Affine Symmetric Spaces Research Articles

Anosov groups: local mixing, counting and equidistribution

Let $G$ be a connected semisimple real algebraic group, and $\Gamma<G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients $\langle (\exp tv). f_1, f_2\rangle$ in $L^2(\Gamma\backslash G)$ as $t\to \infty$ for any $f_1, f_2\in C_c(\Gamma\backslash G)$ and any vector $v$ in the interior of the limit cone of $\Gamma$. These asymptotics involve higher rank analogues of Burger-Roblin measures which are introduced in this paper. As an application, for any affine symmetric subgroup $H$ of $G$, we obtain a bisector counting result for $\Gamma$-orbits with respect to the corresponding generalized Cartan decomposition of $G$. Moreover, we obtain analogues of the results of Duke-Rudnick-Sarnak and Eskin-McMullen for counting discrete $\Gamma$-orbits in affine symmetric spaces $H\backslash G$.

On the orbit decomposition of some affine symmetric spaces

On the orbit decomposition of some affine symmetric spaces

Branching laws for discrete series of some affine symmetric spaces

In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the {\it period integrals}, studied in number theory, and also closely connected to the {\it symmetry-breaking operators}, introduced by T.~Kobayashi. We exhibit non-vanishing symmetry breaking operators for the restriction of a representation $\Pi$ in the discrete spectrum for real hyperboloids to representations of smaller orthogonal groups. In the last part we discuss some conjectures for the restriction of representations in Arthur packets containing the representation $\Pi$ and the corresponding Arthur-Vogan packets to smaller orthogonal groups; these are inspired by the Gross-Prasad conjectures.

Shortest and straightest geodesics in sub-Riemannian geometry

There are several different, but equivalent definitions of geodesics in a Riemannian manifold, based on two characteristic properties: geodesics as shortest curves and geodesics as straightest curves. They are generalized to sub-Riemannian manifolds, but become non-equivalent. We give an overview of different approaches to the definition, study and generalization of sub-Riemannian geodesics and discuss interrelations between different definitions. For Chaplygin transversally homogeneous sub-Riemannian manifold Q, we prove that straightest geodesics (defined as geodesics of the Schouten partial connection) coincide with shortest geodesics (defined as the projection to Q of integral curves (with trivial initial covector) of the sub-Riemannian Hamiltonian system). This gives a Hamiltonization of Chaplygin systems in non-holonomic mechanics.We consider a class of homogeneous sub-Riemannian manifolds, where straightest geodesics coincide with shortest geodesics, and give a description of all sub-Riemannian symmetric spaces in terms of affine symmetric spaces.

Effective equidistribution of shears and applications

A “shear” is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete subgroup of $${\text {PSL}}(2,\mathbb {R})$$ , possibly of infinite co-volume. We prove the regularized equidistribution of shears under large translates with effective (that is, power saving) rates. We also give applications to weighted second moments of GL(2) automorphic L-functions, and to counting lattice points on locally affine symmetric spaces.

Editors' Note

Editors' Note

Local Generalized Symmetries and Locally Symmetric Parabolic Geometries

We investigate (local) automorphisms of parabolic geometries that generalize geodesic symmetries. We show that many types of parabolic geometries admit at most one generalized geodesic symmetry at a point with non-zero harmonic curvature. Moreover, we show that if there is exactly one symmetry at each point, then the parabolic geometry is a generalization of an affine (locally) symmetric space.

The harmonic analysis of lattice counting on real spherical spaces

By the collective name of \it lattice counting we refer to a setup introduced in [DRS] that aims to establish a relationship between arithmetic and randomness in the context of affine symmetric spaces. In this paper we extend the geometric setup from symmetric to real spherical spaces and continue to develop the approach with harmonic analysis which was initiated in [DRS].

QUANDLE VARIETIES, GENERALIZED SYMMETRIC SPACES, AND φ-SPACES

We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation ⊳. It can also be seen as an analogue of a generalized affine symmetric space or a regular s-manifold in algebraic geometry. Assume that Q is normal as an algebraic variety and that the action of its inner automorphism group Inn(Q) has a dense orbit. Then we show that there is an algebraic group G acting on Q with the same orbits as Inn(Q) such that each G-orbit is isomorphic to the quandle (G/H, ⊳φ) associated to the group G, an automorphism φ of G and a subgroup H of Gφ.

Orthogonal symmetric affine Kac-Moody algebras

Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues, known as affine Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side; more precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification.

Affine Kac-Moody symmetric spaces related with $A_{1}^{(1)}\hbox{,}\, A_{2}^{(1)}\hbox{,}\, A_{2}^{(2)}$A1(1),A2(1),A2(2)

Symmetric spaces associated with Lie algebras and Lie groups which are Riemannian manifolds have recently got a lot of attention in various branches of Physics for their role in classical/quantum integrable systems, transport phenomena, etc. Their infinite dimensional counter parts have recently been discovered which are affine Kac-Moody symmetric spaces. In this paper we have (algebraically) explicitly computed the affine Kac-Moody symmetric spaces associated with affine Kac-Moody algebras \documentclass[12pt]{minimal}\begin{document}$A_{1}^{(1)}, A_{2}^{(1)}, A_{2}^{(2)}$\end{document}A1(1),A2(1),A2(2). We hope these types of spaces will play similar roles as that of symmetric spaces in many physical systems.

Double exponential map on symmetric spaces

We establish an asymptotic formula for the double exponential map operator on affine symmetric spaces. This operator plays an important role in the geometric calculus of symbols of (pseudo)differential operators on manifolds with connection, whose foundations were laid by Sharafutdinov. To obtain this result, we essentially use the structural theory of symmetric spaces and techniques of the Lie group theory. One of the key moments is an application of the Campbell-Hausdorff series in Dynkin form.

Strong wavefront lemma and counting lattice points in sectors

We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and, more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz.

On the exponentiality of affine symmetric spaces

An affine symmetric space G/H is said to be exponential if every two points of this space can be joined by a geodesic and weakly exponential if the union of all geodesics issuing from one point is everywhere dense in G/H. For the group space (G × G)/Gdiag of a Lie group G, these properties are equivalent to the exponentiality and weak exponentiality of G, respectively. We generalize known theorems on the image of the exponential mapping in Lie groups to the case of affine symmetric spaces. We prove the weak exponentiality of the symmetric spaces of solvable Lie groups, and in the semisimple case we obtain criteria for exponentiality and weak exponentiality.

Radon transform on symmetric matrix domains

Let K = R, C, H be the field of real, complex or quaternionic numbers and Mp,q(K) the vector space of all p × q-matrices. Let X be the matrix unit ball in Mn-r,r(K) consisting of contractive matrices. As a symmetric space, X = G/K = O(n - r, r)/O(n - r) × O(r), U(n - r, r)/U(n - r) × U(r) and respectively Sp(n - r,r)/Sp(n - r) × Sp(r). The matrix unit ball y0 in Mr'-r,r with r' ≤ n - 1 is a totally geodesic submanifold of X and let Y be the set of all G-translations of the submanifold y0. The set Y is then a manifold and an affine symmetric space. We consider the Radon transform Rf(y) for functions f ∈ C ∞ 0(X) defined by integration of f over the subset y, and the dual transform Rt F(x),x ∈ X for functions F(y) on Y. For 2r < n, 2r ≤ r' with a certain evenness condition in the case K = R, we find a G-invariant differential operator M and prove it is the right inverse of RtR, RtR Mf = cf, for f ∈ C∞ 0(X), c ≠ 0. The operator f → RtR f is an integration of f against a (singular) function determined by the root systems of X and y0. We study the analytic continuation of the powers of the function and we find a Bernstein- Sato type formula generalizing earlier work of the author in the set up of the Berezin transform. When X is a rank one domain of hyperbolic balls in Kn-1 and y0 is the hyperbolic ball in Kr'-1, 1 < r' < n we obtain an inversion formula for the Radon transform, namely MRtR f = cf. This generalizes earlier results of Helgason for non-compact rank one symmetric spaces for the case r' = n - 1.

Complex structures, totally real and totally geodesic submanifolds of compact 3-symmetric spaces, and affine symmetric spaces

We construct invariant complex structures of a compact 3-symmetric space by means of the canonical almost complex structure of the underlying manifold and some involutions of a Lie group. Moreover, by making use of graded Lie algebras and some invariant structures of affine symmetric spaces, we classify half dimensional, totally real and totally geodesic submanifolds of a compact 3-symmetric space with respect to each invariant complex structure.

Bol loops as sections in semi-simple Lie groups of small dimension

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most nine-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are isotopic to direct products of Bruck loops of hyperbolic type or to Scheerer extensions of Lie groups by Bruck loops of hyperbolic type.

On closures of cycle spaces of flag domains

Open orbits D of noncompact real forms G 0 acting on flag manifolds Z = G/Q of their semisimple complexifications G are considered. Given D and a maximal compact subgroup K 0 of G 0, there is a unique complex K 0–orbit in D which is regarded as a point \(C_{0}\in \mathcal {C}_q(D)\) in the space of q-dimensional cycles in D. The group theoretical cycle space \(\mathcal M_{D}\) is defined to be the connected component containing C 0 of the intersection of the G–orbit G(C 0) with \(\mathcal {C}_q(D)\). The main result of the present article is that \(\mathcal {M}_D\) is closed in \(\mathcal {C}_q(D)\). This follows from an analysis of the closure of the universal domain \(\mathcal {U}\) in any G-equivariant compactification of the affine symmetric space G/K, where K is the complexification of K 0 in G.

Spin Calogero models associated with Riemannian symmetric spaces of negative curvature

The Hamiltonian symmetry reduction of the geodesics system on a symmetric space of negative curvature by the maximal compact subgroup of the isometry group is investigated at an arbitrary value of the momentum map. Restricting to regular elements in the configuration space, the reduction generically yields a spin Calogero model with hyperbolic interaction potentials defined by the root system of the symmetric space. These models come equipped with Lax pairs and many constants of motion, and can be integrated by the projection method. The special values of the momentum map leading to spinless Calogero models are classified under some conditions, explaining why the B C n models with two independent coupling constants are associated with SU ( n + 1 , n ) / S ( U ( n + 1 ) × U ( n ) ) as found by Olshanetsky and Perelomov. In the zero curvature limit our models reproduce rational spin Calogero models studied previously and similar models correspond to other (affine) symmetric spaces, too. The construction works at the quantized level as well.

Horospherical model for holomorphic discrete series and horospherical Cauchy transform

We define a complex horospherical transform on an affine symmetric space $X=G/H$ of Hermitian type and show that it has no kernel on the representations of the $H$ -spherical holomorphic discrete series.