English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $G$ be a compact, connected simple Lie group and $\mathfrak{g}$ its Liealgebra. It is known that if $\mu $ is any $G$-invariant measure supported onan adjoint orbit in $\mathfrak{g}$, then for each integer $k$, the $k$% -foldconvolution product of $\mu $ with itself is either singular or in $% L^{2}$.This was originally proven by computations that depended on the Lie type of$\mathfrak{g}$, as well as properties of the measure. In this note, we observethat the validity of this dichotomy is a direct consequence of theDuistermaat-Heckman theorem from symplectic geometry and that, in fact, anyconvolution product of (even distinct) orbital measures is either singular orin $L^{2+\varepsilon }$ for some $\varepsilon >0$. An abstract transferenceresult is given to show that the $L^{2}$-singular dichotomy holds for certainof the $G$-invariant measures supported on conjugacy classes in $G.$

An abstract proof of the L2-singular dichotomy for orbital measures on Lie algebras and groups