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Distal actions and shifted convolution property

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Abstract

A locally compact group G is said to have shifted convolution property (abbr. as SCP) if for every regular Borel probability measure µ on G, either supx∈G µn(Cx) → 0 for all compact subsets C of G, or there exist x ∈ G and a compact subgroup K normalised by x such that µn x −n → ωK, the normalised Haar measure on K. We first consider distality of factor actions of distal actions. It is shown that this holds in particular for factors by compact groups invariant under the action and for factors by the connected component of the identity. We then characterize groups having SCP in terms of a readily verifiable condition on the conjugation action (pointwise distality). This gives some interesting corollaries to distality of certain actions and Choquet-Deny measures which actually motivated SCP and pointwise distal groups. We also relate distality of actions on groups to that of the extensions on the space of probability measures.

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Authors and Affiliations

  1. Stat-Math Unit, Indian Statistical Institute (ISI), 8th Mile Mysore Road, Bangalore, 560 059, India

    C. R. E. Raja

  2. School of Physical Sciences (SPS), Jawaharlal Nehru University (JNU), New Delhi, 110 067, India

    Riddhi Shah

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  1. C. R. E. Raja
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  2. Riddhi Shah
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Correspondence to C. R. E. Raja.

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Dedicated to Professor S. G. Dani on his 60th birthday

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Raja, C.R.E., Shah, R. Distal actions and shifted convolution property. Isr. J. Math. 177, 391–412 (2010). https://doi.org/10.1007/s11856-010-0052-7

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  • DOI: https://doi.org/10.1007/s11856-010-0052-7

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