A classification of multiplicity free representations

Abstract: Let $G$ be a connected reductive linear algebraic group over $\C$ and let $(\rho,V)$ be a regular representation of $G$. There is a locally finite representation $(\hat \rho, \C[V])$ on the affine algebra $\C[V]$ of $V$ defined by $\hat \rho (g) f(v) = f(g^{-1} v)$ for $f \in \C[V]$. Since $G$ is reductive, $(\hat \rho,\C[V])$ decomposes as a direct sum of irreducible regular representations of $G$. The representation $(\rho, V)$ is said to be multiplicity free if each irreducible representation of $G$ occurs at most once in $(\hat \rho, \C[V])$. Kac has classified all irreducible multiplicity free representations. In this paper, we classify arbitrary regular multiplicity free representations, and for each new multiplicity free representation we determine the monoid of highest weights occurring in its affine algebra.

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