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The \(L^ 2\)-Lefschetz numbers of Hecke operators. (English) Zbl 0692.22004
Let \(\underline{G}\) be a connected semisimple algebraic group defined over \({\mathbb{Q}}\) and let \(\Gamma\) be an arithmetic subgroup of \bG defined by congruence conditions. Then \(\Gamma\) is a discrete subgroup of the group \(G=\underline G({\mathbb{R}})\) of real points of \(\underline{G}\). Let \(L^ 2(\Gamma \setminus G)\) be the space of complex valued square integrable functions on \(\Gamma\setminus G\), viewed as usual as a unitary G-module via right translations. In the case when \(\Gamma\setminus G\) is non-compact the space \(L^ 2(\Gamma \setminus G)\) is the direct sum of the discrete spectrum \(L^ 2_ d(\Gamma \setminus G)\) and the continuous spectrum \(L^ 2_{ct}(\Gamma \setminus G)\). The latter space is a Hilbert direct sum of continuous integrals each of which is a continuous sum of unitarily induced representations. The discrete spectrum \(L^ 2_ d(\Gamma \setminus G)\) in \(L^ 2(\Gamma \setminus G)\) decomposes into a direct Hilbert sum of closed irreducible subspaces \(H_{\pi}\) with finite multiplicities \(m(\pi,\Gamma)\) \[ (1)\quad L^ 2_ d(\Gamma \setminus G)=\oplus_{\pi \in \hat G}H_{\pi}^{m(\pi,\Gamma)}. \] It is one of the basic problems to compute the multiplicities.
The inclusion of the space of \(C^{\infty}\)-vectors in the discrete spectrum into \(C^{\infty}(\Gamma \setminus G)\) induces a natural map \(j_ d\) in \(({\mathfrak g},K)\)-cohomology with \({\mathfrak g}=Lie(G)\) and K a maximal compact subgroup of G. The decomposition (1) gives rise to a decomposition \[ H^*({\mathfrak g},K; L^ 2_ d(\Gamma \setminus G)\otimes E)=\oplus_{\pi \in \hat G}H^*({\mathfrak g},K; H\otimes E)^{m(\pi,\Gamma)} \] as a finite algebraic sum; here E denotes the space of an irreducible finite dimensional representation \(\tau\) of G. The image in \(H^*(\Gamma,E)=H^*({\mathfrak g},K; C^{\infty}(\Gamma \setminus G)\otimes E)\) of this cohomology space under \(j_ d\) is the so called square-integrable cohomology of the arithmetic group \(\Gamma\), to be denoted by \(H^*_{(2)}(\Gamma,E).\)
We assume that \(rk_{{\mathbb{R}}}G=rk_{{\mathbb{R}}}K\) holds, i.e. G has discrete series representations. Let \(\hat G_ d\) be the set of equivalence classes of discrete series representations of G and fix (\(\tau\),E) as above. We denote by \(\hat G_{d,E}\) the subset of \(\hat G_ d\) characterized by the condition that the infinitesimal character of the representation \(\pi\) coincides with the one of the contragredient representation \(E^*\) of E. Then the set \(\hat G_{d,E}\) has order \(W_ G/W_ K\) where \(W_ G\) (resp. \(W_ K)\) denotes the Weyl group of G (resp. K), and given a discrete series representation \(\pi\) of G with \([\pi]\in \hat G_{d,E}\) one has \(H^ d({\mathfrak g},K; H_{\pi}\otimes E)={\mathbb{C}}\) with \(d=(1/2) \dim G/K\) and vanishes otherwise, i.e. the relative Lie algebra cohomology of such representations is concentrated in the middle dimension of the underlying symmetric space \(X=G/K.\)
In the paper under review a formula is given for the sum \[ (2)\quad \sum_{[\pi]\in \hat G_{d,E}}m(\pi,\Gamma) \] under a weak regularity condition on \((\tau,E)\). More generally, Hecke operators on \(L^ 2(\Gamma \setminus G)\) are considered. These operators commute with the action of G and its restriction \(r(\pi,h)\) to the subspace that decomposes discretely according to \(\pi\) is defined. A formula for the sum \[ (3)\quad \sum_{[\pi]\in \hat G_{d,E}} \text{trace}(r(\pi,h)) \] of the traces is given.
The expressions (2) and (3) have a cohomological interpretation in terms of the square-integrable cohomology of \(\Gamma\). For example, it turns out that (2) is equal (up to a sign) with the alternating sum of the dimensions dim \(H^ q_{(2)}(\Gamma,E^*)\), i.e. one obtains a formula for the \(L^ 2\)-Euler characteristic of \(\Gamma\). More generally, the formula for expression (3) is one for the \(L^ 2\)-Lefschetz number of a Hecke operator.
The formulas obtained are derived from the trace formula. This is best done in an adelic setting. The formula for the Lefschetz number is a sum over the set of Levi subgroups M of G which contain a fixed minimal one. An individual summand is again a sum over elements \(\gamma\in (M({\mathbb{Q}}))\). Each term has as its ingredients an expression provided by the formulas for the characters of averaged discrete series, an orbital integral of h at \(\gamma\) and a constant which is closely related to the Euler characteristics of the locally symmetric spaces of M. The sums are both finite and the terms can be written down explicitly, at least in principle.
In case of rank one groups there is a topological approach to obtain a Lefschetz formula for the action of Hecke operators on the cohomology of arithmetic groups [cf. J. Bewersdorff, Bonn. Math. Schr. 164 (1985; Zbl 0589.12013)]. There is some hope to generalize this to groups of higher rank. It might even give some information where the group G does not have discrete series representations, e.g. in the case \(SL_ 3\).
Reviewer: J.Schwermer

22E40 Discrete subgroups of Lie groups
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F11 Holomorphic modular forms of integral weight
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