Topology on the unitary dual of completely solvable Lie groups

It was one of great successes of Kirillov's orbit method to see that the unitary dual of an exponential Lie group is in bijective correspondence with the orbit space associated with the linear dual of the Lie algebra of the group in question. To show that this correspondence is an homeomorphism turned out to be unexpectedly difficult. Only in 1994 H. Leptin and J. Ludwig gave a proof using the notion of variable groups. In this article their proof in the case of completely solvable Lie group is reorganized, some “philosophy” and some new arguments are added. The purpose is to contribute to a better understanding of this proof.

OPERATOR KERNELS FOR IRREDUCIBLE REPRESENTATIONS OF EXPONENTIAL LIE GROUPS

A nine-dimensional exponential Lie group G and a linear form ℓ on the Lie algebra of G are presented such that for all Pukanszky polarizations 𝔭 at ℓ the canonically associated unitary representation ρ=ρ(ℓ,𝔭) of G has the property that ρ(ℒ1(G)) does not contain any nonzero operator given by a compactly supported kernel function. This example shows that one of Leptin’s results is wrong, and it cannot be repaired.

Représentations irréductibles bornées des groupes de Lie exponentiels

Let G be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations (T, ) of G on a Banach space by giving a G-orbit in n* (n being the nilradical of g), a topologically irreducible representation of L1(ℝn, ω), for a certain weight ω and a certain n ∈ ℕ, and a topologically simple extension norm. If G is not symmetric, i.e., if the weight ω is exponential, we get a new type of representations which are fundamentally different from the induced representations.

Primitive ideals with bounded approximate units in L1-algebras of exponential lie groups

Let G be an exponential Lie group. We study primitive ideals (i.e. kernels of irreducible *-representations of L1(G)), with bounded approximate units (b.a.u.). We prove a result relating the existence of b.a.u. in certain primitive ideals with the geometry of the corresponding Kirillov orbits. This yields for a solvable group of class 2, a characterization of the primitive ideals with b.a.u.