## Elements of the Representation Theory of the Jacobi GroupThe Jacobi group is a semidirect product of a symplectic group with a Heisenberg group. It is an important example for a non-reductive group and sets the frame within which to treat theta functions as well as elliptic functions - in particular, the universal elliptic curve. This text gathers for the first time material from the representation theory of this group in both local (archimedean and non-archimedean) cases and in the global number field case. Via a bridge to Waldspurger's theory for the metaplectic group, complete classification theorems for irreducible representations are obtained. Further topics include differential operators, Whittaker models, Hecke operators, spherical representations and theta functions. The global theory is aimed at the correspondence between automorphic representations and Jacobi forms. This volume is thus a complement to the seminal book on Jacobi forms by M. Eichler and D. Zagier. Incorporating results of the authors' original research, this exposition is meant for researchers and graduate students interested in algebraic groups and number theory, in particular, modular and automorphic forms. |

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### Contents

The Jacobi Group | 1 |

12 G³ as an algebraic group | 3 |

13 The Lie algebra of G³ | 7 |

14 G³ over the reals | 9 |

Basic Representation Theory of the Jacobi Group | 15 |

22 The Schrodinger representation | 18 |

23 Mackeys method for semidirect products | 21 |

24 Representations of G³ with trivial central character | 22 |

53 Representations of the metaplectic group | 112 |

54 Induced representations | 115 |

55 Supercuspidal representations | 119 |

56 Intertwining operators | 121 |

57 Whittaker models | 127 |

58 Summary and Classification | 132 |

59 Unitary representations | 135 |

Spherical Representations | 137 |

25 The SchrodingerWeil representation | 24 |

26 Representations of GJ with nontrivial central character | 28 |

Local Representations The Real Case | 31 |

31 Representations of g𝐜J | 32 |

32 Models for infinitesimal representations and unitarizability | 39 |

33 Representations induced from BJ | 48 |

34 Representations induced from KJ and the automorphic factor | 51 |

35 Differential operators on X H x C | 59 |

36 Representations induced from NJ and Whittaker models | 63 |

The Space L2TJGJR and its Decomposition | 75 |

41 Jacobi forms and more general automorphic forms | 76 |

42 The cusp condition for GJR | 83 |

43 The discrete part and the duality theorem | 88 |

44 The continuous part | 94 |

Local Representations The padic Case | 105 |

52 Whittaker models for the SchrodingerWeil representation | 107 |

61 The Hecke algebra of the Jacobi group | 138 |

62 Structure of the Hecke algebra in the good case | 140 |

63 Spherical representations in the good case | 148 |

64 Spherical Whittaker functions | 153 |

65 Local factors and the spherical dual | 163 |

66 The EichlerZagier operators | 167 |

Global Considerations | 173 |

71 Adelization of G³ | 174 |

72 The global Schrodinger Weil representation | 176 |

73 Automorphic representations | 179 |

74 Lifting of Jacobi forms | 183 |

75 The representation corresponding to a Jacobi form | 193 |

Bibliography | 201 |

207 | |

211 | |

### Other editions - View all

Elements of the Representation Theory of the Jacobi Group Rolf Berndt,Ralf Schmidt Limited preview - 2012 |

Elements of the Representation Theory of the Jacobi Group Rolf Berndt,Ralf Schmidt Limited preview - 2013 |

Elements of the Representation Theory of the Jacobi Group Rolf Berndt,Ralf Schmidt Limited preview - 2012 |

### Common terms and phrases

adelic admissible representation Assume automorphic factor automorphic forms automorphic representations central character ipm cocycle compute Corollary corresponding coset cusp condition cuspidal decomposition defined definition denotes differential operators discrete series representation eigenform Eisenstein series element equation equivalent finite formula Fourier given GJ(A GJ(F GJ(k GJ(R global H(GJ Hecke algebra Hecke operators Heisenberg group Hence Hilbert space holomorphic induced representation infinitesimal integral invariant ip-m isomorphism Jacobi forms Jacobi group Jk,m Lemma Lie algebra lift lowest weight metaplectic group modular forms non-trivial central character number field polynomial positive Weil representation principal series representation Proof Proposition reductive groups Remark representation TT representations of GJ resp result right translation scalar Schrodinger representation Schrodinger-Weil representation Schwartz function semidirect product smooth representation spherical representations spherical vector Stone-von Neumann theorem subspace transformation trivial TT™w unique unitary representations unramified Waldspurger Weil representation Whittaker functional Whittaker models