Quantization on Nilpotent Lie Groups by Veronique Fischer;Michael Ruzhansky PDF

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In this case we write π1 ∼ π2 or, more precisely sometimes, π 1 ∼ A π2 and denote their equivalence class by [π1 ] = [π2 ]. For unitary representations, A is assumed to be unitary as well. 1) is sometimes called an intertwining operator or intertwiner. 1) is denoted by Hom(π1 , π2 ). Note that for any representation π, Hom(π, π) contains at least λIHπ , λ ∈ C, where IHπ is the identity mapping on Hπ . We now assume that the group G is topological. A representation π of G is continuous if the mapping G × Hπ (x, v) −→ Hπ −→ π(x)v 18 Chapter 1.

It is automatically left-invariant. On the other hand, if the form on g is Ad-invariant, then the extended metric is also right-invariant. Thus, we can conclude that the Killing form induces a biinvariant metric on G. By the last property above, if G is semi-simple, the Killing form is non-degenerate, and hence the corresponding metric is pseudo-Riemannian. Moreover, if G is a connected semi-simple compact Lie group, the positive-definite form −B induces the bi-invariant Riemannian metric on G. For the basis {Xj }nj=1 as above, let us define Rij := B(Xi , Xj ).

Let φ, ψ ∈ S(G). Then the operator π(φ)π(ψ)∗ is trace class for any π ∈ Rep G, and its trace is constant on the equivalence class of π. The function G π → Tr (π(φ)π(ψ)∗ ) is integrable against μ and (φ, ψ)L2 (G) = G φ(x)ψ(x)dx = G Tr (π(φ)π(ψ)∗ ) dμ(π). 2 Plancherel theorem and group von Neumann algebras In this section we describe the concept of the group von Neumann algebra that becomes handy in associating symbols with convolution kernels of invariant operators on G. For the details of the constructions described below we refer to Dixmier’s books [Dix77, Dix81] and to Section B in the appendix of this monograph.

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