The latter question is the main topic of the present text (Chapters 3 and 4). At …rst one is led to restrict to those **symmetric** **spaces**, called here « special» , for which the method applies in the same way as for Lie groups. But, when considering more general **spaces**, one needs to introduce a real- valued function e(X; Y ) of two tangent vectors X; Y : identically 1 for spe- cial **spaces**, this function embodies the modi…cation of the Kashiwara-Vergne method required for a general **symmetric** space. It can be constructed from its in…nitesimal structure (the corresponding Lie triple system) and contains by itself much information about invariant analysis on the **symmetric** space, when transferred to its tangent space at the origin via the exponential map- ping (hence the notation « e» ): invariant di¤erential operators, mean values, spherical functions are related to e. The search for such relations between analysis and in…nitesimal properties is our main guiding line.

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THE FIRST EIGENVALUE OF THE DIRAC OPERATOR ON COMPACT OUTER SPIN **SYMMETRIC** **SPACES**
JEAN-LOUIS MILHORAT
Abstract. In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin **symmetric** **spaces**, providing, for sym- metric **spaces** of “inner” type, a formula giving this first eigenvalue in terms of the algebraic data of the groups involved. We conclude here that study by giving the explicit expression of the first eigenvalue for “outer” compact spin **symmetric** **spaces**.

DIMENSIONAL HERMITIAN **SYMMETRIC** **SPACES**
BRUNO DUCHESNE, JEAN LÉCUREUX, AND MARIA BEATRICE POZZETTI
A BSTRACT . We define a Toledo number for actions of surface groups and complex hyper- bolic lattices on infinite dimensional Hermitian **symmetric** **spaces**, which allows us to define maximal representations. When the target is not of tube type we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite dimensional totally geodesic subspace on which the action is maximal. In the opposite direction we construct examples of geometrically dense maximal representation in the infinite dimensional Hermitian **symmetric** space of tube type and finite rank. Our approach is based on the study of boundary maps, that we are able to construct in low ranks or under some suitable Zariski-density assumption, circumventing the lack of local compactness in the infinite dimensional setting.

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1. Introduction
It is well-known that **symmetric** **spaces** provide examples where detailed infor- mation on the spectrum of Laplace or Dirac operators can be obtained. Indeed, for those manifolds, the computation of the spectrum can be (theoretically) done using group theoretical methods. However the explicit computation is far from being simple in general and only few examples are known. On the other hand, many results require some information about the first (nonzero) eigenvalue, so it seems interesting to get this eigenvalue without computing all the spectrum. In that direction, the aim of this paper is to prove the following formula for the first eigenvalue of the Dirac operator:

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−id.
If (M, H, J, θ, gθ) is a **symmetric** strictly pseudoconvex CR manifold then M = G/K where G is the closed subgroup of P sH(M, θ) generated by all the pseudo-Hermitian symmetries ψ(x0), x0 ∈ M, and K is the isotropy subgroup at a base point. Also M is an homogeneous strictly pseudoconvex CR manifold. Note that the contact sub-**symmetric** **spaces** torsionless are **symmetric** Sasakian manifolds.

6 Conclusion
Across the paper, we have seen how wrapped models can be constructed on manifolds with an exponential map. We have analysed the case of the Lie group SE(n) and mentioned Riemannian manifolds, though such wrapped models can be considered on any manifold with an affine connection. The main limitation of these models is that the underlying manifold should have an exponential map, a log map and a Jacobian which can be computed at a reasonable cost. Note that this is often the case when the manifold is a homogeneous space. Future efforts should focus on three problems. The first one is to obtain convergence results for the density estimation using wrapped models. The second one is to understand and characterize the shape of the injectivity domains in the different possible settings, such as Lie groups and Riemannian **symmetric** **spaces**, see [12]. And the third problem is to identify on which manifolds with an affine connection the Jacobian of the exponential map can be computed explicitly. As shown in section 5, **symmetric** **spaces** seem to be an interesting setting to study this property. Note however that there are known examples outside of this class where the Jacobian is explicit, see for instance the Wasserstein metric on Gaussian distributions [3].

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In this paper, we build on a work of Gavrilov [Gav06] on the Taylor expansion of the composition of two exponentials in affine connection **spaces** to establish in Section 1 the approximation provided by one step of the pole ladder up to order 5. It is remarkable that the scheme is of order 3, thus much higher than the first order of the other parallel transport schemes. This makes pole ladder a very attractive alternative to Schild’s ladder in general affine manifolds. Moreover, the fourth-order term involves the covariant derivative of the curvature only. Since a vanishing covariant derivative is the characteristic of a locally **symmetric** space, this suggests that the error could vanish in this case. Thus, we investigate in Section 2 local and global **symmetric** **spaces** with affine or Riemannian structure. We show that pole ladder is actually locally exact in one step, and even almost surely globally exact in Riemannian **symmetric** manifolds. The key feature is that the differential of the symmetry is the negative of the parallel transport, so that the parallel transport along a geodesic segment can be realized using the composition of two symmetries (a transvection). Even if this result appears to be new for the geometric computing community, is was already known in mathematics (the construction is pictured for instance on [Tro94, Fig.5, p.168]). The contribution is thus in the connection between the different communities here.

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ccsd-00004039, version 1 - 24 Jan 2005
OPERATOR ON COMPACT SPIN **SYMMETRIC** **SPACES**
JEAN-LOUIS MILHORAT
Abstract. Let G/K be a simply connected spin compact inner irreducible **symmetric** space, endowed with the metric induced by the Killing form of G sign-changed. We give a formula for the square of the first eigenvalue of the Dirac operator in terms of a root system of G. As an example of application, we give the list of the first eigenvalues for the spin compact irreducible **symmetric** **spaces** endowed with a quaternion-K¨ ahler structure.

result of a least squares minimization and a recursive algorithm. In particular, we will focus on a special class of Riemannian **symmetric** **spaces**: the special orthogonal group SO(n). Indeed, working in such Riemannian manifold allow nice properties to solve the issues above. The key point to give explicit solution for the interpolation problem and ensures the C 2 diﬀerentiablity condition at joint points is the use of global symmetries in

The proof of the hyperbolic dispersion estimates is quite technical, and occupies Sections 3, 4, 5. It uses a version of the pseudodifferential calculus adapted to the geometry of locally **symmetric** **spaces**, based on Helgason’s version of the Fourier transform for this **spaces**, and inspired by the work of Zelditch in the case of G = SL(2, R) [27]. We point out the fact that an alternative proof of Theorem 1.12 is given in [2], based on more conventional Fourier analysis. The reader might prefer to read [2] instead of Sections 3, 4, 5, however we feel that the two techniques have an interest of their own.

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well-posedness, Theorem 1.3 and Corollary 1.4 , tough, only r α , α ≥ 1 and e Cr with
positive C are admissible behavior at infinity.
The basic advantage in having a spherically **symmetric** manifold relies on the fact that it is possible to decompose the Dirac operator (and analogously its flow) in a sum of ”radial” Dirac operators (see section 3 for details) and so, somehow, handle the geometric term after a change of variable as a potential perturbation. This strategy is strongly inspired by [ 2 ], in which the authors obtain local and global in time Strichartz estimates for the Schr¨ odinger flow on spherically **symmetric** manifolds. However, we should stress the two main differences with respect to their case, which also represent the two main difficulties here: first, the ”radial” decomposition for the Dirac operator is much more subtle, and forces to work on 2-dimensional angular **spaces**, due to the fact that the Dirac operator does not preserve radial spinors. Second, this approach naturally produces, as we will see, scaling critical potential perturbations, and while for the Schr¨ odinger equation with inverse square potential dispersive estimates are well known (see [ 6 ], [ 7 ]), for the Dirac equation with a Coulomb-type potential only some weak local smoothing effect has been proved (see [ 12 ]) but nothing is known at the level of Strichartz estimates.

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This completely describes the topological **spaces** of IRSs in simple higher-rank Lie groups. On the other hand, if G is a group of real rank one then there are always more IRSs than those described in this theorem. For example, every cocompact lattice in G is a Gromov-hyperbolic group and hence contains plenty of normal infinite subgroups of infinite index, see Theorem 17.2.1 of [45]. Taking a random conjugate of one of the latter we obtain an IRS which violates the conclusion of the theorem above. More generally we can induce from IRSs in lattices, which yields further examples as many lattices in SO(n, 1) (and a few in SU(2, 1)) surject onto nonabelian free groups, which have plenty of IRSs by [14]. It turns out that in the case where G = SO(n, 1) the wealth of available IRSs goes well beyond these examples, as hyperbolic geometry yields constructions of ergodic IRSs which are not induced from a lattice. These constructions are the main object of this paper and we will describe them in some detail in the next section.

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4 /Spin 9 , Ω being in this
case the canonical 8-form on manifolds with holonomy Spin 9 .
2. Preliminaries for the proof
2.1. Spectrum of the Dirac operator on spin compact irreducible symmet- ric **spaces**. We consider a spin compact simply connected irreducible **symmetric** space G/K of “type I”, where G is a simple compact and simply-connected Lie group and K is the connected subgroup formed by the fixed elements of an involu- tion σ of G. This involution induces the Cartan decomposition of the Lie algebra g of G into

Remark 2.5. It is interesting to note that the definition of eligible pairs is more complicated for the space SO(p, p) than for the **spaces** SO(p, q) with p < q (recall that the latter **spaces** have a much richer root structure). As for the **spaces** SO(p, q) with p < q, the number of zeroes on the diagonal of D X is important. Unlike in the case SO(p, q)
with p < q, this only becomes a factor when the number of zeroes is greater than 1 (as opposed to greater than 0). In [6], we showed that if p < q and X and Y ∈ a were such that the D X and D Y have no zero diagonal elements

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7. CONCLUSION
The paper is a survey of the work by the author and collaborators on a geometric consensus theory. While ele- mentary, the proposed theory enables precise connections between several popular models of distributed computa- tion, by viewing them as realizations of the same geometric update on different nonlinear configuration **spaces**. Phase synchronization is viewed as a consensus algorithm on the circle, while coordination on Lie groups is viewed as consensus of invariant velocities in the Lie algebra. An essential difference between linear consensus algo- rithms and their nonlinear extensions is the non-convex nature of **symmetric** **spaces** like the circle. This property is what makes the convergence analysis graph dependent on nonlinear **spaces**. From a design viewpoint, it is of interest to reformulate consensus algorithms on nonlinear **spaces** in such a way that they converge (almost) globally under the same assumptions as linear consensus algorithms, and several solutions were proposed on the circle.

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countable. First the base has to be ordered such that the norm of the rest of the decomposition, i.e., P
|j|>N hf, e j i e j , decreases as fast as possible for
regular functions. Second, in order to process locations and directions in- differently, the basis functions must present regularity properties regarding the metric. Eigenfunctions of the Laplacian operator fill all the criteria. Indeed the Fourier transform on R n , or the Fourier–Helgason transform on **symmetric** **spaces**, are highly related to the convolution by isotropic kernel. The orthogonal series density estimator is then equivalent to the kernel den- sity estimator. When the e j functions are Fourier functions, the estimation

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Geodesic completeness is quite mild (avoiding examples as trees with leafs), while the other are strong assumptions (for example non-branching rules out trees and visibility rules out products). Another possible assumption, that we shall not use directly, is for X to have rank one, meaning that it admits no isometric embedding of the Euclidean plane. It is a weaker condition than visibility. More generally the rank of a space Y is the maximal dimension of an isometrically embedded Eulidean space, and it has been proved a very important invariant in the study of **symmetric** **spaces**. For example the hy- perbolic **spaces** RH n , CH n , HH n and OH 2 are the only rank one **symmetric** **spaces** of non-compact type.

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dimension of secant varieties of Grassmannians, while for our purpose we need the whole description of all the elements in V• V annihilating the given tensor. The idea of extending the apolarity to contexts different from the classical **symmetric** one has been pursued by various authors especially in the multi-homogeneous con- text (e.g. [27, 28, 4]) but we are not aware of any other more than [17] where it has been investigated in the skew-**symmetric** context.

The ﬁrst attempt to mix the two requirements was done with **symmetric** Boolean functions. They deserve ﬁrstly attention in cryptography since they guarantee that no input variables has greater or lesser signiﬁcance [Bru84]. Moreover, Shannon has shown in his early works [Sha49] that synthesising them requires at most n 2 gates. Following those results, their