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THE
${L}^{2} $-SINGULAR DICHOTOMY FOR EXCEPTIONAL LIE GROUPS AND ALGEBRAS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 95 / Issue 3 / December 2013
- Published online by Cambridge University Press:
- 24 July 2013, pp. 362-382
- Print publication:
- December 2013
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- Article
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- You have access
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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.
