Bibliography

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4 Index Adjointness formulas 102, 103 Bengston Bessel function Bessel function Bessel-Fourier series 105 Chains of matrices 134 Changing variables 152 Character 50 Completed Lambda function 100 Convergence of Eisenstein series 129 d 50 D 148 Decomposition of Haar measure 25 Determinant character 50 Dual lattice 97 EF HZ 141 Eigenfunction of Hecke zeta operator 140 Eigenvalue 89, 118, 119 Eisenstein series 107, 128, , , , 157, 160, 162 Eisenstein trace 108 Epstein zeta function Equivalent chains of matrices 137 Estimate of Lambda function 107 First order Iwasawa decomposition 5, 6 Fourier series Fourier transform 70, 95 Full Iwasawa coordinates 125 Fun conditions 7 Functional equation of Eisenstein 157, 160 Functional equation of theta 98, 149 Fundamental Domain 1, 6 Gamma function Γ n 55 Gamma integral 55, 118 Gamma kernel 88 Gamma point pair invariant 88 Gamma transform 57 Grenier fundamental domain 6, 17 Haar measure 25 Hecke zeta operator 140 Incomplete gamma integral 101 Inductive coordinates 14 Integral matrices 134 Invariant differential operators 90 Invariant differential operators and polynomials on A 90 Invariant polynomials 75, 91 Iwasawa coordinates 126 Iwasawa decomposition 2, 16 Jacobian 31 K-Bessel function 59 Λ-function Lie algebra generators 84 Lower Bengston function 73 Maass Selberg generators 78

5 168 Index Maass Selberg operators 80 Measure of fundamental domain 47 Measure on SPos 36 Mellin transform 55, 70 Metric 121, 122 Minkowski fundamental domain 7 Minkowski Measure of fundamental domain 47 Minkowski-Hlawka 44 Newton polynomials 77 Non-singular theta series 111 Normal decomposition 93 Normalized primitive chain 140 Parabolic subgroup 132 Partial determinant character 140 Partial Iwasawa decomposition 15, 114 Poisson formula 97 Polar coordinates 32 Polar Haar measure 33 Polynomial expression 81 Primitive chain 139 Primitive Eisenstein series 107, 128, 143 Projection on A 92 Radius of discreteness 125 Regularizing differential operator 122, 148 Reversing matrix 51 Riemann zeta fudge factor 141 Selberg Eisenstein series 143 Selberg power function 143 Siegel set 20, 23 Siegel s formula 41 Standard coordinates 15 Strict fundamental domain 1 Subdeterminants 53 Theta series 102, 111 Trace 149 Trace scalar product 97 Transpose of differential operator 87 Triangular coordinates 26 Triangularization 135, 138 Tubular neighborhood 92 Twists of theta series 99 Unipotent trace 108 Upper Bengston function 71 Weight of polynomial 83 Weyl group 75 Xi function