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Representation theory and automorphic functions. Transl. from the Russian by K. A. Hirsch. Reprint. (English) Zbl 0718.11022

Boston, MA etc.: Academic Press, Inc. xviii, 426 p. $ 39.50 (1990).
This is a reprint of the 1969 translation – without changes except for two added pages of introduction giving a tiny summary of developments in the field since 1969. The first review was not very detailed and emphasized the theory of automorphic forms as a method to prove the Riemann hypothesis. I do not believe that this particular application has been made successfully yet. (Cf. the review of the Russian original of 1966 in Zbl 0138.07201, as to the English translation of 1969 see Zbl 0177.18003.)
Nevertheless this text is certainly a valuable one. It provides a wonderful introduction to the methods of representation theory as applied to the theory of automorphic forms and zeta functions, making the subject accessible to a beginning graduate student. The exposition is concrete and minimizes the prerequisites. It is independent of the earlier volumes on generalized functions. One can reduce the prerequisites even further by considering the case of \(\mathrm{SL}(2)\) over a finite field which appears in several footnotes. My only complaints concern the number of typos or mistranslations. But still there are probably more typos in my own books. And it is often true that the more “perfect” the book the less one can learn from it.
Let me give a brief summary for the non-experts. Certainly the experts have all used this book and need no introduction. The main topic of Chapter 1 is the representation theory of the special linear group \(G = \mathrm{SL}(2,{\mathbb{R}})\) of \(2\times 2\) matrices with real entries and determinant 1. For any discrete subgroup \(\Gamma\) of \(G\), such hat the quotient \(\Gamma\setminus G\) is compact, the Selberg trace formula is given. This is viewed as a formula involving on one side the multiplicities of the irreducible unitary representations of \(G\) contained in the induced representation \(T(g)\) from some representation \(\chi\) of \(\Gamma\). Here we assume \(\chi\) to be trivial. And the other side of the formula is a sum over conjugacy classes \(\gamma\) of \(\Gamma\). The terms in the sum involve orbital integrals over \(G_{\gamma}\setminus G\), where \(G_{\gamma}\) is the centralizer of \(\gamma\) in \(G\). Then the various series of representations of \(G\) are described – continuous and discrete. A duality theory is derived showing that the dimension of the space of analytic modular forms of a certain integral weight for \(\Gamma\) equals the multiplicity of the discrete series representation \(T_n(g)\) in \(T(g)\).
There is also an analogous result for Maass wave forms which are \(\Gamma\)-invariant eigenfunctions of the non-Euclidean Laplacian for the Poincaré upper half plane. Examples of \(\Gamma\)’s satisfying the hypotheses are given in an appendix. These are called quaternion groups, although the word “quaternion” appears almost everywhere in the discussion. Congruence subgroups of the modular group \(\mathrm{SL}(2,\mathbb{Z})\) are also discussed in the appendix. One example is \(\Gamma_n\) which is the group of determinant one integer \(2\times 2\) matrices which are congruent to a diagonal matrix modulo \(n\). Such congruence subgroups have quotients which are non compact but of finite volume. The chapter also presents the preliminaries for obtaining the Selberg trace formula for this case; i.e., the separation of the discrete part of the spectrum using the method of horospheres (or horocycles); i.e., the cuspidal part of the spectrum. Applications are given; e.g., to the derivation of the asymptotics of the representations occurring in the decomposition of \(T\). The trace formula for \(\mathrm{SL}(2,\mathbb{C})\) is sketched in an appendix.
Chapter 2 gives a nice quick development of the basic theory of locally compact fields such as the field \(\mathbb{Q}_p\) of \(p\)-adic numbers. This is the completion of the rationals with respect to the \(p\)-adic absolute value defined for a prime \(p\) by \(| x|_p = p^{-e}\), where \(p^e\) is the power of \(p\) contained in \(x\); i.e., \(x=p^e n/m\), where \(n\) and \(m\) are integers not divisible by \(p\). Besides providing a list of the irreducible unitary representations of \(G= \mathrm{SL}(2,\mathbb{Q}_p)\), this chapter gives a fascinating discussion of special functions such as gamma, beta, spherical and Bessel functions for \(p\)-adic groups. And footnotes give information on the analogs for finite fields which are Gauss sums, Kloosterman sums, etc. The construction of the discrete series representations of \(\mathrm{SL}(2,K)\), for \(p\)-adic and finite \(K\) have been of interest to many people; e.g., Shintani. Recently, we found applications of the finite Bessel functions to the study of adjacency operators of graphs associated to finite analogs of the Poincaré upper half plane [see N. Celniker et al., A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math. 143, 65–88 (1993; Zbl 0792.05065)]. This chapter ends with the Plancherel theorem on the decomposition of the regular representations of \(G\) into irreducibles. It can be viewed as an inversion formula for the Fourier transform on the \(p\)-adic group.
Chapter 3 introduces the adele ring \(\mathbb{A}\) for \(\mathbb{Q}\) via the character group of \(\mathbb{Q}\). It studies the Riemann zeta function as an adelic object. There is sometimes a typo in the chapter heading of part of Section 4, which should read adele group of unimodular matrices of order 2. You can find on pages 356–361, an adelic translation of the theory of automorphic forms for congruence subgroups \(\Gamma_n\). And an adelic formulation is given of the famous Ramanujan-Petersson conjecture on Fourier coefficients of holomorphic modular forms of even integral weight for congruence subgroups (with a misspelling of Petersson). This result has since been proved by Deligne. A. Lubotzky, R. Phillips and P. Sarnak [Combinatorica 8, 261–277 (1988; Zbl 0661.05035)] have used the Ramanujan-Petersson conjecture to show that certain graphs that they have constructed are Ramanujan and thus good expander graphs.
Most of Chapter 3 is devoted to proving the finiteness of the trace of the \(T_{\phi}\) on the cuspidal functions on \(G_{\mathbb{Q}}\setminus G_{\mathbb{A}}\). That is the functions whose integrals vanish over all horospheres. This result is established for more general linear algebraic group \(G\). It is the beginning of the work one must do in order to get a Selberg trace formula for such groups. There has been a vast amount of effort devoted to such results. See, for example, the papers of J. Arthur [Number theory, trace formulas and discrete groups, Oslo 1987, 11–27 (1989; Zbl 0671.10026) and R. P. Langlands, ibid., 125–155 (1989; Zbl 0671.10025)]. There are now many more references for these subjects, but few as charming as the book under review. I cannot resist listing a few more recent references for the non-adelic theory. The original work on the trace formula can be found in A. Selberg [Collected Papers, Vol. I (1989; Zbl 0675.10001)]. If you want to see more about the classical theory of modular forms, there are many references listed in my book “Harmonic analysis on symmetric spaces and applications”, vol. I (1985; Zbl 0574.10029) and II (1988; Zbl 0668.10033). See also D. Hejhal et al. (eds.), The Selberg trace formula and related topics, Contemp. Math. 53 (1986; Zbl 0583.00006), P. Sarnak, Some applications of modular forms, D. I. Wallace (Dorothy Andreoli), The Selberg trace formula for \(\mathrm{SL}(3,\mathbb{Z})\setminus \mathrm{SL}(3,\mathbb{R})/\mathrm{SO}(3,\mathbb{R})\) (preprint).
Modern physics and chemistry make great use of representations of real and complex Lie groups. Even \(p\)-adic groups and symmetric spaces appear in applied mathematics. For example, there are applications of the \(p\)-adic numbers to modeling fish populations in a river [see S. Sawyer, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 42, 279–292 (1978; Zbl 0362.60075)]. Are there any such applications of the adelic theory? I don’t know of any. The theory was invented by Chevalley and others to get around problems with the analytic theory of algebraic numbers when the class number is greater than one. H. M. Stark has described a different way of dealing with this problem in which vectors with only a finite number of components are considered [Number theory, Proc. Conf., Montreal 1985, Conf. Proc. 7, 421–455 (1987; Zbl 0624.10016)]. The adeles also arise naturally in the theory of quadratic forms. The concept of adele group was developed by Ono, Weil, Tamagawa and others and provides a way of formulating Siegel’s main theorem on quadratic forms. At this point in time, the majority of the work on automorphic forms is adelic. Langlands philosophy rules the subject. It is an imposing edifice. The present book provides a door into the subject.

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
22-02 Research exposition (monographs, survey articles) pertaining to topological groups
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
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