Contents ........................................................ v
Acknowledgements ............................................. viii
Introduction .................................................... 1
Notation and Terminology ........................................ 5
Chapter 1 Estimates on SLn Parabolics
1 The hermitian norm on SLn and Siegel sets .................... 9
2 Volume and lattice point estimates .......................... 16
3 Estimates of A-projections .................................. 20
4 Standard reduced parabolics ................................. 23
5 Characters on parabolics .................................... 29
6 Estimates of Aρ-projections ................................. 33
7 Parabolic integral formulas ................................. 35
Chapter 2 Eisenstein Series
1 The character Eisenstein series ............................. 41
2 Twists of character Eisenstein series ....................... 46
3 Two-character Eisenstein series ............................. 51
4 The Gauss space ............................................. 53
5 The parabolic Eisenstein integration formula ................ 58
Chapter 3 Adjointness and Inversion Relations
1 Adjointness formulas and F-cuspidality ...................... 61
2 Adjointness and initial conditions formulas ................. 70
3 P-cuspidality and heat Eisenstein series .................... 72
4 The family of anticuspidal operators JP,Г,ε,t ................. 80
Chapter 4 Applications of the Heat Equation
1 Parabolics and the (a, n)-characters ........................ 85
2 Direct image of Casimir on parabolics ....................... 87
3 The differential equation for ЈP,Г,K and EP,K# ................ 91
4 Convolution of TrГ(Kx) and the Eisenstein
series ...................................................... 95
5 The P-anticuspidal semigroup property ....................... 96
6 The P-anticuspidal operator JpPГ,Pρ and the conjectured
spectral expansion ......................................... 100
7 Onward ..................................................... 104
Chapter Appendix. The Heat Kernel
1 Dodziuk's uniqueness theorem ............................... 107
2 The fundamental solution and the heat kernel ............... 109
3 Properties of the heat kernel .............................. 113
4 Compact manifolds .......................................... 114
Bibliography .................................................. 119
Index ......................................................... 123
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