132 INDEX

Laplace–Beltrami operator, 2, 28

light cone, 11

Mehler kernel, 19

Meijer’s G-distribution, 89

Meijer’s G-function, 22, 23, 115

Meijer’s G-transform, 24

Mellin transform, 81

—, of Bessel function, 110

Mellin–Barnes type integral, 90, 94, 115

metaplectic group, 15

minimal representation, 2, 13, 16, 31

Minkowski space, 19

M¨ obius transform, 17

modified Bessel function

—, of the first kind, 107

—, of the third kind, 108

multiplicity free, 21

multiplicity-free decomposition, 81

N-picture, 17, 27, 120

oscillator representation, 15

Parseval type formula, 21

Plancherel formula

—, for Meijer’s G-transform, 24

—, FC , 18, 78

—, G-function, 61

Radon transform, 8, 79, 81

reciprocal formula, 18

—, for Meijer’s G-transform, 24

—, FC , 18, 78

—, G-function, 61

Riesz distribution, 105

Riesz potential, 82

Schr¨ odinger, 78

Schr¨ odinger model, 2, 15–17, 33, 120

Schwartz space, 6

Segal–Shale–Weil representation, 15

Siegel parabolic subgroup, 15

smooth vector, 7, 35

solution model, 20

spherical harmonics, 23, 113

spherical representation, 16

symplectic group, 15

symplectic Lie algebra, 9

tempered distribution, 17, 19, 63

theta correspondence, 16, 31

ultra-hyperbolic equation, 8, 16

uncertainty relation, 20

unitary inversion operator, 8, 20, 46, 77

wave front set, 14

Weil representation, 3, 9, 15, 33, 60, 78

Weyl algebra, 4, 37

Yamabe operator, 17, 28

Y -Bessel function, 108