The trace class conjecture for arithmetic groups
Let \(G\) a Lie group, real locus of a reductive algebraic group defined over \(\mathbb Q\) and \(\Gamma\) an arithmetic subgroup of \(G(\mathbb Q)\). The right \(G\)-module \({\mathcal H}=L^2(\Gamma\backslash G)\) decomposes as \({\mathcal H}={\mathcal H}_{d}\oplus {\mathcal H}_{c}\) where \({\mathcal H}_{d}=\oplus_{\pi\in\widehat G} m_\pi\pi\) is the smallest subspace of \({\mathcal H}\) containing all subrepresentations of \({\mathcal H}\) and \({\mathcal H}_c\) is a Hilbert continuous integral of irreducibles. \textit{A. Borel} and \textit{H. Garland} [Am. J. Math. 105, 309-335 (1983; Zbl 0572.22007)] proved the finiteness of each \(m_\pi\) and \textit{J. Arthur} [Ann. Math. (2) 100, 326-385 (1974; Zbl 0289.20027)] asked about the trace class property of the integrated representation \(R_d(\alpha)\) defined for \(\alpha\in{\mathcal C}^\infty_0(G)\) as the restriction on \({\mathcal H}_d\) of \[ R(\alpha)f(\Gamma g)=\int_G\alpha(h)f(\Gamma gh) dh, \quad g\in G . \] This trace class property enters in the Arthur trace formula, generalization to a wide class of groups of the Selberg trace formula [\textit{A. Selberg}, J. Indian Math. Soc., New Ser. 20, 47-87 (1956; Zbl 0072.08201)] proved originally for arithmetic lattices of \(G=SL(2,{\mathbb R})\). The trace class conjecture has been proved for \({\mathbb Q}\)-rank one groups firstly by \textit{H. Donnelly} [Mich. Math. J. 31, 349-357 (1984; Zbl 0591.58033)] and later by \textit{R. P. Langlands} [Isr. Math. Conf. Proc. 3, 111-125 (1990; Zbl 0729.11038)] and in the general case, but with the restriction of \(K\)-finiteness for \(\alpha\) (\(K\) is a maximal compact connected subgroup of \(G\)), by \textit{W. Müller} [Ann. Math. (2) 130, No. 3, 473-529 (1989; Zbl 0701.11019)]. In the paper under review, the author proves the conjecture for \(\alpha\) in the Harish-Chandra-Schwartz space of integrable functions without any \(K\)-finiteness restriction; the author studies the case of \(K\)-finite functions (related to Weyl upper bounds for locally homogeneous bundles on the locally symmetric space \(\Gamma\backslash G/ K\)) in a companion paper [J. Differ. Geom. 51, No.1, 97-148 (1999)], to which the reviewed paper refers abundantly. The author considers here only semisimple groups, but the general case of non connected reductive groups is an easy corollary of the semisimple case. The general trace class conjecture has been proved also by \textit{W. Müller} in a simultaneous and independent paper [Geom. Funct. Anal. 8, No. 2, 315-355 (1998)]. By a classical argument due to \textit{G. Warner} [Studies in Algebra and Number Theory, Adv. Math., Suppl. Stud. 6, 1-142 (1979; Zbl 0466.10018)], the traceability of \(R_d(\alpha)\) results from polynomial upper-bounds for the eigenvalues counting function of the Laplace-Beltrami operator \(\Delta_G\), which operates discretely on \({\mathcal H}_d\). Here the operator \(\Delta_G\) is the Laplacian determined by a \(G\)-invariant metric and coincides with the action of \(-\Omega_G+2\Omega_K\) where \(\Omega_G\) and \(\Omega_K\) are the respective Casimir of the groups \(G\) and \(K\). The spectrum of \(\Delta_G\) decomposes in the cuspidal spectrum and the residual spectrum (generated by iterated residues of cuspidal Eisenstein functions introduced by Langlands in his harmonic analysis of \({\mathcal H}\)). For the analysis of both spectra, pseudo-Laplacians \(\Delta_T\), similar to those introduced by \textit{P. D. Lax} and \textit{R. S. Phillips} [Ann. Math. Studies. No. 87 (1976; Zbl 0362.10022)] and \textit{Y. Colin de Verdière} [C. R. Acad. Sci., Paris, Sér. I 293, 361-363 (1981; Zbl 0478.30035)] (related to the truncation operator \(\Lambda_T\) of Arthur), play a key rôle, as their spectra are strongly related to parts of the spectrum of \(\Delta_G.\) These pseudo-Laplacians are defined on spaces associated to rational parabolic subgroups \(P\) of \(G\). For each such \(P\), the author introduces a boundary space \(X_P\) of \(X=G/K\) (\(X_P\) is a component of the reductive Borel-Serre compactification of \(X\)) and a \(K\)-bundle \(B_P\) over \(X_P\); spectral decomposition of the locally homogeneous space \(\Gamma_P\backslash B_P\) (of finite volume) is done and Weyl estimates on its discrete spectrum are established. While refining results of the companion work, analysis of these boundary pseudo-Laplacians is the heart of the reviewed paper. Then estimates for the residual spectrum are derived first for \({\mathbb Q}\)-rank one groups; analysis of the iterate Eisenstein functions and scattering matrices (which are product of matrices for \({\mathbb Q}\)-rank one groups) gives the polynomial upper bound for the residual spectrum of general \(G\).
scientific article; zbMATH DE number 1187447
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