Minimal ideals of group algebras
We first study the behavior of weights on a simply connected nilpotent Lie group G. Then for a subalgebra A of L1(G) containing the Schwartz algebra S(G) as a dense subspace, we characterize all closed two-sided ideals of A whose hull reduces to one point which is a character. Introduction. Let G be a simply connected nilpotent Lie group, g its Lie algebra, and A a subalgebra of L1(G). To every character χl of A we will associate a finite-dimensional translation invariant subspace Pl of the vector space P(G) of complex polynomials on G and we will show that the set of closed two-sided ideals of A with hull {Kerχl} is in bijection with the set of nonzero translation-invariant subspaces of Pl. As an example of A we can take the weighted algebra Lw(G) where w is a weight with polynomial growth. Such weights appear in a natural way in the following manner: let π be a unitary continuous irreducible representation of G in a Hilbert space Hπ. We denote by U(g) the enveloping algebra of g. Fix a nonzero integer k and denote by U(g)k the vector space generated by the elements of U(g) with degree less than k. Let H π be the space of k times differentiable vectors in Hπ, i.e. H π = {ξ ∈ Hπ | ∀z ∈ U(g)k : dπ(z)ξ ∈ Hπ}. Fix a basis (z)|i|≤k of U(g)k. We equip H π with the norm