A Classiication of Multiplicity Free Representations 1. Facts about Multiplicity Free Representations

Let G be a connected reductive linear algebraic group over C and let (; V) be a regular representation of G. There is a locally-nite representation (^ ; C V ]) on the aane algebra C V ] of V deened by ^ (g)f(v) = f(g ?1 v) for f 2 C V ]. Since G is reductive, (^ ; C V ]) decomposes as a direct sum of irreducible regular representations of G. The representation (; V) is said to be multiplicity free if each irreducible representation of G occurs at most once in (^ ; C V ]). Kac has classiied 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 aane algebra. Throughout this paper G will denote a connected reductive linear algebraic group over the complex numbers C. B will denote a Borel subgroup of G, and we will write B = HN to denote the decomposition of B into a maximal torus H and a unipotent radical N. We will suppose (; V) is a regular representation of G and let (^ ; C V ]) denote the representation of G on the aane algebra C V ] of V deened by (^ (g)(f))(v) = f((g ?1)v). Since G is reductive and (^ ; C V ]) is locally regular, the aane algebra decomposes as a direct sum of irreducible G modules. (; V) also induces the structure of an irreducible aane G{variety on V, and the following deenition makes sense for any aane G{variety V : Deenition 1.1. The representation (; V) is said to be multiplicity free provided that the decomposition of (^ ; C V ]) into a direct sum of irreducible G modules contains no irreducible module more than once. We let (or G) denote the set of highest weights of G. Let SV ] = L 2 SV ] be the decomposition of the symmetric algebra SV ] = PV ] into its isotypic components and deene (V) = f 2 jSV ] 6 = 0g h. Likewise,