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Elements of the representation theory of the Jacobi group. (English) Zbl 0931.11013

Progress in Mathematics (Boston, Mass.). 163. Basel: Birkhäuser. xiii, 213 p. (1998).
The book under review collects and regroups results on the representation theory of the Jacobi group of lowest degree mostly due to R. Berndt and his coworkers J. Homrighausen and R. Schmidt. The book is very well written and gives an up to date collection of the results known. It will be quite useful for everyone working in the field.
The first chapter introduces the Jacobi group  \(G^J\), a semi-direct product of \(SL(2)\) with a Heisenberg group, and describes different possible coordinates on the group, the Haar measure, the Lie algebra, as well as, over the reals, a (generalized) Iwasawa decomposition and the associated homogeneous space.
The second chapter starts with basic representation theory of the Jacobi group over a local field of characteristic zero or the ring of adeles of a number field. The essential ingredient is the Stone-von Neumann theorem, which describes all unitary representations of the Heisenberg group with non-trivial central character in terms of the so called Schrödinger representation. This theorem is listed without proof but a complete description of the Schrödinger representation via induction is given. If one assumes that the central character is trivial this theorem together with Mackey’s method, which is described in some detail, already gives a list of all irreducible representations in terms of certain representations of the factors (Theorem 2.4.2). If the central character is non-trivial, the more interesting case, one needs to combine the Schrödinger representation of the Heisenberg group with the Weil representation of \(SL(2)\) into the so called Schrödinger-Weil representation of the semi-direct product of the Heisenberg and the metaplectic group. Tensoring this representation with truly metaplectic representations of the metaplectic group, one gets an isomorphism between the two sets of representations involved (Theorem 2.6.1-3). Strictly speaking, Chapter 2 deals with this only in the local case, the global case being moved to Chapter 7.
Chapters 3 and 4 treat exclusively the theory over the real numbers. Compared to the general approach described before the authors describe an infinitesimal approach. Using results of J. L. Waldspurger [J. Math. Pures Appl., IX. Sér. 60, 375-484 (1981; Zbl 0468.10014)] on the metaplectic group the first main theorem gives a complete description of the irreducible representations of the complexified Lie algebra in terms of principal and discrete series (Theorem 3.1.9). Then models and unitarizability are addressed leading to a complete description of the irreducible unitary representations up to infinitesimal equivalence (Theorem 3.2.12 and Table 3.1). Several realizations are constructed by induction from the “Borel” and the “maximal compact” subgroup, leading also to the canonical factor of automorphy. Finally differential operators and Whittaker models are discussed.
Chapter 4 describes the spectral decomposition of \(L^2(\Gamma^J G^J)\) due to R. Berndt [Abh. Math. Sem. Univ. Hamburg 65, 301-305 (1995; Zbl 0849.11040)] and a joint paper of R. Berndt and S. Böcherer [Math. Z. 204, 13-44 (1990; Zbl 0695.10024)]. First it is shown that the representation on the subspace of cusp forms decomposes into a direct sum of irreducible subspaces (Theorem 4.3.1) and that the multiplicity is equal to the dimension of a certain space of automorphic forms (Theorem 4.3.4). The discussion of the continuous part is restricted to the “theta transform” and the functional equation of the Eisenstein series.
The theory in the non-archimedean case (in characteristic zero) starts with Chapter 5. After Jacquet-Langlands the usual approach is to classify the admissible irreducible representations first and look for unitarizable ones among these next, and the authors follow this path. The main results are Theorem 5.8.3 classifying the irreducible admissible representations and Theorem 5.9.2 describing the pre-unitary ones among these. To achieve this Whittaker models, induced representations and intertwining operators are discussed. This leads to a complete description in terms of principal and special series and supercuspidal ones as in Jacquet-Langlands with some specialties coming in, e.g. the positive and negative Weil representations. Here J. L. Waldspurger’s results from the above mentioned paper are essential again.
Chapter 6 discusses spherical representations, which are quite important for global considerations, just as in the \(GL(2)\) case. It is shown (Theorem 6.3.10) that the only spherical representations among those given in the classification are the principal series and the positive Weil representation. This is only true under a certain assumption on the (fixed) central character, namely in the so called good case, which is the only one considered. The most important tool is again the Hecke algebra, more precisely the Hecke algebra with central character, which appeared already in P. Cartier’s Corvallis paper. This is essential for only in this case the Hecke algebra has a nice structure. The theory is developed up to the Satake isomorphism (Theorem 6.2.8) and a rationality theorem (Corollary 6.2.7) via coset calculations. This is related to results of A. Murase [J. Reine Angew. Math. 401, 122-156 (1989; Zbl 0671.10023)] and the reviewer [Result. Math. 31, 75-94 (1997; Zbl 0880.11044)] using different methods. Then spherical Whittaker functions, i.e. the spherical vectors of the spherical representations in the Whittaker model, and their zeta integrals are calculated. This leads to local \(L\)-factors which are important in the global theory. To conclude the authors describe the relation of their Hecke algebra with the classical Hecke operators used by M. Eichler and D. Zagier [The theory of Jacobi forms, Prog. Math. 55, Birkhäuser (1985; Zbl 0554.10018)].
The final Chapter 7 considers the Jacobi group over the adele ring of some number field. Following the procedure in Chapter 2 the (global) Stone-von Neumann theorem is used to construct a (global) Schrödinger-Weil representation. Thus the results from Chapter 2 carry over to the global setting, but the goal here is of course different, since one wants to know how automorphic representations correspond. To achieve this the authors explicitly construct, loosely speaking, an isomorphism between the tensor product of the right regular representation of the metaplectic group on the usual \(L^2\)-space and the Schrödinger-Weil representation (as a representation of the Jacobi group) with the right regular representation of the Jacobi group. This has important applications (Corollaries 7.3.4-6), e.g. automorphic representations do indeed correspond under the abstract isomorphism. The final two sections deal with the field of rational numbers. The goal is to prove that the subrepresentation of the \(L^2\)-space generated by a cuspidal Hecke eigenform lifted to the adele group in the usual way, is irreducible and to describe (almost) all local components in the tensor product decomposition in terms of the weight, the index and the eigenvalues (Theorem 7.5.5) of the eigenform. The basic ingredient is a strong multiplicity-one result for the Jacobi group (Theorem 7.5.4), which follows essentially from the corresponding theorem for the metaplectic group due (again) to J. L. Waldspurger [Forum Math. 3, 219-307 (1991; Zbl 0724.11024)].

MSC:

11F50 Jacobi forms
11-02 Research exposition (monographs, survey articles) pertaining to number theory
22E50 Representations of Lie and linear algebraic groups over local fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F55 Other groups and their modular and automorphic forms (several variables)
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