A Classification of Multiplicity Free Representations

Let G be a connected reductive linear algebraic group over C and let (ρ, V ) be a regular representation of G . There is a locally finite representation (ρ̂,C[V ]) on the affine algebra C[V ] of V defined by ρ̂(g)f(v) = f(g−1v) for f ∈ C[V ] . Since G is reductive, (ρ̂,C[V ]) decomposes as a direct sum of irreducible regular representations of G . The representation (ρ, V ) is said to be multiplicity free if each irreducible representation of G occurs at most once in (ρ̂,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. 1. Facts about Multiplicity Free Representations 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 (ρ, V ) is a regular representation of G and let (ρ̂,C[V ]) denote the representation of G on the affine algebra C[V ] of V defined by (ρ̂(g)(f))(v) = f(ρ(g−1)v). Since G is reductive and (ρ̂,C[V ]) is locally regular, the affine algebra decomposes as a direct sum of irreducible G modules. (ρ, V ) also induces the structure of an irreducible affine G–variety on V, and the following definition makes sense for any affine G–variety V : Definition 1.1. The representation (ρ, V ) is said to be multiplicity free provided that the decomposition of (ρ̂,C[V ]) into a direct sum of irreducible G modules contains no irreducible module more than once. We let Λ (or ΛG ) denote the set of highest weights of G. Let S[V ] = ⊕ χ∈Λ S[V ]χ be the decomposition of the symmetric algebra S[V ] = P [V ∗] into its isotypic components and define Λ(V ) = {χ ∈ Λ|S[V ]χ 6= 0} ⊆ h∗ . Likewise, let P [V ] = S[V ∗] = ⊕ χ∈Λ S[V ∗]χ be the decomposition of P [V ] into its isotypic ISSN 0949–5932 / $2.50 C © Heldermann Verlag