ extends continuously to 1 if and only if sup

∈ + | ( )| < ∞;
according to [START_REF] Macdonald | Spherical functions on a group of -adic type[END_REF]Theorem 4.7.1] this happens if and only if Re belongs to the convex hull of . log 1 2 0 . On the other hand ℎ is Hermitian if and only if ★ ( ) = ( ) for every ∈ + ; as ★ ( ) = (-) and ( ) = ( ), we deduce that -∈ . + 2 . In summary, the continuous Hermitian characters of 1 are parametrized by the set

(6) Σ = ∈ co( . log 1 2
0 ) + : -∈ . + 2 modulo ⋉ 2 ( ), which we call the spectrum of . Finally, the Gelfand-Fourier transform extends to a continuous homomorphism from 1 into C (Σ).

M

In [START_REF] Semenov-Tjan-Šanskiȋ | Harmonic analysis on Riemannian symmetric spaces of negative curvature, and scattering theory[END_REF] Semenov-Tian-Shansky introduced on Riemannian symmetric space = / of non-compact the following multitemporal wave equation on Riemannian symmetric spaces = / of non-compact type: [START_REF] Bourbaki | Éléments de mathématique: groupes et algèbres de Lie[END_REF] Γ( ) ( , ) = ( , ) for all ∈ D, ∈ and ∈ .

Here Γ denotes the Harish-Chandra isomorphism between the algebra D of -invariant differential operators on , and the algebra ( ) of -invariant polynomials on the Cartan subspace . This equation was further studied in [START_REF] Phillips | Scattering theory for symmetric spaces of noncompact type[END_REF], [START_REF] Helgason | Integral geometry and multitemporal wave equations[END_REF], [17, Ch. V, § § 5.7-5.11]; see the latter reference for more details.

In this section we introduce a discrete analog of (7) on affine buildings of reduced type, and produce a particular non-trivial solution enjoying two basic properties: finite propagation speed and uniform spacetime exponential decay. Specifically, the algebra D of invariant differential operators on is replaced by the algebra 0 of averaging operators on and the Harish-Chandra isomorphism Γ by the isomorphism induced by ↦ → ( ∈ + ) between 0 and the algebra of -invariant trigonometric polynomials on . Thus (7) becomes In particular, ( , •) belongs to 0 . We claim that solves (8) which writes [START_REF] Sh | Graph eigenfunctions and quantum unique ergodicity[END_REF] , ∈ .

( + , ˆ ) = ( ) ( , ˆ ) for all ∈ + and ∈ , Notice that we may therefore restrict to ∈ co( . log If ( , ) = , then by the inversion formula (4) we obtain

( , ) = 1 2 ( -1 ) | | ∫ ( , ˆ ) ( ) d |c( )| 2 = 1 2
( -1 ) | . | ∫ ( ) ( ) d |c( )| 2 .
Now, using the explicit formula for Macdonald functions (2), we get

( , ) = 0 ( ) -1 2 1 | . | ∈ 1 2 ∫ ( ) -. , d c( . ) = 0 ( ) -1 2 | | | . | | | 1 2 ∫ ( ) -, d c( ) .
The last integral vanishes if ; in other words, ( , •) is supported in

∈ : ( , ) .
This property is a higher rank analog of finite propagation speed for the wave equation on homogeneous trees (see [START_REF] Anker | The shifted wave equation on Damek-Ricci spaces and on homogeneous trees[END_REF]Section 4] or [9, Lemma 2]). Next, we write

1 --∨ 1 --1 -∨ = 1 + (1 -) ∞ =1 - - ∨ , thus 1 c = ∈ + ( ) -
where for ∈ + we have set

( ) = ( : ∈Φ + ) ∈ P ( ) ∈Φ + >0 (1 -) - with P ( ) = ( : ∈ Φ + ) ∈ N |Φ + | : ∈Φ + ∨ = .
Therefore,

( , ) = 0 ( ) -1 2 | | | . | 1 2 ∈ + ( ) ∫ ( ) -+ , d = 0 ( ) -1 2 | | | . | ∈ + 1 | .( + )| ( ) , + = 0 ( ) -1 2 | | | . | ∈ .
( . -). [START_REF]Non-localisation of eigenfunctions on large regular graphs[END_REF] Our next aim is to estimate

0 ( ) -1 2 ∈ . | ( . -)|.
Let us recall that the condition .

implies that . -belongs to the cone + generated by the positive co-roots. Proof.

Let = max ∨ , ˜ : ∈ Φ + .
Since 1, we have

∈Φ + 1 ∈Φ + ∨ , ˜ = 1 , ˜ .
Hence, ( )

| ( )| ( ) ∈ P ( ) : >0 ( -1) - 1 , ˜
# ( ) ∈ P ( ) :

∈Φ + = -2
where 2 = min{ : ∈ Φ + }. Consequently,

| ( )| 1 , ˜ #Φ + -2 -, ˜
and the lemma follows.

Lemma 2.2.

There are > 0 and > 0 such that for all , ∈ + ,

0 ( ) -1 2 ∈ . | ( . -)| -| | .
We write if there is > 0 such that .

(iii) for all ∈ and ∈ Σ, we have

F ( , •) ( ) -1 wℎ F ( , •) ( 0 ) .
Proof. We are going to define a kernel with desired properties by describing it on the Gelfand-Fourier side. In view of Theorem 2.3, the convenient building blocks are the elementary symmetric functions which essentially reduces the proof to showing (iii).

Let and be two positive integers whose values will be specified later. Let , be the complex function on C defined by [START_REF] Cartwright | Isotropic random walks in a building of type ˜[END_REF] , ( ) =

1 1 + 4| | 4 =0 ( ) 2 -1.

It is clear that

, is invariant under the action of as well as it is invariant under translations by vectors from 2

. Given ∈ Σ, there are ˜ ∈ and ℓ ∈ such that ˜ . + 2 ℓ = -. We claim that for all ∈ N, (13) ( ) ∈ R. Indeed, we have

( ) = ∈ , . = ∈ -˜ . -2 ℓ, . = ∈ , ˜ -1 0 . = ( )
where 0 is the longest element of and 0 . = -. Using [START_REF] Einsiedler | Invariant measures on /Γ for split simple Lie groups[END_REF] we easily deduce that , -1 on Σ. Our next step is to show that there are ( 0 ), ( 0 ) ∈ N such that for all integers ( 0 ) and ( 0 ) there is an integer such that [START_REF] Einsiedler | Invariant measures and the set of exceptions to Littlewood's conjecture[END_REF] , ( 0 ) . First, let us see how [START_REF] Einsiedler | Invariant measures and the set of exceptions to Littlewood's conjecture[END_REF] allows us to complete the proof. Using the inversion formula (4), we define the kernel : × → C by

( , ) = 1 2 ( -1 ) | | ∫ , ( ) ( ) d |c( )| 2
where = ( , ). It follows from the above that ( , ) satisfies (iii). Next, we deduce from [START_REF] Cartwright | Isotropic random walks in a building of type ˜[END_REF] that ∫ , ( ) ( )

d |c( )| 2 vanishes whenever 8 . Hence, ( , ) ≠ 0 implies that ( , ) 8
, which yields (i) because 2 . Thus it remains to prove (ii). Observe that, according to Corollary 2.3, there are > 0 and > 0 such that for all

∈ + , F -1 ∞ -| | . We claim that (15) , = ∈ + 0≺ 8 with | | 1. For the proof, notice that 4 =0 ( )
is the Euclidean Fourier transform of the characteristic function of the set

= . : ∈ 0 , 0 4 .

Consequently,

, is Euclidean Fourier transform of

1 1 + 4| | 1 * 1 -0
where 0 denotes the Dirac measure at 0. Therefore,

0 1 ℓ 1 1 ℓ ∞ = 1 1 + 4| | as stated.
Lastly, by [START_REF] Einsiedler | Diagonal actions on locally homogeneous spaces, Homogeneous flows, moduli spaces and arithmetic[END_REF], we get

sup , ∈ | ( , )| ∈ + 0≺ 8 F -1 ∞ ′ 8 | | =1 -1 - = O ( -)
which is O ( -). This proves (ii).

It remains to show [START_REF] Einsiedler | Invariant measures and the set of exceptions to Littlewood's conjecture[END_REF]. The proof is divided into three cases. Case 1: If 0 = 0, then for all ∈ N,

( 0 ) = ∈ 0 , . = 1 2 ∈ 0 , . + 1 2 ∈ 0 , 0 . = ∈ cosh 0 , . | |.
Thus ( 16) , ( 0 )

1 1 + 4| | 4 =0 ( 0 ) 2 -1 4 | |
for every , ∈ N. In this case we set = . Case 2: Assume that 0 = 0. We start by the following observation: if

0 < | | 8 + 1 , then 4 ( ) = 4 =-4 = sin(4 + 1 2 ) sin 1 2 = (8 + 1) sin(4 + 1 2 ) (4 + 1 2 ) 1 2 sin 1 2 2 8 + 1 . If = 0 then 4 ( ) = 4 =-4 = 8 + 1. Therefore 4 ( ) 2 (8 + 1) for ∈ -8 + 1 , 8 + 1 .
Using the last inequality, we obtain

4 =0 ( ) = 1 - 1 2 | | + 1 2 ∈ 4 ( , . ) 1 - 1 2 | | + 8 | | + 1 | | 1 + 2| | (17) provided that , . ∈ -8 + 1 , 8 + 1 for all ∈ .
Next, we are going to find such that

(18) 0 , . ∈ -8 + 1 , 8 + 1 + 2 Z for all ∈ . Let (19) = 2ℎ(8 + 1)
.

By Dirichlet's approximation theorem, there is ′ ∈ {1, 2, . . . , } such that

′ 0 , ∨ ∈ 0, 2 + 2 Z for all ∈ {1, 2, . . . , }. Hence, ′ 0 , ∨ ∈ 0, ht( ∨ ) 2 + 2 Z for all ∈ Φ + ,
and so

2 ′ 0 , . ∈ - 2 ℎ , 2 ℎ + 2 Z for all ∈ . Let = 2 ′ ⌈ /(2 ′ )⌉. Then 2 2 ′ . Moreover, if 2 ′ then = 2 ′ 2 .
Otherwise 2 ′ < and + 2 ′ < 2 2 . Hence, we get [START_REF] Lindenstrauss | Invariant measures and arithmetic quantum unique ergodicity[END_REF]. Now, in view of [START_REF]Geometric analysis on symmetric spaces[END_REF],

, ( 0 ) = 1 1 + 4| | 4 =0 ( 0 ) 2 -1 1 1 + 4| | 1 + 2| | 2 -1 1 2 | |
for every ∈ N. Case 3: It remains to consider 0 = 0 + 0 ∈ Σ with 0 ∈ + {0} and 0 ∈ . In fact this case requires more work. First, let us observe that 0 , > 0. As 0 ∈ Σ, there are 1 ∈ and ℓ ∈ such that 1 . 0 = -0 and 1 . 0 + 2 ℓ = 0 . Let us consider

( 0 ) = ∈ : . 0 = 0 .
We claim the following Claim 3.2.

( 0 ) = 0 ( 0 ) 1 = ∈ : . 0 , = 0 , .

To prove the first equality, let ∈ ( 0 ). Then 0 1 . 0 , = . 0 , = 0 , where we have used 1 . 0 = -0 and 0 . = -. To justify the second equality, it is enough to prove the inclusion ⊇. Since 0 ∈ + \ {0}, for each ∈ , there are 0 such that

(20) 0 -. 0 = =1 .
Thus if . 0 , = 0 , , then

0 = 0 -. 0 , = =1 ,
and so 1 = . . . = = 0, that is ∈ ( 0 ), proving the claim For future use, let us observe that

(21) = min 0 -. 0 , : ∈ \ ( 0 ) > 0.
Since for each ∈ N and ∈ Z, we have

∈ ( 0 ) 0 , . = ∈ ( 0 ) 0 , -1 1 0 . = ∈ ( 0 ) - 0 , . ,
we obtain

∈ ( 0 ) 0 , . = 0 , ∈ ( 0 ) 0 , . = 0 , 1 2 ∈ ( 0 ) 0 , . + ∈ ( 0 ) - 0 , . = 0 , ∈ ( 0 ) cos 0 , . . Now, using cos = 1 -2 sin 2 2
, we arrive at

∈ ( 0 ) 0 , . = | ( 0 )| 0 , -2 0 , ∈ ( 0 )
sin 2 0 , .

2

.

In order to estimate , ( 0 ), we write

(22) 4 =0 ( 0 ) = 1 + 1 -2 + 3 + 4 + 5
where we have set

(23) 1 = | ( 0 )| 4 0 , ,
Our goal is to explain how to find an -group H as above in order to see as closely related to therational points of H for some Archimedean completion of , and also how to construct -arithmetic groups projecting onto lattices Γ in . We will also discuss the issue of cocompactness of Γ and explain why, in view of the classification of non-Archimedean Lie groups, this condition either implies restrictions on the absolute root system Φ or implies to work with several Archimedean places. A convenient reference for all the material below is the first chapter in Margulis' book [START_REF] Margulis | Discrete subgroups of semisimple Lie groups[END_REF].

First, if is a locally compact field containing , it is well-known (see [21, Chapter I, Proposition 2.3.6]) that we have the following result, relating an algebraic property for an algebraic group and a topological one for its rational points in this case.

Theorem (Anisotropy is compactness). The topological group of rational points H( ) is compact if and only if the algebraic group H has -rank 0.

Recall that, for any field containing the ground field , the -rank of H, denoted by rk (H), is the dimension of a maximal -split torus in H; these tori are all conjugate by H( ).

We denote by (H) the set of places of such that the group H( ) is compact where is the completion of with respect to . In other words, (H) is the set of places for which H is -anisotropic: It is known to be a finite subset of the set ℛ of all places of . Finally, we denote by ℛ ∞ (resp. ℛ fin ) the set of Archimedean (resp. non-Archimedean) places of .

If we pick a finite subset in ℛ, we denote by ( ) the set of -integral elements in , i.e. the set of elements ∈ such that | | 1 for each absolute value | • | attached to ∈ ℛ fin . From now on, we assume that the group H is given together with an embedding in some general linear group, which we denote by H < GL . This allows us to define the subgroup H ( ) to be

H ( ) = H( ) ∩ GL ( ) .
This is well-defined up to commensurability in the sense that another embedding H < GL would lead to a commensurable subgroup H( ) ∩ GL ( ) , i.e. a subgroup containing a subgroup of finite index in both H( ) ∩ GL ( ) and H( ) ∩ GL ( ) . Any subgroup of the form H ( ) (or, more generally, commensurable with such a subgroup) is called an -arithmetic subgroup of H( ).

The key result to construct lattices in products of Lie groups is the following theorem, due to Borel-Harish-Chandra and Mostow-Tamagawa (see [21, Chapter I, Theorem 3.2.5]).

Theorem (Arithmetic groups are lattices). For any finite set of places of such that ℛ ∞ (H) ⊂ ⊂ ℛ, the -arithmetic group H ( ) is a lattice in the topological group defined to be product group = ∈ H( ). We shall see in the next section, how to construct from the above statements homogeneous spaces Γ\ in the purely Archimedean Lie group setting. In addition, this result is complemented by Godement's compactness criterion, see [ It is clear that for H to be -anisotropic, it is sufficient that H be -anisotropic, hence equivalently that H( ) be compact, for some place of . With this remark in mind, since we are mainly targeting a suitable (cocompact) lattice inclusion Γ < , a very standard trick consists then in finding an -group H as above such that H(R) is for some Archimedean completion R of , and such that H( ) is compact for some other completion , with Archimedean or not. One limitation of this trick is the fact that there are very few compact non-Archimedean simple Lie groups, namely (see e.g. [34, Classification tables]):

Theorem (Very few compact non-Archimedean groups). The only anisotropic simple groups over non-Archimedean local fields are the groups of inner type , in other words division algebras over non-Archimedean local fields.

This classification result explains in particular why quaternion algebras, which are fields over Q, are used to construct cocompact arithmetic lattices in PSL 2 (R). On the contrary, according to Weyl's unitarian trick [7, § 3 Théorème 1 p. 19], any simple real Lie group has a (unique) anisotropic R-form which we denote by H cpt . Recall that an -form of H is a linear algebraic -group ℋ such that H ⊗ and ℋ ⊗ are isomorphic -groups.

The last important result we would like to mention in this abstract general context addresses the freedom of choice that is available when one wants to find an -group as above with prescribed behavior at finitely many places of . The following statement is implied by a deep result in Galois cohomology due to Borel and Harder (see [START_REF] Borel | Existence of discrete cocompact subgroups of reductive groups over local fields[END_REF]Theorem B]).

Theorem (Freedom in prescribing local forms). Let Φ be an irreducible reduced root system. Let be a finite set of places of and, for each ∈ , and let H be a simply connected -group of absolute root system Φ. Then there exists a simply connected -group H such that H ⊗ and H are isomorphic -groups for each ∈ .

The same statement holds with "adjoint" replacing "simply connected" everywhere. It is precious in the sense that, provided a natural compatibility condition is satisfied (namely, sharing the same absolute root system) we can find a suitable -group H in order to construct an -arithmetic lattice H ( ) in as above; moreover, this can be done while prescribing that at least one factor for some ∈ in be compact, ensuring that the lattice inclusion H ( ) < is cocompact. To reformulate the limitation mentioned after the previous theorem, one restriction is that if we choose a compact factor to be non-Archimedean, then the group -group H must be an -form of SL ; otherwise, if we choose the compact factor to be Archimedean, then all absolute root systems are allowed but then must be a number field ≠ Q since several Archimedean places are needed.

Potentially new situations for AQUE.

Using the theoretical framework of the previous subsection, we recall situations when AQUE, or at least positive entropy statements, are known to hold. Then we exhibit situations where the kernels from Theorem 3.1 could be used to prove new positive entropy results. We freely use here the notation used in the introduction as well as the one of the previous subsection.

We first consider the case when adelic Hecke conditions are imposed, i.e. when the functions are eigenfunctions of the full Hecke algebra ℋ (and form a non-degenerate sequence in the sense of [START_REF] Silberman | On quantum unique ergodicity for locally symmetric spaces[END_REF]). Then it is known that AQUE holds for congruence cocompact lattices in ∞ = SL 2 (R), see [START_REF] Lindenstrauss | Invariant measures and arithmetic quantum unique ergodicity[END_REF], and more generally in ∞ = SL (R) for prime [START_REF]Entropy bounds and quantum unique ergodicity for Hecke eigenfunctions on division algebras, Probabilistic methods in geometry, topology and spectral theory[END_REF]. In the same adelic context, AQUE is also known to hold in the non-cocompact case for ∞ = SL 2 (R), since in [START_REF] Soundararajan | Quantum unique ergodicity for SL 2 (Z)[END_REF] it is shown that there is no escape of mass.

We now turn to the case when local Hecke conditions are imposed, i.e. when ( : ∈ N) is a nondegenerate sequence of eigenfunctions of a local Hecke algebra ℋ with respect to some specific prime . Then AQUE holds for congruence cocompact lattices in ∞ = SL 2 (R), and cocompact irreducible lattices in ∞ = SL 2 (R) × SL 2 (R), see [START_REF] Sh | Graph eigenfunctions and quantum unique ergodicity[END_REF], and more generally for congruence cocompact lattices in ∞ = SL (R) for prime, see [START_REF] Silberman | Arithmetic quantum chaos on locally symmetric spaces[END_REF]. In the same local context, AQUE is also known to hold in the non-cocompact case (up to proportionality, since then escape of mass is not disproved yet) for Γ = SL (Z) in ∞ = SL (R) for prime, see [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF]. The latter reference contains a positive entropy result elaborating on ideas from [START_REF] Sh | Graph eigenfunctions and quantum unique ergodicity[END_REF]. We mention below that thanks to our kernels and by using Shem-Tov's ideas on SL (Z) \ SL (R) (where is any integer 2), the positive entropy result [28, Theorem 1.1] can be generalized from the case ∞ = SL (R) to the case when ∞ is any split simple real Lie group.

As it was already mentioned in the introduction, deducing an AQUE result from a positive entropy one requires a precise understanding of measures on the homogeneous spaces Γ\ ∞ . In the higher rank case, the results in this vein that are used in the references above usually come from [START_REF] Einsiedler | Invariant measures and the set of exceptions to Littlewood's conjecture[END_REF] and [START_REF] Einsiedler | Invariant measures on /Γ for split simple Lie groups[END_REF].

Finally, let us explain in which general context our kernels shall be used. Let ∞ be a non-compact simple linear real Lie group whose Lie algebra ∞ is absolutely simple (i.e. ∞ ⊗ R C is simple). We assume that ∞ is the group of real points of an absolutely almost simple simply connected R-group G.

• As a first step, by [START_REF] Borel | Compact Clifford-Klein forms of symmetric spaces[END_REF]Proposition 3.8] and after choosing a totally real number field , we can extend G to an -group, still denoted by G, with G( ) = ∞ , and such that the twisted groups G ( ) are compact for all non-trivial embeddings : → R. Then the group Γ = G( ) is a cocompact lattice in ∞ × ∞ where ∞ is the product of the compact Lie groups G ( ). • As a second step, by the Borel-Harder Theorem above and after choosing a non-Archimedean place ∈ ℛ fin , we can even assume that G is split at the completion of with respect to the place . In particular, the Bruhat-Tits building of = G( ) carries a Brooks-Lindenstrauss kernel as constructed in the present paper, since its (spherical) root system is reduced. Setting = ℛ ∞ ∪ { } we see that that the -integers consist of the elements in

[ 1 ] where is the ring of integers of the number field . Using the notation Γ

[ 1 ] = G( [ 1 ]
), and = G( ) where is the valuation ring of the local field , we have Γ = Γ[ 1 ] ∩ . In addition, the maximal compact subgroup is actually a maximal proper subgroup since it is a maximal parahoric subgroup of an affine Tits system in the simply connected Chevalley group G( ). This provides the equality G(

) = Γ[ 1 ] which together with Γ = Γ[ 1 ] ∩
allows us to identify (in a similar way as in the Introduction after Theorem B) the spaces shows that the characteristic functions 1 provide a basis for the space L ( , ) of compactly supported bi--invariant functions :

Γ\( ∞ × ∞ ) and Γ[ 1 ]\( ∞ × ∞ × )/ .
→ C; the latter space is a unital, commutative algebra for convolution (we pick the Haar measure on for which has volume 1). It follows from a classical computation that for each ∈ + the map ↦ → * 1 coincides with the averaging operator on continuous functions → C which are right -invariant. We will use this remark to write ∈ L ( , ). To go back to the quotient Γ\ ∞ , we will call the operator on 2 (Γ\ ∞ ) defined by the partial convolution as ↦ →

∫

( ∞ , ℎ -1 ) dℎ, the Hecke operator attached to ∈ + . We will denote it again by . We also write ∈ L ( , ) in this context. The connection between eigenvalues of the Hecke algebra and the spectrum Σ is explained in Appendix A. 4.3. Proof of the positive entropy theorem for all R-split simple groups. We can now to turn to an effective application using the kernels constructed in Theorem 3.1. We use the notation of Theorem 4.1 with equal to ∞ above.

We claim that Shem-Tov's proof of [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF]Theorem 1.1] works in the latter context, once one uses our kernels; in particular, the requested generalizations of [28, Proposition 3.1] (suitable kernel) and [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF]Lemma 4.7] (Cauchy-Schwarz and integration on double classes mod ) hold. Moreover, the criterion for positive entropy [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF]Proposition 4.6] is relevant to pure dynamical systems with an abstract ambient space endowed with a probability measure stabilized by a measurable automorphism (i.e. * = ).

The remaining ingredient in the proof is [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF]Lemma 4.1]. This lemma provides the partition of = Γ\ ∞ needed in the assumptions of [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF]Proposition 4.6]. The proof of the lemma itself uses four other lemmas, and we finish by arguing why these are still available. Lemma 4.2 is [31, Lemma 4.4] and holds after using a linear embedding G < GL . Lemma 4.3 is a construction borrowed from [15, 7.51] and valid for general simple real Lie groups. Lemma 4.4 is also a general lemma valid for general simple real Lie groups; its proof uses [15, 7.51] which holds for the same reason. Finally, Lemma 4.5 is also a result which can be used in our context after using a linear embedding G < GL .

A

A. H Σ

In this appendix we retain the notation introduced in Section 4. Suppose that ∈ 2 (Γ\ ∞ ) is a joint eigenfunction of the Hecke algebra, that is there is : + → C, such that for all ∈ + , * = ( ) . We assume that 2 = 1. Then is a homomorphism from L ( , ) into C, thus in view of [19, Propositions 1.2.2 & 1.2.3], there is ∈ C such that ( ) = ( ).

We claim that ( ) is positive definite as a function of ∈ + .

Since both groups ∞ and are of type I, see e.g. [4, Theorem 6.E.19], we can use direct integral decompositions. Hence,

2 Γ 1 \ ∞ × = ∫ ⊕ ∞ × H Γ[ 1 ] (d ).
In particular,

( ) = * , = ∫ ⊕ × Tr ( * ) ( ) ★ Γ[ 1 ] (d ).
In the formulas above ∞ × = ∞ × denotes the unitary dual. To compute the trace, let be an irreducible representation of ∞ × on H , which contains a non-trivial vector fixed by (1, ) for each ∈ . We have

JA 1 . 1 .

 11 .-Ph. Anker and B. Trojan thank École normale supérieure de Lyon for invitations; B. Rémy and B. Trojan thank Université d'Orléans for the same reason. B. Trojan acknowledges financial support from CNRS for a research trimester in 2023 in Orléans. 1. Buildings. A family of non-empty finite subsets of some set is an abstract simplicial complex if for all ∈ , each subset ⊆ also belongs to . The elements of are called simplices. The dimension of a simplex is # -1. Zero dimensional simplices are called vertices. The set ( ) = ∈ is the vertex set of . The dimension of the complex

  For instance = =1 is a strongly dominant co-weight. Notice that = 1 2 ∈Φ ++ ∨ where Φ ++ denotes the set of indivisible positive roots in Φ, see [8, Proposition 10.29, §1, VI] Let H be the family of affine hyperplanes, called walls, being of the form ; = ∈ : , = where ∈ Φ + and ∈ Z. Each wall determines two half-apartments -; = ∈ : , and + ; = ∈ : , .

  coefficients) of monomial symmetric functions = ∈ .

  ) = ( , ) for all , ∈ + and ∈ where the coefficients , are defined by[START_REF] Anker | The shifted wave equation on Damek-Ricci spaces and on homogeneous trees[END_REF].Next, let us consider the function : + × → C whose partial Gelfand-Fourier transform with respect to the second variable is given by( , ˆ ) = ∈ . ,= | | ( ) for all ∈ + and ∈ .

1 2 0

 12 ) ∩ + + .after taking the partial Gelfand-Fourier transform. Indeed,

Lemma 2 . 1 .

 21 There are > 0 and > 1 such that for each ∈ + ,

1 ,

 1 ( , ) d( , ) (1, ℎ -1 ) dℎ = ( ) ∫ (ℎ -1 ) (1, ℎ -1 ) dℎ .The representation is a product ∞ × acting on H ∞ ⊗ H , where both ( ∞ , H ∞ ) and ( , H ) are irreducible unitary representations of ∞ and , respectively. Because = Let us observe that for ∈ , we have( )Λ = ⨏ ( ℎ) dℎ = Λ,and Λ = Λ ( ). Therefore, Λ preserves H , the subspace of -fixed vectors of H , which is 1dimensional, see[START_REF] Macdonald | Spherical functions on a group of -adic type[END_REF] Theorem 1.4.4(ii)]. Moreover, there is∈ C such that if H = C with , =1, then for each ∈ ˜ , , (ℎ) = ˜ ( ).

• Classification of homogeneous measures. The

  next step consists in exploiting results from a wide program which is interesting in itself, namely the classification of invariant and ergodic probability measures on homogeneous spaces such as Γ\ . This program is far from being completed and it is usually the main limiting step in proving instances of AQUE. Some intermediate results often state that measures satisfying suitable positivity entropy conditions are measures associated to homogeneous spaces of algebraic subgroups of : the latter measures are usually called algebraic.

  21, Chapter I, Theorem 3.2.4 (b)] Theorem (Cocompactness amounts to anisotropy). The -arithmetic lattice H ( ) in the topological group is cocompact if and only if the algebraic group H is -anisotropic.

  Note that not insisting on cocompactness of Γ in ∞ allows one to skip the first step and to work with a Q-group G, in which case is merely a prime number .We finally use the identification Γ\ ∞ ≃ Γ[ 1 ]\( ∞ × )/ to see the averaging operators defined in Section 1.4 as Hecke operators acting on 2 (Γ\ ∞ ). More precisely, at the non-Archimedean place the Cartan decomposition =

	∈ +

Using [START_REF]Non-localisation of eigenfunctions on large regular graphs[END_REF] and Lemma 2.2, we can easily deduce the following consequence.

Theorem 2.3. Let be the real function on + × with Fourier-Gelfand transform satisfying F ( , ) = | | ( ) for all ∈ + and ∈ .

Then for each ∈ + , the function

. Moreover, there are > 0 and > 0 such that

for all ∈ + and ∈ .

Remark 2.4. In the case of ˜ 2 buildings we obtained sharper estimates than [START_REF]Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF]: Namely, there is > 0 such that for all ∈ + and ∈ ,

where + 1 denotes the thickness of the building.

Remark 2.5. It would be interesting to extend Theorem 2.3 to all fundamental solutions of [START_REF]Chapters 4-6, Elements of Mathematics[END_REF]. We intend to return to this question in the future.

K

In this section we prove our main result stated in the introduction as Theorem A which holds for any (regular, thick, locally finite) affine building of reduced type. The proof, inspired by the kernel construction of Brooks and Lindenstrauss in [9, Section 2], [10, Section 3] and [START_REF]Joint quasimodes, positive entropy, and quantum unique ergodicity[END_REF]Section 3.1], uses the solution of the multitemporal wave equation described in Theorem 2.3.

We claim that 1 is the leading term. To see this, let ( 0 ) and ( 0 ) be two positive integers such that the following inequalities holds for all ( 0 ) and

where is defined in [START_REF] Margulis | Discrete subgroups of semisimple Lie groups[END_REF], and

The estimate for 5 is straightforward:

To bound 4 we proceed as follows:

4

( 0 , -) ′ 1

8 .

For 3 we have

which is smaller than 1 8 if . To control 2 , we show that there is a positive integer , such that ( 26)

For this purpose we resume the arguments in Case 2 based on the Dirichlet approximation theorem. Let

Hence,

We again set = 2 ′ ⌈ /(2 ′ )⌉. Then min 2 ′ , 2 , and

which implies [START_REF] Rudnick | The behaviour of eigenstates of arithmetic hyperbolic manifolds[END_REF]. Combining all the estimates on , we deduce that

Hence, , ( 0 ) 4| | . This completes the proof of ( 14), and the theorem follows.

A

In what follows, we describe some potentially new situations where arithmetic quantum unique ergodicity (AQUE) could be proved, using the kernel constructed in Section 3 and taking advantage of the fact that we are no more limited by the type of the underlying (reduced) root system. This would require additional progress in various steps of the known strategies for AQUE, and more particularly in the strategy elaborated by Brooks and Lindenstrauss (for more details, see the introduction). Nevertheless, an already available result is a positive entropy theorem as proved by Shem-Tov in [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF].

Theorem 4.1. Let be a simple real Lie group which is R-split, let Γ be an arithmetic subgroup of and let be an -invariant probability measure on the homogeneous space = Γ\ . Assume that = lim →∞ | | 2 d is a weak-★ limit where > 0 and are some 2 -normalized joint eigenfunctions of both ( ) and the Hecke algebra at some fixed prime . Then any regular element ∈ acts with positive entropy on each ergodic component of . This statement is merely a generalization of [28, Theorem 1.1], passing from SL (R) to any split simple R-group. Its proof consists in checking that Shem-Tov's arguments in [START_REF] Shem-Tov | Positive entropy using Hecke operators at a single place[END_REF]Section 4] can be used, once we replace his kernels constructed in [28, Section 3.1] by our kernels from Theorem 3.1.

Before going into this specific result, we first describe the general standard context which will allow us to address some additional potential applications.

The general setting.

In what follows, H denotes a linear algebraic group defined over a number field which is assumed to be absolutely almost simple and simply connected. We denote by Φ the (absolute) root system of H where is an algebraic closure of . The root system Φ is reduced since H is split; its Dynkin diagram is connected since H is almost simple.

The starting point is relevant to real homogeneous spaces. Let be a non-compact simple linear real Lie group. Its associated Riemannian symmetric space is = / where is any maximal compact subgroup of . The desired quantum unique ergodicity results are expected to hold on the homogeneous space Γ\ where Γ is a torsion-free lattice, and have geometric interpretations on the locally symmetric space = Γ\ / .

By [


Alwin Stegeman
email: a.w.stegeman@rug.nl

Pierre Comon
email: pcomon@unice.fr

SUBTRACTING A BEST RANK-1 APPROXIMATION MAY INCREASE TENSOR RANK

Keywords: tensor rank, low-rank approximation, tensor decomposition, multi-way, Candecomp, Parafac. AMS subject classifications: 15A03, 15A22, 15A69, 49M27, 62H25

Is has been shown that a best rank-R approximation of an order-k tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using Candecomp/Parafac and related models. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and substracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximation generally does not decrease tensor rank. In this paper, we provide a mathematical treatment of this property for real-valued 2 × 2 × 2 tensors, with symmetric tensors as a special case. Regardless of the symmetry, we show that for generic 2 × 2 × 2 tensors (which have rank 2 or 3), subtracting a best rank-1 approximation will result in a tensor that has rank 3 and lies on the boundary between the rank-2 and rank-3 sets. Hence, for a typical tensor of rank 2, subtracting a best rank-1 approximation has increased the tensor rank.

INTRODUCTION

Tensors of order d are defined on the outer product of d linear spaces, S ℓ , 1 ≤ ℓ ≤ d. Once bases of spaces S ℓ are fixed, they can be represented by d-way arrays. For simplicity, tensors are usually assimilated with their array representation. We assume throughout the following notation: bold italic uppercase for tensors e.g. , bold uppercase for matrices e.g. T, bold lowercase for vectors e.g. a, calligraphic for sets e.g. S , and plain font for scalars e.g. X i jk , T i j or a i , will be distinguished thanks to their font.

Let be a 3rd order tensor defined on the tensor product S 1 ⊗ ⊗ ⊗ S 2 ⊗ ⊗ ⊗ S 3 . If a change of bases is performed in the spaces S 1 , S 2 , S 3 by invertible matrices S, T, U, then the tensor representation is transformed into def = (S, T, U) •

whose coordinates are given by X i jk = ∑ pqr S ip T jq U kr X pqr . This is known as the multi-lnearity property enjoyed by ten-sors. Matrices, which can be associated with linear operators, are tensors of order 2.

The rank of a tensor is defined as the smallest number of outer product tensors whose sum equals , i.e. the smallest R such that

= R ∑ r=1 a r ⊗ ⊗ ⊗ b r ⊗ ⊗ ⊗ c r . (2)
Hence a rank-1 tensor is the outer product of vectors a, b, c and has entries X i jk = a i b j c k . The decomposition of a tensor into a sum of outer products of vectors and the corresponding notion of tensor rank were first introduced and studied by [START_REF] Hitchcock | The expression of a tensor or a polyadic as a sum of products[END_REF] [START_REF] Hitchcock | Multiple invariants and generalized rank of a p-way matrix or tensor[END_REF].

The multilinear rank of a 3rd order tensor is a triplet (r 1 , r 2 , r 3 ), where r i denotes the rank of the set of mode-i vectors. A mode-i vector is obtained by varying the ith index and keeping other indices fixed.

Usefulness. Tensors play a wider and wider role in numerous application areas including blind techniques for Telecommunications [START_REF] Sidiropoulos | Blind PARAFAC receivers for DS-CDMA systems[END_REF] [10] [START_REF] Almeida | Parafac-based unified tensor modeling for wireless communication systems[END_REF], Arithmetic Complexity [START_REF] Kruskal | Three-way arrays: Rank and uniqueness of trilinear decompositions[END_REF] [START_REF] Strassen | Rank and optimal computation of generic tensors[END_REF] [1] [START_REF] Stegeman | On kruskal's condition for the candecomp/parafac decomposition[END_REF], or Data Analysis [START_REF] Smilde | Multi-Way Analysis[END_REF]. For instance, Independent Component Analysis was originally introduced for symmetric tensors whose rank did not exceed dimension [START_REF] Comon | Independent Component Analysis[END_REF] [2]. Now, it has become possible to estimate more factors than the dimension [START_REF] Donoho | Uncertainty principles and ideal atomic decompositions[END_REF] [16] [START_REF] De Lathauwer | A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization[END_REF]. In some applications, tensors may be symmetric only in some modes [START_REF] Comon | Blind identification of under-determined mixtures based on the characteristic function[END_REF], or may not be symmetric nor have equal dimensions [3] [22] [START_REF] Kroonenberg | Applied Multiway Data Analysis[END_REF]. In most of these applications, the decomposition of a tensor into a sum of rank-1 terms is relevant, since tensors entering the models to fit have a reduced rank.

Matrix algebra is insufficient. The manipulation of tensors remains difficult, because of major differences between their properties when we go from second order to higher. Several of these differences have already been underlined in the past [START_REF] Comon | Symmetric tensors and symmetric tensor rank[END_REF], e.g. (i) tensor rank often exceeds dimensions, (ii) tensor rank can be different in real and complex fields, (iii) maximal tensor rank is not generic, and is still unknown in general, (iv) computing the rank of a tensor is very difficult, (v) a tensor may not have a best low-rank approximate [23] [24] [25] [START_REF] Lim | Tensor rank and the illposedness of the best low-rank approximation problem[END_REF] [18] [START_REF] Stegeman | A method to avoid diverging components in the Candecomp/Parafac model for generic IxJx2 arrays[END_REF].

It has been observed numerically in [17, section 7] that a best or "good" rank-R approximation cannot be obtained by consecutively computing and substracting R best rank-1 approximations (which always exist). The reason for this is that subtracting a best rank-1 approximation generally does not decrease tensor rank. Hence, the deflation technique practiced for matrices (via the Singular Value Decomposition) cannot generally be extended to higher-order tensors. A special case where this deflation technique works is when the tensor is diagonalizable by orthonormal multilinear transformation; see [17, section 7].

In this paper, we provide a mathematical treatment of the (in)validity of a rank-1 deflation procedure for higher-order tensors. We consider 2 × 2 × 2 tensors over the real field, with symmetric tensors as a special case. First, however, we discuss the problem of finding a best rank-1 approximate to a 3rd order tensor. The proofs of our results will be available in a forthcoming full-length version of this paper.

BEST RANK-1 APPROXIMATION

Finding the best rank-1 approximate consists of minimizing the criterion

Ψ = 1 2 || -a⊗ ⊗ ⊗ b⊗ ⊗ ⊗ c|| 2 (3)
with respect to vectors a, b and c. The solution will likely depend on the norm, and we shall restrict to the Frobenius norm:

|| || 2 = ∑ i jk |X i jk | 2 .
Obviously there is a scale indeterminacy in this problem, and we could impose two of these three vectors to be unit norm. We shall not do that here because the presentation would be slightly longer. Let • ℓ denote the summation over the ℓth index (that is the contraction operator in the ℓth space). For instance, the product AB T between two matrices can be written as A • 2 B; if is a 3rd order tensor, • 1 a is a matrix, and

• 1 a • 2 b is a vector.
And let us rewrite criterion (3) as:

Ψ = 1 2 || || 2 -• 1 a • 2 b • 3 c + 1 2 ||a|| 2 ||b|| 2 ||c|| 2 . (4)
Proceeding as in [START_REF] Comon | Tensor decompositions[END_REF], gradients with respect to the three vectors can be obtained:

dΨ a = -• 2 b • 3 c + a||b|| 2 ||c|| 2 dΨ b = -• 1 a • 3 c + b||a|| 2 ||c|| 2 dΨ c = -• 1 a • 2 b + c||a|| 2 ||b|| 2
Concerning the uniqueness of a best rank-1 approximate, one may ask the following question: are there tensors for which the solution defined by

a = ||b|| -2 ||c|| -2 • 2 b • 3 c
and dΨ b = dΨ c = 0 is not unique up to scale? We exhibit in this section a family of such tensors.

If we plug the expression of a back in the equation of stationary values of b, we get that (

• 3 c) • 1 ( • 3 c) • 2 b = λ b, where λ = ||a|| 2 ||b|| 2 ||c|| 4 , which means that b is an eigenvector of the matrix ( • 3 c) • 1 ( • 3 c).
If the latter matrix is proportional to the identity for any c, then any b is an eigenvector. Analogously, substituting the expression for a into dΨ c = 0, we get that (

• 2 b) • 1 ( • 2 b) • 3 c = µ c, where µ = ||a|| 2 ||b|| 4 ||c|| 2 . If ( • 2 b) • 1 ( • 2 b
) is also proportional to the identity for any b, then it follows that any (a, b, c) with a given by dΨ a = 0 is a stationary point. Substituting the expression of a into the criterion (4) then yields a criterion function in (b, c) for which any (b, c) is a stationary point. Hence, the function is constant and any (b, c) is a minimizer. This yields the following proposition Proposition 1 If a tensor is such that the matrix ( • 3 c) is orthogonal for any vector c, and ( • 2 b) is orthogonal for any vector b, then has infinitely many best rank-1 approximates.

In accordance with the usual practice, we shall represent a p × p × 2 tensor with two p × p matrix slices, X 1 and X

2 , as [X 1 | X 2 ]. ⊲ EXAMPLE 1. Let = 1 0 0 -1 0 1 1 0 .
Then for any choice of non zero vector c, the matrix ( • 3 c), obtained by linear combination of the above two matrix slices, is orthogonal. Also, for any non zero vector b, the matrix ( • 2 b) is orthogonal. Hence has infinitely many rank-1 approximates.

⊳ Most tensors have multiple locally best rank-1 approximates, with one of them being better than the others (i.e., a unique best rank-1 approximate), as pointed out in Section 3. Examples will illustrate this fact.

Remark. The tensor in Example 1 has rank 3. Ten Berge et al. [START_REF] Berge | Explicit Candecomp/Parafac solutions for a contrived 2x2x2 array of rank three[END_REF] showed that has no best rank-2 approximation, the infimum of || -|| 2 over of rank at most 2 being 1. A more general result was obtained by De Silva and Lim [START_REF] Lim | Tensor rank and the illposedness of the best low-rank approximation problem[END_REF] who showed that no 2 × 2 × 2 tensor of rank 3 has a best rank-2 approximation. Stegeman [START_REF] Stegeman | Degeneracy in Candecomp/Parafac explained for p × p × 2 arrays of rank p + 1 or higher[END_REF] showed that any sequence of rank-2 approximations (n) for which || -(n) || 2 converges to the infimum of 1, features diverging components. It is shown in [START_REF] De Lathauwer | On the best rank-1 and rank-(r 1 , r 2 , . . . , r n ) approximation of higher-order tensors[END_REF] that the stationary points of the 2 × 2 × 2 symmetric best rank-1 approximation problem are obtained as the roots of a 3rd degree polynomial.

BEST RANK-1 SUBTRACTION

From now on, we restrict our discussion to tensors in the real field. De Silva and Lim [START_REF] Lim | Tensor rank and the illposedness of the best low-rank approximation problem[END_REF]Section 7] showed that 2 × 2 × 2 tensors (over the real field) can be transformed by invertible multilinear matrix multiplications into eight distinct canonical forms. This partitions the space R 2×2×2 into eight distinct orbits under the action of invertible transformations in each of the 3 modes. Table 1 lists the canonical forms for each orbit as well as their rank and multilinear rank. These quantities are invariant under the transformations defining an orbit. This kind of classification is better known for symmetric tensors or multivariate polynomials [START_REF] Comon | Tensor decompositions[END_REF]. Recall the following result stated by De Silva and Lim [START_REF] Lim | Tensor rank and the illposedness of the best low-rank approximation problem[END_REF]:

Lemma 2 Let be a 2 × 2 × 2 tensor with matrix slices X 1 and X 2 . (i) If X 2 X -1 1 or X 1 X -1 2 has real eigenvalues and is diago- nalizable, then is in orbit G 2 . (ii) If X 2 X -1 1 or X 1 X -1
2 has two identical real eigenvalues with only one associated eigenvector, then is in orbit

D 3 . (iii) If X 2 X -1 1 has complex eigenvalues, then is in orbit G 3 .
We shall use this lemma to verify the orbit of 2-dimensional 3rd order tensors. ⊲ EXAMPLE 2. Consider the tensor

= 1 0 0 -2 0 1 1 0 . ( 5
)
Since X 2 X -1 1 has complex eigenvalues, is in orbit G 3 . It can be verified that has a unique best rank-1 approximation

= 0 0 0 -2 0 0 0 0 . (6)
When is subtracted from we end up in orbit D 3 , since

= -= 1 0 0 0 0 1 1 0 , (7)
can be transformed to the canonical form of orbit D 3 by swapping rows within each slice. ⊳

Canonical

Tensor Multilinear form rank rank D 0 : 0 0 0 0 0 0 0 0 0 (0, 0, 0)

D 1 : 1 0 0 0 0 0 0 0 1 (1, 1, 1) D 2 : 1 0 0 0 0 1 0 0 2 (2, 2, 1) D ′ 2 :
1 0 0 1 0 0 0 0 2 (1, 2, 2)

D ′′ 2 : 1 0 0 0 0 0 1 0 2 (2, 1, 2) G 2 : 1 0 0 0 0 0 0 1 2 (2, 2, 2)
D 3 : 0 1 1 0 1 0 0 0 3 (2, 2, 2) G 3 : -1 0 0 1 0 1 1 0 3 (2, 2 , 2)
Table 1: Canonical forms of 2 × 2 × 2 tensors for the eight orbits under the action of invertible multilinear matrix multiplications over the real field. The letters D and G stand for "degenerate" (zero volume set in the 8-dimensional space of 2 × 2 × 2 tensors) and "typical" (positive volume set), respectively.

For tensors in the orbits of Table 1, we would like to know in which orbitis contained, where is a best rank-1 approximation of . We have the following result for the degenerate orbits of ranks 1 and 2.

Proposition 3 Let

be a 2 × 2 × 2 tensor, and let be a best rank-1 approximation of .

(i) If is in orbit D 1 , then -is in orbit D 0 . (ii) If is in orbit D 2 , D ′ 2 , or D ′′ 2 , then -is in orbit D 1 .
For in orbit G 2 or D 3 , the tensoris not restricted to a single orbit. ⊲ EXAMPLE 3. For the canonical tensor of orbit G 2 in Table 1, it can be seen thatis in D 1 . On the other hand, consider

= 0 1 1 0 1 0 0 2 . ( 8
)
For this tensor, X 2 X -1 1 has two distinct real eigenvalues. Hence, by Lemma 2, the tensor is in G 2 . It can be shown that has a unique best rank-1 approximation such that equals the canonical tensor of orbit D 3 in Table 1. ⊳ ⊲ EXAMPLE 4. Next, consider tensors in orbit D 3 : 0 1 2 0 1 0 0 0 , 1 0 0 1 0 0 2 0 , 1 0 0 2 0 0 1 0 . We have no deterministic result for tensors in orbits G 2 and G 3 , but we still have the following result, verified almost everywhere (hence the word "generic"):

Proposition 5 Let be a generic 2 × 2 × 2 tensor,

and be a best rank-1 approximation of . Then almost all tensors

are in orbit D 3 .

Hence, for typical tensors in orbit G 2 , subtracting a best rank-1 approximate increases the rank to 3. For typical tensors in orbit G 3 , subtracting a best rank-1 approximate does not affect the rank. However, some non typical tensors of rank 2 may have a different behavior, as now shown.

Proposition 6 Let be a 2 × 2 × 2 rank-2 tensor with diagonal slices, and let be a best rank-1 approximation of . Thenis in orbit D 1 .

⊲ EXAMPLE 5. Let = a 0 e 0 0 d 0 h . Then = 1 0 ⊗ ⊗ ⊗ 1 0 ⊗ ⊗ ⊗ a e + 0 1 ⊗ ⊗ ⊗ 0 1 ⊗ ⊗ ⊗ d h
Then it can be seen that = (I, I, T) • Á, where I denotes the identity matrix and Á the diagonal tensor tensor with ones on its diagonal:

Á = 1 0 0 0 0 0 0 1 , and T = a d e h
This shows that is is orbit G 2 , and Proposition 6 implies that this is an exception to Proposition 5. ⊳ Proposition 5 states that such exceptions form a set of null measure.

SYMMETRIC TENSORS

A tensor is symmetric if its entries are invariant under arbitrary permutations of its indices. There is a bijection between the space of symmetric I × I × I tensors and the space of homogeneous polynomials of degree 3 in I variables. A symmetric I × I × I tensor can be associated with the polynomial p(s 1 , . . . , s I ) = ∑ i jk x i jk s i s j s k .

The symmetric rank of an order-3 symmetric tensor is the minimal number R such that [START_REF] Comon | Symmetric tensors and symmetric tensor rank[END_REF]:

= R ∑ r=1 a r ⊗ ⊗ ⊗ a r ⊗ ⊗ ⊗ a r . (10)
The orbits of symmetric 2×2×2 tensors are given in Table 2.

canonical form polynomial sym. rank D 0 : 0 0 0 0 0 0 0 0 0 0

D 1 : 1 0 0 0 0 0 0 0 s 3 1 1 G 2 : 1 0 0 0 0 0 0 1 s 3 1 + s 3 2 2 D 3 : 0 1 1 0 1 0 0 0 3 s 2 1 s 2 3 G 3 : -1 0 0 1 0 1 1 0 -s 3 1 + 3 s 1 s 2 2 + s 3 2 3
Table 2: Canonical forms of symmetric 2 × 2 × 2 tensors and associated polynomials, for the three orbits under the action of invertible multilinear transformations over the real field. The letters D and G stand for "degenerate" (zero volume set in the 4-dimensional space of symmetric 2 × 2 × 2 tensors) and "typical" (positive volume set), respectively.

The symmetric rank of symmetric tensors of dimension 2 can be obtained from the Sylvester Theorem, at any order. This Theorem is formulated below in the case of third order tensors.

Theorem 7 (Sylvester) A symmetric 2 × 2 × 2 tensor with associated polynomial

p(s 1 , s 2 ) = γ 3 s 3 1 + 3 γ 2 s 2 1 s 2 + 3 γ 1 s 1 s 2 2 + γ 0 s 3 2 , ( 11
)
has a symmetric rank-R decomposition [START_REF] Castaing | Tensor-based techniques for the blind separation of DS-CDMA signals[END_REF] if and only if there exists a vector g = (g 0 , . . . , g R ) T such that

       γ 0 . . . γ R γ 1 . . . γ R+1 . . . . . . γ 3-R . . . γ 3        g = 0 , (12)
and if the polynomial q(s 1 , s

2 ) = g R s R 1 + g R-1 s R-1 1 s 2 + • • • + g 1 s 1 s R-1 2 + g 0 s R
2 has R distinct real roots. ⊲ EXAMPLE 6. Using Sylvester's Theorem, one can obtain the following decomposition for the representative of orbit D 3 given in Table 2:

6 s 2 1 s 2 = (s 1 + s 2 ) 3 + (-s 1 + s 2 ) 3 -2 s 3 2 . (13)
In other words, the associated tensor can be written

= a ⊗ ⊗ ⊗ 3 + b ⊗ ⊗ ⊗ 3 -2 c ⊗ ⊗ ⊗ 3 , where a = [1, 1] T , b = [-1, 1] T and c = [0, 1] T .
⊳ We have the following analogue of Lemma 2 to verify the orbit of symmetric tensors of dimension 2. Lemma 8 Let be a symmetric 2 × 2 × 2 tensor with matrix slices X 1 and X 2 .

(i) If X 2 X -1 1 or X 1 X -1 2 has distinct real eigenvalues, then is in orbit G 2 . (ii) If X 2 X -1 1 or X 1 X -1 2 has two identical real eigenvalues, then is in orbit D 3 . (iii) If X 2 X -1 1 has complex eigenvalues, then is in orbit G 3 .
Next, we present an example of a symmetric 2 × 2 × 2 tensor in orbit G 3 , that has a unique best symmetric rank-1 approximation , such thatis in orbit D

3 . ⊲ EXAMPLE 7. Let = 0 1 1 1 1 1 1 0 . (14)
We have

X 2 X -1 1 = 0 1 -1 1 , (15)
which has complex eigenvalues. Hence, by Lemma 8 (iii) the tensor is in orbit G 3 .

Next, we compute the best symmetric rank-1 approximation to , which has the form

= x 3 1 x 2 1 x 2 x 2 1 x 2 x 1 x 2 2 x 2 1 x 2 x 1 x 2 2 x 1 x 2 2 x 3 2 . (16)
After some manipulations, it can be shown that the minimum of || -|| 2 is obtained for x 3 1 = x 3 2 = 3/4, that is

By subtraction, we obtain

= -= 1 4 -3 1 1 1 1 1 1 -3 , (18)
and

Z 2 Z -1 1 = 0 1 -1 -2 , (19)
which has a double eigenvalue -1. Hence, by Lemma 8 (ii) the tensor is in orbit D 3 . ⊳ In our next example, the symmetric 2 × 2 × 2 tensor is in orbit G 2 , and has a unique best symmetric rank-1 approximation , such thatis in orbit D 3 . ⊲ EXAMPLE 8. Let

= 3 1 1 1 1 1 1 3 . (20)
We have

X 2 X -1 1 = 0 1 -1 4 , (21)
which has real and distinct eigenvalues. Hence, by Lemma 8 (i) the tensor is in orbit G 2 .

Next, we compute the best symmetric rank-1 approximation to . It can be shown that the minimum of || -|| 2 is obtained for

= 3 2 1 1 1 1 1 1 1 1 . (22)
By subtraction, we obtain

= -= 1 2 3 -1 -1 -1 -1 -1 -1 3 , (23)
and

Z 2 Z -1 1 = 0 1 -1 -2 , ( 24
)
which has a double eigenvalue -1. Hence, by Lemma 8 (ii) the tensor is in orbit D 3 . ⊳ Finally, we have the following analogue of Proposition 5.

Proposition 9

Let be a generic symmetric 2 × 2 × 2 tensor, and be a best rank-1 approximation of . Then almost all tensorsare in orbit D 3 .

Hence, for typical symmetric 2 × 2 × 2 tensors with symmetric rank 2, subtracting a best rank-1 approximate increases the symmetric rank to 3. For typical symmetric tensors with symmetric rank 3, subtracting a best rank-1 approximate does not affect the symmetric rank.

Table 1 ,

 1 Subtracting the best rank-1 approximation from these tensors amounts to replacing the element 2 by zero. Hence, is in orbit D 2 , D ′ 2 , and D ′′ 2 , respectively. ⊳ On the other hand, it can be verified numerically or analytically that for equal to the canonical tensor of orbit D 3 inis also in orbit D 3 . Moreover, numerical experiments show that for a generic in orbit D 3 , we have in orbit D 3 as well. This suggests the following Conjecture 4 If is in D 3 and is the best rank-1 approximate of , then almost all tensorsare in D 3 . Tensors given in Examples 1 or 2 were both in orbit G 3 , and we have seen thatis in orbit D 3 in Example 2. For Example 1, this can be proven for any of the infinite best rank-1 approximates of . Numerically and analytically, we have not found any in orbit G 3 for whichis not in orbit D 3 .

This work has been partially supported by contract ANR-06-BLAN-0074 "Decotes". A. Stegeman is supported by the Dutch Organisation for Scientific Research (NWO), VENI grant 451-04-102 and VIDI grant 452-08-001.