The Selberg-Arthur trace formula. Based on lectures by James Arthur
There are at least two approaches to the trace formula. Approach \(A\) involves automorphic representations of adele groups. Approach \(C\) involves the spectral theory of the Laplace and other \(G\)-invariant differential operators on a symmetric space \(G/K\) and the classical theory of automorphic forms. The present monograph is clearly type \(A\). It comes from 1987 lectures of J. Arthur. Let's consider a few examples. Example 1. Finite Groups [from p. 18 of the monograph under review]. Suppose that \(G\) is a finite group. Then a (finite-dimensional) unitary representation \(\pi\) of \(G\) is a group homomorphism \(\pi: G\to U(d_ \pi)\). Here \(U(n)\) is the unitary group of \(n\times n\) complex matrices \(g\) such that \(g^* g=I\), where \(g^*\) denotes the transpose conjugate of \(g\). The unitary representation \(\pi\) is irreducible if it is not uniformly block diagonalizable. Two representations are equivalent if one can be obtained from the other by uniform change of basis. Let \(\widehat G\) be a complete set of inequivalent irreducible unitary representations of \(G\). If \(f: G\to\mathbb{C}\), then the Fourier transform is \(\widehat f(\pi)=\sum_{g\in G} f(g) \pi(g)\). Note that \(\widehat f(\pi)\) is a \(d_ \pi\times d_ \pi\) matrix. Next let's discuss the Selberg trace formula for a finite group \(G\) with subgroup \(H\). Suppose for simplicity that \(\omega\) is a one-dimensional representation of \(H\). Let \(\rho\) be the induced representation of \(G\); i.e. \(\rho=\text{Ind}^ G_ H \omega\), acting on the space \[ V_ \omega=\bigl\{\phi: G\to\mathbb{C}\mid\phi(hx)=\omega(h)\phi(x),\;\text{for all }x\in G,\;h\in H\bigr\} \] by the right action \(\rho(g)\phi(x)=\phi(xg)\), for \(x,g\in G\). We can extend the definition of the Fourier transform above to arbitrary representations, not just irreducible ones. Then we find that for \(\phi\in V_ \omega\) we have \[ \bigl[\widehat f(\rho)\phi\bigr](x)=\sum_{y\in G} f(y) \phi(xy)=\sum_{y\in H\backslash G}\sum_{h\in H} f(x^{-1} hy) \omega(h) \phi(y). \] This means that the trace of the operator on \(V_ \omega\) is \(\text{Tr}\bigl(\widehat f(\rho)\bigr)=\sum_{x\in H\backslash G}\sum_{h\in H} f(x^{-1}hx) \omega(h)\). On the other hand, the representation \(\rho\) is a direct sum of integer multiples \(m(\pi,\rho)=m_ G(\pi,\rho)\) of \(\pi\in\widehat G\). So we find another formula for the trace which gives the pre-trace formula: \[ \sum_{\pi\in\widehat G} m(\pi,\rho) \text{Tr}\bigl(\widehat f(\pi)\bigr)=\sum_{x\in H\backslash G}\sum_{h\in H} f(x^{-1} hx) \omega(h). \] For example, setting \(f(x)=\text{Tr }\overline{\tau(x)}\), for \(\tau\in\widehat G\), we obtain the Frobenius reciprocity law \[ m_ G(\tau,\text{Ind}^ G_ H\omega)=m_ H(\omega,\text{Res}^ G_ H\tau),\quad\text{where }\text{Res}^ G_ H\tau=\text{restriction of }\tau\in\widehat G\text{ to } H. \] One can also deduce the Frobenius formula for the character of the induced representation. The trace formula can be seen to specialize to Poisson's summation when \(G\) is Abelian. Selberg rewrote the right hand side of this formula in terms of conjugacy classes \(\{h\}\) in \(H\): \[ \sum_{\{h\}}\omega(h){| G_ h|\over | H_ h|}\sum_{x\in G_ h\backslash G} f(x^{-1} hx), \] where \(G_ h\) is the centralizer of \(h\) in \(G\) and \(H_ h\) the centralizer of \(h\) in \(H\). The inner sum is an analog of an orbital integral in the real case. It is interesting to carry this out explicitly for \(G=GL(2,\mathbb{F}_ q)\) and \(H=GL(2,\mathbb{F}_ p)\), where \(\mathbb{F}_ q\) is the finite field with \(q\) elements and \(q=p^ r\). See \textit{J. Angel}, \textit{S. Poulos}, \textit{C. Trimble}, \textit{A. Terras} and \textit{E. Velasquez}, Spherical functions and transforms on finite upper half planes: eigenvalues of the Laplacian, chaotic pictures, uncertainty, and traces, (preprint). Here one can see that the terms are quite analogous to those that occur over the real field. In fact, let \(K\) be the subgroup of \(G\) consisting of matrices \(\begin{pmatrix} a &b\delta\\ b &a\end{pmatrix}\), for \(\delta\) a non-square in \(\mathbb{F}_ q\). This subgroup \(K\) is analogous to the orthogonal group. Then \(G/K\) can be viewed as an analog of the Poincaré upper half plane (or a 2-fold cover thereof). It is possible to associate graphs to \(G/K\) which can be shown to be Ramanujan and thus expander graphs [see \textit{J. Angel} et al., Contemp. Math. 138, 1-26 (1992)]. Example 2. Selberg's Trace Formula for \(SL(2,\mathbb{Z})\). Here we take \(G=SL(2,\mathbb{R})\), \(K=SO(2,\mathbb{R})\), \(\Gamma=SL(2,\mathbb{Z})\) -- or some congruence subgroup. [See \textit{A. Selberg}, Collected Papers. I (Springer 1989; Zbl 0675.10001), pp. 626-674.] This is the case that \(G/K\) can be identified with the Poincaré upper half plane \(H\) and \(\Gamma\backslash H\) is a non-compact Riemann surface. Selberg used the spectral theory of Hilbert-Schmidt operators with finite trace to obtain the Selberg trace formula. There is no mention of representation theory. Instead of \(\text{Tr}\bigl(\widehat f(\pi)\bigr)\), Selberg wrote down an integral transform, which can be viewed as the spherical or Helgason transform of the \(K\)-bi-invariant function \(f\). [See \textit{A. Terras}, Harmonic analysis on symmetric spaces and applications. I (Springer 1985; Zbl 0574.10029), p. 262.] The continuous spectrum of \(\Delta=y^ 2\bigl(\partial^ 2/\partial x^ 2+\partial^ 2/\partial y^ 2\bigr)\) on \(\Gamma\backslash G/K\) is indexed by \(s\in\mathbb{C}\) via \(\Delta y^ s=s(s-1)y^ s\). Summing over \(\Gamma\) produces Eisenstein series \(E_ S(z)\), \(z\in H\), -- convergent for \(\text{Re} s>1\) and then continued to all \(s\in\mathbb{C}\) as a meromorphic function. This is a Maass wave form. There are also cusp forms which approach 0 at the cusps of the fundamental domain. The cusp forms \(v_ n\), \(n=1,2,\dots\), are more mysterious. They give the discrete spectrum of \(\Delta\) on \(\Gamma\backslash H\). But we do not appear to have an explicit construction for them. One last element of the spectrum is \(v_ 0=\) constant. Start with the Roelcke-Selberg spectral decomposition of \(\Delta\) on \(L^ 2(\Gamma\backslash G/K)\): \[ \phi(z)=\sum^ \infty_{n=0} (\phi,v_ n)v_ n(z)+{1\over 4\pi i}\int_{\text{Re} s=1/2} (\phi,E_ S) E_ S(z) ds. \] Consider the operator given by convolution with \(f\in C^ \infty_ 0(K\backslash G/K)\) and attempt to obtain a formula for the trace of this operator on \(L^ 2(\Gamma\backslash G/K)\). We really want the sum of the eigenvalues of the operator over the discrete spectrum. Since this operator has a continuous spectrum, you must truncate integrals at the cusp of the fundamental domain and cancel the continuous spectrum terms against those from the parabolic conjugacy classes in \(\Gamma=SL(2,\mathbb{Z})\); e.g. those containing upper triangular matrices like \(\begin{pmatrix} 1 &n\\0&1\end{pmatrix}\), \(n\neq 0\). After some effort, Selberg obtained the trace formula. See \textit{A. Selberg} (loc. cit.), \textit{A. Terras} (loc. cit., pp. 285-286). There are many applications of the Selberg trace formula for \(SL(2,\mathbb{R})\). Some of these can be found in \textit{A. Terras} (loc. cit. pp. 290-295). For example, one can derive the Weyl law for the asymptotics of eigenvalues of the non-Euclidean Laplacian for \(L^ 2(\Gamma\backslash H)\). [See \textit{M. C. Gutzwiller}, Chaos in classical and quantum mechanics (Springer 1990; Zbl 0727.70029), for a discussion from the point of view of mathematical physics.] The trace formula provides a kind of duality between the eigenvalues of \(\Delta\) and the closed geodesics in \(\Gamma\backslash G/K\), which correspond to hyperbolic conjugacy classes in \(\Gamma\). This is analogous to that between zeros of the Riemann zeta function and primes. [See \textit{P. Buser}, Geometry and spectra of compact Riemann surfaces (Birkhäuser, 1992; Zbl 0770.53001).] Example 3. \(SL(n,\mathbb{Z})\). The next question that Selberg began to consider was the trace formula for higher rank groups \(\Gamma\) like \(SL(n,\mathbb{Z})\), \(n>2\). Here the theory becomes much more complicated. The Eisenstein series involve \(n-1\) complex variables, or a mixture of complex variables and lower rank cusp forms. Selberg did not publish many details of his work on this subject. See \textit{H. Maass}, Siegel's modular forms and Dirichlet series (Lecture Notes Math. 216) (1971; Zbl 0224.10028) and \textit{A. Terras} (loc. cit., Vol. II, 1988; Zbl 0668.10033), \textit{Harish-Chandra} [Automorphic forms on semisimple Lie groups (Lecture Notes Math. 62) (1968; Zbl 0186.047)]\ and \textit{R. P. Langlands} [On the functional equations satisfied by Eisenstein series (Lecture Notes Math. 544) (1976; Zbl 0332.10018)]\ present the analytic continuation of Eisenstein series for very general groups \(G\). We should also mention \textit{S.-T. Wong}, The meromorphic continuation and functional equations of cuspidal Eisenstein series for maximal cuspidal groups, Mem. Am. Math. Soc. 83 (1990; Zbl 0691.10016) and \textit{M. S. Osborne} and \textit{G. Warner}, The theory of Eisenstein systems (Pure Appl. Math. 99) (Academic Press, 1981; Zbl 0489.43009) (see the review of \textit{R. P. Langlands}, Bull. Am. Math. Soc. 9, 351-361 (1983)). Mirroring the complications from the multitudes of Eisenstein series for \(SL(n)\), are the multitudes of conjugacy classes and the complications of fundamental domains for \(SL(n,\mathbb{Z})\backslash SL(n,\mathbb{R})/\) \(SO(n,\mathbb{R})\). Such fundamental domains have been studied since Minkowski. See \textit{Terras} (loc. cit., II, Section 4.4). In the present context one needs to envision adelic fundamental domains. The monograph under review gives an outline of the theory of Arthur's truncation operators needed to truncate the kernels associated to equivalence classes in \(\Gamma\) as well as truncation operators associated to Eisenstein series. Once this mechanism is successfully accomplished, the trace formula looks deceptively simple: \[ \sum_{{\mathfrak o}\in{\mathfrak D}} J^ T_{{\mathfrak o}}(f)=\sum_{\chi\in\Omega} J^ T_ \chi(f), \] as in formula (7.6) on p. 78 of the monograph under review. Here both sides are polynomials in \(T\). Extra work is required to obtain distributions invariant under conjugation. Many of the details of the proofs are to be found in Arthur's papers. There are many other references on higher rank groups such as \(SL(n,\mathbb{R})\). \textit{Dorothy Wallace} [Trans. Am. Math. Soc. 327, 781-793 (1991; Zbl 0742.11032)] gives an explicit generalization of the Selberg trace formula to \(SL(3,\mathbb{R})\). Here one finds terms as explicit as those in Selberg's original paper. Other work on higher rank groups like \(Sp(n,\mathbb{R})\) obtaining dimensions of spaces of Siegel modular forms can be found in, for example, \textit{R. Tsushima} [Adv. Stud. Pure Math. 15, 41-64 (1989; Zbl 0698.10017)]. See also the review of \textit{M. Eie}, M. R. 91e:11050. The monograph under review mentions many sorts of applications of the trace formula. For example, the multiplicities of irreducible representations have topological meaning and thus the trace formula can give information on Betti numbers. There are applications to index theory, the arithmetic of Shimura varieties, non-Abelian class field theory. To summarize, these lecture notes give a quick and interesting outline presentation of Arthur's trace formula for \(GL(n)\). Some of the topics are: Arthur's reduction theory, kernel functions, adelic weighted orbital integrals, the Langlands-Arthur formula for inner products of truncated Eisenstein series, non-invariant and invariant versions of the Arthur trace formula.
inner products of truncated Eisenstein series
scientific article; zbMATH DE number 53220