The Weil representation, Maslov index and theta series.

*(English)*Zbl 0444.22005
Progress in Mathematics, 6. Boston - Basel - Stuttgart: Birkhäuser. viii, 337 p. SFr. 30.00 (1980).

This book is divided into two parts. The first, written jointly by the two authors, gives an account of the Maslov index from the point of view of representation theory. The general idea is that a symplectic vector space \(V_n\) is naturally associated with the Heisenberg group \(N_n\); that the Heisenberg group has (up to scale factors) only one interesting representation; that symplectic automorphisms of \(V_n\) lead to a projective representation of \(\mathrm{Sp}(n,\mathbb R)\), the Shale-Weil (or oscillator) representation; and that the cocycle of this representation is closely related to the Maslov index. A by-product of this approach is that one can define the Maslov index in the \(p\)-adic case.

The second part is entirely the work of the second author. It starts with an account of modular forms on the Siegel upper half-plane. The general idea is that modular forms correspond to highest weight vectors in projective representation of \(\mathrm{Sp}(n,\mathbb R)\), and that these vectors are easily found by using the oscillator representation. This idea is extended in later sections to give modular forms corresponding to discrete subgroups of orthogonal groups; here, one uses results of Rallis-Schiffmann and Howe to imbed representations of orthogonal groups in the oscillator representation. One consequence is a reinterpretation of some results of Hecke and Zagier. The last few sections use this same general approach to obtain results on correspondences among modular forms.

The second part is entirely the work of the second author. It starts with an account of modular forms on the Siegel upper half-plane. The general idea is that modular forms correspond to highest weight vectors in projective representation of \(\mathrm{Sp}(n,\mathbb R)\), and that these vectors are easily found by using the oscillator representation. This idea is extended in later sections to give modular forms corresponding to discrete subgroups of orthogonal groups; here, one uses results of Rallis-Schiffmann and Howe to imbed representations of orthogonal groups in the oscillator representation. One consequence is a reinterpretation of some results of Hecke and Zagier. The last few sections use this same general approach to obtain results on correspondences among modular forms.

Reviewer: Lawrence J. Corwin

##### MSC:

22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

53D12 | Lagrangian submanifolds; Maslov index |

11F27 | Theta series; Weil representation; theta correspondences |

11F37 | Forms of half-integer weight; nonholomorphic modular forms |

11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |

22E40 | Discrete subgroups of Lie groups |

11F32 | Modular correspondences, etc. |