| In order to study the asymptotic distribution of geometric or spectral data associated with quotients of a reductive group by a lattice, one needs a trace formula for test functions on that group with noncompact support. Arthur has proved a trace formula for compactly supported test functions on reductive groups of arbitrary rank. We show that the coarse geometric expansion in his formula converges for rapidly decreasing functions. |
| Weighted orbital integrals are distributions
on reductive groups over local fields appearing both in the local and global
trace formulas. There are associated invariant distributions, which play
the same role in the invariant trace formulas. In the case of real groups,
the Fourier transforms of these distributions satisfy a system of differential
equations.
As a step towards determining those Fourier transforms, we show that this system is holonomic and has a simple singularity at infinity. We deduce that any solution has a series expansion and is a linear combination of certain canonical solutions. For some groups of small rank, we solve the recursion formula for the coefficients explicitly. |
| Let Γ be a rank-one arithmetic subgroup of a semisimple Lie group G. For fixed K-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of G, whose discrete part encodes the dimensions of the spaces of square-integrable Γ-automorphic forms. It is shown that this distribution converges to the Plancherel measure of G when Γ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices Γ follows from results of DeGeorge-Wallach and Delorme. |
| We prove a uniform upper estimate on the number of cuspidal eigenvalues of the Γ-automorphic Laplacian below a given bound when Γ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each Γ in the family is assumed to contain a principal congruence subgroup whose index in Γ does not exceed a fixed number. The bound we prove depends linearly on the covolume of Γ and is deduced from the analogous result about the cut-off Laplacian. The proof generalizes the heat-kernel method which has been applied by Donnelly in the case of a fixed lattice Γ. |
| We prove a non-adelic invariant trace formula for rank one lattices. Unlike Arthur's adelic invariant trace formula for groups of general rank, our formula applies to non-congruence lattices, too, and the geometric terms are made explicit. The contribution of the continuous spectrum to the trace is given in terms of certain generalised Hecke operators, and the convergence problem with this contribution is resolved. |
| Weighted orbital integrals and the associated invariant distributions IM appear in the Selberg-Arthur trace formula. We calculate explicitly the invariant Fourier transforms of the IM on semisimple Lie groups of real rank one, thereby reaching a goal aimed at by G. Warner in J. Funct. Anal. 64 (1985). We also present IM as an integral transform of ordinary orbital integrals. |
| The aim of the present article is to prove the Selberg trace formula for Hecke operators and vector-valued automorphic forms of arbitrary real weight. The analogous formula without Hecke operators has been proved by D. Hejhal in Lecture Notes in Math., vol. 1001. Our trace formula is invariant in the sense of J. Arthur, i.e., the trace is given in terms of the invariant Fourier transform of the test function. |
| We prove that the basic property of cuspidal functions being rapidly decreasing is still true for nonarithmetic lattices provided the structure of their cusps has a certain property (D) defined in terms of Diophantine approximation. In order to show that a condition of this kind is necessary, we give examples of locally-symmetric spaces of finite volume for which the number of the eigenvalues of the Laplacian below a given bound grows arbitrarily fast with that bound. |
| We derive the trace formula for Hecke operators acting on the completely G-decomposable subspace of L2(G/Γ), where G is a real reductive Lie group and Γ is a lattice of rank one in G. If Γ is arithmetic this means that G has Q-rank one. The trace is given in terms of (weighted) orbital integrals and the usual intertwining and residual terms. |
Last modified: 12.1.2005