Elements of the Representation Theory of the Jacobi Group

Front Cover
Springer Science & Business Media, 1998 - Mathematics - 213 pages
0 Reviews
The 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.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

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
Index of Notations
207
Index
211
Copyright

Other editions - View all

Common terms and phrases

References to this book

All Book Search results »

About the author (1998)

Berndt, University of Hamburg, Germany.

Schmidt, University of Hamburg, Germany.

Bibliographic information