1 Heisenberg Groups. 11 1.1 The Heisenberg Group H(n,m)R . 11 1.2 Theta Functions . 15 1.3 Induced Representations . 25 1.4 The Schrodinger Representation . 27 1.5 Fock Representations . 32 1.6 Lattice Representations . 49 1.7 Coadjoint Orbits . 59 1.8 Hermite Operators . 66 1.9 Harmonic Analysis on the Quotient Space of H(n,m)R . 75 2 Theta Functions and the Weil Representation . 83 2.1 The Symplectic Group . 83 2.2 Some Geometry on Siegel Space . 92 2.3 The Weil Representation . 101 2.4 Covariant Maps for the Weil Representation . 110 2.5 Theta Series with Quadratic Forms . 122 2.6 Theta Series in Spherical Harmonics . 129 2.7 Relation between Theta Series and the Weil Representation  . 139 2.8 Spectral Theory on the Abelian Variety  . 145 References . 153

Department of Mathematics Inha University Incheon 402-751 Republic of Korea

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