Unitary Representation Theory of Exponential Lie Groups

L 1-determined ideals in group algebras of exponential Lie groups

“…Recall that there exists a g f -invariant Pukanszky polarization p 0 ⊂ h at f 0 ∈ h * because g is exponential, see §4, Chapter I of [19] and Chapter 5 of [1]. We shall verify that p = g f + p 0 ⊂ g defines a Pukanszky polarization at f ∈ g * : Clearly…”

“…In order to prepare the proof of Theorem 2.6 we recall the well-known restrictioninduction-lemma of Fell, see Theorem 3.1 and Lemma 4.2 of [11]. A proof can also be found on p. 32 of [19]. We presume the definition of induced representations.…”

“…connected, simply connected, solvable Lie groups G such that the exponential map exp : g −→ G is a global diffeomorphism). We will use the construction of irreducible representations π = K(f ) = ind G P χ f via Pukanszky / Vergne polarizations p at f and the bijectivity of the Kirillov map K : g * / Ad * (G) −→ G, see Chapters 4 and 6 of [1], and Chapter 1 of [19]. Mostly we regard K as a map from g * onto G which is constant on coadjoint orbits.…”

“…First we recall how to compute the C * -kernel of π | N in the Kirillov picture, compare Theorem 9 in Section 5 of Chapter 1 in [19]. Note that the linear projection r : g * → −→ n * given by restriction is Ad * (G)-equivariant so that r(Ad * (G)f ) = Ad * (G)l.…”

Nilpotent Lie Groups: Fourier Inversion and Prime Ideals

“…This may be handled most easily by the concept of variable Lie structures. Such structures were already considered in [5], [11], [10] and [9], among others.…”