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1 Symmetric spaces 1

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Contents

1 Symmetric spaces 1

1.1 General features of symmetric spaces . . . . 2

1.2 Rank of symmetric Lie algebras . . . . 6

1.3 Pseudo-Riemannian symmetric spaces . . . . 7

1.3.1 Pseudo-Riemannian symmetric spaces with constant sectional cur- vature . . . 11

1.3.2 Semisimple pseudo-Riemannian symmetric spaces . . . 17

1.4 Classication of indecomposable Riemannian symmetric spaces . . . 20

1.4.1 Isotropic Riemannian symmetric spaces . . . 23

1.5 Classication of indecomposable Lorentzian symmetric spaces . . . 24

1.5.1 Cahen-Wallach spaces . . . 27

1.6 Symplectic symmetric spaces with Ricci-type curvature . . . 29

1.6.1 Classication of Ricci-type symplectic symmetric spaces . . . 32

1.7 Totally geodesic submanifolds . . . 45

1.7.1 Totally geodesic submanifolds in pseudo-Riemannian symmetric spaces 47 1.7.2 Totally geodesic submanifolds in symplectic symmetric spaces . . . . 48

2 Integral geometry on Riemannian symmetric spaces 53 2.1 Framework of homogeneous spaces in duality . . . 55

2.2 Radon transforms on Riemannian homogeneous spaces . . . 58

2.2.1 Main dual Radon transform . . . 58

2.2.2 Shifted dual Radon transform . . . 61

2.3 Totally geodesic Radon transforms on isotropic Riemannian symmetric spaces 61 2.3.1 Spherical integrals . . . 64

2.3.2 Inversion formulas on Riemannian symmetric spaces with constant sectional curvature . . . 68

2.3.3 Inversion formulas on isotropic Riemannian symmetric spaces of com- pact type . . . 77

2.3.4 Inversion formulas on isotropic Riemannian symmetric spaces of non- compact type . . . 80

2.4 Spherical transform and Fourier transform . . . 83

2.4.1 Spherical functions and spherical transform . . . 84

2.4.2 Fourier transform on Riemannian symmetric spaces of non-compact type . . . 86

xvii

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xviii Contents

3 Integral geometry on Lorentzian symmetric spaces 89

3.1 Pseudo-spherical integrals on the pseudo-Euclidean vector space . . . 90 3.2 Orbital integrals on semisimple pseudo-Riemannian symmetric spaces . . . . 91 3.3 Orbital integrals on Cahen-Wallach spaces . . . 93 3.4 Limit formulas on even-dimensional Lorentzian symmetric spaces with con-

stant sectional curvature . . . 95 3.5 Limit formulas on rank-one semisimple pseudo-Riemannian symmetric spaces101 3.6 Limit formulas on indecomposable Lorentzian symmetric spaces . . . 110 3.6.1 Limit formulas on Cahen-Wallach spaces . . . 111 3.6.2 Conclusion in the Lorentzian case . . . 116

4 Integral geometry on symplectic symmetric spaces 117

4.1 Orbital integrals on Ricci-type symplectic symmetric spaces . . . 118 4.1.1 Limit formulas on Ricci-type symplectic symmetric spaces . . . 121 4.2 Double brations in a symplectic framework . . . 124 4.2.1 Orbits of connected symplectic totally geodesic submanifolds . . . . 125 4.2.2 Orbits of connected Lagrangian totally geodesic submanifolds . . . . 130 4.3 Symplectic totally geodesic Radon transforms on P

n

( C ) and H

n

( C ) . . . 135 4.3.1 Inversion formulas on complex projective spaces . . . 136 4.3.2 Inversion formulas on complex hyperbolic spaces . . . 139

Conclusion and prospects 143

A Invariant dierential operators 149

A.1 Invariant dierential operators on reductive homogeneous spaces . . . 149 A.2 Invariant dierential operators on pseudo-Riemannian symmetric spaces . . 150 A.2.1 On quasi-isotropic pseudo-Riemannian symmetric spaces . . . 152 A.2.2 On Cahen-Wallach spaces . . . 159

B Riemann-Liouville integrals and Riesz potentials 165

B.1 Riemann-Liouville integrals . . . 165 B.2 Riesz potentials . . . 166

Bibliography 169

Références

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