Mathematical Research Letters

Volume 5 (1998)

Number 3

Spectral transfer and pointwise ergodic theorems for semi-simple Kazhdan groups

Pages: 305 – 325

DOI: http://dx.doi.org/10.4310/MRL.1998.v5.n3.a5

Author

Amos Nevo (Technion-Israel Institute of Technology)

Abstract

Let $G$ be a connected semi-simple Lie group with finite center and no compact factors, and $(X, \Cal B, m)$ a $G$-space with $\sigma$-finite $G$-invariant measure $m$. For each probability measure $\mu$ on $G$ consider the operator $\pi(\mu) : L^2(X)\to L^2(X)$, given by $\pi(\mu)f=\int_G\pi(g)f d\mu(g)$. The explicit spectral estimates (“quantitative property T”) of M. Cowling \cite{Co2} and R. Howe \cite{H} (see also \cite{H-T}\cite{Li}\cite{Mo}\cite{Oh}) are used to obtain explicit estimates of $\Vert\pi_0(\mu)\Vert$, where $\pi_0$ is the representation on the space orthogonal to the space of $G$-invariant functions, provided $\pi_0$ has a spectral gap. In particular, for actions of Kazhdan groups, the norm estimate is uniform and does not depend on the action. The norm estimates can be viewed as a spectral transfer principle, analogous to the transfer principle for amenable groups (see \cite{W}\cite{Ca} \cite{C-W}\cite{Hz1}\cite{Co3}\cite{Co4}). The spectral estimates are used to derive exponential-maximal inequalities for natural families of averages on the group, as well as pointwise ergodic theorems in $L^p$ for these averages. The pointwise convergence of the averages to the ergodic mean is exponentially fast with an explicit rate. This phenomenon in the case of bi-$K$-invariant measures was established in \cite{M-N-S}, and here we discuss non-radial averages, which may be absolutely continuous, singular or discrete. Some other topics discussed are averages supported on lattice points, almost orthogonality, and best possible estimates of convolution norms and exponential rate of convergence to the ergodic mean.

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