If the space Hπ is finite dimensional, the representation π is said to be finite dimensional and its dimension/degree is defined by. The set of bounded linear maps A with bounded inverse satisfying the relation (1.1) is denoted by Hom(π1, π2). A representationπ of Gi is called strongly continuous if the map π :G → L(Hπ) is continuous for the strong operator topology in L(Hπ), dat.

If the representation π is finite dimensional, then the Fourier coeﬃcientf(π) after a choice of a basis in the representation space Hπ can also be considered as a matrixf(π)∈Cdπ×dπ. Conversely, given X(e)∈TeG, the vector field X defined by (1.6) is automatically smooth and, by definition, left-invariant. This gives an identification between the universal enveloping algebraU(g) and the space of left-invariant differential operators onG.

## Distributions and Schwartz kernel theorem

The smallest closed set in which u is supported is called the support of u and is denoted by suppu. The space of compactly supported distributions on M is denoted by E(M), and the duality between E(M) and C∞(M) will still be denoted by·,·. We write u∈ Dj(Ω) for the space of distributions of order j on Ω, which means that for any compact subsetK of Ω,.

## Convolutions

One can easily check the following simple properties:. if one of org is continuous onGthenf ∗g is continuous onG;. convolution is commutative if and only ifGis commutative;. Conversely, Schwartz's kernel integral theorem implies that if Ais is left-invariant, it can be written as a right convolution Af =f ∗κ, and if A is right-invariant, it can be written as a left convolutionAf =f ∗ κ, see Section 1.4 and later Corollary 3.2.1. With our choice of definition of convolution and Fourier transform in (1.2), one can easily check that forf, g∈L1(G), we have.

We say that an operator A is of the weak type (p, p) if there is a constant C >0 so that for every λ >0 we have. If we assume a compactly supported distribution v∈ E(G) in Definition 1.5.3, we arrive at the definition of the composition u∗v foru∈ D(G) env∈ E(G), given by the same formula as in Definition 1.5.4. A warning must be said about the convolution of distributions, namely that it is generally not associative to distributions, although it is associative to functions.

## Nilpotent Lie groups and algebras

Its Lie algebra hno is R2no+1 equipped with the Lie bracket given by the commutator relations of its canonical basis{X1,. In caseno= 1, we will often simplify the notation and write X, Y, T for the base of h1, etc. It can be proved that any (simply connected) nilpotent Lie group can be realized as an iTno subgroup.

The basis is {Ei,j,1≤i < j ≤no}, where Ei,j is a matrix with all zero entries except for the i-th row and j-th column, which is 1. Let G be a connected simple nilpotent Lie group with Lie algebra g. a). The exponential mapping expG is a diﬀomorphism from gon to G. b) If G is identified with gviaexpG, the group law (x, y)→xy is a polynomial mapping. Xn} for g, Proposition 1.6.6, part (a), implies that the group Gi is identified with Rn via an exponential mapping; that is, the pointx= (x1, . . . , xn)∈Rn is identified with the point.

Indeed, in the nilpotent case, since it is ad nilpotent, the Baker-Campbell-Hausdorff formula is finite and holds for any two elements of the Lie algebra. We will see in Section 3 that in the special case of homogeneous Lie groups by choice of basis in Section 3.1.3, this fact together with some additional homogeneous properties is proved in Proposition 3.1.24. In the special case of homogeneous Lie groups by choice of basis in Section 3.1.3, this fact, together with some additional homogeneous properties, is proved in Lemma 3.1.47.

Discarding the abelian case (Rn,+), we use the multiplicative notation for the group law of any other connected simply connected nilpotent Lie group G. As a result of the Baker-Campbell-Hausdorf formula (see Theorem 1.3. 2), the inverse of an element is in fact its opposite, that is, with the notation above,.

## Smooth vectors and inﬁnitesimal representations

The function f is of class Ck if and only if all its partial derivatives of order 1,2,. Since π is continuous, we have by using the identification of g with the space of left-invariant vector fields. Let G be a Lie group with Lie algebra and letπ be a strongly continuous representation of G on a Hilbert space Hπ.

Let G be a Lie group with Lie algebragan and let π be a strong continuous unitary representation of Gon as a Hilbert space H π. For the infinitesimal representation dπ of gonH∞π Eachdπ(X) for X∈gis is skew-Hermitian: dπ(X)∗=−dπ(X). ii) The space H∞π of smooth vectors is invariant under π(x) for every x ∈G, and. For (ii) we first see that the map x→π(x)π(xo)v is the composite of x→xxo and x→π(x)v.

For (iii), under the assumption for v ∈ S the map Fv : G x → π(x)v is differentiable at the neutral element e, the partial derivative in the direction X ∈ g. If Fv is of class Ck fork∈N, then maptax→XFv(x) =π(x)XFv(e) is of class Ck and Fv must be of class Ck+1. In the following proposition, we show that the space of smooth vectors is dense in the space of a strongly continuous representation.

It turns out that the G˚arding subspace is not only in the subspace H∞π, but is in fact equal to H∞π. The space H∞π of smooth vectors is spanned by all vectors of the form π(φ)v for v ∈ H∞π and φ ∈ D(G).

## Plancherel theorem

### Orbit method

In this section, we briefly discuss the idea of the orbit method and its implications for our analysis. The orbit method describes a way to associate to a given linear functional on g a set of unitary, irreducible representations of G that are all unitary equivalent. Consequently, one can associate to any element of the dualg org an equivalence class of unitary irreducible representations.

In the case of the Heisenberg group Hno presented in Example 1.6.4, it is a family of representatives of all co-joined orbits. Thus, by the orbit method, the unitary dual G is 'concretely' described as a subset of some Euclidean space. The dual G is then equipped with a measure μ, called Plancher measures, which satisfies the following property for any φ∈ S(G).

The operator π(φ)≡φ(π) is the trace class for any strongly connected unitary irreducible representation π∈RepG and Tr(π(φ)) depends only on the π class; the function Gπ→Tr (π(φ)) is integrable against μ and the following formula applies:. The latter equality can be seen using the same argument as above applied to the function f(x·). The subset G formed by 1-dimensional representations is negligible with respect to the Plancherel measure.

The function Gπ→ π(φ)2HS is integrable against μand. 1.27) The formula (1.27) can be uniformly extended to apply to any φ∈L2(G), which allows the definition of the Fourier transform of the group of squarely integrable functions on G. Then the operator π(φ)π(ψ)∗ is the sequence class for any π∈RepG and its trace is constant on the equivalence class π.

### Plancherel theorem and group von Neumann algebras

Instead, we present its consequences applicable to our setting, starting with the existence of Plancherel mass. We begin by describing the part of Plancherel's theorem that deals with the Plancherel formula. Here we use the usual identiﬁcations of a strongly continuous irreducible unitary representation by RepG with its equivalence class onG, and of a field of operators onG with its equivalence class with respect to the Plancherel measure μ.

We now present the parts of the Plancherel theorem (relevant to our subsequent analysis) with respect to the description of the group von Neumann algebra. Here we use the usual identification of a strongly continuous irreducible unitary representation of RepG with its equivalence class inG, and of a field of operators onG with its equivalence class with respect to the Plancherel measure μ. One can make sure that being in L∞(G) does not depend on a particular representative ofπand of the field of operators.

The Plancherel theorem implies that this map is in fact a bijection and an isometry, and thus a von Neumann algebra isomorphism. If κ∈ D(G) is such that the corresponding convolution operator D(G)f →f∗κ extends to a bounded operatorTκonL2(G) then Tκ∈LL(L2(G)), and we extend the definition of the Fourier transform of the group by setting . The part of the Plancherel theorem that we have already presented implies that the space K(G) is a von Neumann algebra isomorphic to LL(L2(G)) and to L∞(G).

In particular, the Fourier transform of the Dirac mass group, which will destroy the neutral element, is the identity operator. In general, the Fourier group transform of the Dirac mass δxo in the elementinxo∈Gis.

### Fields of operators acting on smooth vectors

Unless stated otherwise, all operator G-fields are assumed to be measurable and operators are defined from smooth vectors. Note that we do not require the domain of each operator to be the entire representation space Hπ1, but only the space of smooth vectors. A measurable G-field of operators acting on the smooth vectors is a measurable G-field of operatorsσ ={σπ :H∞π → Hπ, π ∈G} so that for everyπ1∈RepG we have that.

We will often consider measurable field of operatorsσπ, collapsing on smooth vectors and parametrized not only by G but also by another setS. If T ∈ U(g) then {π(T), π ∈ G} yields a measurable field of operators that act on smooth vectors and are parametrized by G (see also Theorem 1.7.3). If φ1, φ2 ∈ D(G), then the composition of φ1 with φ2 as fields of operators acting on smooth vectors is φ2∗φ1.

If we assume a field of operatorsσ={σπ: H∞π → Hπ, π ∈ G} defined on smooth vectors, we cannot always extend this to . On a compact Lie group, every G-field of operators is measurable and the operators act on smooth vectors. However, for a non-compact Lie group we cannot limit ourselves to the case of G-fields acting on smooth vectors in general, since a non-compact Lie group can have infinite dimensional (strongly continuous irreducible) representations with no -smooth vectors and we can then find fields in L2(G) that do not act on smooth vectors.

Gvπ2HS(Hπ)dμ(π)<∞, then construct a field of operators {vπ⊗v∗π, π∈G} in L2(G) that does not act on smooth vectors. As an application of Lemma 1.8.19, we see that the field δxo given at the end of Example 1.8.10 acts on smooth vectors:.