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THE ${L}^{2} $ -SINGULAR DICHOTOMY FOR EXCEPTIONAL LIE GROUPS AND ALGEBRAS

  • K. E. HARE (a1), D. L. JOHNSTONE (a1), F. SHI (a2) and W.-K. YEUNG (a2)

Abstract

We show that every orbital measure, ${\mu }_{x} $ , on a compact exceptional Lie group or algebra has the property that for every positive integer either ${ \mu }_{x}^{k} \in {L}^{2} $ and the support of ${ \mu }_{x}^{k} $ has non-empty interior, or ${ \mu }_{x}^{k} $ is singular to Haar measure and the support of ${ \mu }_{x}^{k} $ has Haar measure zero. We also determine the index $k$ where the change occurs; it depends on properties of the set of annihilating roots of $x$ . This result was previously established for the classical Lie groups and algebras. To prove this dichotomy result we combinatorially characterize the subroot systems that are kernels of certain homomorphisms.

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References

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