Lefschetz formulae for arithmetic varieties
The paper under review is a continuation of the author's work on \(L^2\)-cohomology (with \textit{L. Saper} [Ann. Math., II. Ser. 132, No. 1, 1-69 (1990; Zbl 0722.14009)]) and \(L^2\)-index theory [Invent. Math. 96, No. 2, 231-282 (1989; Zbl 0694.58039), J. Differ. Geom. 31, No. 3, 771-789 (1990; Zbl 0715.53039)] for arithmetic varieties. Its purpose is to prove a Lefschetz formula for Hecke operators on, in general, noncompact arithmetic varieties. More precisely, consider an arithmetic variety \(X_\Gamma = \Gamma \setminus G/K\), where the \(\mathbb{Q}\)-irreducible Hermitean symmetric space \(G/K\) is a quotient of the principal component \(G\) of the group of real points of a semisimple algebraic group over \(\mathbb{Q}\) by a maximal compact subgroup \(K\), and \(\Gamma\) is an arithmetic subgroup of \(G\). Given a finite dimensional complex representation \((\rho, E)\) of \(G\), one obtains a flat vector bundle \(\mathcal E\) over \(X_\Gamma\), and a signature operator \(D_\rho = d + d^*\) acting on smooth square integrable \(\mathcal E\)-valued forms. With respect to an involution \(\tau\) on the space of square integrable \(\mathcal E\)-valued forms anticommuting with \(D_\rho\) and associated to the signature complex, the Lefschetz number of a Hecke operator \(T_G (\alpha)\) is defined by \[ L(\alpha, \tau, E) = \text{tr } \tau T_G (\alpha) |_{\text{ker}} D_\rho. \] For \(\alpha\) an element of the commensurator of \(\Gamma\) the Hecke operator \(T_G (\alpha)\) is defined on \(L^2 (\Gamma \setminus G)\), and extended trivially to commute with \(\tau.\) The problem to find a formula for \(L(\alpha, \tau, E)\) has been approached in different ways by \textit{H. Moscovici} [Lect. Notes Math. 1077, 321-258 (1984; Zbl 0561.58046)] for \(X_\Gamma\) compact, and by \textit{J. Arthur} [Invent. Math. 97, No. 2, 257-290 (1989; Zbl 0692.22004)] for \(X_\Gamma\) non-compact and \(\tau\) associated to the Gauss-Bonnet-complex using the trace formula, and by M. Goresky and R. MacPherson using intersection cohomology. The present paper uses the heat equation method already employed by the author to prove \(L^2\) index theorems for generalized signature operators on locally symmetric spaces. It gives an integral formula consisting of three ingredients, a local contribution from coincidence loci, and two global contributions from boundary components (with respect to the Baily-Borel-Satake compactification) in form of eta and zeta terms. The eta terms can be computed by Lefschetz numbers on the boundary components. The zeta terms occur only in degenerate cases for the signature complex and are missing in the analogous Lefschetz formula for the Gauss-Bonnet-complex which is also deduced. As a special case of the latter, the corresponding \(L^2\)-Gauss-Bonnet-theorem is stated, and an explicit computation of the \(L^2\) Euler characteristic is done for \(G = \text{SL} (4,\mathbb{R})\). A Lefschetz formula for the Dolbeault complex is sketched, too.
scientific article; zbMATH DE number 549845
scientific article