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OPERATOR KERNELS FOR IRREDUCIBLE REPRESENTATIONS OF EXPONENTIAL LIE GROUPS

Part of: Abstract harmonic analysis Lie groups

Published online by Cambridge University Press:  01 October 2008

DETLEV POGUNTKE
DETLEV POGUNTKE*
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100 131, 33501 Bielefeld, Germany (email: poguntke@math.uni-bielefeld.de)
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Abstract

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A nine-dimensional exponential Lie group G and a linear form ℓ on the Lie algebra of G are presented such that for all Pukanszky polarizations 𝔭 at ℓ the canonically associated unitary representation ρ=ρ(ℓ,𝔭) of G has the property that ρ(ℒ1(G)) does not contain any nonzero operator given by a compactly supported kernel function. This example shows that one of Leptin’s results is wrong, and it cannot be repaired.

Keywords

concrete realizations of representations of solvable Lie groupssmooth kernelspolarizationsBeurling algebras

MSC classification

Secondary: 43A20: $L^1$-algebras on groups, semigroups, etc. 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 2 , October 2008 , pp. 301 - 316
Copyright
Copyright © 2008 Australian Mathematical Society

References

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