Abstract
A compact Lie group G and a faithful complex representation V determine the Sato-Tate measure μG,V on C, defined as the direct image of Haar measure with respect to the character map g 7→ tr(g|V ). We give a necessary and sufficient condition for a Sato-Tate measure to be an isolated point in the set of all Sato-Tate measures, regarded as a subset of the space of distributions on C. In particular we prove that if G is connected and semisimple and V is irreducible, then μG,V is an isolated point.
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