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**Manifolds with cusps of rank one. Spectral theory and \(L^ 2\)-index theorem.**
*(English)*
Zbl 0632.58001

Lecture Notes in Mathematics, 1244. Subseries: Mathematisches Institut der Universität und Max-Planck-Institut für Mathematik, Bonn, Vol. 9. Berlin etc.: Springer-Verlag. XI, 158 p.; DM 28.50 (1987).

Let \(M=\Gamma \setminus G/K\), where G is a connected real semisimple Lie group, K a maximal compact subgroup and \(\Gamma\) a discrete, torsion-free and not cocompact subgroup of finite covolume of G. If V,W are finite dimensional unitary K-modules, let \(\tilde E,\tilde F\) be the respective induced vector bundles over \(G/K\) and let \(E=\Gamma \setminus \tilde E\), \(F=\Gamma \setminus \tilde F\) be the pushdown bundles over M. Also, let \(D: C^{\infty}(\tilde E)\to C^{\infty}(\tilde F)\) be a G-invariant elliptic differential operator and \(D: C^{\infty}(E)\to C^{\infty}(F)\), the induced elliptic operator. By a result of Moscovici such D has a well defined \(L^ 2\)-index which depends only on \(ch(V)- ch(W)\). By using the Selberg trace formula D. Barbasch and H. Moscovici [(*) J. Funct. Anal. 53, 151-201 (1983; Zbl 0537.58039)] have derived an explicit formula for the \(L^ 2\)-index of D in the case when G has \({\mathbb{R}}\)-rank one.

In the present work the author obtains an explicit formula for the \(L^ 2\)-index in the case of a locally symmetric space of Q-rank one. In fact he considers, roughly speaking, manifolds which are locally symmetric near infinity with ends and obtains a formula for the \(L^ 2\)-index of a class of elliptic 1st-order differential operators (called by the author Dirac chiral operators). As seen in (*) this suffices to compute the \(L^ 2\)-index of a locally invariant elliptic operator in the case of a locally symmetric space. The main contribution of the cusps in this index formula is given by a special value of a certain L-series associated to the locally symmetric structure of the ends of these manifolds. This generalizes the L-series arising in a previous paper by the author [J. Differ. Geom. 20, 55-119 (1984; Zbl 0575.10023)] where he investigated the signature operator on Hilbert modular varieties \(X=\Gamma \setminus H^ n\), where \(\Gamma =Sl(2,O_ F)\) is the Hilbert modular group of a totally real number field of degree n.

In the present work the author obtains an explicit formula for the \(L^ 2\)-index in the case of a locally symmetric space of Q-rank one. In fact he considers, roughly speaking, manifolds which are locally symmetric near infinity with ends and obtains a formula for the \(L^ 2\)-index of a class of elliptic 1st-order differential operators (called by the author Dirac chiral operators). As seen in (*) this suffices to compute the \(L^ 2\)-index of a locally invariant elliptic operator in the case of a locally symmetric space. The main contribution of the cusps in this index formula is given by a special value of a certain L-series associated to the locally symmetric structure of the ends of these manifolds. This generalizes the L-series arising in a previous paper by the author [J. Differ. Geom. 20, 55-119 (1984; Zbl 0575.10023)] where he investigated the signature operator on Hilbert modular varieties \(X=\Gamma \setminus H^ n\), where \(\Gamma =Sl(2,O_ F)\) is the Hilbert modular group of a totally real number field of degree n.

Reviewer: R.J.Miatello

### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58J20 | Index theory and related fixed-point theorems on manifolds |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |