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This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed.
Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.
Introduction 1
0.1 The trace formula as a Lefschetz formula 1
0.2 A short history of the hypoelliptic Laplacian 2
0.3 The hypoelliptic Laplacian on a symmetric space 3
0.4 The hypoelliptic Laplacian and its heat kernel 4
0.5 Elliptic and hypoelliptic orbital integrals 5
0.6 The limit as b ? 0 5
0.7 The limit as b ? +∞: an explicit formula for the orbital integrals 6
0.8 The analysis of the hypoelliptic orbital integrals 6
0.9 The heat kernel for bounded b and the Malliavin calculus 7
0.10 The heat kernel for large b, Toponogov, and local index 9
0.11 The hypoelliptic Laplacian and the wave equation 9
0.12 The organization of the book 9
1 Clifford and Heisenberg algebras 12
1.1 The Clifford algebra of a real vector space 12
1.2 The Clifford algebra of V + V* 14
1.3 The Heisenberg algebra 15
1.4 The Heisenberg algebra of V + V* 17
1.5 The Clifford-Heisenberg algebra of V + V* 18
1.6 The Clifford-Heisenberg algebra of V + V* when V is Euclidean 19
2 The hypoelliptic Laplacian on X = G/K 22
2.1 A pair (G, K) 23
2.2 The flat connection on TX + N 25
2.3 The Clifford algebras of g 25
2.4 The flat connections on Λ (T* X + N*) 25
2.5 The Casimir operator 27
2.6 The form κg 28
2.7 The Dirac operator of Kostant 30
2.8 The Clifford-Heisenberg algebra of g + g* 32
2.9 The operator Db 33
2.10 The compression of the operator Db 34
2.11 A formula for D2b 34
2.12 The action of Db on quotients by K 35
2.13 The operators LX and LXb 39
2.14 The scaling of the form B 41
2.15 The Bianchi identity 41
2.16 A fundamental identity 41
2.17 The canonical vector fields on X 45
2.18 Lie derivatives and the operator LXb 46
3 The displacement function and the return map 48
3.1 Convexity, the displacement function, and its critical set 49
3.2 The norm of the canonical vector fields 50
3.3 The subset X (γ) as a symmetric space 54
3.4 The normal coordinate system on X based at X (γ) 57
3.5 The return map along the minimizing geodesies in X (γ) 62
3.6 The return map on X 64
3.7 The connection form in the parallel transport trivialization 65
3.8 Distances and pseudodistances on X and X 67
3.9 The pseudodistance and Toponogov's theorem 68
3.10 The flat bundle (TX + N) (γ) 75
4 Elliptic and hypoelliptic orbital integrals 76
4.1 An algebra of invariant kernels on X 77
4.2 Orbital integrals 78
4.3 Infinite dimensional orbital integrals 81
4.4 The orbital integrals for the elliptic heat kernel of X 84
4.5 The orbital supertraces for the hypoelliptic heat kernel 84
4.6 A fundamental equality 85
4.7 Another approach to the orbital integrals 86
4.8 The locally symmetric space Z 87
5 Evaluation of supertraces for a model operator 92
5.1 The operator <$$> and the function <$$> 92
5.2 A conjugate operator 94
5.3 An evaluation of certain infinite dimensional traces 95
5.4 Some formulas of linear algebra 103
5.5 A formula for <$$> 110
6 A formula for semisimple orbital integrals 113
6.1 Orbital integrals for the heat kernel 113
6.2 A formula for general orbital integrals 114
6.3 The orbital integrals for the wave operator 116
7 An application to local index theory 120
7.1 Characteristic forms on X 120
7.2 The vector bundle of spinors on X and the Dirac operator 122
7.3 The McKean-Singer formula on Z 124
7.4 Orbital integrals and the index theorem 125
7.5 A proof of (7.4.4) 126
7.6 The case of complex symmetric spaces 130
7.7 The case of an elliptic element 131
7.8 The de Rham-Hodge operator 134
7.9 The integrand of de Rham torsion 136
8 The case where <$$> 138
8.1 The case where G = K 138
8.1 Thecase <$$> 139
8.3 The case where G = SL2 (R) 140
9 A proof of the main identity 142
9.1 Estimates on the heat kernel <$$> away from <$$> 142
9.2 A rescaling on the coordinates (f, Y) 145
9.3 A conjugation of the Clifford variables 147
9.4 The norm of α 150
9.5 A conjugation of the hypoelliptic Laplacian 150
9.6 The limit of the rescaled heat kernel 152
9.7 A proof of Theorem 6.1.1 153
9.8 A translation on the variable YTX 153
9.9 A coordinate system and a trivialization of the vector bundles 156
9.10 The asymptotics of the operator <$$> 158
9.11 A proof of Theorem 9.6.1 159
10 The action functional and the harmonic oscillator 161
10.1 A variational problem 162
10.2 The Pontryagin maximum principle 164
10.3 The variational problem on an Euclidean vector space 166
10.4 Mehler's formula 173
10.5 The hypoelliptic heat kernel on an Euclidean vector space 175
10.6 Orbital integrals on an Euclidean vector space 177
10.7 Some computations involving Mehler's formula 182
10.8 The probabilistic interpretation of the harmonic oscillator 183
11 The analysis of the hypoelliptic Laplacian 187
11.1 The scalar operators <$$> on X 188
11.2 The Littlewood-Paley decomposition along the fibres TX 189
11.3 The Littlewood-Paley decomposition on X 192
11.4 The Littlewood Paley decomposition on X 193
11.5 The heat kernels for <$$> 201
11.6 The scalar hypoelliptic operators on X 205
11.7 The scalar hypoelliptic operator on X with a quartic term 206
11.8 The heat kernel associated with the operator <$$> 210
12 Rough estimates on the scalar heat kernel 212
12.1 The Malliavin calculus for the Brownian motion on X 214
12.2 The probabilistic construction of exp (<$$>) over X 217
12.3 The operator <$$> and the wave equation 219
12.4 The Malliavin calculus for the operator <$$> 222
12.5 The tangent variational problem and integration by parts 223
12.6 A uniform control of the integration by parts formula as 6 → 0 226|2 12.7 Uniform rough estimates on <$$> for bounded b 228
12.8 The limit as b → 4 230
12.9 The rough estimates as b → +∞ 237
12.10 The heat kernel <$$> 241
12.11 The heat kernel <$$> 244
13 Refined estimates on the scalar heat kernel for bounded b 248
13.1 The Hessian of the distance function 248
13.2 Bounds on the scalar heat kernel on X for bounded b 251
13.3 Bounds on the scalar heat kernel on X for bounded b 260
14 The heat kernel <$$> for bounded b 262
14.1 A probabilistic construction of exp (<$$>) 263
14.2 The operator <$$> and the wave equation 263
14.3 Changing Y into - Y 264
14.4 A probabilistic construction of exp (<$$>) 265
14.5 Estimating V 266
14.6 Estimating W 267
14 7 A proof of (4.5.3) when E is trivial 268
14.8 A proof of the estimate (4.5.3) in the general case 270
14.9 Rough estimates on the derivatives of <$$> for bounded b 274
14.10 The behavior of V. as b → 0 280
14.11 The limit of <$$> as b → 0 287
15 The heat kernel <$$> for b large 290
15.1 Uniform estimates on the kernel <$$> over X 291
15 2 The deviation from the geodesic flow for large b 292
15.3 The scalar heat kernel on X away from <$$> 294
15.4 Gaussian estimates for <$$> near <$$> 299
15 5 The scalar heat kernel on X away from <$$> 299
15.6 Estimates on the scalar heat kernel on X near <$$> 306
15.7 A proof of Theorem 9.1.1 310
15.8 A proof of Theorem 9.1.3 311
15.9 A proof of Theorem 9.5.6 312
15.10 A proof of Theorem 9.11.1 313
Bibliography 317
Subject Index 323
Index of Notation 325
Overview
This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir ...