Zeta Integrals, Schwartz Spaces and Local Functional Equations

Preface......Page 6
Contents......Page 7
  1.1.1 From Tate\'s Thesis to Godement–Jacquet Theory......Page 9
  1.1.2 The Doubling Method......Page 11
  1.1.3 Braverman–Kazhdan: L-Monoids......Page 12
  1.1.4 Integral Representations and Spherical Varieties......Page 13
  1.1.5 Other Integrals......Page 14
  1.2.1 More General Zeta Integrals......Page 15
  1.2.2 Local Functional Equation......Page 17
  1.2.3 L2-Theory......Page 18
  1.2.4 On the Global Case......Page 21
 1.3 Positive Results......Page 22
 1.4 Structure of This Book......Page 24
  1.5.1 Convex Geometry......Page 25
  1.5.3 Varieties......Page 26
  1.5.6 Vector Spaces......Page 27
  1.5.7 Representations......Page 28
 2.1 Review of Spherical Varieties......Page 29
 2.2 Boundary Degenerations......Page 33
 2.3 Cartan Decomposition......Page 36
 2.4 Geometric Data......Page 39
 3.1 Integration of Densities......Page 43
 3.2 Direct Integrals and L2-Spectral Decomposition......Page 46
 3.3 Gelfand–Kostyuchenko Method......Page 48
 3.4 Hypocontinuity and Barreled Spaces......Page 51
 4.1 Coefficients of Smooth Representations......Page 53
 4.2 The Group Case......Page 56
 4.3 Auxiliary Definitions......Page 58
 4.4 Schwartz Spaces: Desiderata......Page 61
 4.5 Model Transitions: The Local Functional Equation......Page 65
 4.6 Connection with L2 Theory......Page 68
 5.1 Cellular Decompositions......Page 73
 5.2 Smooth Asymptotics......Page 76
 5.3 Proof of Convergence......Page 80
 6.1 Fourier Transform of Half-Densities......Page 83
 6.2 Review of Prehomogeneous Vector Spaces......Page 87
 6.3 Local Functional Equation......Page 91
 6.4 Local Godement–Jacquet Integrals......Page 95
 7.1 Geometric Set-up......Page 101
 7.2 The Symplectic Case......Page 105
 7.3 Doubling Zeta Integrals......Page 111
 7.4 Relation to Reductive Monoids......Page 115
 7.5 Remarks on the General Case......Page 121
 8.1 Basic Vectors......Page 123
 8.2 Theta Distributions......Page 127
 8.3 Relation to Periods......Page 129
 8.4 Global Functional Equation and Poisson Formula......Page 135
References......Page 140
Index......Page 144