Pseudo-Differential Operators and Symmetries

Pseudo-diﬀerential operators and symmetries

Michael Ruzhansky and Ville Turunen

September 2009

Part IV

Non-commutative symmetries

541

In this part, we develop a non-commutative quantization of pseudo-differential operators on compact Lie groups. The idea is that it can be constructed in a way to run more or less parallel to the Kohn–Nirenberg quantization of operators on Rn that was presented in Chapter 2, and to the toroidal quantization of operators on Tn that was developed in Chapter 4. The main advantage of such approach is that once the basic notions and definitions are understood, one can see and enjoy a lot of features which are already familiar from the commutative analysis. The introduced matrix-valued full symbols turn out to have a number of interesting properties. The main difference with the toroidal quantization here is that due to the non-commutativity of the group symbols become matrix-valued with sizes depending on the dimensions of the unitary irreducible representations of the group, which are all finite dimensional because if the group is compact. Among other things, the introduced approach provides a characterisation of the Hörmander class of pseudo-differential operators on a compact Lie group G using a global quantization of operators, thus relying on the representation theory rather than on the usual expressions in local coordinate charts. This yields a notion of the full symbol of an operator as a mapping defined globally on G×G, where G is the unitary dual of G. As such, this presents an advantage over the local theory where only the notion of the principal symbol can be defined globally. In the case of the torus G = Tn, we naturally have G × G =∼ Tn × Zn, and we recapture the notion of a toroidal symbol introduced in Chapter 4, where symbols are scalar- valued (or 1 × 1 matrix-valued) because all the unitary irreducible representations of the torus are one-dimensional. As an important example, the approach developed here will give us quite detailed information on the global quantization of operators on the 3-dimensional sphere S3. More generally, we note that if we have a closed simply-connected 3- dimensional manifold M, then by the recently resolved Poincaréconjecture there is a global diffeomorphism M ≃ S3 ≃ SU(2) that turns M into a Lie group with a group structure induced by S3 (or by SU(2)). Thus, we can use the approach developed for SU(2) to immediately obtain the corresponding global quantization of operators on M with respect to this induced group product. In fact, all the formulae remain completely the same since the unitary dual of SU(2) (or S3 in the quaternionic R4) is mapped by this diffeomorphism as well; for an example of this construction in the case of S3 ≃ SU(2) see Section 12.5 . The choice of the group structure on M may be not unique and is not canonical, but after using the machinery that we develop for SU(2), the corresponding quantization can be described entirely in terms of M, for an example compare Theorem 12.5.3 for S3 and Theorem 12.4.3 for SU(2). In this sense, as different quantizations of operators exist already on Rn depending on the choice of the underlying structure (e.g. Kohn–Nirenberg quantization, Weyl quantizations, etc.), the possibility to choose different group products on M resembles this. Due to space limitations, we postpone the detailed analysis of operators on the higher dimensional spheres Sn ≃ SO( n+1) /SO( n) viewed as homogeneous spaces. However, we will introduce a general machinery on how to obtain the global quantization on homogeneous 542 spaces using the one on the Lie group that acts on the space. Although we do not have general analogues of the diffeomorphic Poincaréconjecture in higher dimensions, this will cover cases when M is a convex surface or a surface with positive curvature tensor, as well as more general manifolds in terms of their Pontryagin class, etc. Thus, the cases of the 3-dimensional sphere S3 and Lie group SU(2) are analysed in detail in Chapter 12 . There we show that pseudo-differential operators from Hörmander’s classes Ψ m(SU(2)) and Ψ m(S3) have matrix-valued symbols with a remarkable rapid off-diagonal decay property. In Chapter 11 we develop the necessary foundations of this analysis on SU(2) which together with Chapter 12 provides a more detailed example of the quantization from Chapter 10 . Finally, in Chapter 13 we give an application of these constructions to analyse pseudo-differential operators on homogeneous spaces. Chapter 10

Pseudo-diﬀerential operators on compact Lie groups

10.1 Introduction

In this chapter we develop a global theory of pseudo-diﬀerential operators on m Rn Rn ∞ Rn Rn general compact Lie groups. As usual, S1,0( × ) ⊂ C ( × ) refers to the Euclidean space symbol class, deﬁned by the symbol inequalities

α β m−| α| ∂ξ ∂x p(x, ξ ) ≤ C (1 + |ξ|) , (10.1)

Nn N N for all multi-indices α, β ∈ 0 , 0 = {0}∪ , where the constant C is independent of x, ξ ∈ Rn but may depend on α, β, p, m . On a compact Lie group G we define the class Ψ m(G) to be the usual Hörmander class of pseudo-differential operators of order m. Thus, the operator A belongs to Ψ m(G) if in (all) local coordinates operator A is a pseudo-differential operator on Rn with some symbol p(x, ξ ) satisfying estimates ( 10.1 ), see Definition 5.2.11 . Of course, symbol p depends on the local coordinate systems. It is a natural idea to build pseudo-differential operators out of smooth families of convolution operators on Lie groups. In this work, we strive to develop the convolution approach into a symbolic quantization, which always provides a much more convenient framework for the analysis of operators. For this, our analysis of operators and their symbols is based on the representation theory of Lie groups. This leads to the description of the full symbols of pseudo-differential operators on Lie groups as sequences of matrices of growing size equal to the dimension of the corresponding representation of the group. Moreover, the analysis is global and is not confined to neighbourhoods of the neutral element since it does not rely on the exponential map and its properties. We also characterise, in terms of the introduced quantizations, standard Hörmander’s classes Ψ m(G) on Lie groups. One of the advantages of the presented approach is that we obtain a notion of full 544 Chapter 10. Pseudo-differential operators on compact Lie groups

(global) symbols compared with only principal symbols available in the standard theory via localisations. In our analysis on a Lie group G, at some point we have to make a choice whether to work with left- or right-convolution kernels. Since left-invariant operators on C∞(G) correspond to right-convolutions f &→ f ∗ k, once we decide to identify the Lie algebra g of G with the left-invariant vector fields on G, it becomes most natural to work with right-convolution kernels in the sequel, and to define symbols as we do in Definition 10.4.3 . It is also known that globally defined symbols of pseudo-differential operators can be introduced on manifolds in the presence of a connection which allows one to use a suitable globally defined phase function, see e.g. [ 148 , 97 , 106 ]. However, on compact Lie groups the use of the group structure allows one to develop a theory parallel to those of Rn and Tn owing to the fact that the Fourier analysis is well adopted to the underlying representation theory. Some elements of such a theory were discussed in [ 125 , 126 ] as well as in the PhD thesis of V. Turunen [ 136 ]. However, here we present the finite dimensional symbols and we do not rely on the exponential mapping thus providing a genuine global analysis in terms of the Lie group itself. We also note that the case of the compact commutative group Tn is also recovered from this general point of view, however a more advanced analysis is possible in this case because of the close relation between Tn and Rn. Unless specified otherwise, in this chapter G will stand for a general compact Lie group, and dµ G will stand for the (normalised) Haar measure on G, i.e. the unique regular Borel probability measure which is left-translation-invariant:

f(x) d µG(x) = f(yx ) d µG(x) !G !G for all f ∈ C(G) and y ∈ G. Then also

−1 f(x) d µG(x) = f(xy ) d µG(x) = f(x ) d µG(x), !G !G !G see Remark 7.4.4 . Usually we abbreviate d µG(x) to d x since this should cause no confusion.

10.2 Fourier series on compact Lie groups

We begin with the Fourier series on a compact group G. Definition 10.2.1 (Rep (G) and G). Let Rep( G) denote the set of all strongly continuous irreducible unitary representations of G. In the sequel, whenever we mention unitary representations (of a compact Lie group G), we always mean strongly continuous irreducible unitary representations, which are then automat- ically smooth. Let G denote the unitary dual of G, i.e. the set of equivalence classes of irreducible unitary representations from Rep( G), see Definitions 6.3.18 10.2. Fourier series on compact Lie groups 545 and 7.5.7 . Let [ ξ] ∈ G denote the equivalence class of an irreducible unitary representation ξ : G → U (H ); the representation space H is finite-dimensional since ξ ξ G is compact (see Corollary 7.5.6 ), and we set dim( ξ) = dim Hξ.

We will always equip a compact Lie group G with the Haar measure µG, i.e. the uniquely determined bi-invariant Borel regular probability measure, see p p Remark 7.4.4 . For simplicity, we will write L (G) for L (µG), G f dx for G f dµG, etc. First we collect several definitions scattered over previous" chapters in" different forms. Definition 10.2.2 (Fourier coefficients). Let us define the Fourier coefficient f(ξ) ∈ End( H ) of f ∈ L1(G) by ξ

f(ξ) := f(x) ξ(x)∗ dx; (10.2) !G more precisely,

(f(ξ)u, v ) = f(x) ( ξ(x)∗u, v ) dx = f(x) ( u, ξ (x)v) dx Hξ Hξ Hξ !G !G for all u, v ∈ H ξ, where ( ·, ·)Hξ is the inner product of Hξ. Remark 10.2.3 . Notice that ξ(x)∗ = ξ(x)−1 = ξ(x−1). Remark 10.2.4 ( Fourier coeﬃcients on Tn as a group) . Let G = Tn. Let us naturally identify End( C) with C, and U(C) with {z ∈ C : |z| = 1 }. For each k ∈ Zn, i2 πx ·k we deﬁne ek : G → U (C) by ek(x) := e . Then

−i2 πx ·k f(k) := f(ek) = f(x) e dx !Tn is the usual Fourier coeﬃcient of f ∈ L1(Tn). Remark 10.2.5 ( Intertwining isomorphisms) . Let U ∈ Hom( η, ξ ) be an intertwining isomorphism , i.e. let U : Hη → H ξ be a bijective unitary linear mapping such that Uη (x) = ξ(x)U for all x ∈ G. Then we have

−1 f(η) = U f(ξ) U ∈ End( Hη). (10.3) Proposition 10.2.6 (Inner automorphisms). For u ∈ G, consider the inner automorphisms −1 φu = ( x &→ u xu ) : G → G. Then for all ξ ∈ Rep( G) we have

∗ f ◦ φu(ξ) = ξ(u) f(ξ) ξ(u) . (10.4) 546 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

Proof. We can calculate

−1 ∗ f ◦ φu(ξ) = f(u xu ) ξ(x) dx !G = f(x) ξ(uxu −1)∗ dx !G = ξ(u) f(x) ξ(x)∗ dx ξ (u)∗ !G = ξ(u) f(ξ) ξ(u)∗, which gives ( 10.4 ). Proposition 10.2.7 (Convolutions). If f, g ∈ L1(G) then

f!∗ g = g f. Proof. If ξ ∈ Rep( G) then

f!∗ g(ξ) = (f ∗ g)( x) ξ(x)∗ dx !G = f(xy −1)g(y) d y ξ (x)∗ dx !G !G = g(y) ξ∗(y) f(xy −1) ξ(xy −1)∗ dx dy !G !G = g(ξ) f(ξ), completing the proof. Remark 10.2.8 . The product g f in Proposition 10.2.7 usually differs from f g because f(ξ), g(ξ) ∈ End( H ) are operators unless G is commutative when they are ξ scalars (see Corollary 6.3.26 ) and hence commute. This order exchange is due to the definition of the Fourier coefficients ( 10.2 ), where we chose the integration of the function with respect to ξ(x)∗ instead of ξ(x). This choice actually serves us well, as we chose to identify the Lie algebra g with left-invariant vector fields on the Lie group G: namely, a left -invariant continuous linear operator A : C∞(G) → C∞(G) can be presented as a right -convolution operator Ca = ( f &→ f ∗ a), resulting in convenient expressions like

CaCbf = a b f. However, in Remark 10.4.13 we will still explain what would happen had we chosen another deﬁnition for the Fourier transform.

Proposition 10.2.9 (Diﬀerentiating the convolution). Let Y ∈ g and let DY : C∞(G) → C∞(G) be deﬁned by d D f(x) = f(x exp( tY )) . Y dt t=0

10.2. Fourier series on compact Lie groups 547

∞ Let f, g ∈ C (G). Then DY (f ∗ g) = f ∗ DY g. Proof. We have

d D (f ∗ g)( x) = f(y) g(y−1x exp( tY )) dy = f ∗ D g(x). Y ! Y G dt t=0

We now summarise properties of the Fourier series as a corollary to the Peter–Weyl theorem 7.5.14 : Corollary 10.2.10 (Fourier series). If ξ : G → U( d) is a unitary matrix representation then f(ξ) = f(x) ξ(x)∗ dx ∈ Cd×d !G has matrix elements

f(ξ)mn = f(x) ξ(x)nm dx ∈ C, 1 ≤ m, n ≤ d. !G If here f ∈ L2(G) then

f(ξ)mn = ( f, ξ (x)nm)L2(G), (10.5) and by the Peter–Weyl Theorem 7.5.14 we have

f(x) = dim( ξ) Tr ξ(x) f(ξ) # $ % [ξ]∈G d

= dim( ξ) ξ(x)nm f(ξ)mn (10.6) # # [ξ]∈G m,n =1 for almost every x ∈ G, where the summation is understood so that from each class [ξ] ∈ G we pick just (any) one representative ξ ∈ [ξ]. The particular choice of a representation from the representation class is irrelevant due to formula ( 10.3 ) and the presence of the trace in ( 10.6 ). The convergence in ( 10.6 ) is not only pointwise almost everywhere on G but also in the space L2(G). Example. For f ∈ L2(Tn), we get

f(x) = ei2 πx ·k f(k), #Zn k∈ −i2 πx ·k C where f(k) = Tn f(x) e dx is as in Remark 10.2.4 . Here was identiﬁed with C1×1. " Finally, we record a useful formula for representations: 548 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

Remark 10.2.11 . Let e ∈ G be the neutral element of G and let ξ be a unitary matrix representation of G. The unitarity of the representation ξ implies the identity

−1 −1 δmn = ξ(e)mn = ξ(x x)mn = ξ(x )mk ξ(x)kn = ξ(x)km ξ(x)kn . #k #k Similarly,

δmn = ξ(x)mk ξ(x)nk . #k

Here, as usual, δmn is the Kronecker delta: δmn = 1 for m = n, and δmn = 0 otherwise.

10.3 Function spaces on the unitary dual

In this section we lay down a functional analytic foundation concerning the function spaces that will be useful in the sequel. In particular, distribution space S′(G) is of importance since it provides a distributional interpretation for series on G. 10.3.1 Spaces on the group G

We recall Deﬁnition 8.3.45 of the Laplace operator LG on a Lie group G:

Remark 10.3.1 ( Laplace operator L = LG). The Laplace operator LG is a second order negative definite bi-invariant partial differential operator on G corresponding to the Casimir element of the universal enveloping algebra U(g) of the Lie algebra g of G. If G is equipped with the unique (up to a constant) bi-invariant Riemannian metric, LG is its Laplace–Beltrami operator. We will often denote the Laplace operator simply by L if there is no need to emphasize the group G in the notation. We refer to Section 8.3.2 for a discussion of its main properties. In Definition 5.2.4 we defined Ck mappings on a manifold. These can be also characterised globally: n Exercise 10.3.2. Let n = dim G and let {Yj}j=1 be a basis of the Lie algebra g of k α α α1 αn G. Show that f ∈ C (G) if and only if ∂ f ∈ C(G) for all ∂ = Y1 · · · Yn for all |α| ≤ k, or if and only if Lf ∈ C(G) for all L ∈ U (g) of degree ≤ k. ∞ k Exercise 10.3.3. Show that f ∈ C (G) if and only if ( −L G) f ∈ C(G) for all k ∈ N. Show that f ∈ C∞(G) if and only if Lf ∈ C(G) for all L ∈ U (g). (Hint: use a-priori estimates from Theorem 2.6.9 .) We can recall from Remark 5.2.15 the definition of the space D′(M) of distributions on a compact manifold M. Let us be more precise in the case of a compact Lie group G: Definition 10.3.4 (Distributions D′(G)). We define the space of distributions D′(G) as the space of all continuous linear functionals on C∞(G). This means that u ∈ D′(G) if it is a functional u : C∞(G) → C such that 10.3. Function spaces on the unitary dual 549

1. u is linear, i.e. u(αϕ + βψ ) = αu (ϕ) + βu (ψ) for all α, β ∈ C and all ϕ, ψ ∈ C∞(G); ∞ 2. u is continuous, i.e. u(ϕj) → u(ϕ) in C whenever ϕj → ϕ in C (G), as j → ∞. 1 ∞ 2 α α α Here ϕj → ϕ in C (G) if e.g. ∂ ϕj → ∂ ϕ for all ∂ ∈ U (g), as j → ∞. ′ ′ We deﬁne the convergence in the space D (G) as follows. Let uj, u ∈ D (G). ′ We will say that uj → u in D (G) as j → ∞ if uj(ϕ) → u(ϕ) in C as j → ∞, for all ϕ ∈ C∞(G). ′ ∞ Deﬁnition 10.3.5 (Duality ,· , ·- G). Let u ∈ D (G) and ϕ ∈ C (G). We denote

,u, ϕ -G := u(ϕ). If u ∈ Lp(G), 1 ≤ p ≤ ∞, we can identify u with a distribution in D′(G) (which we will continue to denote by u) in a canonical way by

,u, ϕ -G = u(ϕ) := u(x) ϕ(x) d x, !G where d x is the Haar measure on G. p Exercise 10.3.6. Let 1 ≤ p ≤ ∞. Show that if uj → u in L (G) as j → ∞ then ′ uj → u in D (G) as j → ∞. Remark 10.3.7 ( Derivations in D′(G)). Similar to operations on distributions in Rn described in Section 1.3.2 , we can define different operations on distributions on G. For example, for Y ∈ g, we can differentiate u ∈ D ′(G) with respect to the vector field Y by defining (Y u )( ϕ) := −u(Y ϕ ), ∞ for all ϕ ∈ C (G). Here the derivative Y ϕ = DY ϕ is as in Proposition 10.2.9 . Similarly, if ∂α ∈ U (g) is a differential operator of order |α|, we define

(∂αu)( ϕ) := ( −1) |α|u(∂αϕ), for all ϕ ∈ C∞(G). Exercise 10.3.8. Show that ∂αu ∈ D ′(G) and that ∂α : D′(G) → D ′(G) is continuous. Deﬁnition 10.3.9 (Sobolev space Hs(G)). First let us note that the Laplacian 1/2 s L = LG is symmetric and I − L is positive. Denote Ξ := ( I − L ) . Then Ξ ∈ L(C∞(G)) and Ξ s ∈ L (D′(G)) for every s ∈ R. Let us deﬁne

s s ∞ (f, g )Hs(G) := (Ξ f, Ξ g)L2(G) (f, g ∈ C (G)) .

1For a general setting on compact manifolds see Remark 5.2.15 . 2Exercise 10.3.2 provides more options. 550 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

∞ 1/2 The completion of C (G) with respect to the norm f &→ . f.Hs(G) = ( f, f )Hs(G) gives us the Sobolev space Hs(G) of order s ∈ R. This is the same space as that in Deﬁnition 5.2.16 on general manifolds, or as the Sobolev space obtained using any smooth partition of unity on the compact manifold G, by Corollary 5.2.18 . The operator Ξ r is a Sobolev space isomorphism Hs(G) → Hs−r(G) for every r, s ∈ R. Exercise 10.3.10. Show that if Y ∈ g, then the diﬀerentiation with respect to Y is a bounded linear operator from Hk(G) to Hk−1(G) for all k ∈ N. Extend this to k ∈ R. Remark 10.3.11 ( Sobolev spaces and C∞(G)). We have

H2k(G) = C∞(G). (10.7) k&∈N

2k k This can be seen locally, since H (G) = domain(( −L G) ) and LG is elliptic, so that ( 10.7 ) follows from the local a-priori estimates in Theorem 2.6.9 . Since the analysis of Sobolev spaces is closely intertwined with spaces on G, we study them also in the next section, in particular we refer to Remark 10.3.24 for their characterisation on the Fourier transform side.

10.3.2 Spaces on the dual G Since we will be mostly using the “right” Peter–Weyl theorem (Theorem 7.5.14 ), we may simplify the notation slightly, also adopting it to the analysis of pseudo- diﬀerential operators from the next sections. Thus, the space

dim( ξ) dim( ξ) ξij : ξ = ( ξij )i,j =1 , [ξ] ∈ G '( ) is an orthonormal basis for L2(G), and the space

ξ 2 H := span {ξij : 1 ≤ i, j ≤ dim( ξ)} ⊂ L (G) is πR-invariant, ξ ∼ πR|Hξ , and

L2(G) = Hξ. [ξ*]∈G

By choosing an unitary matrix representation from each equivalence class [ ξ] ∈ G, we can identify Hξ with Cdim( ξ)×dim( ξ) by choosing a basis in the linear space Hξ. ξ Remark 10.3.12 ( Spaces H and Hξ). We would like to point out a diﬀerence ξ between spaces H and Hξ to eliminate any confusion. Recall that if ξ ∈ Rep( G), then ξ is a mapping ξ : G → U (Hξ), where Hξ is the representation space of ξ, ξ 2 with dim Hξ = dim( ξ). On the other hand, the space H ⊂ L (G) is the span of 10.3. Function spaces on the unitary dual 551 the matrix elements of ξ and dim Hξ = (dim( ξ)) 2. In the notation of right and left Peter–Weyl theorem in Theorem 7.5.14 and Remark 7.5.16 , we have

dim( ξ) dim( ξ) ξ ξ ξ H = Hi, · = H·,j . *i=1 *j=1

ξ Informally, spaces Hξ can be viewed as “columns/rows” of H , because for example ξ(x)v ∈ H ξ for every v ∈ H ξ. We recall the important property of the Laplace operator on spaces Hξ from Theorem 8.3.47 : Theorem 10.3.13 (Eigenvalues of the Laplacian on G). For every ξ ∈ G the space ξ H is an eigenspace of L and −L | ξ = λ I, for some λ ≥ 0. G G H ξ ξ Exercise 10.3.14. Show that Hξ ⊂ C∞(G) for all ξ ∈ Rep( G). (Hint: Use Theorem 10.3.13 and the ellipticity of LG.) From Definition 7.6.11 we recall the Hilbert space L2(G) which we can now describe as follows. But first, we can look at the space of all mappings on G: Definition 10.3.15 (Space M(G)). The space M(G) consists of all mappings

∞ m×m F : G → L(Hξ) ⊂ C , + + [ξ]∈G m=1 satisfying F ([ ξ]) ∈ L (Hξ) for every [ ξ] ∈ G. In matrix representations, we can view F ([ ξ]) ∈ Cdim( ξ)×dim( ξ) as a dim( ξ) × dim( ξ) matrix. The space L2(G) consists of all mappings F ∈ M(G) such that

2 2 || F || := dim( ξ) .F ([ ξ]) . < ∞, L2(G) HS [ξ#]∈G where ∗ || F ([ ξ]) || HS = Tr( F ([ ξ]) F ([ ξ]) ) ( stands for the Hilbert–Schmidt norm of the linear operator F ([ ξ]), see Deﬁnition B.5.42 . Thus, the space

2 L (G) = F : G → L(Hξ), F : [ ξ] &→ L (Hξ) :  + [ξ]∈G  2 2 || F || := dim( ξ) .F ([ ξ]) . < ∞ (10.8) L2(G) HS  [ξ#]∈G  552 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups is a Hilbert space with the inner product

∗ (E, F )L2(G) := dim( ξ) Tr( E([ ξ]) F ([ ξ]) ). [ξ#]∈G

Remark 10.3.16 ( Mappings on G or on Rep (G)?) . We note that because the representations in G are all unitary, and because of the Hilbert–Schmidt norm that we use, we can write F (ξ) instead of F ([ ξ]), so that F is deﬁned on Rep( G) instead of G. Thus, to simplify the notation we can write F (ξ) with a convention that F ∈ M(G) if F (ξ) = F (η) whenever ξ ∼ η (i.e. whenever [ ξ] = [ η]). Proposition 10.3.17 (Parseval’s identity). Let f, g ∈ L2(G). Then we have

∗ 2 (f, g )L (G) = dim( ξ) Tr f(ξ) g(ξ) = ( f(ξ), g(ξ)) L2(G). # $ % [ξ]∈G Consequently, 2 || f|| L (G) = || f|| L2(G). Proof. Writing f and g as Fourier series ( 7.3) in Corollary 7.6.10 , we obtain

(f, g )L2(G) =

= dim( ξ) dim( η) Tr ξ(x) f(ξ) Tr( η(x) g(η)) d x ! # G $ % [ξ],[η]∈G = dim( ξ) dim( η) Tr ξ(x) f(ξ) Tr( η(x)∗ g(η)∗) d x ! # G $ % [ξ],[η]∈G dim( ξ) dim( η) = dim( ξ) dim( η) ξ(x) f(ξ) η(x) g(η)∗ dx ! ij ji lk lk # G i,j#=1 k,l#=1 [ξ],[η]∈G dim( ξ) dim( η) ∗ = dim( ξ) dim( η) (ξ(x)ij , η (x)lk )L2(G)f(ξ)ji g(η)lk . # i,j#=1 k,l#=1 [ξ],[η]∈G From this, by Lemma 7.5.12 , we obtain

dim( ξ) ∗ (f, g )L2(G) = dim( ξ) f(ξ)ji g(ξ)ij # i,j#=1 [ξ]∈G = dim( ξ) Tr f(ξ) g(ξ)∗ , # $ % [ξ]∈G ﬁnishing the proof. 10.3. Function spaces on the unitary dual 553

We now define a scale ,ξ- on Lie groups that is useful in measuring the growth on G and that is associated to the eigenvalues of the Laplace operator L = LG from Remark 10.3.1 and Theorem 10.3.13 . Definition 10.3.18 (Definition of ,ξ- on Lie groups). Let ξ ∈ Rep( G), so that vw ξ : G → U (Hξ). Given v, w ∈ H ξ, the function ξ : G → C defined by

vw ξ (x) := ,ξ(x)v, w -Hξ is not only continuous but even C∞-smooth 3. Let span( ξ) denote the linear span vw 4 of {ξ : v, w ∈ H ξ} . If ξ ∼ η then span( ξ) = span( η); consequently , we may write span[ ξ] := span( ξ) ⊂ C∞(G). It follows from Theorem 10.3.13 that

vw vw −L ξ (x) = λ[ξ]ξ (x), where λ[ξ] ≥ 0, and we denote

1/2 ,ξ- := (1 + λ[ξ]) . (10.9) A relation between the dimension of the representation ξ and the weight ,ξ- is given by Proposition 10.3.19 (Dimension and eigenvalues). There exists a constant C > 0 such that the inequality dim G dim( ξ) ≤ C,ξ- 2 holds for all ξ ∈ Rep( G). Proof. We note by Theorem 10.3.13 that ,ξ- is an eigenvalue of the ﬁrst order 1/2 ξ elliptic operator (1 − L G) . The corresponding eigenspace H has the dimension dim( ξ)2. Denoting n = dim G, the Weyl formula for the counting function of the 1/2 eigenvalues of (1 − L G) yields

2 n n−1 dim( ξ) = C0λ + O(λ ) %ξ#&≤ λ as λ → ∞. This implies the estimate dim( ξ)2 ≤ C,ξ-n for large ,ξ-, implying the statement. Deﬁnition 10.3.20 (Space S(G)). The space S(G) consists of all mappings H ∈ N M(G) such that for all k ∈ we have k pk(H) := dim( ξ) ,ξ- || H(ξ)|| HS < ∞. (10.10) [ξ#]∈G

3See Exercise 10.3.14 . 4In matrix representations this is the space Hξ. 554 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

We will say that Hj ∈ S (G) converges to H ∈ S (G) in S(G), and write Hj → H N in S(G) as j → ∞, if pk(Hj − H) → 0 as j → ∞ for all k ∈ , i.e. if k dim( ξ) ,ξ- || Hj(ξ) − H(ξ)|| HS → 0 as j → ∞, [ξ#]∈G for all k ∈ N. We can take diﬀerent families of seminorms on S(G). In particular, the following equivalence will be of importance since it provides a more direct relation with Sobolev spaces on G:

Proposition 10.3.21 (Seminorms on S(G)). For H ∈ M(G), let pk(H) be as in (10.10 ), and let us deﬁne a family q (H) by k 1/2 k 2 qk(H) :=  dim( ξ) ,ξ- || H(ξ)|| HS  . # [ξ]∈G 

Then pk(H) < ∞ for all k ∈ N if and only if qk(H) < ∞ for all k ∈ N.

Proof. Let H ∈ M(G). We claim that pk(H) ≤ Cq 4k(H) if k is large enough, and that q (H) ≤ p (H). Indeed, by the Cauchy–Schwartz inequality (i.e. H¨older’s 2k k inequality similar to that in Lemma 3.3.28 where we use the discreteness of G) we can estimate 1/2 −k 1/2 2k pk(H) = (dim( ξ)) ,ξ- (dim( ξ)) ,ξ- || H(ξ)|| HS ' )' ) [ξ#]∈G 1/2 1/2 −2k 4k 2 ≤  dim( ξ) ,ξ-   dim( ξ) ,ξ- || H(ξ)|| HS  # # [ξ]∈G  [ξ]∈G  ≤ Cq 4k(H), if we choose k large enough and use Proposition 10.3.19 . Conversely, we have

2 2 2k 2 pk(H) = (dim( ξ)) ,ξ- || H(ξ)|| HS + [ξ]=[#η]∈G k k + dim( ξ) dim( η) ,ξ- ,η- || H(ξ)|| HS || H(η)|| HS [ξ#](=[ η] 2k 2 ≥ dim( ξ) ,ξ- || H(ξ)|| HS [ξ#]∈G 2 = q2k(H) . 10.3. Function spaces on the unitary dual 555

As a corollary of Proposition 10.3.21 we have Corollary 10.3.22. For H ∈ M(G), we have −k dim( ξ) ξ! || H(ξ)|| HS < ∞ [ξ ]∈G for some k ∈ N if and only if

−l 2 dim( ξ) ξ! || H(ξ)|| HS < ∞ [ξ ]∈G for some l ∈ N. Let us now summarise properties of the Fourier transform on L2(G). Theorem 10.3.23 (Fourier inversion). Let G be a compact Lie group. The Fourier 2 2 transform f #→ F Gf = f deﬁnes a surjective isometry L (G) → L (G). The inverse Fourier transform is given by −1 (FG H)( x) = dim( ξ) Tr( ξ(x) H(ξ)) , (10.11) [ξ ]∈G and we have −1 −1 FG ◦ F G = id and FG ◦ F G = id 2 2 on L (G) and L (G), respectively. Moreover, the Fourier transform FG is unitary, F ∗ = F −1, and for any H ∈ S (G) we have G G ∗ −1 ∗ −1 −1 (FG H)( x) = FG (H )( x ) = FG H (x) ! " for all x ∈ G, where H∗(ξ) := H(ξ)∗ for all ξ ∈ Rep( G). Proof. In Theorem 7.6.13 we have already shown that the Fourier transform is a surjective isometry L2(G) → L2(G). By Corollary 7.6.10 the inverse Fourier transform is given by ( 10.11 ). Let us show the last part. Let f ∈ C∞(G) and H ∈ S (G). Then we have ∗ 2 (f, FG H)L (G) = ( FGf, H )L2(G) = dim( ξ) Tr f(ξ) H(ξ)∗ # $ [ξ]∈G = dim( ξ) Tr f(x) ξ∗(x) d x H (ξ)∗ %&G ' [ξ ]∈G

= f(x)  dim( ξ) Tr ξ(x−1) H(ξ)∗  dx &G   [ξ ]∈G ! "

−1 ∗ −1  = f(x) FG (H )( x ) d x, &G 556 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

∗ −1 ∗ −1 which implies FG H(x) = FG (H )( x ). Finally, the unitarity of the Fourier transform follows from continuing the calculation:

∗ ∗ ∗ (f, FG H)L2(G) = dim( ξ) Tr f(x) ξ (x) d x H (ξ) %&G ' [ξ ]∈G

= f(x)  dim( ξ) Tr( ξ∗(x) H(ξ)∗) dx &G   [ξ ]∈G   = f(x)  dim( ξ) Tr( ξ(x) H(ξ))  dx &G   [ξ ]∈G   = f(x) F −1H (x) d x. & G G ! "

Remark 10.3.24 ( Sobolev space Hs(G)). On the Fourier transform side, in view of Theorem 10.3.23 , the Sobolev space Hs(G) can be characterised by

Hs(G) = {f ∈ D ′(G) : ξ!sf(ξ) ∈ L2(G)}.

We also note that since G is compact, for s = 2 k and k ∈ N, we have that 2k k 2 f ∈ H (G) if ( −L G) f ∈ L (G). By Theorem 10.3.25 , the Fourier transform FG is a continuous bijection from H2k(G) to the space

2k 2 F ∈ M(G) : dim( ξ) ξ! || F (ξ)|| HS < ∞.   [ξ]∈G   We now analyse the Fourier transforms on C∞(G) and S(G) in preparation for their extension to the spaces of distributions. Theorem 10.3.25 (Fourier transform on C∞(G) and S(G)). The Fourier transform F : C∞(G) → S (G) and its inverse F −1 : S(G) → C∞(G) are continuous, G G satisfying F −1 ◦ F = id and F ◦ F −1 = id on C∞(G) and S(G), respectively. G G G G ∞ 2k Proof. By ( 10.7 ) in Remark 10.3.11 , writing C (G) = k∈N H (G), the smoothness f ∈ C∞(G) is equivalent to f ∈ H2k(G) for all k.∈ N, by Remark 10.3.24 . This means that 2k 2 dim( ξ) ξ! || f|| HS < ∞ [ξ]∈G ∞ for all k ∈ N. Hence FGf ∈ S (G). Consequently, fj → f in C (G) implies that f → f in H2k(G) for all k ∈ N. Taking the Fourier transform we see that j 10.3. Function spaces on the unitary dual 557

q2k(FGfj − F Gf) → 0 as j → ∞ for all k ∈ N, which implies that FGfj → F Gf in S(G). Inverting this argument implies the continuity of the inverse Fourier transform F −1 from S(G) to C∞(G). The last part of the theorem follows from G Theorem 10.3.23 . Corollary 10.3.26 (S(G) is a Montel nuclear space). The space S(G) is a Montel space and a nuclear space. Proof. This follows from the same properties of C∞(G) because S(G) is homeo- morphic to C∞(G), a homeomorphism given by the Fourier transform. See Exer- cises B.3.35 and B.3.51 .

Deﬁnition 10.3.27 (Space S′(G)). The space S′(G) of slowly increasing or tempered distributions on the unitary dual G is deﬁned as the space of all H ∈ M(G) for which there exists some k ∈ N such that −k dim( ξ) ξ! || H(ξ)|| HS < ∞. [ξ ]∈G

′ ′ The convergence in S (G) is deﬁned as follows. We will say that Hj ∈ S (G) ′ ′ N converges to H ∈ S (G) in S (G) as j → ∞, if there exists some k ∈ such that

−k dim( ξ) ξ! || Hj(ξ) − H(ξ)|| HS → 0 [ξ ]∈G as j → ∞. Lemma 10.3.28 (Trace and Hilbert–Schmidt norm). Let H be a Hilbert space and 5 let A, B ∈ S2(H) be Hilbert–Schmidt operators . Then

|Tr( AB )| ≤ || A|| HS || B|| HS .

Proof. By the Cauchy–Schwartz inequality for the (Hilbert–Schmidt) inner product on S2(H) we have

∗ |Tr( AB )| = | A, B !HS | ≤ || A|| HS || B|| HS , proving the required estimate.

′ Deﬁnition 10.3.29 (Duality · , ·! G). Let H ∈ S (G) and h ∈ S (G). We denote H, h !G := dim( ξ) Tr( H(ξ) h(ξ)) . (10.12) [ξ ]∈G

5Recall Deﬁnition B.5.42 . 558 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

The sum is well-deﬁned in view of

H, h !G ≤ dim( ξ) | Tr( H(ξ) h(ξ)) | [ξ ]∈G

≤ dim( ξ) (H(ξ)(HS (h(ξ)(HS [ξ ]∈G 1/2 −k 2 ≤  dim( ξ) ξ! (H(ξ)(HS  × [ξ]∈G  1/2 k 2 × dim( ξ) ξ! (h(ξ)(HS  [ξ]∈G  < ∞, which is ﬁnite in view of Proposition 10.3.21 and Corollary 10.3.22 . Here we used Lemma 10.3.28 and the Cauchy–Schwarz inequality (see Lemma 3.3.28 ) in the ′ estimate. The bracket · , ·! G in ( 10.12 ) introduces the duality between S (G) and ′ S(G) so that S (G) is the dual space to S(G). Proposition 10.3.30 (Sequential density of S(G) in S′(G)). The space S(G) is sequentially dense in S′(G). Proof. We use the standard approximation in sequence spaces by cutting the sequence to obtain its approximation. Thus, let H ∈ S ′(G). For ξ ∈ G we deﬁne

H(ξ), if ξ! ≤ j, H (ξ) := j 3 0, if ξ! > j.

Let k ∈ N be such that

−k dim( ξ) ξ! || H(ξ)|| HS < ∞. (10.13) [ξ ]∈G

′ Clearly Hj ∈ S (G) for all j, and Hj → H in S (G) as j → ∞, because

−k dim( ξ) ξ! || Hj(ξ) − H(ξ)|| HS = [ξ ]∈G −k = dim( ξ) ξ! || H(ξ)|| HS → 0 [ξ]∈G, $ξ%>j as j → 0, in view of the convergence of the series in ( 10.13 ) and the fact that the eigenvalues of the Laplacian on G are increasing to inﬁnity. 10.3. Function spaces on the unitary dual 559

We now establish a relation of the duality brackets · , ·! G and · , ·! G with the Fourier transform and its inverse. This will allow us to extend the actions of the Fourier transforms to the spaces of distributions on G and G. Proposition 10.3.31 (Dualities and Fourier transforms). Let ϕ ∈ C∞(G), h ∈ S(G). Then we have the identities −1 F Gϕ, h !G = ϕ, ι ◦ F G h G 4 5 and −1 FG h, ϕ G = h, FG(ι ◦ ϕ)!G, where (ι ◦ ϕ)( x) = ϕ(x−1)4. 5 Proof. We can calculate

F Gϕ, h !G = dim( ξ) Tr( ϕ(ξ) h(ξ)) [ξ ]∈G = dim( ξ) Tr ϕ(x) ξ∗(x) d x h (ξ) %&G ' [ξ ]∈G

= ϕ(x) dim( ξ) Tr ξ(x−1) h(ξ)  dx &G   [ξ ]∈G ! "

(10 .11 )  −1 −1  = ϕ(x) ( FG h)( x ) d x &G −1 = ϕ, ι ◦ F G h G. 4 5 Similarly, we have

F −1h, ϕ = (F −1h)( x) ϕ(x) d x G G & G 4 5 G

=  dim( ξ) Tr( ξ(x) h(ξ))  ϕ(x) d x &G   [ξ ]∈G   = dim( ξ) Tr ξ(x−1) ϕ(x−1) d x h(ξ) %3 &G 6 ' [ξ ]∈G

= dim( ξ) Tr ξ∗(x) ( ι ◦ ϕ)( x) d x h(ξ) %3 &G 6 ' [ξ ]∈G = dim( ξ) Tr( ι ◦ ϕ(ξ) h(ξ)) [ξ ]∈G 7 = h, FG(ι ◦ ϕ)!G, completing the proof. 560 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

Proposition 10.3.31 motivates the following definitions: Definition 10.3.32 (Fourier transforms on D′(G) and S′(G)). For u ∈ D ′(G), we define F u ≡ u ∈ S ′(G) by G −1 F Gu, h !G := u, ι ◦ F G h G, (10.14) 4 5 ′ −1 ′ for all h ∈ S (G). For H ∈ S (G), we define FG H ∈ D (G) by −1 FG H, ϕ G := H, FG(ι ◦ ϕ)!G, (10.15) 4 5 for all ϕ ∈ C∞(G). Theorem 10.3.33 (Well-defined and continuous). For u ∈ D ′(G) and H ∈ S ′(G), their forward and inverse Fourier transforms F u ∈ S ′(G) and F −1H ∈ D ′(G) G G are well-defined. Moreover, the mappings F : D′(G) → S ′(G) and F −1 : S′(G) → G G D′(G) are continuous, and −1 −1 FG ◦ F G = id and FG ◦ F G = id on D′(G) and S′(G), respectively. −1 ∞ Proof. For h ∈ S (G), its inverse Fourier transform satisfies FG h ∈ C (G) by −1 ∞ Theorem 10.3.25 . Therefore, ι ◦ F G h ∈ C (G), so that the definition in ( 10.14 ) −1 ∞ ∞ makes sense. Moreover, the mappings FG : S(G) → C (G) and ι : C (G) → ∞ C (G) are continuous, so that FGu is a continuous functional on S(G), implying that F u ∈ S ′(G). Let us now show that F : D′(G) → S ′(G) is continuous. G G Indeed, if u → u in D′(G), then u , ι ◦ F −1h → u, ι ◦ F −1h as j → ∞ for j j G G G G ′ every h ∈ S (G), implying that FG4uj → F Gu in 5 S (G4). 5 The proof that F −1H ∈ D ′(G) for every H ∈ S ′(G) is similar, as well as the G continuity of F −1 : S′(G) → D ′(G), and are left as Exercise 10.3.34 . G To show that F −1 ◦ F = id on D′(G), we take u ∈ D ′(G) and ϕ ∈ C∞(G), G G and calculate

−1 (10 .15 ) (FG ◦ F G)u, ϕ G = F Gu, FG(ι ◦ ϕ)!G 4 5 (10 .14 ) −1 = u, ι ◦ F G (FG(ι ◦ ϕ)) G Theorem 10 .3.25 4 5 = u, ϕ !G. Similarly, for H ∈ S ′(G) and h ∈ S (G), we have

−1 (10 .14 ) −1 −1 (FG ◦ F G )H, h G = FG H, ι ◦ F G h G 4 5 (10 .15 ) 4 −51 = H, FG ι ◦ ι ◦ F G h G Theorem 10 .3.25 4 ! "5 = H, h !G, completing the proof. 10.3. Function spaces on the unitary dual 561

−1 Exercise 10.3.34. Complete the proof of Theorem 10.3.33 by showing that FG H ∈ ′ ′ −1 ′ ′ D (G) for every H ∈ S (G) and that FG : S (G) → D (G) is continuous. Corollary 10.3.35 (Sequential density of C∞(G) in D′(G)). The space C∞(G) is sequentially dense in D′(G).

Proof. The statement follows from Proposition 10.3.30 saying that the space S(G) is sequentially dense in S′(G), and properties of the Fourier transform from The- orems 10.3.25 and 10.3.33 . 10.3.3 Spaces Lp(G) Deﬁnition 10.3.36 (Spaces Lp(G)). For 1 ≤ p < ∞, we will write Lp(G) ≡ 2 1 p p( − 2 ) ′ ℓ G, dim p for the space of all H ∈ S (G) such that # $ 1/p p 2 − 1 p ( p 2 ) || H|| Lp(G) :=  (dim( ξ)) || H(ξ)|| HS  < ∞. [ξ]∈G 

For p = ∞, we will write L∞(G) ≡ ℓ∞ G, dim −1/2 for the space of all H ∈ S ′(G) # $ such that −1/2 || H|| L∞(G) := sup (dim( ξ)) || H(ξ)|| HS < ∞. [ξ]∈G

p 2 − 1 We will usually write Lp(G) but the notation ℓp G, dim ( p 2 ) can be also used # $ to emphasize that these spaces have a structure of weighted sequence spaces on the discrete set G, with the weights given by the powers of the dimensions of the representations. Exercise 10.3.37. Prove that spaces Lp(G) are Banach spaces for all 1 ≤ p ≤ ∞.

Remark 10.3.38 . Two important cases of L2(G) = ℓ2 G, dim 1 and L1(G) = # $ ℓ1 G, dim 3/2 are deﬁned by the norms # $ 1/2 2 || H|| L2(G) :=  dim( ξ) || H(ξ)|| HS  , [ξ]∈G  which is already familiar from ( 10.8 ), and by

3/2 || H|| L1(G) := (dim( ξ)) || H(ξ)|| HS . [ξ ]∈G 562 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

We now discuss several properties of spaces Lp(G). We ﬁrst recall a result on the interpolation of weighted spaces from [ 14 , Theorem 5.5.1]: Theorem 10.3.39 (Interpolation of weighted spaces). Let us denote dµ0(x) = p p w0(x) d µ(x), dµ1(x) = w1(x) d µ(x), and write L (w) = L (w dµ) for the weight w. Suppose that 0 < p 0, p 1 < ∞. Then

p0 p1 p (L (w0), L (w1)) θ,p = L (w), where 0 < θ < 1, 1 = 1−θ + θ , and w = wp(1 −θ)/p 0 wpθ/p 1 . p p0 p1 0 1 From this we obtain: p Proposition 10.3.40 (Interpolation of L (G) spaces). Let 1 ≤ p0, p 1 < ∞. Then Lp0 (G), L p1 (G) = Lp(G), # $θ,p where 0 < θ < 1 and 1 = 1−θ + θ . p p0 p1

p 2 − 1 Proof. The statement follows from Theorem 10.3.39 if we regard Lp(G) = ℓp G, dim ( p 2 ) 2 #1 $ p − as a weighted sequence space over G with the weight given by dim( ξ) ( p 2 ). Lemma 10.3.41 (Hilbert–Schmidt norm of representations). If ξ ∈ Rep( G), then

|| ξ(x)|| HS = dim( ξ) 8 for every x ∈ G. Proof. We have

∗ 1/2 1/2 || ξ(x)|| HS = ( Tr( ξ(x) ξ(x) )) = Tr Idim( ξ) = dim( ξ), ! " 8 by the unitarity of the representation ξ. Proposition 10.3.42 (Fourier transforms on L1(G) and L1(G)). The Fourier transform F is a linear bounded operator from L1(G) to L∞(G) satisfying G 1 || f|| L∞(G) ≤ || f|| L (G).

−1 1 The inverse Fourier transform FG is a linear bounded operator from L (G) to L∞(G) satisfying −1 ∞ ||F G H|| L (G) ≤ || H|| L1(G).

∗ Proof. Using f(ξ) = G f(x) ξ(x) dx, by Lemma 10.3.41 we get 9 ∗ 1/2 || f(ξ)|| HS ≤ |f(x)| || ξ(x) || HS dx ≤ (dim( ξ)) || f|| L1(G). &G 10.3. Function spaces on the unitary dual 563

Therefore,

−1/2 1 || f|| L∞(G) = sup (dim( ξ)) || f(ξ)|| HS ≤ || f|| L (G). [ξ]∈G −1 On the other hand, using ( FG H)( x) = [ξ]∈G dim( ξ) Tr( ξ(x) H(ξ)), by Lemma 10.3.28 and Lemma 10.3.41 we have :

−1 |(FG H)( x)| ≤ dim( ξ) || ξ(x)|| HS || H(ξ)|| HS [ξ ]∈G 3/2 = (dim( ξ)) || H(ξ)|| HS [ξ ]∈G

= || H|| L1(G),

−1 ∞ from which we get ||F G H|| L (G) ≤ || H|| L1(G). 1 1 Theorem 10.3.43 (Hausdorﬀ–Young inequality). Let 1 ≤ p ≤ 2 and p + q = 1 . p p −1 p q Let f ∈ L (G) and H ∈ L (G). Then || f|| Lq (G) ≤ || f|| L (G) and ||F G H|| L (G) ≤

|| H|| p . L (G) Theorem 10.3.43 follows from the L1 → L∞ and L2 → L2 boundedness in Proposition 10.3.42 and in Proposition 10.3.17 , respectively, by the following interpolation theorem in [ 14 , Corollary 5.5.4] (which is also a consequence of Theorem 10.3.39 ):

Theorem 10.3.44 (Stein–Weiss interpolation). Let 1 ≤ p0, p 1, q 0, q 1 < ∞ and let

p0 q0 p1 q1 T : L (U, w 0 dµ) → L (V, w0 dν), T : L (U, w 1 dµ) → L (V, w1 dν), with norms M0 and M1, respectively.; Then ;

T : Lp(U, w dµ) → Lq(V, w dν) with norm M ≤ M 1−θM θ, where 1 = 1−θ + θ , 1 =<1−θ + θ , w = wp(1 −θ)/p 0 wpθ/p 1 0 1 p p0 p1 q q0 q1 0 1 p(1 −θ)/p 0 pθ/p 1 and w = w0 w1 . p

H, h !G := dim( ξ) Tr( H(ξ) h(ξ)) . [ξ ]∈G 564 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

Assume ﬁrst 1 < p < ∞. Then, if H ∈ Lp(G) and h ∈ Lq(G), then using Lemma 10.3.28 we get

H, h !G ≤ dim( ξ) (H(ξ)(HS (h(ξ)(HS = = = = [ξ]∈G 2 1 2 1 p − 2 q − 2 = (dim( ξ)) (H(ξ)(HS (dim( ξ)) (h(ξ)(HS [ξ ]∈G 1/p 2 1 p( p − 2 ) p ≤  (dim( ξ)) (H(ξ)(HS  × [ξ]∈G  1/q 2 1 q( q − 2 ) q × (dim( ξ)) (h(ξ)(HS  [ξ]∈G  = || H|| Lp(G) || h|| Lq (G), where we also used the discrete H¨older’s inequality (Lemma 3.3.28 ). Let now p = 1. In this case we have

3/2 −1/2 H, h !G ≤ (dim( ξ)) (H(ξ)(HS (dim( ξ)) (h(ξ)(HS = = = = [ξ]∈G ≤ || H|| L1(G) || h|| L∞(G).

We leave the other part of the proof as an exercise.

p α Remark 10.3.46 ( Sobolev spaces Lk(G)). If we use difference operators △ from Definition 10.7.1 , we can also define Sobolev spaces Lp(G), k ∈ N, on the unitary k dual G by p p α p Lk(G) = H ∈ L (G) : △ H ∈ L (G) for all |α| ≤ k . > ? 10.4 Symbols of operators

Let G be a compact Lie group. Let us endow D(G) = C∞(G) with the usual test function topology (which is the uniform topology of C∞(G); we refer the reader to Section 10.12 for some additional information on the topics of distributions and Schwartz kernels if more introduction is desirable). For a continuous linear ∞ ∞ ′ operator A : C (G) → C (G), let KA, L A, R A ∈ D (G × G) denote respectively 10.4. Symbols of operators 565 the Schwartz, left-convolution and right-convolution kernels, i.e.

Af (x) = KA(x, y ) f(y) d y &G −1 = LA(x, xy ) f(y) d y &G −1 = f(y) RA(x, y x) d y (10.16) &G in the sense of distributions. To simplify the notation in the sequel, we will often write integrals in the sense of distributions, with a standard distributional interpretation. Proposition 10.4.1 (Relations between kernels). We have

−1 RA(x, y ) = LA(x, xyx ), (10.17) as well as

−1 −1 RA(x, y ) = KA(x, xy ) and LA(x, y ) = KA(x, y x), with the standard distributional interpretation. Proof. Equality ( 10.17 ) follows directly from ( 10.16 ). The proof of the last two equalities is just a change of variables. Indeed, ( 10.16 ) implies that KA(x, y ) = −1 −1 −1 −1 RA(x, y x). Denoting v = y x, we have y = xv , so that KA(x, xv ) = −1 RA(x, v ). Similarly, KA(x, y ) = LA(x, xy ) from ( 10.16 ) and the change w = −1 −1 −1 xy yield y = w x, and hence KA(x, w x) = LA(x, w ). We also note that left-invariant operators on C∞(G) correspond to right-convolutions f #→ f ∗k. Since we identify the Lie algebra g of G with the left-invariant vector ﬁelds on G, it will be most natural to study right-convolution kernels in the sequel. Let us explain this in more detail: Remark 10.4.2 ( Left or right?) . For g ∈ D ′(G), deﬁne the respective left-convolution and right-convolution operators l(f), r (f) : C∞(G) → C∞(G) by

l(f)g := f ∗ g, r(f)g := g ∗ f.

In this notation, the relation between left- and right-convolution kernels of these convolution operators in the notation of ( 10.16 ) becomes Ll(f)(x, y ) = f(y) = Rr(f)(x, y ). Also, if Y ∈ g, then Proposition 10.2.9 implies that DY l(f) = l(f)DY . Let the respective left and right regular representations of G be denoted by πL, π R : G → U (L2(G)), i.e.

−1 πL(x)f(y) = f(x y),

πR(x)f(y) = f(yx ). 566 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

Operator A is

left − invariant if πL(x) A = A π L(x),

right − invariant if πR(x) A = A π R(x), for every x ∈ G. Notice that A is left − invariant ⇐⇒ right − convolution , right − invariant ⇐⇒ left − convolution . Indeed, we have, for example,

[πR(x)l(f)g]( z) = ( f ∗ g)( zx ) = f(y) g(y−1zx ) d y &G −1 = f(y) ( πR(x)g)( y z) d y &G = [ f ∗ πR(x)g]( z)

= [ l(f)πR(x)g]( z), so that πR(x)l(f) = l(f)πR(x), and similarly πL(x)r(f) = r(f)πL(x). The Lie algebras are often (but not always) identiﬁed with left-invariant vector ﬁelds, which are right-convolutions , that is why our starting choice in the sequel are right-convolution kernels . We refer to Remark 10.4.13 for a further discussion of left and right.

10.4.1 Full symbols We now deﬁne symbols of operators on G.

Deﬁnition 10.4.3 (Symbols of operators on G). Let ξ : G → U (Hξ) be an irreducible unitary representation. The symbol of a linear continuous operator A : C∞(G) → C∞(G) at x ∈ G and ξ ∈ Rep( G) is deﬁned as

σA(x, ξ ) := rx(ξ) ∈ End( Hξ), where rx(y) = RA(x, y ) is the right-convolution kernel of A as in ( 10.16 ). Hence

∗ σA(x, ξ ) = RA(x, y ) ξ(y) dy &G in the sense of distributions, and by Corollary 10.2.10 the right-convolution kernel can be regained from the symbol as well:

RA(x, y ) = dim( ξ) Tr ( ξ(y) σA(x, ξ )) , (10.18) [ξ ]∈G where this equality is interpreted distributionally. We now show that operator A can be represented by its symbol: 10.4. Symbols of operators 567

Theorem 10.4.4 (Quantization of operators). Let σA be the symbol of a continuous linear operator A : C∞(G) → C∞(G). Then

Af (x) = dim( ξ) Tr ξ(x) σA(x, ξ ) f(ξ) . (10.19) # $ [ξ]∈G for every f ∈ C∞(G) and x ∈ G. ∞ Proof. Let us deﬁne a right-convolution operator Ax0 ∈ L (C (G)) by its kernel

RA(x0, y ) = rx0 (y), i.e. by

−1 Ax0 f(x) := f(y) rx0 (y x) d y = ( f ∗ rx0 )( x). &G Thus

σAx0 (x, ξ ) = rx0 (ξ) = σA(x0, ξ ), so that by ( 10.6 ) we have @

Ax0 f(x) = dim( ξ) Tr ξ(x) Ax0 f(ξ) [ξ ]∈G # $

= dim( ξ) Tr ξ(x) σA(x0, ξ ) f(ξ) , # $ [ξ]∈G ! where we used that f ∗ rx0 = rx0 f by Proposition 10.2.7 . This implies the result, because Af (x) = Axf(x), for each ﬁxed x. @ Deﬁnition 10.4.3 and Theorem 10.4.4 justify the following notation:

Definition 10.4.5 (Pseudo-differential operators). For a symbol σA, the corresponding operator A defined by ( 10.19 ) will be also denoted by Op( σA). The operator defined by formula ( 10.19 ) will be called the pseudo-differential operator with symbol σA. If we fix representations to be matrix representations we can express all the formulae above in matrix components. Thus, if ξ : G → U(dim( ξ)) are irreducible unitary matrix representations then

dim( ξ) dim( ξ)

Af (x) = dim( ξ) ξ(x)nm  σA(x, ξ )mk f(ξ)kn  , m,n =1 k =1 [ξ]∈G   and as a consequence of ( 10.18 ) and Corollary 10.2.10 we also have formally:

dim( ξ)

RA(x, y ) = dim( ξ) ξ(y)nm σA(x, ξ )mn . (10.20) [ξ ]∈G m,n =1 568 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

Alternatively, setting Aξ (x)mn := ( A(ξmn ))( x), we have

dim( ξ)

σA(x, ξ )mn = ξkm(x) ( Aξ kn )( x), (10.21) k =1

1 ≤ m, n ≤ dim( ξ), which follows from the following theorem:

Theorem 10.4.6 (Formula for the symbol via representations). Let σA be the symbol of a continuous linear operator A : C∞(G) → C∞(G). Then for all x ∈ G and ξ ∈ Rep( G) we have ∗ σA(x, ξ ) = ξ(x) (Aξ )( x). (10.22) Proof. Working with matrix representations ξ : G → U(dim( ξ)), we have

dim( ξ)

ξkm(x) ( Aξ kn )( x) k =1 (10 .19 ) = ξkm(x) dim( η) Tr η(x) σA(x, η ) ξkn (η) k # $ [η]∈G @ = ξkm(x) dim( η) η(x)ij σA(x, η )jl ξkn (η)li k i,j,l [η]∈G @ = ξkm(x) ξ(x)kj σA(x, ξ )jn k,j Rem 10 .2.4 = σA(x, ξ )mn , where we take η = ξ if η ∈ [ξ] in the sum, so that ξkn (η)li = ξkn , η il !L2 by ( 10.5 ), which equals 1 if ξ = η, k = i and n = l, and zero otherwise. dim( ξ) @ Remark 10.4.7 ( Formula for symbol on Tn). Since in the case of the torus G = Tn n i2 πx ·k n by Remark 10.2.4 representations of T are given by ek(x) = e , k ∈ Z , formula ( 10.22 ) gives the formula for the toroidal symbol

−i2 πx ·k i2 πx ·k σA(x, k ) := σA(x, e k) = e (Ae )( x), k ∈ Zn, as in Theorem 4.1.4 . Remark 10.4.8 ( Symbol of the Laplace operator) . We note that by Theorem 10.3.13 the symbol of the Laplace operator L = LG on G is σL(x, ξ ) = −λ[ξ]Idim ξ, where Idim ξ is the identity mapping on Hξ and λ[ξ] are the eigenvalues of −L . Remark 10.4.9 ( Symbol as a mapping on G × G). The symbol of A ∈ L (C∞(G)) is a mapping σA : G × Rep( G) → End( Hξ), ξ∈Rep(A G) 10.4. Symbols of operators 569

where σA(x, ξ ) ∈ End( Hξ) for every x ∈ G and ξ ∈ Rep( G). However, it can be viewed as a mapping on the space G × G. Indeed, let ξ, η ∈ Rep( G) be equivalent via an intertwining isomorphism U ∈ Hom( ξ, η ): i.e. such that there exists a linear unitary bijection U : Hξ → H η such that U η (x) = ξ(x) U for every x ∈ G, that is η(x) = U −1 ξ(x) U. Then by Remark 10.2.5 we have f(η) = U −1 f(ξ) U, and hence also −1 σA(x, η ) = U σA(x, ξ ) U. Therefore, taking any representation from the same class [ ξ] ∈ G leads to the same operator A in view of the trace in formula ( 10.19 ). In this sense we may think that symbol σA is deﬁned on G × G instead of G × Rep( G). Remark 10.4.10 ( Symbol of right-convolution) . Notice that if A = ( f #→ f ∗ a) then RA(x, y ) = a(y) and hence

σA(x, ξ ) = a(ξ), and hence Af (ξ) = a(ξ) f(ξ) = σA(x, ξ ) f(ξ).

Proposition@ 10.4.11 (Symbol of left-convolution). If B = ( f #→ b ∗ f) is the left- convolution operator, then

−1 −1 LB(x, y ) = b(y), R B(x, y ) = LB(x, xyx ) = b(xyx ), and the symbol of B is given by

∗ σB(x, ξ ) = ξ(x) b(ξ) ξ(x). Exercise 10.4.12. Prove Proposition 10.4.11 . (Hint: use ( 10.4 ) and ( 10.17 ).) Remark 10.4.13 ( What if we started with left-convolution kernels?) . What if we had chosen right -invariant vector ﬁelds and corresponding left -convolution operators as the starting point of the Fourier analysis? Let us deﬁne another “Fourier transform” by

πf (ξ) := f(x) ξ(x) d x. &G ∞ ∞ Then πf∗g = πf πg, and a continuous linear operator A : C (G) → C (G) can be presented by

Af (x) = dim( ξ) Tr ξ(x) σA(x, ξ ) f(ξ) # $ [ξ]∈G ∗ = dim( ξ) Tr ( ξ(x) σA(x, ξ ) πf (ξ)) , [ξ ]∈G ; where

σA(x, ξ ) = πy&→LA(x,y )(ξ) = ξ(x) ( A(ξ∗))( x). ; 570 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

In the coming symbol considerations this left–right choice is encountered e.g. as follows:

σAB (x, ξ ) ∼ σA(x, ξ ) σB(x, ξ ) + · · · if we use right-convolutions

σAB (x, ξ ) ∼ σB(x, ξ ) σA(x, ξ ) + · · · if we use left-convolutions .

There isB an explicit link between the left–right cases. We refer to Section 10.11 for a further discussion of these issues for the operator-valued symbols. At the same time, we note that these choices already determine the need to work with right actions on homogeneous spaces in Chapter 13 , so that the homogeneous spaces there are K\G instead of G/K , see Remark 13.2.5 for the discussion of this issue.

Remark 10.4.14 . We have now associated a unique full symbol σA to each continu- ∞ ∞ ous linear operator A : C (G) → C (G). Here σA(x, ξ ) : Hξ → H ξ is a linear operator for each x ∈ G and each irreducible unitary representation ξ : G → U (Hξ). The correspondence A "→ σA is linear in the sense that

σA+B(x, ξ ) = σA(x, ξ ) + σB(x, ξ ) and σλA (x, ξ ) = λσ A(x, ξ ), where λ ∈ C. However, σAB (x, ξ ) is not usually σA(x, ξ )σB(x, ξ ) (unless B is a right-convolution operator, so that the symbol σB(x, ξ ) = b(ξ) does not depend on the variable x ∈ G). A composition formula will be established in Theorem 10.7.8 below.

10.4.2 Conjugation properties of symbols In the sequel, we will need conjugation properties of symbols which we will now analyse for this purpose. Definition 10.4.15 (φ–pushforwards). Let φ : G → G be a diffeomorphism, f ∈ C∞(G), A : C∞(G) → C∞(G) continuous and linear. Then the φ-pushforwards ∞ ∞ ∞ fφ ∈ C (G) and Aφ : C (G) → C (G) are defined by −1 fφ := f ◦ φ , −1 −1 Aφf := A(fφ ) φ = A(f ◦ φ) ◦ φ . ! " Notice that

Aφ◦ψ = ( Aψ)φ . Exercise 10.4.16. Using the local theory of pseudo-diﬀerential operators show that µ µ A ∈ Ψ (G) if and only if Aφ ∈ Ψ (G).

Deﬁnition 10.4.17. For u ∈ G, let uL, u R : G → G be deﬁned by

uL(x) := ux and uR(x) := xu. −1 −1 −1 −1 Then ( uL) = ( u )L and ( uR) = ( u )R. The inner automorphism φu : G → −1 G deﬁned in Proposition 10.2.6 by φu(x) := u xu satisﬁes −1 −1 φu = uL ◦ uR = uR ◦ uL . 10.4. Symbols of operators 571

Proposition 10.4.18. Let u ∈ G, B = AuL , C = AuR and F = Aφu . Then we have the following relations between symbols:

−1 σB(x, ξ ) = σA(u x, ξ ), ∗ −1 σC (x, ξ ) = ξ(u) σA(xu , ξ ) ξ(u), ∗ −1 σF (x, ξ ) = ξ(u) σA(uxu , ξ ) ξ(u).

Especially, if A = ( f "→ f ∗ a), i.e. σA(x, ξ ) = a(ξ), then

σB(x, ξ ) = a(ξ), σ (x, ξ ) = ξ(u)∗ a(ξ) ξ(u) C = σ (x, ξ ). F

−1 Proof. We notice that F = C(u )L , so it suﬃces to consider only operators B and

C. For the operator B = AuL , we get

−1 f(z) RB(x, z x) f(z) d z = Bf (x) #G −1 = A(f ◦ uL)( uL (x)) −1 −1 −1 = f(uy ) RA(u x, y u x) d y #G −1 −1 = f(z) RA(u x, z x) d z, #G

−1 so RB(x, y ) = RA(u x, y ), yielding

−1 σB(x, ξ ) = σA(u x, ξ ).

For the operator C = AuR , we have

−1 f(z) RC (x, z x) d z = Cf (x) #G −1 = A(f ◦ uR)( uR (x)) −1 −1 −1 = f(yu ) RA(xu , y xu ) f(yu ) d y #G −1 −1 −1 = f(z) RA(xu , uz xu ) d z, #G 572 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

−1 −1 so that RC (x, y ) = RA(xu , uyu ), yielding

∗ σC (x, ξ ) = RC (x, y ) ξ(y) dy #G −1 −1 ∗ = RA(xu , uyu ) ξ(y) dy #G −1 −1 ∗ = RA(xu , z ) ξ(u zu ) dz #G −1 ∗ ∗ = RA(xu , z ) ξ(u) ξ(z) ξ(u) d z #G ∗ −1 = ξ(u) σA(xu , ξ ) ξ(u) and completing the proof.

Let us now record how push-forwards by translation aﬀect vector ﬁelds.

Lemma 10.4.19 (Push-forwards of vector ﬁelds). Let u ∈ G, Y ∈ g and let E = ∞ ∞ DY : C (G) → C (G) be deﬁned by

d D f(x) = f(x exp( tY )) . (10.23) Y dt $ $t=0 $ $ Then

−1 EuR = Eφu = Du Y u , i.e. −1 −1 DY (f ◦ uR)( xu ) = DY (f ◦ φu)( uxu ) = Du−1Y u f(x).

Proof. We have

−1 EuR f(x) = E(f ◦ uR)( xu ) d = (f ◦ u )( xu −1 exp( tY )) dt R t=0 $ d $ = f(xu −1 exp( tY )u) dt t=0 $ d $ = f(x exp( tu −1Y u ) dt t=0 $ = Du−1Y u f(x). $

Due to the left-invariance, we have EuL = E, so that

−1 −1 Eφu = ( E )uR = EuR = Du Y u . uL 10.5. Boundedness of operators on L2(G) 573

For more transparency, we also calculate directly:

−1 Eφu f(x) = E(f ◦ φu)( uxu ) d = (f ◦ φ )( uxu −1 exp( tY )) dt u t=0 $ d $ = f(xu −1 exp( tY )u) dt t=0 $ d $ = f(x exp( tu −1Y u )) dt t=0 $ = Du−1Y u f(x), $ yielding the same result.

Remark 10.4.20 ( Symbol of iDY can be diagonalised) . Notice ﬁrst that the complex vector ﬁeld i DY is symmetric:

(i DY f, g )L2(G) = (i DY f)( x) g(x) d x #G

= −i f(x) DY g(x) d x #G

= ( f, iDY g)L2(G) .

Hence it is always possible to choose a representative ξ ∈ Rep( G) from each [ ξ] ∈ λ1  .  G such that σiDY (x, ξ ) is a diagonal matrix .. , with diagonal  λ   dim( ξ) entries λj ∈ R, which follows because symmetric matrices can be diagonalised by unitary matrices. Notice that then also the commutator of symbols satisfy

[σiDY , σ A]( x, ξ )mn = ( λm − λn) σA(x, ξ )mn .

10.5 Boundedness of operators on L2(G)

In this section we will state some natural conditions on the symbol of an operator A : C∞(G) → C∞(G) to guarantee its boundedness on L2(G). Recall first that the Hilbert–Schmidt inner product of matrices is defined as a special case of Definition B.5.42 : Definition 10.5.1 (Hilbert–Schmidt inner product). The Hilbert–Schmidt inner product of A, B ∈ Cm×n is

m n ∗ &A, B 'HS := Tr( B A) = Bij Aij , +i=1 +j=1 574 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

1/2 with the corresponding norm (A(HS := &A, A 'HS , and the operator norm n×1 (A(op := sup (Ax (ℓ2 : x ∈ C , (x(ℓ2 ≤ 1 = (A(ℓ2→ℓ2 ,

n ,2 1/2 - where (x(ℓ2 = ( j=1 |xj| ) is the usual Euclidean norm. Let A, B ∈.Cn×n. Then by Theorem 12.6.1 proved in Section 12.6 we have

(AB (HS ≤ ( A(op (B(HS . Moreover, we also have

n×n (A(op = sup (AX (HS : X ∈ C , (X(HS ≤ 1 . , - By this, taking the Fourier transform of the convolution and using Plancherel’s formula (Corollary 7.6.10 ), by Proposition 10.2.7 we get Proposition 10.5.2 (Operator norm of convolutions). We have

(g "→ f ∗ g(L(L2(G)) = (g "→ g ∗ f(L(L2(G)) = sup (f(ξ)(op . (10.24) ξ∈Rep( G)

We also note that (f(ξ)(op = (f(η)(op if [ξ] = [ η] ∈ G. We now extend this property to operators that are not necessarily left- or right-invariant. First we introduce derivatives of higher order on the Lie group G: α dim( G) Deﬁnition 10.5.3 (Operators ∂ on G). Let {Yj}j=1 be a basis for the Lie algebra of G, and let ∂j be the left-invariant vector ﬁelds corresponding to Yj, n α α1 αn ∂j = DYj , as in ( 10.23 ). For α ∈ N0 , let us denote ∂ = ∂1 · · · ∂n . Sometimes α we denote these operators by ∂x .

Remark 10.5.4 ( Orderings) . We note that unless G is commutative, operators ∂j do not in general commute. Thus, when we talk about “all operators ∂α”, we mean that we take these operators in all orderings. However, if we ﬁx a certain ordering α α of Yj’s, the commutator of a general ∂ with ∂ taken in this particular ordering is an operator of lower order (this can be easily seen either for the simple properties of commutators in Exercise D.1.5 or from the general composition Theorem 10.7.9 ). The commutator is again a combination of operators of the form ∂β with |β| ≤ |α| − 1. Thus, since usually we require some property to hold for example “for all ∂α with |α| ≤ N”, we can rely iteratively on that the assumption is already satisﬁed for ∂β, thus making this ordering issue less important. Theorem 10.5.5 (Boundedness of operators on L2(G)). Let G be a compact Lie group of dimension n and let k be an integer such that k > n/ 2. Let σA be the symbol of a linear continuous operator A : C∞(G) → C∞(G). Assume that there is a constant C such that α (∂x σA(x, ξ )(op ≤ C for all x ∈ G, all ξ ∈ Rep( G), and all |α| ≤ k. Then A extends to a bounded operator from L2(G) to L2(G). 10.6. Taylor expansion on Lie groups 575

Proof. Let Af (x) = ( f ∗ rA(x))( x), where rA(x)( y) = RA(x, y ) is the right- convolution kernel of A. Let Ayf(x) = ( f ∗ rA(y))( x), so that Axf(x) = Af (x). Then 2 2 2 (Af (L2(G) = |Axf(x)| dx ≤ sup |Ayf(x)| dx, #G #G y∈G and by an application of the Sobolev embedding theorem we get

2 α 2 sup |Ayf(x)| ≤ C |∂y Ayf(x)| dy. y∈G # |α+|≤ k G

Therefore, using Fubini theorem to change the order of integration, we obtain

2 α 2 (Af ( 2 ≤ C |∂ Ayf(x)| dx dy L (G) # # y |α+|≤ k G G

10.6 Taylor expansion on Lie groups

As Taylor polynomial expansions are useful in obtaining symbolic calculus on Rn, we would like to have analogous expansions on a group G. Here, the Taylor expansion formula on G will be obtained by embedding G into some Rm, using the Taylor expansion formula in Rm, and then restricting it back to G. Let U ⊂ Rm be an open neighbourhood of some point 'e ∈ Rm. The Nth m ∞ order Taylor polynomial PN f : R → C of f ∈ C (U) at 'e is given by

1 α α PN f('x ) = ('x − 'e ) ∂x f('e ). Nm α! α∈ 0+: |α|≤ N

Then the remainder EN f := f − PN f satisﬁes

α EN f('x ) = ('x − 'e ) fα('x ) |α|+=N+1 576 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

∞ for some functions fα ∈ C (U). In particular,

N+1 EN f('x ) = O(|'x − 'e | ) as 'x → 'e.

Let G be a compact Lie group; we would like to approximate a smooth function u : G → C using a Taylor polynomial type expansion nearby the neutral element e ∈ G. By Corollary 8.0.21 we may assume that G is a closed subgroup of GL( n, R) ⊂ Rn×n, the group of real invertible ( n × n)-matrices, and thus a closed submanifold of the Euclidean space of dimension m = n2. This embedding of G into Rm will be denoted by x "→ 'x , and the image of G under this embedding will be still denoted by G. Also, if x ∈ G, we may still write x for 'x to simplify the notation. Let U ⊂ Rm be a small enough open neighbourhood of G ⊂ Rm such that for each 'x ∈ U there exists a unique nearest point p('x ) ∈ G (with respect to the Euclidean distance). For u ∈ C∞(G) we deﬁne f ∈ C∞(U) by

f('x ) := u(p('x )) .

The effect is that f is constant in the directions perpendicular to G. As above, we m m may define the Euclidean Taylor polynomial PN f : R → C at e ∈ G ⊂ R . Let us define PN u : G → C as the restriction,

PN u := PN f|G.

∞ We call PN u ∈ C (G) a Taylor polynomial of u of order N at e ∈ G. Then for x ∈ G, we have α u(x) − PN u(x) = (x − e) uα(x) |α|+=N+1 ∞ α α for some functions uα ∈ C (G), where we set ( x−e) := ( 'x −'e ) . There should be no confusion with this notation because there is no substraction on the group G, so substracting group elements means substracting them when they are embedded in a higher dimensional linear space. Taylor polynomials on G are given by 1 P u(x) = (x − e)α ∂(α)u(e), N α! x |α+|≤ N where we set (α) α ∂x u(e) := ∂x f('e ). (10.25) Remark 10.6.1 . We note that in this way we can obtain diﬀerent forms of the Taylor series. For example, it may depend on the embedding of G into GL( n, R), on the choice of the coordinates in Rn × Rn, etc. Let us now consider the example of G = SU(2). Recall the quaternionic identiﬁcation

4 (x01 + x1i + x2j + x3k "→ (x0, x 1, x 2, x 3)) : H → R , 10.7. Symbolic calculus 577 to be discussed in detail in Section 11.4 . Moreover, there is the identiﬁcation H ⊃ S3 =∼ SU(2), given by

x0 + i x3 x1 + i x2 x11 x12 'x = ( x0, x 1, x 2, x 3) "→ = = x. /−x1 + i x2 x0 − ix30 /x21 x22 0 Hence we identify (1 , 0, 0, 0) ∈ R4 with the neutral element of SU(2). Remark 10.6.2 . Notice that the functions

q+(x) = x12 = x1 + i x2,

q−(x) = x21 = −x1 + i x2,

q0(x) = x11 − x22 = 2i x3 also vanish at the identity element of SU(2). A function u ∈ C∞(S3) can be extended to f ∈ C∞(U) = C∞(R4 \ { 0}) by f('x ) := u('x/ ('x (). ∞ 3 Therefore, we obtain PN u ∈ C (S ), 1 P u('x ) := ('x − 'e )α ∂αf('e ), N α! x |α+|≤ N where 'e = (1 , 0, 0, 0). Expressing this in terms of x ∈ SU(2), we obtain Taylor polynomials for x ∈ SU(2) in the form 1 P u(x) = (x − e)α ∂(α)u(e), N α! x |α+|≤ N

(α) α where we write ∂x u(e) := ∂x f('e ), and where (x − e)α := := ( 'x − 'e )α

α1 α2 α3 α4 = ( x0 − 1) x1 x2 x3 x + x α1 x − x α2 x + x α3 x − x α4 = 11 22 − 1 12 21 12 21 11 22 . / 2 0 / 2 0 / 2i 0 / 2i 0 This gives an example of possible Taylor monomials on SU(2).

10.7 Symbolic calculus

In this section, we study global symbols of pseudo-differential operators on compact Lie groups, as defined in Definition 10.4.3 . We also derive elements of the calculus in quite general classes of symbols. For this, we first introduce difference operators acting on symbols in the ξ-variable. These are analogues of the n n ∂ξ-derivatives in R and of the difference operators △ξ on T , and are obtained by the multiplication by “coordinate functions” on the Fourier transform side. 578 Chapter 10. Pseudo-differential operators on compact Lie groups

10.7.1 Difference operators As explained in Section 10.6 , smooth functions on a group G can be approxi- mated by Taylor polynomial type expansions. More precisely, there exist partial (α) ∞ differential operators ∂x of order |α| on G such that for every u ∈ C (G) we have 1 u(x) = q (x−1) ∂(α)u(e) + q (x−1) u (x) α! α x α α |α+|≤ N |α|+=N+1 1 ∼ q (x−1) ∂(α)u(e) (10.26) α! α x α+≥0 ∞ ∞ in a neighbourhood of e ∈ G, where uα ∈ C (G), and qα ∈ C (G) satisfy (α) qα+β = qαqβ, and ∂x are as in ( 10.25 ). Moreover, here q0 ≡ 1, and qα(e) = 0 if |α| ≥ 1. α α Definition 10.7.1 (Difference operators △ξ ). Let us define difference operators △ξ α α+β α β acting on Fourier coefficients by △ξ f(ξ) := qαf(ξ). Notice that △ξ = △ξ △ξ . −1 Remark 10.7.2 . The technical choice of writing1 qα(x ) in ( 10.26 ) is dictated by our desire to make the asymptotic formulae in Theorems 10.7.8 and 10.7.10 look similar to the familiar Euclidean formulae in Rn, and by an obvious freedom in se- lecting among different forms of Taylor polynomials qα, see Remark 10.6.1 . For example, on SU(2), if we work with operators ∆ +, ∆−, ∆0 defined in ( 12.14 )-( 12.16 ), we can choose the form of the Taylor expansion ( 10.26 ) adapted to functions −1 −1 q+, q −, q 0. On SU(2), we can observe that q+(x ) = −q−(x), q −(x ) = −q+(x), −1 −1 q0(x ) = −q0(x), so that for |α| = 1 the functions qα(x) and qα(x ) are linear combinations of q+, q −, q 0. In terms of the quaternionic identification, these are functions from Remark 10.6.2 . Taylor monomials ( x − e)α from the previous section, when restricted to SU(2), can be expressed in terms of functions q+, q −, q 0. For an argument of this type we refer to the proof of Lemma 12.4.5 . Remark 10.7.3 ( Differences reduce the order of symbols) . In Theorem 12.3.6 we will apply the differences on the symbols of specific differential operators on SU(2). In general, on a compact Lie group G, a difference operator of order |γ| applied to a symbol of a partial differential operator of order N gives a symbol of order N − | γ|. More precisely: Proposition 10.7.4 (Differences for symbols of differential operators). Let

α D = cα(x) ∂x (10.27) |α+|≤ N

∞ α be a partial differential operator with coefficients cα ∈ C (G), and ∂x as in Def- inition 10.5.3 . For q ∈ C∞(G) such that q(e) = 0 , we define difference operator △q acting on symbols by △qf(ξ) := qf (ξ). 2 10.7. Symbolic calculus 579

Then we obtain

α |β| β △qσD(x, ξ ) = cα(x) (−1) (∂ q)( e) σ α−β (x, ξ ), (10.28) /β0 x ∂x |α+|≤ N β+≤α which is a symbol of a partial diﬀerential operator of order at most N − 1. More precisely, if q has a zero of order M at e ∈ G then Op( △qσD) is of order N − M. ∞ Proof. Let D in ( 10.27 ) be be a partial diﬀerential operator, where cα ∈ C (G) α ∞ and ∂x : D(G) → D (G) is left-invariant of order |α|. If |α| = 1 and φ, ψ ∈ C (G) then we have the Leibniz property

α α α ∂x (φψ ) = φ (∂x ψ) + ( ∂x φ) ψ, leading to α α 0 = φ(x) ∂x ψ(x) d x + ∂x φ(x) ψ(x) d x. #G #G ∞ ′ More generally, for |α| ∈ N0, φ ∈ C (G) and f ∈ D (G), we have for the distributional derivatives

α |α| α φ(x) ∂x f(x) d x = ( −1) ∂x φ(x) f(x) d x, #G #G with a standard distributional interpretation. Recall that the right-convolution ′ kernel RA ∈ D (G × G) of a continuous linear operator A : D(G) → D (G) satisﬁes

−1 Aφ (x) = φ(y) RA(x, y x) d y. #G For instance, informally

−1 φ(x) = φ(y) δe(y x) d y = φ(y) δx(y) d y, #G #G ′ where δp ∈ D (G) is the Dirac delta distribution at p ∈ G. Notice that

α |α| α −1 ∂x φ(x) = (−1) (∂y φ)( xy ) δe(y) d y #G −1 α = φ(xy ) ∂y δe(y) d y. #G The right-convolution kernel of the operator D from ( 10.27 ) is given by

α RD(x, y ) = cα(x) ∂y δe(y). |α+|≤ N

∞ ∞ Let Dq : C (G) → C (G) be deﬁned by

σDq (x, ξ ) := △qσD(x, ξ ), 580 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups i.e.

RDq (x, y ) = q(y) RD(x, y ).

Then Dq = Op( σDq ) is a diﬀerential operator:

−1 α Dqφ(x) = φ(xy ) q(y) cα(x) ∂ δe(y) d y # y G |α+|≤ N

|α| α −1 = (−1) cα(x) ∂ φ(xy ) q(y) δe(y) d y # y |α+|≤ N G ! " α = (−1) |α| c (x) (−1) |α−β| (∂βq)( e) ∂α−βφ(x). α /β0 x x |α+|≤ N β+≤α

Thus

α |β| β △qσD(x, ξ ) = cα(x) (−1) (∂ q)( e) σ α−β (x, ξ ). /β0 x ∂x |α+|≤ N β+≤α

Hence if q has a zero of order M at e ∈ G then Dq is of order N − M.

Exercise 10.7.5. Provide the distributional interpretation of all the steps in the proof of Proposition 10.7.4 .

10.7.2 Commutator characterisation

m Definition 10.7.6 (Operator classes Ak (M)). For a compact closed manifold M, let m ∞ ∞ A0 (M) denote the set of those continuous linear operators A : C (M) → C (M) m 2 m m which are bounded from H (M) to L (M). Recursively define Ak+1 (M) ⊂ A k (M) m m such that A ∈ A k (M) belongs to Ak+1 (M) if and only if [ A, D ] = AD − DA ∈ m Ak (M) for every smooth vector field D on M. We now recall a variant of the commutator characterisation of pseudo-differential operators given in Theorem 5.3.1 which assures that the behaviour of commutators in Sobolev spaces characterises pseudo-differential operators:

Theorem 10.7.7. A continuous linear operator A : C∞(M) → C∞(M) belongs to m ∞ m Ψ (M) if and only if A ∈ k=0 Ak (M). 3 We note that in such a characterisation on a compact Lie group M = G, it suﬃces to consider vector ﬁelds of the form D = Mφ∂x, where Mφf := φf is ∞ multiplication by φ ∈ C (G), and ∂x is left-invariant. Notice that

[A, M φ∂x] = Mφ [A, ∂ x] + [ A, M φ] ∂x, 10.7. Symbolic calculus 581

where [ A, M φ]f = A(φf ) − φAf . Hence we need to consider compositions MφA, AM φ, A ◦ ∂x and ∂x ◦ A. First, we observe that

σMφA(x, ξ ) = φ(x) σA(x, ξ ), (10.29)

σA◦∂x (x, ξ ) = σA(x, ξ ) σ∂x (x, ξ ), (10.30)

σ∂x◦A(x, ξ ) = σ∂x (x, ξ ) σA(x, ξ ) + ( ∂xσA)( x, ξ ), (10.31)

where σ∂x (x, ξ ) is independent of x ∈ G. Here ( 10.29 ) and ( 10.30 ) are straightforward and ( 10.31 ) follows by the Leibniz formula:

∂x ◦ Af (x) = ∂x dim( ξ) Tr ξ(x) σA(x, ξ ) f(ξ) + 4 5 [ξ]∈G

= dim( ξ) Tr (∂xξ)( x) σA(x, ξ ) f(ξ) + 4 5 [ξ]∈G

+ dim( ξ) Tr ξ(x) ∂xσA(x, ξ ) f(ξ) , + 4 5 [ξ]∈G ∗ where we used that σ∂x (x, ξ ) = ξ(x) (∂xξ)( x) by Theorem 10.4.6 to obtain ( 10.31 ). Next we claim that we have the formula 1 σ (x, ξ ) ∼ △ασ (x, ξ ) ∂(α)φ(x), AMφ α! ξ A x α+≥0

(α) where ∂x are certain partial diﬀerential operators of order |α|. This follows from the general composition formula in Theorem 10.7.8 .

10.7.3 Calculus Here we discuss elements of the symbolic calculus of operators. First we recall the fundamental quantity &ξ' from Deﬁnition 10.3.18 that will allow us to introduce the orders of operators. We note that this scale &ξ' on G is determined by the eigenvalues of the Laplace operator L on G. We now formulate the result on compositions:

Theorem 10.7.8 (Composition formula I). Let m1, m 2 ∈ R and ρ > δ ≥ 0. Let A, B : C∞(G) → C∞(G) be continuous and linear, their symbols satisfy

△ασ (x, ξ ) ≤ C &ξ'm1−ρ|α|, ξ A op α 6 6 6 β 6 m2+δ|β| ∂x σB(x, ξ ) op ≤ Cβ &ξ' , 6 6 6 6 for all multi-indices α and β, uniformly in x ∈ G and [ξ] ∈ G. Then 1 σ (x, ξ ) ∼ (△ασ )( x, ξ ) ∂(α)σ (x, ξ ), (10.32) AB α! ξ A x B α+≥0 582 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups where the asymptotic expansion means that for every N ∈ N we have

1 6σ (x, ξ ) − (△ασ )( x, ξ ) ∂(α)σ (x, ξ )6 ≤ 6 AB α! ξ A x B 6 6 |α+|

−1 ABf (x) = (Bf )( xz ) RA(x, z ) d z #G −1 −1 = f(xy ) RB(xz, yz ) d y R A(x, z ) d z, #G #G where we use the standard distributional interpretation of integrals. Hence

σAB (x, ξ )

∗ = RAB (x, y ) ξ(y) dy #G −1 −1 ∗ ∗ = RA(x, z ) ξ(z ) RB(xz, yz ) ξ(yz ) dz dy #G #G 1 −1 −1 −1 ∗ = RA(x, z ) qα(z ) ξ(z ) × α! # # |α+|

m1+δN (uα(x, ξ )(op ≤ C&ξ' since uα(x, y ) is the remainder2 in the Taylor expansion of RB(x, y ) in x only and so it satisﬁes similar estimates to those of σB with respect to ξ. This completes the proof. A similar proof yields another version of the composition formula:

Theorem 10.7.9 (Composition formula II). Let m1, m 2 ∈ R and ρ > δ ≥ 0. Let A, B : C∞(G) → C∞(G) be continuous and linear, their symbols satisfying

∂β△ασ (x, ξ ) ≤ C &ξ'm1−ρ|α|+δ|β|, x ξ A op α 6 6 6∂β△ασ (x, ξ )6 ≤ C &ξ'm2−ρ|α|+δ|β|, x ξ B op β 6 6 6 6 10.7. Symbolic calculus 583 for all multi-indices α and β, uniformly in x ∈ G and [ξ] ∈ G. Then

1 σ (x, ξ ) ∼ (△ασ )( x, ξ ) ∂(α)σ (x, ξ ), AB α! ξ A x B α+≥0 where the asymptotic expansion means that for every N ∈ N we have

1 △γ ∂β σ (x, ξ ) − (△ασ )( x, ξ ) ∂(α)σ (x, ξ ) ≤ ξ x  AB α! ξ A x B  |α| δ ≥ 0. Let A : C∞(G) → C∞(G) be continuous and linear, with symbol σA satisfying

△α∂βσ (x, ξ ) ≤ C #ξ$m−ρ|α|+δ|β|, ξ x A op α for all multi-indices α, uniformly in x ∈ G and [ξ] ∈ G. Then the symbol of A∗ is

1 α (α) ∗ σ ∗ (x, ξ ) ∼ △ ∂ σ (x,% ξ ) , A α! ξ x A α"≥0 where the asymptotic expansion means that for every N ∈ N we have

1 △γ ∂β σ (x, ξ ) − △α∂(α)σ (x, ξ )∗ ≤ ξ x  A α! ξ x A  |α|

∗ −1 A g(y) = g(x) RA∗ (y, x y) d x, &G we get the relation −1 −1 RA∗ (y, x y) = RA(x, y x) between kernels, which means that

−1 −1 RA∗ (x, v ) = RA(xv , v ). 584 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

From this we ﬁnd

∗ σA∗ (x, ξ ) = RA∗ (x, v ) ξ(v) dv &G −1 −1 ∗ = RA(xv , v ) ξ(v) dv &G 1 (α) −1 ∗ = qα(v) ∂x RA(x, v ) ξ(v) dv + RN (x, ξ ) α! G |α"|

Above we considered the symbol of the adjoint. Since KAt (x, y ) = KA∗ (x, y ), Theorem 10.7.10 provides a follow-up: Corollary 10.7.11 (Transpose). Let A : C∞(G) → C∞(G) be as in Theorem 10.7.10 . Then the symbol of the transpose At is

1 α (α) ∗ σ t (x, ξ ) ∼ △ ∂ σ (x, ξ ) , A α! ξ x A α"≥0 ∞ ∞ where the right-convolution kernel of A : C (G) → C (G) is deﬁned by RA(x, y ) := RA(x, y ), and where the asymptotic expansion is interpreted as in Theorem 10.7.10 . We postpone the discussion of the asymptotic expansion of the parametrix for elliptic operators in Theorem 10.9.10 to after we introduce symbol classes.

10.7.4 Leibniz formula For studying products of pseudo-differential symbols, it would be beneficial to have a Leibniz-like formula for the “derivatives with respect to the dual variable ξ ∈ G”. Classically for smooth functions σA, σ B : R → C, the Leibniz formula is the familiar ′ ′ ′ % (σAσB) (ξ) = σA(ξ) σB(ξ) + σA(ξ) σB(ξ). In the context of pseudo-differential calculus on the torus T, we have T =∼ Z; for functions σA, σ B : Z → C a useful Leibniz-like formula reads % △ξ(σAσB) = ( △ξσA) σB + σA (△ξσB) + ( △ξσA) ( △ξσB) , 10.8. Boundedness on Sobolev spaces Hs(G) 585

where the diﬀerence operator is deﬁned by △ξσ(ξ) = σ(ξ + 1) − σ(ξ). Let G be a compact Lie group. Given q ∈ C∞(G) and f ∈ D ′(G), let

△qf(ξ) := qf (ξ). (10.33)

Let σA(ξ) = a(ξ) and σB(ξ) = b(ξ).% Then )

% △q(σAσB)( ξ)% = q(b ∗ a)( ξ)

= b ∗ (qa )( ξ) + Rb (ξ)

= ( △qσA(ξ)) σB(ξ) + Rb (ξ), ) ′ ′ where the remainder operator R : D (G) → D (G) is given) by Rb = q(b ∗ a) − b ∗ (qa ), that is Rb (x) = b(y) q(x) − q(y−1x) a(y−1x) d y. &G Thus by the Taylor expansion from* Section 10.6 we+ have

Rb (x) ∼ b(y) c(q, α )( x) y(α) a(y−1x) d y, G   & |α"|≥ 0   where 1 (α) −1 c(q, α )( x) := ∂z q(x) − q(z x) . α! z=e , As usual, according to ( 10.33 ), the diﬀerence* △ is +, c(q,α ) ,

△c(q,α )f(ξ) = c(q, α )f(ξ). (10.34)

Notice that c(q, 0)( x) = q(x) − q(x%) = 0, so that we get an asymptotic Leibniz formula:

Theorem 10.7.12 (Asymptotic Leibniz formula). For symbols σA and σB, we have

α △q(σAσB)( ξ) ∼ (△qσA(ξ)) σB(ξ) + △c(q,α ) σA(ξ) △ξ σB(ξ) . |α|>0 " * * ++ 10.8 Boundedness on Sobolev spaces Hs(G)

In this section we show conditions on the symbol for operators to be bounded on Sobolev spaces Hs(G). The Sobolev space Hs(G) of order s ∈ R can be defined via a smooth partition of unity of the closed manifold G, and there are other definitions as well, in particular in terms of the Laplace operator on G, see 586 Chapter 10. Pseudo-differential operators on compact Lie groups

Deﬁnition 10.3.9 . We also recall Deﬁnition 10.3.18 of the quantity #ξ$ on the Lie group G, measuring the order of operators compared to that of the Laplacian L: vw vw if −L ξ (x) = λ[ξ]ξ (x), then

1/2 #ξ$ := (1 + λ[ξ]) . according to ( 10.9 ). Now we can formulate the result on the Sobolev space boundedness: Theorem 10.8.1 (Boundedness of operators on Sobolev spaces). Let G be a compact Lie group. Let A be a continuous linear operator from C∞(G) to C∞(G) and let σA be its symbol. Assume that there are constants µ, C α ∈ R such that

α µ )∂x σA(x, ξ ))op ≤ Cα #ξ$ holds for all x ∈ G, ξ ∈ Rep( G) and all multi-indices α. Then A extends to a bounded operator from Hs(G) to Hs−µ(G), for all s ∈ R. Remark 10.8.2 . Notice that we may easily show a special case of this result with s = µ. Namely, if σA is as in Theorem 10.8.1 , then

α −µ ∂x σA(x, ξ )#ξ$ op ≤ Cα

* −µ+ for every multi-index α. Here σA(x, ξ )#ξ$ = σA◦Ξ−µ (x, ξ ), and thus Theo- rem 10.5.5 implies that A◦Ξ−µ is bounded on L2(G), so that A ∈ L (Hµ(G), L 2(G)).

Proof of Theorem 10.8.1 . Observing the continuous mapping Ξ s : Hs(G) → L2(G), we have to prove that operator Ξ s−µ ◦A◦Ξ−s is bounded from L2(G) to L2(G). Let −s −s us denote B = A ◦ Ξ , so that the symbol of B satisﬁes σB(x, ξ ) = #ξ$ σA(x, ξ ) for all x ∈ G and ξ ∈ Rep( G). Since Ξ s−µ ∈ Ψs−µ(G), by ( 10.24 ) and Lemma 10.9.3 its symbol satisﬁes

α ′ s−µ−| α| )△ ξ σΞs−µ (x, ξ ))op ≤ Cα #ξ$ . (10.35)

Now we can observe that the asymptotic formula in Theorem 10.7.8 works for the composition Ξ s−µ ◦ B in view of ( 10.35 ), and we obtain

β 1 α −s (α) β ∂ σ s−µ (x, ξ ) ∼ △ σ s−µ (x, ξ ) #ξ$ ∂ ∂ σ (x, ξ ). x Ξ ◦B α! ξ Ξ x x A α≥0 " * + It follows that β ′′ s−µ ∂x σΞ ◦B(x, ξ ) op ≤ Cβ , s−µ 2 so that Ξ ◦ B is bounded on L (G) by Theorem 10.5.5 . This completes the proof. 10.9. Symbol classes on compact Lie groups 587

Remark 10.8.3 ( Functional analytic argument) . The boundedness of operators on the Sobolev spaces follows also by a purely functional analytic argument if G is compact. Indeed, because the space C∞(G) is nuclear (by adopting Exercise ∞ ∞ B.3.51 ), the tensor product space C (G) ⊗π ℓ (G) can be endowed with the α Fréchet topology given by the seminorms sup x∈G, [ξ]∈G |∂x a(x, ξ )|. Hence, as soon α % as the operator satisfies sup x, ∈G, [ξ]∈G |∂x a(x, ξ )| < ∞ for finitely many α, it is bounded on the corresponding space Hs(G).

Exercise 10.8.4. Work out the details of this argument.

10.9 Symbol classes on compact Lie groups

The goal of this section is to describe the pseudo-differential symbol inequalities on compact Lie groups that yield Hörmander’s classes Ψ m(G). On the way to characterise the usual Hörmander classes Ψ m(G) in Theorem 10.9.6 , we need some properties concerning symbols of pseudo-differential operators that we will establish in the next paragraphs.

10.9.1 Some properties of symbols of Ψm(G) Given an operator from Hörmander’s class Ψ m(G), we can derive some information about its full symbol as defined in Definition 10.4.3 . A precise characterisation of the class Ψ m(G) will be given in Theorem 10.9.6 .

Lemma 10.9.1. Let A ∈ Ψm(G). Then there exists a constant C < ∞ such that

m )σA(x, ξ ))op ≤ C#ξ$ for all x ∈ G and ξ ∈ Rep( G). Also, if u ∈ G and if B is an operator with symbol m σB(x, ξ ) = σA(u, ξ ), then B ∈ Ψ (G). Proof. First, B ∈ Ψm(G) follows from the local theory of pseudo-diﬀerential operators, by studying

−1 Bf (x) = KA(u, ux y) f(y) d y. &G Hence the right-convolution operator B is bounded from Hs(G) to Hs−m(G), m implying )σA(u, ξ )) ≤ C#ξ$ .

Exercise 10.9.2. Provide the details for the proof of Lemma 10.9.1 .

m α β m−| α| Lemma 10.9.3. Let A ∈ Ψ (G). Then Op( △ξ ∂x σA) ∈ Ψ (G) for all α, β . 588 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

m α β Proof. First, given A ∈ Ψ (G), let us deﬁne σB(x, ξ ) = △ξ ∂x σA(x, ξ ). We must show that B ∈ Ψm−| α|(G). If here |β| = 0, we obtain

−1 −1 Bf (x) = f(xy ) qα(y) RA(x, y ) d y = qα(y x) KA(x, y ) f(y) d y. &G &G Moving to local coordinates, we need to study

˜ Bf (x) = φ(x, y ) KA˜(x, y ) f(y) d y, Rn & ˜ m Rn Rn ∞ Rn Rn where A ∈ Ψ ( × ) with φ ∈ C ( × ), the kernel KA˜ being compactly supported. Let us calculate the symbol of B˜:

2πi( y−x)·ξ σB˜ (x, ξ ) = e φ(x, y ) KA˜(x, y ) d y Rn & 1 γ 2πi( y−x)·ξ γ ∼ ∂z φ(x, z )|z=x e (y − x) KA˜(x, y ) d y γ! G γ"≥0 & 1 = ∂γ φ(x, z )| Dγ σ (x, ξ ). γ! z y=x ξ A˜ γ"≥0 ˜ m Rn Rn α m−| α| This shows that B ∈ Ψ ( × ). We obtain Op( △ξ σA) ∈ Ψ (G) if A ∈ Ψm(G). β m Next we show that B = Op( ∂x σA) ∈ Ψ (G). We may assume that |β| = 1. β Left-invariant vector ﬁeld ∂x is a linear combination of terms of the type c(x)Dx, ∞ where c ∈ C (G) and Dx is right-invariant. By the previous considerations on B˜, we may remove c(x) here, and consider only C = Op (DxσA). Since RA(x, y ) = −1 KA(x, xy ), we get

DxRA(x, y ) = ( Dx + Dz)KA(x, z )|z=xy −1 , leading to

−1 Cf (x) = f(xy ) DxRA(x, y ) d y = f(y) ( Dx + Dy)KA(x, y ) d y. &G &G Thus, we study local operators of the form

˜ β β Cf (x) = f(y) φ(x, y )∂x + ψ(x, y )∂y KA˜(x, y ) d y, Rn & * + where the kernel of A˜ ∈ Ψm(Rn × Rn) has compact support, φ, ψ ∈ C∞(Rn × Rn), and φ(x, x ) = ψ(x, x ) for every x ∈ Rn. Let C˜ = D˜ + E˜, where

˜ β β Df (x) = f(y) φ(x, y ) ∂x + ∂y KA˜(x, y ) d y, Rn & * + ˜ β Ef (x) = f(y) ( ψ(x, y ) − φ(x, y )) ∂y KA˜(x, y ) d y. Rn & 10.9. Symbol classes on compact Lie groups 589

By the above considerations about B˜, we may assume that φ(x, y ) ≡ 1 here, and β ˜ m Rn Rn obtain σD˜ (x, ξ ) = ∂x σA˜(x, ξ ). Thus D ∈ Ψ ( × ). Moreover, 1 Ef˜ (x) ∼ ∂γ (ψ(x, z ) − φ(x, z )) | × γ! z z=x γ"≥0 γ β × f(y) ( y − x) ∂y KA˜(x, y ) d y, Rn & yielding

γ β σE˜ (x, ξ ) ∼ cγ (x) ∂ξ ξ σA˜(x, ξ ) γ≥0 " * + ∞ n for some functions cγ ∈ C (R ) for which c0(x) ≡ 0. Since |β| = 1, this shows ˜ m Rn Rn β m m that E ∈ Ψ ( × ). Thus Op( ∂x σA) ∈ Ψ (G) if A ∈ Ψ (G). Lemma 10.9.4. Let A ∈ Ψm(G) and let D : C∞(G) → C∞(G) be a smooth vector m+1 m ﬁeld. Then Op( σAσD) ∈ Ψ (G) and Op([ σA, σ D]) ∈ Ψ (G).

Proof. For simplicity, we may assume that D = Mφ∂x, where ∂x is left-invariant and φ ∈ C∞(G). Now

σA(x, ξ ) σD(x, ξ ) = φ(x) σA(x, ξ ) σ∂x (ξ) = σMφA◦∂x (x, ξ ),

m+1 and it follows from the local theory that MφA◦∂x ∈ Ψ (G). Thus Op( σAσD) ∈ Ψm+1 (G). Next,

σD(x, ξ ) σA(x, ξ ) = φ(x) σ∂x (ξ) σA(x, ξ ) (10 .31 ) = φ(x) ( σ∂x◦A(x, ξ ) − (∂xσA)( x, ξ ))

= σMφ◦∂x◦A(x, ξ ) − φ(x) ( ∂xσA)( x, ξ ). From this we see that

Op([ σA, σ D]) = MφA ◦ ∂x − Mφ∂x ◦ A + Mφ Op( ∂xσA)

= Mφ[A, ∂ x] + Mφ Op( ∂xσA).

m m Here Op( ∂xσA) ∈ Ψ (G) by Lemma 10.9.3 . Hence Op([ σA, σ D]) belongs to Ψ (G) as a sum of operators from Ψ m(G).

10.9.2 Symbol classes Σm(G) Combined with asymptotic expansion ( 10.32 ) for composing operators, commutator characterisation Theorem 10.7.7 motivates deﬁning the following symbol classes ∞ m m Σ (G) = Σk (G) k-=0 590 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

m m that we will show to characterise Hörmander’s classes Ψ (G). The classes Σ k (G) are defined iteratively in the following way: m R m Definition 10.9.5 (Symbol classes Σ (G)). Let m ∈ . We denote σA ∈ Σ0 (G) if

sing supp ( y /→ RA(x, y )) ⊂ { e} (10.36) and if α β m−| α| )△ ξ ∂x σA(x, ξ ))op ≤ CAαβm #ξ$ , (10.37) for all x ∈ G, all multi-indices α, β , and all ξ ∈ Rep( G), where #ξ$ is deﬁned in m (10.9 ). Then we say that σA ∈ Σk+1 (G) if and only if

m σA ∈ Σk (G), (10.38) m σ∂j σA − σAσ∂j ∈ Σk (G), (10.39) γ m+1 −| γ| (△ξ σA) σ∂j ∈ Σk (G), (10.40) for all |γ| > 0 and 1 ≤ j ≤ dim( G). Let

∞ m m Σ (G) := Σk (G). k-=0 m m We denote A ∈ OpΣ (G) if and only if σA ∈ Σ (G). Theorem 10.9.6 (Equality OpΣ m(G) = Ψ m(G)). Let G be a compact Lie group m m and let m ∈ R. Then A ∈ Ψ (G) if and only if σA ∈ Σ (G).

m Proof. First, applying Theorem 10.7.8 to σA ∈ Σk+1 (G), we notice that [ A, D ] ∈ m ∞ ∞ OpΣ k (G) for any smooth vector ﬁeld D : C (G) → C (G). Consequently, if here A ∈ OpΣ m(G) then also [ A, D ] ∈ OpΣ m(G). By Remark 10.8.2 we obtain OpΣ m(G) ⊂ L (Hm(G), L 2(G)). Consequently, Theorem 10.7.7 implies OpΣ m(G) ⊂ Ψm(G). Conversely, we have to show that Ψ m(G) ⊂ OpΣ m(G). This follows by Lemma 10.9.1 , and Lemmas 10.9.3 and 10.9.4 . More precisely, let A ∈ Ψm(G). Then we have

α β m−| α| Op( △ξ ∂x σA) ∈ Ψ (G), m Op [σ∂j , σ A] ∈ Ψ (G), γ m+1 −| γ| Op (△*ξ σA)σ∂j + ∈ Ψ (G). . / m Moreover, )σA(x, ξ )) ≤ C#ξ$ by Lemma 10.9.1 , and the singular support y /→ RA(x, y ) is contained in {e} ⊂ G. This completes the proof. 10.9. Symbol classes on compact Lie groups 591

Corollary 10.9.7. The set Σm(G) is invariant under x-freezings, x-translations and m ξ-conjugations. More precisely, if (x, ξ ) /→ σA(x, ξ ) belongs to Σ (G) and u ∈ G then also the following symbols belong to Σm(G):

(x, ξ ) /→ σA(u, ξ ), (10.41)

(x, ξ ) /→ σA(ux, ξ ), (10.42)

(x, ξ ) /→ σA(xu, ξ ), (10.43) ∗ (x, ξ ) /→ ξ(u) σA(x, ξ ) ξ(u). (10.44)

Proof. The symbol classes Σ m(G) are defined by conditions ( 10.36 )-( 10.40 ), which are checked for points x ∈ G fixed (with constants uniform in x). Therefore it follows that Σ m(G) is invariant under the x-freezing ( 10.41 ), and under the left and right x-translations ( 10.42 ),( 10.43 ). The x-freezing property ( 10.41 ) would have followed also from Lemma 10.9.1 and Theorem 10.9.6 . From the general local theory of pseudo-differential operators it follows that A ∈ Ψm(G) if and only if m −1 the φ-pullback Aφ belongs to the same class Ψ (G), where Aφf = A(f ◦ φ) ◦ φ . This, combined with the x-translation invariance and Proposition 10.4.18 , implies the conjugation invariance in ( 10.44 ). From Theorem 10.9.6 and Lemma 10.9.3 we also obtain: m α β m−| α| Corollary 10.9.8. If σA ∈ Σ (G) then △ξ ∂x σA ∈ Σ (G). Finally, we formulate a simple relation between convergence of symbols and operators. It is a straightforward consequence of Theorems 10.5.5 and 10.8.1 for L2(G) and Hs(G) cases, respectively. In Corollary 12.4.11 we will give an improve- ment of this result on the group SU(2). Corollary 10.9.9 (Convergence of symbols and operators). Let σ ∈ Σ0(G), and 0 assume that a sequence σk ∈ Σ (G) satisfies inequalities ( 10.37 ) uniformly in k. Let N ∈ N be such that N > dim( G)/2, and assume that for all |β| ≤ N we have the convergence β β ∂x σk(x, ξ ) → ∂x σ(x, ξ ) as k → ∞ (10.45) in the operator norm, uniformly over all x ∈ G and ξ ∈ Rep( G). Then Op σk → Op σ strongly on L2(G). Moreover, if the convergence ( 10.45 ) holds for all multi-indices β, then Op σk → Op σ strongly on Hs(G) for any s ∈ R. From the asymptotic expansion for the composition of pseudo-differential operators in Section 10.7.3 , we get an expansion for a parametrix of an elliptic operator: m−j Theorem 10.9.10 (Parametrix). Let σAj ∈ Σ (G), and let

∞

σA(x, ξ ) ∼ σAj (x, ξ ). j=0 " 592 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

−1 Assume that A is elliptic in the sense that σA0 (x, ξ ) = σB0 (x, ξ ) is an invertible −m matrix for every x and ξ, and that B0 = Op( σB0 ) ∈ Ψ (G). Then there exists −m σB ∈ Σ (G) such that I − BA and I − AB are smoothing operators. Moreover,

∞

σB(x, ξ ) ∼ σBk (x, ξ ), k"=0 −m−k where the operators Bk ∈ Σ (G) are deﬁned recursively by

σBN (x, ξ ) := N−1 N−k 1 = −σ (x, ξ ) △γ σ (x, ξ ) ∂(γ)σ (x, ξ ). (10.46) B0 γ! ξ Bk x Aj j=0 k"=0 " |γ|="N−j−k 0 1 ∞ Proof. If σI ∼ σBA holds for some σB ∼ k=0 σBk , then by Theorem 10.7.8 we have 2 Idim( ξ) = σI (x, ξ ) ∼ σBA (x, ξ ) 1 ∼ △γ σ (x, ξ ) ∂(γ)σ (x, ξ ) γ! ξ B x A γ"≥0 0 1 1 ∞ ∞ ∼ △γ σ (x, ξ ) ∂(γ) σ (x, ξ ). γ! ξ Bk x Aj 3 4 j=0 γ"≥0 k"=0 "

From this, we want to ﬁnd σBk . Now Idim( ξ) = σB0 (x, ξ ) σA0 (x, ξ ), and for |γ| ≥ 1 we can demand that 1 0 = △γ σ (x, ξ ) ∂(γ)σ (x, ξ ). γ! ξ Bk x Aj |γ|="N−j−k 0 1 Then ( 10.46 ) provides the solution to these equations, and the reader may verify −m−N ∞ also that σBN ∈ Σ (G). Thus σB ∼ k=0 σBk . Finally, notice that σI ∼ σBA . 2 −m−N Exercise 10.9.11. Show that σBN ∈ Σ (G) in Theorem 10.9.10 . Exercise 10.9.12. In the notation of Theorem 10.9.10 , let us denote

−1 σC0 (x, ξ ) := σB0 (x, ξ ) = σA0 (x, ξ ) , ∞ and let σC ∼ N=0 σCN , where

2 N−1 N−k 1 σ (x, ξ ) = −σ (x, ξ ) △γ σ (x, ξ ) ∂(γ)σ (x, ξ ). CN C0 γ! ξ Aj x Ck j=0 k"=0 " |γ|="N−j−k 0 1 Check that also C is a parametrix of A. 10.10. Full symbols on compact manifolds 593

10.10 Full symbols on compact manifolds

In this section we discuss how the introduced constructions are mapped by global diffeomorphisms. Let Φ : G → M be a diffeomorphism from a compact Lie group G to a smooth manifold M. Such diffeomorphisms can be obtained for large classes of compact manifolds by the Poincaréconjecture type results. For example, if dim G = dim M = 3 it is now known that such Φ exists for any closed simply- connected 6 manifold, and we can take G =∼ S3 =∼ SU(2). We will now explain how the diffeomorphism Φ induces the quantization of operators on M from that on G. Let us endow M with the natural Lie group structure induced by Φ, i.e. with the group multiplication (( x, y ) /→ x · y) : M × M → M defined by x · y := Φ Φ−1(x) Φ −1(y) . The spaces C∞(G) and C∞(M) are* isomorphic via mappings+

∞ ∞ −1 Φ∗ : C (G) → C (M), f /→ fΦ = f ◦ Φ , ∗ ∞ ∞ Φ : C (M) → C (G), g /→ gΦ−1 = g ◦ Φ, and the Haar integral on M is given by

g dµM ≡ g dx := g ◦ Φ d µG, &M &M &G where µG is the Haar measure on G, because

g(x · y) d x = g(Φ(Φ −1(x) Φ −1(y))) d x &M &M = (g ◦ Φ)(Φ −1(x) Φ −1(y)) d(Φ −1(x)) &G = (g ◦ Φ)( z) d z &G = g(x) d x. &M ∞ ∞ Moreover, Φ ∗ : C (G) → C (M) extends to a linear unitary bijection Φ ∗ : 2 2 L (µG) → L (µM ):

g(x) h(x) d x = (g ◦ Φ) (h ◦ Φ) d µG. &M &G Notice also that there is an isomorphism

Φ∗ : Rep( G) → Rep( M), ξ /→ Φ∗(ξ) = ξ ◦ Φ

6For the definition of simply-connectedness see Definition 8.3.18 . 594 Chapter 10. Pseudo-differential operators on compact Lie groups of irreducible unitary representations. Thus G =∼ M in this sense. This immediately implies that the whole construction of symbols of pseudo-differential operators on M is equivalent to that on G. In Section 12.5 we will give an example of this identification in the case of SU(2) =∼ S3.

10.11 Operator-valued symbols

In this section we discuss another notion of a symbol which we call operator-valued symbols of operator. We recall left- and right-convolution operators from Remark 10.4.2 , which will now play an important role: Deﬁnition 10.11.1 (Convolution operators l(f) and r(f)). For f ∈ D ′(G), we deﬁne the respective left-convolution and right-convolution operators l(f), r (f) : C∞(G) → C∞(G) by

l(f)g := f ∗ g, r(f)g := g ∗ f.

Exercise 10.11.2. To check that we indeed have l(f), r (f) : C∞(G) → C∞(G) one can use a distributional interpretation similar to the one in Section 1.4.2 . Work out the details of this. Remark 10.11.3 . For f ∈ L2(G), in the literature, operator l(f) is sometimes called “global” Fourier transform of f. Such terminology can be found in e.g. W. F. Stinespring [ 117 ], where it is denoted by π(f), and for related integration theory of operators by I. E. Segal, see [ 104 ] and [ 105 ]. However, since we are dealing with the quantization of operators on G that have both left- and right-convolution kernels, we want to keep track of both left- and right-convolution operators. At the same time, it will turn out that the proceeding theory of full symbols is better adapted to the operator r(f) because our starting point was right-convolution kernels. As usual, the left and right regular representations of G are denoted by 2 πL, π R : G → U (L (G)), respectively, i.e.

−1 πL(x)f(y) = f(x y),

πR(x)f(y) = f(yx ).

Exercise 10.11.4. Verify that these representation are indeed unitary. For example, −1 −1 ∗ check that πR(x) = πR(x ) = πR(x) , and similarly for πL(x). 10.11. Operator-valued symbols 595

Keeping in mind right and left Peter–Weyl’s theorems (Theorem 7.5.14 and Remark 7.5.14 ), the Fourier inversion formulae may be viewed in the following form: Proposition 10.11.5 (“Fourier inversion formulae”). Let f ∈ C∞(G). Then we have

∗ r(f) = f(y) πR(y) dy and l(f) = f(y) πL(y) d y. (10.47) !G !G Conversely, for every x ∈ G we have

∗ f(x) = Tr ( r(f) πR(x)) and f(x) = Tr ( l(f) πL(x) ), (10.48) where Tr is the trace functional, see Deﬁnition B.5.37 ; notice that Tr( AB ) = 2 Tr( BA ). These formulae have an almost everywhere extension to L (µG). Proof. We will prove the case of right-convolutions since for left-convolutions it is similar. First, we can write

(r(f)g)( x) = ( g ∗ f)( x)

= g(xy −1)f(y) d y !G −1 = f(y)( πR(y )g)( x) d y, !G −1 ∗ from which we obtain ( 10.47 ) in view of the unitarity πR(y ) = πR(y) . The proof of ( 10.48 ) is somewhat lengthier if we provide the details. First we observe that ( 10.47 ) implies that

∗ −1 r(f)πR(x) = f(y) πR(y) πR(x) d y = f(y) πR(y x) d y. (10.49) !G !G By Peter–Weyl’s theorem (Theorem 7.5.14 ) whose notation we will use in this 2 proof, we know that {dim( φ)φij }φ,i,j is an orthonormal basis in L (µG), so that by Deﬁnition B.5.37 of the trace we have

Tr( r(f) πR(x)) (10.50) dim( φ)

= dim( φ)( r(f)πR(x)φij , φ ij ) [φ"]∈G i,j"=1 dim( φ) (10 .49 ) −1 = dim( φ) f(y)φij (zy x)φij (z) d y dz !G !G [φ"]∈G i,j"=1 dim( φ) −1 = dim( φ) f(xw )φij (zw )φij (z) d w dz, !G !G [φ"]∈G i,j"=1 596 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups where in the last equality we changed the variables w := y−1x. Using φ(zw ) = φ(z)φ(w), we obtain dim( φ)

φ(zw )ij = φik (z)φkj (w) (10.51) k"=1 and dim( φ) dim( φ) −1 −1 φ(y x)jj = φji (y )φij (x) = φij (y)φij (x), (10.52) k"=1 k"=1 to be used later as well. We also notice that

1 φik (z)φij (z) d z = #φik , φ ij $L2 = δkj (10.53) !G dim( φ) in view of Lemma 7.5.12 . Plugging ( 10.51 ) and ( 10.53 ) into ( 10.50 ) we get

Tr( r(f) πR(x)) dim( φ) −1 = f(xw ) φjj (w) d w !G [φ"]∈G i,j"=1 dim( φ) −1 = dim( φ) f(xw ) φjj (w) d w (10.54) !G [φ"]∈G "j=1 dim( φ) −1 y:= xw −1 = dim( φ) f(y) φjj (y x) d y !G [φ"]∈G "j=1 dim( φ) (10 .52 ) = dim( φ) #f, φ ij $L2 φij (x) [φ"]∈G "j=1 = f(x), with the last equality in view of Corollary 7.6.7 .

Corollary 10.11.6 (Character description). For every f ∈ L2(G) we have

f = dim( φ) f ∗ χφ, [φ"]∈G

2 φ where χφ is the character of φ. Thus, the projection of f ∈ L (G) to H is given by f %→ f ∗ χφ. 10.11. Operator-valued symbols 597

Proof. Writing the expression in the line of ( 10.54 ) as a trace, we see that

dim( φ) −1 dim( φ) f(xw )φjj (w) d w !G [φ"]∈G "j=1 = dim( φ)( f ∗ Tr φ)( x) [φ"]∈G

= dim( φ)( f ∗ χφ)( x). [φ"]∈G

Consequently ( 10.54 ) implies the statement. Exercise 10.11.7. Provide details for the proof of Proposition 10.11.5 in the left- convolution case. ∞ Deﬁnition 10.11.8 (Right and left kernels RA(x), L A(x)). Let A : C (G) → ∞ ′ C (G) be a linear continuous operator, and let RA(x, y ), L A(x, y ) ∈ D (G)⊗D (G) be its right-convolution and left-convolution kernels, respectively 7. For every x ∈ G ′ we deﬁne RA(x), L A(x) ∈ D (G) by

[RA(x)]( y) := RA(x, y ) and [ LA(x)]( y) := LA(x, y ).

In this notation we can write

(Af )( x) = [ f ∗ RA(x)]( x) = [ r(RA(x)) f]( x), (10.55) and similarly (Af )( x) = [ LA(x) ∗ f]( x) = [ l(LA(x)) f]( x). (10.56) This motivates the following definition. Definition 10.11.9 (Operator-valued symbols). Let A : C∞(G) → C∞(G) be a linear continuous operator. We define its right operator-valued symbol rA : G %→ L(C∞(G)) by x %→ rA(x) := r(RA(x)) . ∞ Similarly, we define its left operator-valued symbol lA : G %→ L (C (G)) by

x %→ lA(x) := l(LA(x)) .

We observe that ( 10.55 ) and ( 10.56 ) imply the equality

[rA(x)f]( x) = Af (x) = [ lA(x)] f(x). (10.57)

Lemma 10.11.10. For every f ∈ C∞(G) and every x ∈ G we have

r(rA(x)f) = rA(x)r(f) and l(lA(x)f) = lA(x)l(f).

7As deﬁned in ( 10.16 ) in Section 10.4 , and shown in Lemma 10.12.5 in Section 10.12 . 598 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

Proof. We prove only the “right” case since the “left” one is similar. We have a simple calculation

r(rA(x)f)g = g ∗ (rA(x)f)

= g ∗ (f ∗ RA(x)) Prop 7.7.3 = ( g ∗ f) ∗ RA(x)

= r(RA(x))( g ∗ f)

= rA(x)r(f)g, completing the proof. Exercise 10.11.11. Prove the left cases of Lemma 10.11.10 as well as of the following theorem: Theorem 10.11.12 (Operator-valued quantization). Let A : C∞(G) → C∞(G) be a linear continuous operator. Then we have

Af (x) = Tr( rA(x) r(f) πR(x)) ∗ = Tr( lA(x) l(f) πL(x) ), (10.58) for all f ∈ C∞(G) and x ∈ G. Proof. We can write

(10 .57 ) Af (x) = ( rA(x)f)( x) (10 .48 ) = Tr ( r[rA(x)f] πR(x)) Lemma 10 .11 .10 = Tr ( rA(x) r(f) πR(x)) , completing the proof for the right case. The left one is similar. Remark 10.11.13 ( Symbol as a family of convolution operators) . In view of The- orem 10.11.12 the operator-valued symbols lA and rA can be regarded as a family of convolution operators obtained from A by “freezing” it at points x ∈ G. We note that if A ∈ L (D(G)) is a left-invariant operator, i.e. Aπ L(x) = πL(x)A for every x ∈ G, then its right operator-valued symbol is the constant mapping x %→ rA(x) ≡ A (A is a right-convolution operator). Remark 10.11.14 ( Operator-valued symbols and operators) . The quantization A %→ rA, l A is an injective linear mapping. Conversely, starting from an operator- valued symbol we can deﬁne the corresponding operator. Indeed, let ρ : G → L(D′(G)) be a mapping such that ρ(x) = r(s(x)), for some s ∈ D (G) ⊗ D ′(G). Then we can deﬁne Op( ρ) ∈ L (D(G)) by (Op( ρ)f)( x) = ( ρ(x)f)( x), for which s is the right-convolution kernel and ρ is the right operator-valued symbol. 10.11. Operator-valued symbols 599

Corollary 10.11.15 (Decomposition in matrix blocks). According to the Peter– Weyl Theorem, all operators in ( 10.58 ) have direct sum decompositions with corresponding ﬁnite-dimensional square matrix blocks rA(x, ξ ) := rA(x)|Hξ , lA(x, ξ ) := lA(x)|Hξ , πR(x, ξ ) := πR(x)|Hξ , πL(x, ξ ) := πL(x)|Hξ , where ξ ∈ Rep( G) is a representation ξ : G → U (Hξ). In this notation ( 10.58 ) implies

Af (x) = Tr ( πR(x, ξ ) rA(x, ξ ) r(f)|Hξ ) (10.59) ξ"∈G ∗ = Tr ( πL(x, ξ ) lA(x, ξ ) l(f)|Hξ ) . ξ"∈G

The meaning of operators r(f)|Hξ and l(f)|Hξ can be clariﬁed as follow: if ξ ∈ Rep( G), then l(u)ξ(x) = u(ξ)ξ(x) and r(u)ξ(x) = ξ(x)u(ξ).

Let us show the last formula for l(u). Indeed, we have

l(u)ξ(x) = u(y)ξ(y−1x) d y = u(y)ξ(y)∗ dy ξ(x) = u(ξ)ξ(x), !G #!G $ and the calculation for r(u) is similar, where we can commute ξ(x) and the integral since ξ(x) is finite dimensional. We now establish a relation between different quantizations. Namely, for ∞ ∞ every x ∈ G the mapping rA(x) : C (G) → C (G) from Definition 10.11.9 is linear and continuous, so that we can find its symbol σrA(x)(y, ξ ) according to Definition 10.4.3 . Theorem 10.11.16 (Relation between quantizations). Let A : C∞(G) → C∞(G) ∞ be a linear continuous operator and let rA, l A : G → L (C (G)) be its right and left operator-valued symbols, respectively. Then for all x, y ∈ G and ξ ∈ Rep( G) we have

σrA(x)(y, ξ ) = σA(x, ξ ) (10.60) and ∗ ∗ σlA(x)(y, ξ ) = ξ(y) ξ(x)σA(x, ξ )ξ(x) ξ(y). (10.61) The operator-valued symbols take the form

[rA(x)f]( y) = dim( ξ) Tr ξ(y) σA(x, ξ ) f(ξ) (10.62) " % & [ξ]∈G and ∗ [lA(x)f]( y) = dim( ξ) Tr ξ(x) σA(x, ξ ) ξ(x) ξ(y) f(ξ) . (10.63) " % & [ξ]∈G Finally, we also have

Af (x) = Tr (Op( σA(x, ξ )) r(f) πR(x)) . (10.64) 600 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

Remark 10.11.17 . Formulae ( 10.62 ) and ( 10.63 ) when evaluated at y = x give us Af (x), thus recovering equality ( 10.57 ). Formula ( 10.60 ) shows that the operator rA(x) when quantized in terms of symbols from Deﬁnition 10.4.3 becomes a multiplier, with its symbol independent of y. Formula ( 10.61 ) is slightly more complicated as a consequence of the fact that the symbols from Deﬁnition 10.4.3 are better to right-convolutions, see Remark 10.4.2 . In the “left” case additional conjugations in ( 10.61 ) appear in view of Proposition 10.4.11 .

Proof of Theorem 10.11.16 . We start with the right operator-valued symbol rA(x). By Theorem 10.4.6 its symbol can be found as

∗ σrA(x)(y, ξ ) = ξ(y) [rA(x)ξ]( y).

Thus, we calculate

[rA(x)ξ]( y) = [ r(RA(x)) ξ]( y)

= [ ξ ∗ RA(x)]( y)

−1 = ξ(yz )RA(x)( z)dz !G ∗ = ξ(y) ξ(z) RA(x, z )dz !G = ξ(y)σA(x, ξ ), yielding ( 10.60 ). Here, as usual, we used the ﬁnite dimensionality of representations to commute ξ(y) with the integral. This implies ( 10.62 ) by Theorem 10.4.4 :

[rA(x)f]( y) = dim( ξ) Tr ξ(y) σrA(x)(y, ξ ) f(ξ) " % & [ξ]∈G

= dim( ξ) Tr ξ(y) σA(x, ξ ) f(ξ) . " % & [ξ]∈G Moreover, ( 10.64 ) follows from ( 10.60 ) by Theorem 10.11.12 . Similarly, in view of

∗ σlA(x)(y, ξ ) = ξ(y) [lA(x)ξ]( y) we calculate

[lA(x)ξ]( y) = [ l(LA(x)) ξ]( y)

= [ LA(x) ∗ ξ]( y)

−1 = LA(x)( yz )ξ(z)dz !G −1 = LA(x, yz )ξ(z)dz. !G 10.11. Operator-valued symbols 601

−1 −1 −1 Using formula ( 10.17 ), i.e. RA(x, w ) = LA(x, xwx ) with w = x yz x, gives z = xw −1x−1y and −1 −1 −1 LA(x, yz ) = RA(x, x yz x).

Continuing the calculation for [ lA(x)ξ]( y) above, we get

−1 [lA(x)ξ]( y) = LA(x, yz )ξ(z)dz !G −1 −1 = RA(x, x yz x)ξ(z)dz !G −1 −1 = RA(x, w )ξ(xw x y)dz !G

∗ −1 = ξ(x) RA(x, w )ξ(w) dw ξ(x )ξ(y) #!G $ ∗ = ξ(x)σA(x, ξ )ξ(x) ξ(y), yielding ( 10.61 ). This implies ( 10.63 ) by Theorem 10.4.4 . In Theorem 10.5.5 we presented conditions on the L2(G)-boundedness of operators in terms of symbols σ(x, ξ ). We now briefly discuss the same question in terms of the operator-valued symbols. We will do this in terms of the right operator-valued symbols because the left case is the same, see Exercise 10.11.21 . Definition 10.11.18 (Derivations of operator-valued symbols). Let operator A : C∞(G) → C∞(G) be linear and continuous, with right operator-valued symbol ∞ ∞ rA : G → L (C (G)). Let p(D) ∈ L (C (G)) be a partial differential operator. For each x ∈ G we define

(p(D)rA)( x) = p(D)rA(x) := r(RB(x)) , where RB(x)( y) = RB(x, y ) and

RB = ( p(D) ⊗ id) RA.

Operator B deﬁned by Bf (x) = [ r(RB(x)) f]( x) is then also a linear continuous operator from C∞(G)) to C∞(G)) because p(D) ∈ L (C∞(G)), id ∈ L (D′(G)). Theorem 10.11.19 (Boundedness on L2(G)). Let G be a compact Lie group of dimension n. Let A be a linear continuous operator from C∞(G) to C∞(G) and k 2 assume that its right operator-valued symbol rA satisﬁes rA ∈ C (G, L(L (G))) with k > n/ 2. Then A extends to a bounded linear operator from L2(G) to L2(G). Proof. By ( 10.57 ) we have

2 2 (Af (L2(G) = |(rA(x)f)( x)| dx !G 2 ≤ sup |(rA(y)f)( x)| dx, !G y∈G 602 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups and by an application of the Sobolev embedding theorem we get

2 α 2 sup |(rA(y)f)( x)| ≤ Ck |(( ∂y rA)( y)f)( x)| dy. y∈G !G |α"|≤ k

Therefore using Fubini theorem to change the order of integration, we obtain

2 α 2 (Af (L2(G) ≤ C |(( ∂y rA)( y)f)( x)| dx dy !G !G |α"|≤ k

α 2 ≤ C sup |(( ∂ rA)( y)f)( x)| dx y∈G !G |α"|≤ k α 2 = C sup ((∂ rA)( y)f(L2(G) y∈G |α"|≤ k α 2 2 ≤ C sup ((∂ rA)( y)(L(L2(G)) (f(L2(G). y∈G |α"|≤ k

k 2 The proof is complete, because G is compact and rA ∈ C (G, L(L (G))). Theorem 10.11.19 yields the following Sobolev boundedness result: Corollary 10.11.20. Let G be a compact Lie group and let A be a linear continuous ∞ ∞ operator from C (G) to C (G) such that its right operator-valued symbol rA ∞ m 0 satisﬁes rA ∈ C (G, L(H (G), H (G))) , m ∈ R. Then A extends to a bounded linear operator from Hm(G) to H0(G).

Exercise 10.11.21. Show that we can replace rA by the left operator-valued symbol lA in Theorem 10.11.19 and Corollary 10.11.20 for them to remain true. Work out details of the proofs. Remark 10.11.22 ( M. E. Taylor’s characterisation) . Let g be the Lie algebra of a compact Lie group G, and let n = dim( G) = dim( g). By the exponential mapping exp : g → G, a neighbourhood of the neutral element e ∈ G can be identified m m m n n with a neighbourhood of 0 ∈ g. Let X = S1# ⊂ S1,0(R × R ) consist of the m n n x-invariant symbols ( x, ξ ) %→ p(ξ) in S1,0(R × R ) with the usual Fréchet space topology (given by the optimal constants in symbol estimates as seminorms). A distribution k ∈ D ′(G) with a sufficiently small support is said to belong to space X m if sing supp( k) ⊂ { e} and k ∈ X m ⊂ C∞(g′), ' where the Fourier transform k is the usual Fourier transform on g =∼ Rn, and the dual space satisfies g′ =∼ Rn (and we are using the exponential coordinates for k(y) when y ≈ e ∈ G). If k ∈ X m then the convolution operator ' u %→ k ∗ u, k ∗ u(x) = k(xy −1) u(y) d y, !G 10.11. Operator-valued symbols 603 is said to belong to space OP X m, which is endowed with the natural Fréchet space m structure obtained from X . Formally, let k(x, y ) = kx(y) be the left-convolution kernel of a linear operator K : C∞(G) → C∞(G), i.e.

−1 Ku(x) = kx(xy ) u(y) d y. !G In [ 125 ], M. E. Taylor showed that K ∈ Ψm(G) if and only if the mapping

m (x %→ (u %→ kx ∗ u)) : G → OP X

m is smooth; here naturally u %→ kx ∗ u must belong to OP X for each x ∈ G. This approach was pursued in [ 125 ] as well as in [ 133 , 136 ] in the setting of the left-convolution kernels, and relied on the exponential mapping and on the pseudo- differential operator on the Lie algebra (see Remark 10.11.22 ). In this approach many arguments may be restricted to a suitable neighbourhood of the identity element. However, the notion of the symbol σA from Definition 10.4.3 appears to be more practical as it allows a finite-dimension realisation of the symbol and works globally on the group. Remark 10.11.23 ( Operator-valued calculus) . It is possible to construct the calculus of operator-valued symbols, including compositions, adjoints, transposes, and the inverse. In exponential coordinates of the Lie algebra versions of it can be found in [ 125 , 136 ]. However, using our calculus in Section 10.7 together with Corollary 10.11.15 one can obtain this directly on the group without referring to the exponential coordinates in the neighbourhood of the origin. We leave this as an exercise for an interested reader.

10.11.1 Example on the torus Tn We will show here how the construction of this section apply for the operator- valued quantization of operator on the torus Tn. The starting point for this example is the operator-valued quantization formula ( 10.58 ) in Theorem 10.11.12 :

Af (x) = Tr( rA(x) r(f) πR(x)) ∗ = Tr( lA(x) l(f) πL(x) ), (10.65)

2 where G is the unitary dual of G, πL : G → L (L (G)) is the left regular representation of G on L2(G), r , l : G → L (D(G)) are the right and left operator- A A valued symbol of A, and r(f) and l(f) are the right and left convolution operators. According to the Peter–Weyl Theorem, these operators also have direct sum decompositions with corresponding ﬁnite-dimensional square matrix blocks which appear in ( 10.59 ). Let us inspect and summarise what this means in the special case of the n-torus G = Tn. 604 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups

The 1-torus T = R/Z can be identified with the multiplicative group of the unit circle in the plane. Let U(1) be the group of 1 × 1 unitary matrices; U(1) can be n n identified with the unit circle, again. For each ξ ∈ Z , define eξ : T → U(1) by

i2 πx ·ξ eξ(x) = e ;

n up to isomorphism, an irreducible unitary representation of G = T is some eξ. n n Thus G can be identiﬁed with the integer lattice Z , and {eξ : ξ ∈ Z } is an orthonormal basis for L2(G), where the Haar measure on G = Tn is obtained from the Lebesgue measure of Rn. Let us analyse this in terms of the left regular 2 representation πL : G → L (L (G)) which is now

n (πL(x)f)( y) = f(y − x) (for almost every y ∈ T ).

Especially, we notice

∗ i2 πx ·ξ (πL(x) eξ)( y) = ( πL(−x)eξ)( y) = eξ(y + x) = e eξ(y). (10.66)

Let u ∈ C∞(G). Then l(u) ∈ L (L2(G)) is the convolution operator given by

(l(u)f)( x) = ( u ∗ f)( x) = u(x − y) f(y) d y = u(y) f(x − y) d y. !G !G Especially, we have l(u)eξ = u(ξ) eξ, (10.67) n where u(ξ) ∈ C is the usual Fourier coefficient of u : T → C. Let A be a continuous linear operator from C∞(G) to C∞(G). Recall that here G = Tn and G = Zn; let us define σ : G × G → C by A −i2 πx ·ξ σA(x, ξ ) = e (σAeξ)( x), which is the toroidal symbol of A as well as the 1 × 1 matrix symbol, see Remark 10.4.7 . The left operator-valued symbol of A is the convolution operator-valued ∞ mapping lA : G → L (C (G)) that by ( 10.57 ) satisfies

(lA(x)u)( x) = ( Au )( x).

Thereby and by Theorem 10.11.16 we have

(lA(x)u)( y) = eξ(y) σA(x, ξ ) u(ξ). (10.68) ξ"∈Zn In particular, ( 10.66 ) and ( 10.68 ) imply that

∗ ∗ i2 πx ·ξ πL(x) lA(x)eξ = πL(x) eξσA(x, ξ )eξ(ξ) = e σA(x, ξ )eξ. (10.69)

10.12. Appendix: integral kernels 605

The trace of a linear operator B : L2(G) → L2(G) is

Tr( B) = #Be ξ, e ξ$L2(G), ξ"∈Zn

2 where #f, g $L2(G) = G f(y) g(y) d y is the inner product of L (G), see Deﬁnition B.5.37 . Let us now explore( formula ( 10.65 ) in the case of G = Tn:

∗ ∗ Tr ( πL(x) lA(x) l(u)) = #πL(x) lA(x) l(u) eξ, e ξ$L2(G) ξ"∈Zn

(10 .67 ) ∗ = #πL(x) lA(x) u(ξ) eξ, e ξ$L2(G) ξ"∈Zn (10 .69 ) i2 πx ·ξ = # e σA(x, ξ ) u(ξ) eξ, e ξ$L2(G) ξ"∈Zn i2 πx ·ξ = e σA(x, ξ ) u(ξ) #eξ, e ξ$L2(G) ξ"∈Zn i2 πx ·ξ = e σA(x, ξ ) u(ξ) ξ"∈Zn = ( Au )( x), where the last equality is the toroidal quantization of the operator A, given in Theorem 4.1.3 . This is the case on the commutative group G = Tn, where spaces ξ H = span( eξ) are all one-dimensional. On non-commutative compact Lie groups, even using the quantization ( 10.59 ), πL(x, ξ ), σA(x, ξ ) and l(u)|Hξ are no longer numbers (more precisely: no longer 1 × 1 matrices), but they are dim( ξ) × dim( ξ) matrices, where the dimension dim( ξ) ∈ Z+ depends on the corresponding representation ξ ∈ G and is usually greater than 1.

Exercise 10.11.24. Work out the above example for the “right” representation πR and right operator-valued symbols rA. In particular show that on the torus “left” and “right” coincide.

10.12 Appendix: integral kernels

Here we provide a short appendix with more technical explanations about integral kernels of operators on the compact Lie group G. Definition 10.12.1 (Duality between D(G) and D′(G)). Let D(G) be the set C∞(G) equipped with the usual Fréchet space topology defined by seminorms pα(f) = α α max x∈G |∂ f(x)|, with ∂ as in Definition 10.5.3 . Thus, the convergence on D(G) is just the unform convergence of functions and all their derivatives: fk → f in ∞ α α C (G) (or in D(G)) if ∂ fk(x) → ∂ f(x) for all x ∈ G, due to the compactness of G. 606 Chapter 10. Pseudo-differential operators on compact Lie groups

Let D′(G) = L(D(G), C) be its dual, i.e. the set of distributions with D(G) as the test function space. We equip the space of distributions with the weak ∗- topology. The duality D′(G) × D (G) → C is denoted by

#f, φ $ := f(φ),

′ ′ where φ ∈ D (G) and f ∈ D (G), and an embedding D(G) ֒→ D (G), ψ %→ fψ = ψ is given by #ψ, φ $ = ψ(x) φ(x) d y !G (in the same way as on Rn in Remark 1.3.7 ). Deﬁnition 10.12.2 (Transpose and adjoint). The transpose of A ∈ L (D(G)) is At ∈ L (D′(G)) deﬁned by the equality

#Atf, φ $ = #f, Aφ $, and the adjoint A∗ ∈ L (D′(G)) is given by the equality

(A∗f, φ ) = ( f, Aφ ), where ( f, φ ) = #f, φ $, φ(x) := φ(x) is the complex conjugate. Exercise 10.12.3. For f ∈ D ′(G) and for the left-convolution operator l(f)φ := f ∗ φ show that l(f)∗ = l(f˜) where f˜(x) = f(x−1), and l(f)t = l(fˇ) where fˇ(x) = f(x−1). Remark 10.12.4 ( Schwartz kernel) . Let D(G)⊗D ′(G) denote the complete locally convex tensor product of the nuclear spaces D(G) and D′(G) (see Section B.3.1 ). Then K ∈ D (G) ⊗ D ′(G) deﬁnes a linear operator A ∈ L (D(G)) by

#f, Aφ $ := #K, φ ⊗ f$. (10.70)

In fact, the Schwartz kernel theorem states that L(D(G)) and D(G) ⊗ D ′(G) are ′ isomorphic: for every A ∈ L (D(G)) there exists a unique KA ∈ D (G)⊗D (G) such that the duality ( 10.70 ) is satisﬁed with K = KA, which is called the Schwartz kernel of A. Duality ( 10.70 ) gives us also the interpretation for

(Aφ )( x) = KA(x, y ) φ(y) d y. !G For a more general Schwartz kernel theorem see Theorem B.3.55 and Deﬁnition B.3.56 . Lemma 10.12.5. Let A ∈ L (D(G)) , and let

−1 LA(x, y ) := KA(x, y x)

′ in the sense of distributions. Then LA ∈ D (G) ⊗ D (G). 10.12. Appendix: integral kernels 607

Proof. Notice that D(G) ⊗ D (G) =∼ D(G × G). Let us deﬁne the multiplication m : D(G) ⊗ D (G) → D (G), m (f ⊗ g)( x) := f(x)g(x), the co-multiplication

∆ : D(G) → D (G) ⊗ D (G), (∆ f)( x, y ) := f(xy ), and the antipode

S : D(G) → D (G), (Sf )( x) := f(x−1).

These mappings are a part of the (nuclear Fr´echet) Hopf algebra structure of D(G), see Chapter 9 (as well as e.g. [ 1] or [ 123 ]). The mappings are all continuous and linear. The convolution of operators A, B ∈ L (D(G)) is said to be the operator

A ∗ B := m(A ⊗ B)∆ ∈ L (D(G)); it is easy to calculate the Schwartz kernel

−1 KA∗B(x, y ) = KA(x, yz ) KB(x, z ) d z, !G or t KA∗B = ( m ⊗ ∆ )(id ⊗ τ ⊗ id)( KA ⊗ KB), where τ : D′(G) ⊗ D (G) → D (G) ⊗ D ′(G), τ(f ⊗ φ) := φ ⊗ f, and ∆ t : D′(G) ⊗ D′(G) → D ′(G) is the transpose of the co-multiplication ∆, ∆ t(f ⊗ g) = f ∗ g (i.e. ∆ t extends the convolution of distributions). Now ( A∗S)S ∈ L (D(G)), hence ′ K(A∗S)S ∈ D (G) ⊗ D (G) by the Schwartz kernel theorem, and

−1 K(A∗S)S(x, y ) = KA∗S(x, y )

−1 −1 = KA(x, y z ) KS(x, z ) d z !G −1 = KA(x, y x)

= LA(x, y ).

Remark 10.12.6 . Any distribution s ∈ D (G) ⊗ D ′(G) can be considered as a mapping s : G → D ′(G), x %→ s(x), where s(x)( y) := s(x, y ). If D ∈ L (D(G)) and M ∈ L (D′(G)) then ( D ⊗ M)s ∈ D(G) ⊗ D ′(G). For instance, D could be a partial diﬀerential operator, and M a multiplication.

Exercise 10.12.7. Prove Lemma 10.12.5 for the right-convolution kernel RA(x, y ). 608 Chapter 10. Pseudo-diﬀerential operators on compact Lie groups Chapter 11

Fourier analysis on SU (2)

In this chapter we develop elements of the Fourier analysis on the group SU(2) in the form suitable for the consequent development of the theory of pseudo- differential operators on SU(2) in Chapter 12 . Certain results from this chapter can be found in [ 145 ], which, together with [ 151 ] we can recommend for further reading, including for some instances of explicitly calculated Clebsch–Gordan coefficients. However, on this occasion, with pseudo-differential operators in mind and the form of the analysis necessary for us and adopted to Chapter 10 , we present an independent exposition of SU(2) with considerably more direct proofs and different arguments compared to e.g. [ 145 ].

11.1 Preliminaries: groups U(1) , SO(2) , and SO(3)

First, we discuss a simpler model of commutative groups U(1) =∼ SO(2), to treat rotation group SO(3) in a similar manner later. Following that, we study the special unitary group SU(2). For the deﬁnitions of SO( n) and SU( n) we refer to Remarks 6.2.9 and 6.2.10 , respectively. We start by discussing Lie algebras of U(1), SO(2), and SO(3). Deﬁnition 11.1.1 (Group U(1) ). Let g be the real vector space of 1-by-1-matrices

X = X(x) = i2 πx , (x ∈ R).

Of course, one may treat X(x) as a pure imaginary number, but we insist on the matrix interpretation in the light of the future developments. Then

exp( X(x)) = ei2 πx is a matrix belonging to the Lie group G = U(1), the unitary matrix group of dimension 1. Of course, U(1) =∼ {z ∈ C : |z| = 1 } =∼ T1 = R/Z. 610 Chapter 11. Fourier analysis on SU (2)

Definition 11.1.2 (Group SO(2) ). Let g be the real vector space of matrices 0 −x X = X(x) = , (x ∈ R). x 0 ! " Now cos( x) − sin( x) exp( X(x)) = sin( x) cos( x) ! " 2 is a matrix belonging to the Lie group G = SO(2). If |x| < π/ 2 and g = ( gij )i,j =1 = exp( X(x)) then x = arcsin( g21 ) = arcsin (( g21 − g12 )/2) . Definition 11.1.3 (Group SO(3) ). Let the real vector space g consist of matrices of the form 0 −x3 x2 X(x) = x 0 −x ,  3 1 −x2 x1 0 3 3   3 where x = ( xj)j=1 ∈ R . Thus x "→ X(x) identifies R with g, and we equip g 2 2 2 with the Euclidean norm $X(x)$g := $x$R3 = x1 + x2 + x3. Actually, g is a Lie algebra with the Lie product ( A, B ) "→ [A, B ] := AB − BA , since ' x2y3 − x3y2 [X(x), X (y)] = X( x y − x y ).  3 1 1 3 x1y2 − x2y1   It turns out that g is the Lie algebra of the group G = SO(3). Let

X1 := X((1 , 0, 0)) , X 2 := X((0 , 1, 0)) , X 3 = X((0 , 0, 1)) . Then [X1, X 2] = X3, [X2, X 3] = X1 and [ X3, X 1] = X2. 3 If x ∈ R and t := $x$R3 then we have the Rodrigues representation formula sin( t) 1 − cos( t) exp( X(x)) = I + X(x) + X(x)2 t t2 equaling to 2 2 1−cos t sin t 1−cos t sin t 1−cos t 1 + ( x1 − t ) t2 −x3 t + x1x2 t2 x2 t + x1x3 t2 sin t 1−cos t 2 2 1−cos t sin t 1−cos t x3 t + x1x2 t2 1 + ( x2 − t ) t2 −x1 t + x2x3 t2 .  sin t 1−cos t sin t 1−cos t 2 2 1−cos t  −x2 t + x1x3 t2 x1 t + x2x3 t2 1 + ( x3 − t ) t2  3  If here $x$R3 < π and g = ( gij )i,j =1 = exp( X(x)), we obtain the formula g − g t 32 23 x = ( x )3 = g − g , j j=1 2 sin( t)  13 31  g21 − g12   2 where sin( t) = 1 − cos (t) with cos( t) = ( g11 + g22 + g33 − 1) /2. ' 11.1. Preliminaries: groups U(1) , SO(2) , and SO(3) 611

11.1.1 Euler angles on SO(3) Euler angles are useful as local coordinates on SO(3). First we note that rotations 3 of R by an angle t ∈ R around the xj-axis, j = 1 , 2, 3, respectively, are expressed by the matrices ωj(t) = exp( tX j) given by

1 0 0 cos t 0 sin t cos t − sin t 0 0 cos t − sin t , 0 1 0 , sin t cos t 0 . 0 sin t cos t  − sin t 0 cos t  0 0 1        We represent rotations by Euler angles φ, θ, ψ ∈ R. Any g ∈ SO(3) is of the form

g = ω(φ, θ, ψ ) := ω3(φ) ω2(θ) ω3(ψ), where −π < φ, ψ ≤ π and 0 ≤ θ ≤ π. If 0 < θ 1, θ 2 < π then ω(φ1, θ 1, ψ 1) = ω(φ2, θ 2, ψ 2) if and only if θ1 = θ2 and φ1 ≡ φ2 (mod 2 π) and ψ1 ≡ ψ2 (mod 2 π); thus we conclude that the Euler angles provide local coordinates for the manifold SO(3) nearby a point ω(φ, θ, ψ ) whenever θ '≡ 0 (mod π). Let g = ω(φ, θ, ψ ) be the Euler angle representation of g ∈ SO(3), where −π < φ, ψ ≤ π, 0 ≤ θ ≤ π, so that ω(φ, θ, ψ ) is

cos φ cos θ cos ψ − sin φ sin ψ − cos φ cos θ sin ψ − sin φ cos ψ cos φ sin θ sin φ cos θ cos ψ + cos φ sin ψ − sin φ cos θ sin ψ + cos φ cos ψ sin φ sin θ .  − sin θ cos ψ sin θ sin ψ cos θ 

The group SO(3) acts transitively on the space S2, as ω(φ, θ, ψ ) moves the north T 2 pole e3 = (0 , 0, 1) ∈ S to the point

cos φ sin θ ω(φ, θ, ψ )e3 = sin φ sin θ .  cos θ    If 0 < θ < π and −π/ 2 < φ,ψ < π/ 2, then the Euler angles and the exponential coordinates are related by

ω(φ, θ, ψ ) = exp( X(x)) ,

3 where x ∈ R , 0 < t := $x$R3 < π ,

cos t = (cos( φ + ψ)(1 + cos θ) + cos θ − 1) /2, and sin θ(sin ψ − sin φ) t x = sin θ(cos ψ + cos φ) . 2 sin t (1 + cos θ) sin( φ + ψ)   612 Chapter 11. Fourier analysis on SU (2)

11.1.2 Partial derivatives on SO(3) For an element g ∈ SO(3) we now introduce the notation for its components. Thus, if g denotes elements of the group SO(3), we can view it as the identity mapping from SO(3) to SO(3). In its matrix components we can write this as 3 g = ( gij )i,j =1 : SO(3) → SO(3). In particular, if g ∈ SO(3), we can denote its matrix components by gij , 1 ≤ i, j ≤ 3, so that we obtain functions ( g "→ gij ) : ∞ SO(3) → R which are smooth on G: gij ∈ C (G); with this identiﬁcation and 3 notation, g = ( gij )i,j =1 : SO(3) → SO(3) is the identity mapping. Deﬁne

(1 ,0,0) (0 ,1,0) (0 ,0,1) ∂1 = ∂ , ∂ 2 = ∂ , ∂ 3 = ∂ . Notice that d (∂ f)( g) = f(ω (t)g)| . k dt k t=0 In particular, we have ′ ∂kg = ωk(0) g. Here ′ ′ ′ ω1(0) = X1, ω 2(0) = X2, , ω 3(0) = X3, so that ∂1g, ∂2g and ∂3g are respectively

0 0 0 g31 g32 g33 −g21 −g22 −g23 −g −g −g , 0 0 0 , g g g .  31 32 33     11 12 13  g21 g22 g23 −g11 −g12 −g13 0 0 0       11.1.3 Invariant integration on SO(3) We recall that on a compact group G there exists a unique translation-invariant regular Borel probability measure, called the Haar measure µG (see Remark 7.4.4 ); 2 2 customarily L (G) refers to L (G, µ G). Integrations on G are (unless otherwise mentioned) with respect to µG, so we may write

f(x) d x (G instead of G f dµG = G f(x) d µG(x). Using the Euler angle coordinates on G = SO(3), we deﬁne an orthogonal ) 2 ) projection PS2 ∈ L (L (SO(3))) by 1 π (PS2 f)( ω(φ, θ, ψ )) = f(ω(φ, θ, ψ˜)) d ψ˜ 2π (−π for almost all g = ω(φ, θ, ψ ). With the natural interpretation we have PS2 f ∈ 2 2 ∞ ∞ 2 L (S ), and if f ∈ C (SO(3)) then PS2 f ∈ C (S ). Thereby

f(x) d x = PS2 f dσ, S2 (SO(3) ( 11.2. General properties of SU(2) 613 where the measure d σ on the sphere is the normalised angular part of the Lebesgue measure of R3. This yields the Haar integral on SO(3):

f "→ f(x) d x = (SO(3) 1 π π π = f(ω(φ, θ, ψ )) sin( θ) d φ dθ dψ. (11.1) 8π2 (−π (0 (−π Exercise 11.1.4. The reader can check rigorously now that ( 11.1 ) indeed deﬁnes the Haar integral on SO(3).

11.2 General properties of SU(2)

From now on, we shall study the group of 2-dimensional special unitary matrices, denoted by SU(2). In other words, we study

SU(2) = u ∈ C2×2 : det( u) = 1 and u∗u = I ,

1 0 * + where I = ∈ C2×2 is the identity matrix of dimension 2. Indeed, SU(2) is 0 1 ! " a matrix group, since for u, v ∈ SU(2), we have

det( uv ) = det( u) det( v) = 1 , (uv )∗(uv ) = v∗u∗uv = I, det( u∗) = det( u) = 1 , (u∗)∗u∗ = ( uu ∗)∗ = I∗ = I, u−1 = u∗ ∈ SU(2) .

Lemma 11.2.1 (Elements of SU(2) ). The matrix u ∈ C2×2 belongs to SU(2) if and only if it is of the form α β u = , −β α ! " where α, β ∈ C are such that |α|2 + |β|2 = 1 . Moreover, SU(2) is a compact group. Proof. If u ∈ C2×2 is as above, we have det( u) = |α|2 + |β|2 = 1 and

α −β α β |α|2 + |β|2 0 u∗u = = = I β α −β α 0 |β|2 + |α|2 ! " ! " ! " so that u ∈ SU(2). Now suppose

u u u = 11 12 ∈ C2×2 u u ! 21 22 " 614 Chapter 11. Fourier analysis on SU (2)

−1 ∗ belongs to SU(2). Then u = u and 1 = det( u) = u11 u22 − u12 u21 . Speciﬁcally, 1 u −u u −u u−1 = 22 12 = 22 12 det( u) −u21 u11 −u21 u11 ! " ! " and u u u−1 = u∗ = 11 21 . u u ! 12 22 " This yields the desired form for u, with α = u11 , β = u12 . 2 From the proof so far, we see that the mapping ( u "→ (u11 , u 12 )) : SU(2) → C provides a homeomorphism from SU(2) to the Euclidean unit sphere of C2. Thus SU(2) is compact.

11.3 Euler angle parametrisation of SU(2)

The unitary group U(1) = {u ∈ C : u∗u = 1 } is often parametrised by the angle t ∈ R, i.e. u(t) = e it ∈ U(1); of course, the parameter range 0 ≤ t < 2π is sufficient here. These angles provide convenient expressions for group operations: −1 u(t0) u(t1) = u(t0 + t1) and u(t) = u(−t). (11.2) In analogy to this, elements of SU(2) can be endowed with so-called Euler angles , our next topic. Notice that the Euclidean unit sphere of C2 is naturally identified with the Euclidean unit sphere S3 of R4. An easy (non-unique) way to parametrise the 2 points ( u11 , u 12 ) of the unit sphere of C by r, s, t ∈ R is is it u11 = cos( r) e , u 12 = i sin( r) e , resulting in cos( r) e is i sin( r) e it u = . (11.3) i sin( r) e −it cos( r) e −is ! " Putting r = 0 in ( 11.3 ) we obtain a one-parametric subgroup of matrices eis 0 , 0 e −is ! " and with s = 0 = t we get another one-parametric subgroup of matrices cos( r) i sin( r) . i sin( r) cos( r) ! " Multiplying elements of these one-parametric subgroups, let us define eis 0 cos( r) i sin( r) eit 0 u(2 s, 2r, 2t) := 0 e −is i sin( r) cos( r) 0 e −it ! " ! " ! " cos( r) e i( s+t) i sin( r) e i( s−t) = . i sin( r) e −i( s−t) cos( r) e −i( s+t) ! " 11.3. Euler angle parametrisation of SU(2) 615

Clearly, by Lemma 11.2.1 any matrix u ∈ SU(2) is of this form. This leads us to the Euler angle parametrisation of SU(2): Deﬁnition 11.3.1 (Euler’s angles on SU(2) ). Euler’s angles (φ, θ, ψ ) from the parameter ranges 0 ≤ φ < 2π, 0 ≤ θ ≤ π, −2π ≤ ψ < 2π (11.4) correspond to the group element

cos( θ ) e i( φ+ψ)/2 i sin( θ ) e i( φ−ψ)/2 u(φ, θ, ψ ) = 2 2 ∈ SU(2) . i sin( θ ) e −i( φ−ψ)/2 cos( θ ) e −i( φ+ψ)/2 ! 2 2 " Exercise 11.3.2. Check that the parameter ranges for φ, θ, ψ in ( 11.4 ) are suﬃcient. Show that the Euler angle parametrisation is almost injective in the sense that if (φ1, θ 1, ψ 1) '= ( φ2, θ 2, ψ 2) with 0 ≤ φj < 2π, 0 < θ j < π , −2π ≤ ψ < 2π, then u(φ1, θ 1, ψ 1) '= u(φ2, θ 2, ψ 2). The angle parametrisation of U(1) behaves well with respect to the group operations in ( 11.2 ). Unfortunately, the situation with the Euler angles of SU(2) is complicated. Nevertheless, let us study this problem. a b Exercise 11.3.3. Let u(φ, θ, ψ ) = . Verify that c d ! " 2aa = 1 + cos( θ), 2ab = i e iφ sin( θ), −2ab = i e iψ sin( θ).

Notice how these formulae allow one to recover Euler angles φ, θ, ψ from the matrix u(φ, θ, ψ ). Let us examine the multiplication

u(φ, θ, ψ ) = u(φ0, θ 0, ψ 0) u(φ1, θ 1, ψ 1).

Abbreviating 2 rj := θj and s := ψ0 + φ1, we have

u(φ, θ, ψ ) = u(φ0, 0, 0) v u (0 , 0, ψ 1), where

v = u(0 , θ 0, ψ 0) u(φ1, θ 1, 0) cos r eiψ0/2 i sin r e−iψ0/2 cos r eiφ1/2 i sin r eiφ1/2 = 0 0 1 1 i sin r eiψ0/2 cos r e−iψ0/2 i sin r e−iφ1/2 cos r e−iφ1/2 ! 0 0 " ! 1 1 " cos( r ) cos( r )e is − sin( r ) sin( r )e −is . . . = 0 1 0 1 . i sin( r ) cos( r )e is + i cos( r ) sin( r )e −is . . . ! 0 1 0 1 " 616 Chapter 11. Fourier analysis on SU (2)

Notice that there is no need for calculating the second column of the matrix v. a b Applying Exercise 11.3.3 to u = u(φ , 0, 0) v u (0 , 0, ψ ) = , we get 0 1 c d ! " 1 + cos( θ) = 2 aa 2 2 2 2 = 2 cos( r0) cos( r1) + 2 sin( r0) sin( r1)

−4 cos(2 s) cos( r0) sin( r0) cos( r1) sin( r1) 2 2 = (1 + cos( θ ))(1 + cos( θ )) + (1 − cos( θ ))(1 − cos( θ )) 4 0 1 4 0 1 − cos(2 s) sin( θ0) sin( θ1)

= 1 + cos( θ0) cos( θ1) − cos( ψ0 + φ1) sin( θ0) sin( θ1),

ie iφ sin( θ) = 2 ab

iφ0 i2 s 2 −i2 s 2 = 2ie e cos( r0) cos( r1) sin( r1) − e sin( r0) sin( r1) cos( r1) 2 2 + cos( ,r0) sin( r0) cos( r1) − sin( r0) cos( r0) sin( r1) iφ0 = ie [cos(2 s) cos(2 r0) sin(2 r1) + i sin(2 s) sin(2 r1) - + sin(2 r0) cos(2 r1)] iφ0 = ie [cos( ψ0 + φ1) cos( θ0) sin( θ1) + i sin( ψ0 + φ1) sin( θ1)

+ sin( θ0) cos( θ1)] ,

ie iψ sin( θ) = −2ab

iψ1 i2 s 2 −i2 s 2 = 2ie e cos( r0) sin( r0) cos( r1) − e sin( r0) cos( r0) sin( r1) 2 2 + cos( ,r0) cos( r1) sin( r1) − sin( r0) sin( r1) cos( r1) iψ1 = ie [cos(2 s) sin(2 r0) cos(2 r1) + i sin(2 s) sin(2 r0) - + cos(2 r0) sin(2 r1)] iψ1 = ie [cos( ψ0 + φ1) sin( θ0) cos( θ1) + i sin( ψ0 + φ1) sin( θ0)

+ cos( θ0) sin( θ1)] . Let us collect the outcomes:

Proposition 11.3.4. Let u(φ, θ, ψ ) = u0 u1, where uj = u(φj, θ j, ψ j). Then cos( θ) = cos( θ0) cos( θ1)−cos( ψ0+φ1) sin( θ0) sin( θ1), cos( ψ + φ ) cos( θ ) sin( θ ) + i sin( ψ + φ ) sin( θ ) + sin( θ ) cos( θ ) eiφ = e iφ0 0 1 0 1 0 1 1 0 1 , sin( θ) cos( ψ + φ ) sin( θ ) cos( θ ) + i sin( ψ + φ ) sin( θ ) + cos( θ ) sin( θ ) eiψ = e iψ1 0 1 0 1 0 1 0 0 1 . sin( θ) 11.4. Quaternions 617

−1 Exercise 11.3.5. Let u(φ1, θ 1, ψ 1) = u(φ0, θ 0, ψ 0) , where

0 ≤ φk < 2π, 0 ≤ θk ≤ π and − 2π ≤ ψk < 2π.

Express ( φ1, θ 1, ψ 1) in terms of ( φ0, θ 0, ψ 0).

11.4 Quaternions

The quaternion space H is the associative R-algebra with a vector space basis {1, i, j, k}, where 1 ∈ H is the unit and

i2 = j2 = k2 = −1 = ijk .

The mapping 3 x = ( xm)m=0 "→ x01 + x1i + x2j + x3k identiﬁes R4 with H, and the quaternion inner product is given by

(x, y ) "→ ( x, y )H := x0y0 + x1y1 + x2y2 + x3y3, the corresponding norm being

1/2 x "→ $ x$H := (x, x )H . (11.5)

For all x, y ∈ H we have $xy $H = $x$H $y$H; if $x$H = 1 then both y "→ xy and y "→ yx are linear isometries H → H. In particular, the unit sphere S3 ⊂ R4 =∼ H is a multiplicative group. Exercise 11.4.1. Show that i, j, k satisfy the following multiplication rules:

i = jk = −kj , j = ki = −ik and k = ij = −ji .

11.4.1 Quaternions and SU(2) A bijective homomorphism S3 → SU(2) is deﬁned by

x + i x x + i x α β x "→ u(x) = 0 3 1 2 =: , (11.6) −x + i x x − ix −β α ! 1 2 0 3" ! " 2 2 with det u(x) = || x|| H = |x| = 1. Thus from Lemma 11.2.1 we get:

Proposition 11.4.2 (S3 =∼ SU(2) ). We have S3 =∼ SU(2) . 618 Chapter 11. Fourier analysis on SU (2)

The reader may ask why we reject perhaps a more obvious candidate for a homomorphism, namely x + i x x + i x x "→ 0 1 2 3 ; −x + i x x − ix ! 2 3 0 1" the reason is that our choice ﬁts perfectly with the choice of traditional Euler 3 4 angles, where the x3-axis is the “fundamental one”. On S ⊂ R =∼ H, the Euler angle representation is φ+ψ θ x0 cos 2 cos 2 x − sin φ−ψ sin θ x = 1 = 2 2 , x   φ−ψ θ  2 cos 2 sin 2 φ+ψ θ x3  sin cos     2 2      where the ranges of parameters φ, θ, ψ are the same as in ( 11.4 ), i.e. 0 ≤ φ < 2π, 0 ≤ θ ≤ π, −2π ≤ ψ < 2π. Notice that the Euler angle representation is unique if and only if 0 < θ < π , but this is not a true problem for us, since θ ∈ { 0, π } corresponds to a set of lower dimension on S3. Remark 11.4.3 . Recall that for u = u(φ, θ, ψ ) ∈ SU(2), the Euler angle representation is given by ei( φ+ψ)/2 cos θ ei( φ−ψ)/2i sin θ u = 2 2 e−i( φ−ψ)/2i sin θ e−i( φ+ψ)/2 cos θ ! 2 2 " eiφ/ 2 0 cos( θ/ 2) i sin( θ/ 2) eiψ/ 2 0 = , 0 e −iφ/ 2 i sin( θ/ 2) cos( θ/ 2) 0 e −iψ/ 2 ! " ! " ! " where 0 ≤ φ < 2π, 0 ≤ θ ≤ π and −2π ≤ ψ < 2π.

11.4.2 Quaternions and SO(3) Let P : H → H be the orthogonal projection ∼ 3 P (x) := x1i + x2j + x3k = (xm)m=1 . The subspace P (H) ⊂ H is naturally identiﬁed with R3, and the cross product of R3 is a “shadow” of the quaternion product: xy − yx P (P (x)P (y)) = =∼ (x )3 × (y )3 . 2 m m=1 m m=1 Then S2 ⊂ R3 =∼ P (H) ⊂ H consists of points x with cos(( φ + ψ)/2) = 0, i.e. 0 0 θ ˜ ± cos φ sin 2 cos φ sin θ x =  θ  =  ˜ , ± sin φ sin 2 sin φ sin θ  ± cos θ   cos θ˜   2        11.4. Quaternions 619 where 0 ≤ φ < 2π and 0 ≤ θ, θ˜ ≤ π. of S2. The quaternion conjugate of x ∈ H is

∗ x := x01 − x1i − x2j − x3k.

It is easy to see that u(x∗) = u(x)∗. Now ( x "→ x∗) : H → H is a linear isometry ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 such that ( x ) = x, ( xy ) = y x and x x = xx = $x$H1. Consequently, if $x$H = 1 then (y "→ xyx ∗) : H → H is a linear orientation-preserving isometry mapping P (H) → P (H) bijectively; ∗ hence y "→ xyx corresponds to a rotation Rx ∈ SO(3), and x "→ Rx is a group homomorphism S3 → SO(3). This surjective homomorphism is given by

x =∼ u(x) = u(φ, θ, ψ ) "→ Re( α2 + β2) −Im( α2 − β2) 2Im( αβ ) "→ Im( α2 + β2) Re( α2 − β2) −2Re( αβ ) = ω(φ, θ, ψ ), (11.7)   2Im( αβ ) 2Re( αβ ) |α|2 − | β|2   i.e. u(φ, θ, ψ ) "→ ω(φ, θ, ψ ), and its kernel is {1, −1} ∈ S3. Thus we obtain

Theorem 11.4.4 (SO(3) =∼ S3/{± 1}). We have

SO(3) =∼ S3/{± 1}, and the space C∞(SO(3)) consists of smooth functions f ∈ C∞(S3) satisfying f(−x) = f(x).

11.4.3 Invariant integration on SU(2)

Let us consider the doubly covering homomorphism

(u(φ, θ, ψ ) "→ ω(φ, θ, ψ )) : SU(2) → SO(3) , presented in formula ( 11.7 ) in Section 11.4.2 . From this, recalling the Haar integral (11.1 ) of SO(3), we obtain the Haar integral for SU(2):

1 2π π π f "→ f(x) d x = f(u(φ, θ, ψ )) sin( θ) d φ dθ dψ. 16 π2 (SU(2) (−2π (0 (−π Exercise 11.4.5. Deduce the Haar integral of SU(2) by considering the group as the unit sphere in R4 and expressing the Lebesgue measure of R4 in “polar coordinates”. 620 Chapter 11. Fourier analysis on SU (2)

11.4.4 Symplectic groups Here we brieﬂy review the notion of symplectic groups and show that the group SU(2) can be also viewed as the ﬁrst symplectic group Sp(1). In general, the subgroups

O( n) < GL( n, R), U( n) < GL( n, C) and Sp( n) < GL( n, H) have a lot in common, namely all of them are subgroups of unitary elements of the respective general linear groups with respect to their inner products, Euclidean, Hermitian, and symplectic, respectively. For any n ∈ N, the symplectic group Sp( n) is the group of automorphisms of Hn preserving the norm. More precisely, for x, y ∈ Hn we can generalise the inner product ( 11.5 ) from H to Hn by setting

n ∗ (x, y )Hn := xjyj , j=1 0 n so that the norm in H becomes || x|| Hn = (x, x )Hn . The inner product (x, y )Hn is sometimes called the symplectic inner product .

Deﬁnition 11.4.6 (Symplectic groups Sp( n)). The symplectic group is deﬁned as the set

n Sp( n) = {A ∈ GL( n, H) : || Ax || Hn = || x|| Hn for all x ∈ H }.

As an immediate corollary we obtain:

Corollary 11.4.7. We have

3 Sp(1) =∼ {x ∈ H : || x|| H = 1 } =∼ S =∼ SU(2) .

Exercise 11.4.8. Show that if A ∈ Sp( n) then (Ax, Ay )Hn = (x, y )Hn for all x, y ∈ Hn.

Remark 11.4.9 ( Symplectic matrices) . The inner-product preserving identification H = C2 = R4 extends to the inner-produce preserving identification Hn = C2n = R4n. In particular, norms on Hn and C2n coincide. Therefore, the symplectic group Sp( n) can be identified with a subgroup of GL(2 n, C) preserving the Euclidean norm with respect to the matrix multiplication. Thus, Sp( n) can be identified with α β the subgroup of U(2 n) of matrices of the form , where α, β ∈ End( Cn). −β α Such matrices are called symplectic matrices . ! " 11.5. Lie algebra and differential operators on SU(2) 621

Remark 11.4.10 ( Complex symplectic groups) . The complex symplectic group is deﬁned by Sp( n, C) = {A ∈ GL(2 n, C) : AT JA = J},

T ∗ 0 −In where A = A is the transpose matrix of A and J = , where In in In 0 ! 2" the identity matrix in GL( n, C). The matrix J has the property J = I2n so it can be viewed as an extension of the complex identity i 2 = −1 to higher dimensions. The product in GL(2 n, C) deﬁned by a · b := aT Jb is invariant under the action of Sp( n, C). Exercise 11.4.11. Show that Sp( n) = Sp( n, C) ∩ U(2 n). Also show that Sp( n, C) is not compact but that Sp( n) is.

11.5 Lie algebra and diﬀerential operators on SU(2)

Let f ∈ C∞(SU(2)) and u : R → SU(2) be a smooth function. Let φ(t), θ (t), ψ (t) be the Euler angles of u(t). If 0 < φ (0) < 2π, 0 < θ (0) < π and −2π < ψ (0) < 2π then d ∂ ∂ ∂ f(u) = φ′(0) + θ′(0) + ψ′(0) f(u(φ, θ, ψ )) . (11.8) dt ∂φ ∂θ ∂ψ 1t=0 ! " 1t=0 1 1 1 1 This is the derivative1 of f at u(0) ∈ SU(2) in the direction u′(0),1 as expressed in the Euler angles. Of particular interest are those diﬀerential operators that arise naturally from the Lie algebra. Let us consider one-parametric subgroups ω1, ω 2, ω 3 : R → SU(2), where

cos( t/ 2) i sin( t/ 2) ω (t) = , 1 i sin( t/ 2) cos( t/ 2) ! " cos( t/ 2) − sin( t/ 2) ω (t) = , 2 sin( t/ 2) cos( t/ 2) ! " eit/ 2 0 ω (t) = . 3 0 e −it/ 2 ! "

Let us also make a special choice wj := ωj(π/ 2), i.e.

1 i w = 2 −1/2 , 1 i 1 ! " 1 −1 w = 2 −1/2 , 2 1 1 ! " 1+i 0 w = 2 1/2 . 3 0 1 − i ! " 622 Chapter 11. Fourier analysis on SU (2)

Exercise 11.5.1. Find the Euler angles of ωj(t).

′ Deﬁnition 11.5.2 (Basis of su (2) ). Let Yj := ωj(0), i.e.

1 0 i 1 0 −1 1 i 0 Y = , Y = , Y = . 1 2 i 0 2 2 1 0 3 2 0 −i ! " ! " ! "

Matrices Y1, Y 2, Y 3 constitute a basis for the real vector space su (2), the Lie algebra of SU(2).

2 Remark 11.5.3 ( Pauli matrices) . We note that the matrices i Yj, j = 1 , 2, 3, are i known as Pauli (spin) matrices in physics. In our case, the coefficient 2 appears in front of these Pauli matrices because we obtained Yj’s as elements of the Lie algebra su (2) (leading to the coefficient i), and because of the use of the Euler 1 angles (leading to the coefficient 2 ). It can be also noted that k = span {Y3} and p = span {Y1, Y 2} form a Cartan pair of the Lie algebra su (2), but we will not pursue this topic here. Exercise 11.5.4. Check the commutation relations

[Y1, Y 2] = Y3, [Y2, Y 3] = Y1, [Y3, Y 1] = Y2.

State explicitly a Lie algebra isomorphism su (2) → so (3). Exercise 11.5.5. For x, y ∈ R3 let us deﬁne X, Y ∈ su (2) by

3 3 X = xjYj and Y = yjYj. j=1 j=1 0 0 What has [ X, Y ] to do with the vector cross product x × y ∈ R3?

Exercise 11.5.6. Let Y (y) = y1Y1 + y2Y2 + y3Y3. Show that

sin( $y$) u = exp( Y (y)) = I cos( $y$) + Y (y) , $y$ and (provided that $y$ is small enough) that

−i( u + u ) $y$ 12 21 y = u − u . sin( $y$)  21 12  i( u22 − u11 )   Recall that an inner automorphism of a group G is a group isomorphism of the form ( x "→ g−1xg ) : G → G for some g ∈ G. Let us study speciﬁc inner automorphisms of SU(2) conjugating the one-parametric subgroups ω1, ω 2, ω 3 to each other. 11.5. Lie algebra and diﬀerential operators on SU(2) 623

Proposition 11.5.7 (Conjugating one-parametric subgroups to each other). Let wj = ωj(π/ 2) and t ∈ R. Then

−1 w1 ω2(t) w1 = ω3(t), −1 w2 ω3(t) w2 = ω1(t), −1 w3 ω1(t) w3 = ω2(t). The diﬀerential versions of these formulae are

−1 w1 Y2 w1 = Y3, −1 w2 Y3 w2 = Y1, −1 w3 Y1 w3 = Y2. Proof. We can calculate 1 1 −i eit/ 2 0 1 i w−1 ω (t) w = 1 3 1 2 −i 1 0 e −it/ 2 i 1 ! " ! " ! " 1 eit/ 2 + e −it/ 2 ie it/ 2 − ie −it/ 2 = 2 −ie it/ 2 + ie −it/ 2 eit/ 2 + e −it/ 2 ! " cos( t/ 2) − sin( t/ 2) = sin( t/ 2) cos( t/ 2) ! " = ω2(t). We also have 1 1 −1 eit/ 2 0 1 1 w ω (t) w−1 = 2 3 2 2 1 1 0 e −it/ 2 −1 1 ! " ! " ! " 1 eit/ 2 + e −it/ 2 eit/ 2 − ie −it/ 2 = 2 eit/ 2 − ie −it/ 2 eit/ 2 + e −it/ 2 ! " cos( t/ 2) i sin( t/ 2) = i sin( t/ 2) cos( t/ 2) ! " = ω1(t). Finally, we have

−1 w3 ω1(t) w3 1 1+i 0 cos( t/ 2) i sin( t/ 2) 1 − i 0 = 2 0 1 − i i sin( t/ 2) cos( t/ 2) 0 1+i ! " ! " ! " 1 1+i 0 (1 − i) cos( t/ 2) (i − 1)i sin( t/ 2) = 2 0 1 − i (i + 1) sin( t/ 2) (1 + i) cos( t/ 2) ! " ! " cos( t/ 2) − sin( t/ 2) = sin( t/ 2) cos( t/ 2) ! " = ω2(t). 624 Chapter 11. Fourier analysis on SU (2)

The differential version follows immediately by differentiating the above formulae with respect to t at t = 0. Definition 11.5.8 (Left-invariant differential operators). To a vector Y ∈ su (2) we ∞ ∞ associate the left-invariant differential operator DY : C (SU(2)) → C (SU(2)) defined by d D f(u) = f(u exp( tY )) . Y dt 1t=0 1 In the sequel, we denote Dj := DYj . 1 1 Proposition 11.5.9 (Derivatives D1, D 2, D 3 in Euler angles). Expressed in the Euler angles, ∂ sin( ψ) ∂ cos( θ) ∂ D = cos( ψ) + − sin( ψ) , 1 ∂θ sin( θ) ∂φ sin( θ) ∂ψ ∂ cos( ψ) ∂ cos( θ) ∂ D = − sin( ψ) + − cos( ψ) , 2 ∂θ sin( θ) ∂φ sin( θ) ∂ψ ∂ D = , 3 ∂ψ provided that sin( θ) '= 0 , i.e. 0 < θ < π . Proof. The simplest case is d d ∂ D f(u) = f(u(φ, θ, ψ ) ω (t)) = f(u(φ, θ, ψ + t)) = f(u). 3 dt 3 dt ∂ψ 1t=0 1t=0 1 1 1 1 ′ ′ ′ Next we shall deal with D1. According1 to ( 11.8 ), we need to calculate1 φ (0) , θ (0) , ψ (0), where u(φ, θ, ψ ) = u(φ0, θ 0, ψ 0) ω1(t). Let us exploit Proposition 11.3.4 with u(φ1, θ 1, ψ 1) = ω1(t) = u(0 , t, 0), yielding

cos( θ) = cos( θ0) cos( t) − cos( ψ0) sin( θ0) sin( t),

i( φ−φ0) e sin( θ) = cos( ψ0) cos( θ0) sin( t) + i sin( ψ0) sin( t) + sin( θ0) cos( t), iψ e sin( θ) = cos( ψ0) sin( θ0) cos( t) + i sin( ψ0) sin( θ0) + cos( θ0) sin( t). Diﬀerentiating these equalities with respect to t at t = 0, we obtain

′ −θ (0) sin( θ0) = − cos( ψ0) sin( θ0), ′ ′ iφ (0) sin( θ0) + θ (0) cos( θ0) = cos( ψ0) cos( θ0) + i sin( ψ0), iψ0 ′ ′ e [i ψ (0) sin( θ0) + θ (0) cos( θ0)] = cos( θ0).

′ ′ Thereby θ (0) = cos( θ0), φ (0) = sin( ψ0)/ sin( θ0), and

′ −iψ0 iψ (0) sin( θ0) = e − cos( ψ0) cos( θ0) = −i sin( ψ0) cos( θ0),

′ yielding ψ (0) = − sin( ψ0,) cos( θ0)/ sin( θ0-). This proves the expression for D1. Lastly, the D2 formula must be deduced. Again, we use equation ( 11.8 ), where this 11.5. Lie algebra and diﬀerential operators on SU(2) 625

time u(φ, θ, ψ ) = u(φ0, θ 0, ψ 0) ω2(t). Thus let u(φ1, θ 1, ψ 1) = ω2(t) = u(π/ 2, t, −π/ 2), and apply Proposition 11.3.4 : noticing that sin( ψ0 + π/ 2) = cos( ψ0) and cos( ψ0 + π/ 2) = − sin( ψ0), we get

cos( θ) = cos( θ0) cos( t) + sin( ψ0) sin( θ0) sin( t), i( φ−φ0) e sin( θ) = − sin( ψ0) cos( θ0) sin( t) + i cos( ψ0) sin( t)

+ sin( θ0) cos( t), i( ψ+π/ 2) e sin( θ) = − sin( ψ0) sin( θ0) cos( t) + i cos( ψ0) sin( θ0)

+ cos( θ0) sin( t).

Derivating these equalities with respect to t at t = 0, we get

′ −θ (0) sin( θ0) = sin( ψ0) sin( θ0), ′ ′ iφ (0) sin( θ0) + θ (0) cos( θ0) = − sin( ψ0) cos( θ0) + i cos( ψ0), iψ0 ′ ′ ie [i ψ (0) sin( θ0) + θ (0) cos( θ0)] = cos( θ0).

′ ′ In this case, θ (0) = − sin( ψ0), φ (0) = cos( ψ0)/ sin( θ0), and

′ −iψ0 iψ (0) sin( θ0) = sin( ψ0) − ie cos( θ0) = −i cos( ψ0) cos( θ0),

′ , - so that ψ (0) = − cos( ψ0) cos( θ0)/ sin( θ0). This completes the proof. Endowing SU(2) with the Riemannian metric inherited from R4 ⊂ S3 =∼ SU(2), we could show that {Y1, Y 2, Y 3} is orthogonal, so that operators D1, D 2, D 3 form a good choice for generating the first-order left-invariant partial differential operators. However, in order to simplify notation in the sequel, we will work with slightly different operators. Definition 11.5.10 (Creation, annihilation, and neutral operators). Let us define ∞ left-invariant first-order partial differential operators ∂+, ∂ −, ∂ 0 : C (SU(2)) → C∞(SU(2)), called creation, annihilation, and neutral operators, respectively, by

−i ∂+ := i D1 − D2, D1 = 2 (∂− + ∂+) 1 ∂− := i D1 + D2, i.e. D2 = 2 (∂− − ∂+) , ∂0 := i D3, D3 = −i∂0.   The reader may ask why to work with operators ∂+, ∂ −, ∂ 0 instead of vector ﬁelds D1, D 2, D 3. The reason is that many calculations become considerably shorter. Indeed, already expressions in Euler’s angles in Corollary 11.5.13 are quite simpler than those in Proposition 11.5.9 , as well as expressions in Theorem 11.9.3 are simpler than those in Proposition 11.9.2 . The other reason is that the symbols of ∂+, ∂ −, ∂ 0 will turn out to have less non-zero elements, see Theorem 12.2.1 . The terminology of “creation”, “annihilation”, and “neutral” operators is explained in Remark 12.2.3 . 626 Chapter 11. Fourier analysis on SU (2)

2 2 2 Remark 11.5.11 ( Laplacian) . The Laplacian L satisﬁes L = D1 + D2 + D3 and we have [L, D j] = 0 for every j ∈ { 1, 2, 3}. Notice that it can be expressed as

2 L = −∂0 − (∂+∂− + ∂−∂+)/2.

Exercise 11.5.12. Show that the operators ∂+, ∂ −, ∂ 0 satisfy commutator relations

[∂0, ∂ +] = ∂+, [∂−, ∂ 0] = ∂−, [∂+, ∂ −] = 2 ∂0.

The operators ∂+, ∂ −, ∂ 0 coincide with operators Hˆ+, Hˆ−, Hˆ0 in Vilenkin [ 145 , p. 140]. By Proposition 11.5.9 , we also have

Corollary 11.5.13 (Operators ∂+, ∂ −, ∂ 0 in Euler’s angles).

∂ 1 ∂ cos( θ) ∂ ∂ = e −iψ i − + , + ∂θ sin( θ) ∂φ sin( θ) ∂ψ ! " ∂ 1 ∂ cos( θ) ∂ ∂ = e iψ i + − , − ∂θ sin( θ) ∂φ sin( θ) ∂ψ ! " ∂ ∂ = i . 0 ∂ψ

a(u) b(u) DY a D Y b Exercise 11.5.14. For u = ∈ SU(2), let us denote DY u := . c(u) d(u) DY c D Y d Without resorting to Euler! angles, show" that ! "

i b a 1 b −a i a −b D u = , D u = , D u = , 1 2 d c 2 2 d −c 3 2 c −d ! " ! " ! " −b 0 0 −a 1 −a b ∂ u = , ∂ u = , ∂ u = . + −d 0 − 0 −c 0 2 −c d ! " ! " ! " 11.6 Irreducible unitary representations of SU(2)

In this section we shall determine irreducible unitary representations of SU(2) up to unitary equivalence. Our ﬁrst task is to ﬁnd a natural representation to start with. Matrices act naturally on vectors by matrix multiplication, and this is indeed 2 1×2 a good idea: let us identify z = ( z1, z 2) ∈ C with matrix z = z1 z2 ∈ C , and consider

T : SU(2) → GL( C[z1, z 2]) , (T (u)f)( z) := f(zu ), 11.6. Irreducible unitary representations of SU(2) 627

where GL( C[z1, z 2]) is the space of invertible linear mappings on the complex 2 vector space C[z1, z 2] consisting of two-variable polynomials f : C → C. Clearly T is a representation of SU(2) on C[z1, z 2], necessarily reducible as C[z1, z 2] is inﬁnite-dimensional. So, decomposing T is the next step. It is clear that the orders of polynomials f, T (u)f ∈ C[z1, z 2] are the same, so that e.g. polynomials f ∈ C[z1, z 2] of order less than k ∈ N0 form a T -invariant subspace. But these subspaces can clearly be decomposed further. For each l ∈ 1 2 N0, let Vl be the subspace of C[z1, z 2] containing the homogeneous polynomials of order 2 l ∈ N0, i.e.

2l k 2l−k 2l Vl = f ∈ C[z1, z 2] : f(z1, z 2) = akz1 z2 , {ak}k=0 ⊂ C . 6 7 k0=0 Let

Tl : SU(2) → GL( Vl), (Tl(u)f)( z) = f(zu ), denote the restriction of T on the T -invariant subspace Vl of dimension (2 l + 1) ∈ + Z . Our objective is to show that Tl is irreducible, unitary with respect to a natural inner product of Vl, and that (up to unitary equivalence) there are no other irreducible unitary representations for SU(2). By regarding f ∈ Vl as a 2 natural function on SU(2), we shall endow Vl with the L -inner product of the group. A natural basis for the vector space Vl is {plk : k ∈ { 0, 1,..., 2l}} , where

k 2l−k plk (z) = z1 z2 . (11.9)

Theorem 11.6.1. As deﬁned above, Tl is irreducible. Moreover, plk is an eigen- iφ(k−l/ 2) function of Tl(u(φ, 0, 0)) with eigenvalue e .

Proof. By Schur’s Lemma (Corollary 6.3.25 ), irreducibility of Tl follows, if we can show that its intertwining operators are necessarily scalar. So let A ∈ End( Vl) be an intertwining operator for Tl, i.e. such that for every u ∈ SU(2) holds

AT l(u) = Tl(u)A.

Due to the decomposition

u(s, r, t ) = u(s, 0, 0) u(0 , r, 0) u(0 , 0, t ), checking the cases u = u(0 , 0, s ) and u = u(0 , r, 0) suﬃces. First,

Tl(u(2 s, 0, 0)) plk (z) = plk (zu (2 s, 0, 0)) = is −is is(2 k−l) = plk (z1e , z 2e ) = e plk (z), 628 Chapter 11. Fourier analysis on SU (2)

is(2 k−l) so that plk is an eigenvector of Tl(u(2 s, 0, 0)) with eigenvalue e . The same thing is true for the vector Ap lk ∈ Vl:

Tl(u(2 s, 0, 0)) Ap lk (z) = AT l(u(2 s, 0, 0)) plk (z) is(2 k−l) = A z "→ e plk (z) is8(2 k−l) 9 = e Ap lk (z).

Let Ap lk = j aj plj ∈ Vl. Then

: is(2 k−l) e Ap lk = Tl(u(2 s, 0, 0)) aj plj j 0 = aj Tl(u(2 s, 0, 0)) plj j 0 is(2 j−l) = aj e plj , j 0 which is possible only if Ap lk = ak plk . This yields especially

Tl(u(0 , 2r, 0)) Ap l0(z) = Tl(u(0 , 2r, 0)) a0pl0(z)

= a0 pl0(zu (0 , 2r, 0)) l = a0 (z1 sin( r) + cos( r)z2) l = a sin( r)k cos( r)l−kp (z). 0 k lk 0k ! " On the other hand, this coincides with

AT l(u(0 , 2r, 0)) pl0(z) = A (z "→ pl0(zu (0 , 2r, 0))) l = A z "→ (z1 sin( r) + cos( r)z2) l = A sin( r)k cos( r)l−k p (z) k lk 0k ! " l = sin( r)k cos( r)l−k a p (z). k k lk 0k ! "

Choosing r so that sin( r) '= 0 '= cos( r), we see that ak = a0 for each k. Thus A = a0I is scalar.

Next we shall ﬁnd out that, up to equivalence, representations Tl provide a complete set of irreducible representations for SU(2). Lemma 11.6.2 (Decomposition of elements of SU(2) ). For g ∈ SU(2) there is a decomposition g = uhu −1, where u, h ∈ SU(2) such that h is diagonal. 11.6. Irreducible unitary representations of SU(2) 629

a b Proof. Let g = ∈ SU(2). If b = 0 then the claim is trivial with the −b a choice h := g and!u = I," so we assume b '= 0. The characteristic polynomial of g is Pg : C → C, where

Pg(z) = det( zI − g) = ( z − a)( z − a) − b(−b) |a|2+|b|2=1 = z2 − 2Re( a)z + 1 .

Now |a| '= 1, because b '= 0, and therefore Pg has two distinct roots, i.e. g has two diﬀerent eigenvalues z1, z 2 ∈ C. Let z 0 h = 1 , 0 z ! 2" and let the column vectors of v ∈ C2×2 be the corresponding normalised eigenvectors of g. Now g = vhv −1. The eigenvectors corresponding to the diﬀerent eigenvalues are orthogonal. Thus v ∈ U(2), and u := λv ∈ SU(2) with a suitable choice of λ ∈ C, |λ| = 1. Now we have g = uhu −1.

Theorem 11.6.3 (Completeness of the representation series). Let T∞ : SU(2) → GL( V ) be an irreducible representation. Then T∞ is equivalent to Tl, where dim( V ) = 2l + 1 . 1 Proof. Let m ∈ 2 N0 ∪ {∞} . Let χm : SU(2) → C be the character of Tm. By 1 2 Theorem 7.8.6 it is enough to show that (χ∞, χ l)L (SU(2)) '= 0 for some l ∈ 2 N0. Notice that −1 χm(uhu ) = χm(h).

By Lemma 11.6.2 , we may identify χm with fm : R → C deﬁned by eit 0 f (t) := χ (h(t)) , where h(t) = . m m 0 e −it ! " Thereby

(χ∞, χ l)L2(SU(2)) = χ∞(g) χl(g) d µSU(2) (g) (SU(2) 1 2π = f (t) f (t) d t. 2π ∞ l (0 Recall that {plk : 0 ≤ k ≤ 2l} is a basis for vector space Vl, and it −it it(2 k−l) Tl(h(t)) plk (z) = plk (z h (t)) = plk (z1e , z 2e ) = e plk (z). Thus we get 2l l it(2 k−l) itk fl(t) = Tr ( Tl(h(t))) = e = e . k0=0 k0=−l 630 Chapter 11. Fourier analysis on SU (2)

Thereby span {fl : l ∈ N0/2} is dense in the space of 2 π-periodic continuous even functions. Because f∞ : R → C is a 2 π-periodic continuous even function,

2π f∞(t) fl(t) d t '= 0 (0 for some l ∈ N0/2. This completes the proof.

11.6.1 Representations of SO(3) We now brieﬂy review representations of SO(3) and related spherical harmonics. We will only sketch this topic since it is analogous to constructing representations of SU(2). However, to provide an additional insight into the topic we indicate the relation to spherical harmonics. The group SO(3) is a subgroup of the group GL(3 , R) of three dimensional 3 matrices acting on R by the matrix multiplication. In analogy to the spaces Vl in the case of SU(2), we introduce the space Pl to be the complex subspace of the space C[x1, x 2, x 3] containing all the homogeneous polynomials of order l ∈ N0 in three variables. The group SO(3) acts on Pl in the same way as in the case of SU(2), namely we have

Tl : SO(3) → GL( Pl), (Tl(u)f)( x) = f(xu ).

1 Exercise 11.6.4. Show that dim Pl = 2 (l + 1)( l + 2) .

The problem with the spaces Pl is they do not yield irreducible representation spaces for l 2. For example, the space generated by the polynomial x2 + x2 + x2 ≥ 1 2 3 is a non-trivial SO(3)-invariant subspace of P2. Thus, we have to decompose the spaces Pl further which is done in the following exercises. ∂2 ∂2 ∂2 Exercise 11.6.5 (Spherical harmonics). Let = 2 + 2 + 2 be the Laplacian L ∂x 1 ∂x 2 ∂x 3 on R3. A polynomial f P is called a harmonic polynomial of order l if f = 0, ∈ l L and we set Hl := f Pl : f = 0 . Show that dim Hl = 2 l + 1. Restrictions of harmonic polynomials{ ∈ to theL sphere } S2 are called spherical harmonics of order l. Consequently, since harmonic polynomials are homogeneous, the dimension of the space of spherical harmonics of order l is also 2 l + 1. Exercise 11.6.6. Show that is SO(3)-invariant. Consequently, show that H is L l an SO(3)-invariant subspace of Pl. Exercise 11.6.7. Show that the mapping

T : SO(3) GL( H ), (T (u)f)( x) = f(xu ) l → l l is an irreducible representation of SO(3). Moreover, show that an irreducible representation T∞ : SO(3) GL( V ) is equivalent to Tl, where dim( V ) = 2 l + 1 for l N . → ∈ 0 11.7. Matrix elements of representations of SU(2) 631

11.7 Matrix elements of representations of SU(2)

Recalling polynomials in ( 11.9 ), we see that the collection

q : k l, l + 1 , , +l 1, +l { lk ∈ {− − · · · − }} is an orthonormal basis for the representation space Vl, where

l−k l+k zl z2 qlk (z) = . (l k)!( l + k)! − l Let us compute the matrix elements tmn (φ, θ, ψ ) of Tl(u(φ, θ, ψ )) with respect to this basis. Theorem 11.7.1 (Matrix elements of T l). Let u SU(2) be given by ∈ a b ei( φ+ψ)/2 cos θ ei( φ−ψ)/2i sin θ u = u(φ, θ, ψ ) = = 2 2 . c d e−i( φ−ψ)/2i sin θ e−i( φ+ψ)/2 cos θ ! 2 2 ! Then

l tmn (u) = d l−m d l+m (z a + z c)l−n(z b + z d)l+n = 1 2 1 2 , (11.10) dz1 dz2 (l m)!( l + m)!( l n)!( l + n)! ! ! − − where one can see that the right hand side does not depend on z. In Euler’s angles

l −i( mφ +nψ ) l tmn (φ, θ, ψ ) = e Pmn (cos( θ)) , where

(1 x)(n−m)/2 d l−m P l (x) = cl − (1 x)l−n(1 + x)l+n mn mn (1 + x)(m+n)/2 dx − ! " # with l−n n−m l −l ( 1) i (l + m)! cmn = 2 − . (l n)! ( l + n)! $(l m)! − − Moreover, we have

min {l−n,l −m} l (l m)!( l + m)! tmn (u) = − $ (l n)!( l + n)! × − i=max %{0,n −m} (l n)!( l + n)! − aibl−m−icl−n−idn+m+i. (11.11) × i!( l n i)!( l m i)!( n + m + i)! − − − − 632 Chapter 11. Fourier analysis on SU (2)

l l Deﬁnition 11.7.2 (Representations t ). The matrix ( tmn )m,n with indices m, n such that l m, n l and l m, l n Z will be denoted by tl. The usual convention− in≤ all the formulae≤ is− that 0!− = 1.∈ We note that although l, m, n can be half-integers, the spacing between them is such that the diﬀerences between any of them are always integers!

Remark 11.7.3 . Let 0 θ π and x = cos( θ). Notice that ≤ ≤

(1 x)1/2 = 2 1/2 sin( θ/ 2) , − (1 + x)1/2 = 2 1/2 cos( θ/ 2) .

In [ 145 ] the classical orthogonal polynomials of Legendre and Jacobi are connected l to functions Pmn , which consequently could be called e.g. generalised Legendre– Jacobi functions .

Proof of Theorem 11.7.1 . First, notice that if vectors qm form an orthonormal basis of a ﬁnite-dimensional inner product space V , then we can write u = (u, q ) q for all u V . Especially, T q = T q for a linear op- m m V m ∈ n m mn m erator T : V V , where the numbers Tmn = ( T q n, q m) are the matrix elements & → & V of T with respect to the chosen basis. Thus for the linear operator Tl(u) : Vl Vl, l → the matrix coeﬃcients tmn (u) satisfy

l Tl(u) qln (z) = tmn (u) qlm(z) m % l−m l+m l z1 z2 = tmn (u) . m (l m)!( l + m)! % − Since Tl(u)qln (z) = qln (zu ), we have

(l m)!( l + m)! d l−m d l+m tl (u) = − T (u)q (z) mn (l m)!( l + m)! dz dz l ln − 1 ! 2 ! d l−m d l+m (z a + z c)l−n(z b + z d)l+n = 1 2 1 2 , dz1 dz2 (l m)!( l + m)!( l n)!( l + n)! ! ! − − proving ( 11.10 ). Here we can use that the total number of derivatives is 2 l and l−n l+n also that ( z1a + z2c) (z1b + z2d) is a homogeneous polynomial of degree 2l in ( z1, z 2) so that the value of the derivative corresponds to the coeﬃcient of l−m l+m z1 z2 in this polynomial, and the right hand side does not depend on z. So 11.7. Matrix elements of representations of SU(2) 633 we can calculate

d l−m d l+m (z a + z c)l−n(z b + z d)l+n dz dz 1 2 1 2 1 ! 2 ! " l−n l+n # d l−m d l+m l n l + n = − dz dz i j × 1 2 i=0 j=0 ! ! % % ! ! aibjcl−n−idl+n−jzi+jz2l−i−j × 1 2 d l−m l−n l+n l n l + n = (l + m)! − dy i j × i=0 j=0 ! % % ! ! aibjcl−n−idl+n−jyi+j × y=0 d l−m ' = ( l + m)! (ay + c')l−n(by + d)l+n . dy y=0 ! " #

Recalling ad bc = det( u) = 1, we notice that a(by +d) b(ay +c) = 1, inspiring a change of variables:− let x be such that a(by +d) = ( x+1) /−2, i.e. b(ay +c) = ( x 1) /2. Hence d x/ dy = 2 ab , so that −

l 1 tmn = (l + m)! (l m)!( l + m)!( l n)!( l + n)! − − l−m d (ay + c)l−n(by + d)l+n dy y=0 ! " 1 # = (l + m)! (l m)!( l + m)!( l n)!( l + n)! − − l−m l−n l+n d x 1 x + 1 (2 ab )l−m − dx 2b 2a ! ( ! ! )x=2 ad −1 1 = (l + m)! (l m)!( l + m)!( l n)!( l + n)! − − l−m bn−m d 2−l−m (x 1) l−n(x + 1) l+n . am+n dx − x=2 ad −1 ! " #

Here x = 2 ad 1 = 2 cos( θ/ 2) 2 1 = cos( θ), i.e. cos( θ/ 2) 2 = (1 + x)/2 and − − 634 Chapter 11. Fourier analysis on SU (2) sin( θ/ 2) 2 = (1 x)/2, a = e i( φ+ψ)/2 cos( θ/ 2) and b = e i( φ−ψ)/2i sin( θ/ 2). Thus − i( φ−ψ)/2 n−m l (l + m)! −l−m e n−m tmn = 2 i (l m)!( l n)!( l + n)! i( φ+ψ)/2 n+m $ − − *e + l−m sin( θ/ 2) n−m d * + (x 1) l−n(x + 1) l+n cos( θ/ 2) m+n dx − x=cos θ ! " # (l + m)! = 2−l−m e−i( mφ +nψ ) in−m (l m)!( l n)!( l + n)! $ − − (n−m)/2 1−x d l−m 2 (x 1) l−n(x + 1) l+n 1+ x (m+n)/2 dx − x=cos θ * 2 + ! " # * + (l + m)! = 2−l e−i( mφ +nψ ) in−m (l m)!( l n)!( l + n)! $ − − (1 x)(n−m)/2 d l−m − (x 1) l−n(x + 1) l+n . (1 + x)(m+n)/2 dx − x=cos θ ! " # Let us ﬁnally establish formula ( 11.11 ). Again, we note that the total number of l−n l+n derivatives in ( 11.10 ) is 2 l and also that ( z1a + z2c) (z1b + z2d) is a homogeneous polynomial of degree 2 l in ( z1, z 2). Therefore, tl (u) = ( l m)!( l + m)! α, mn − l−m l+m l−n where α is the coeﬃcient of z1 z2 in the polynomial ( z1a + z2c) (z1b + l+n z2d) . Using the binomial formula, we have

l−n l+n l n l + n α = − aibjcl−n−idl+n−j. (11.12) i j i=0 j=0 ! ! % i+j%=l−m The restriction j l + n implies l m = i + j i + l + n, so that i n m. On the other hand, the≤ restriction j −0 implies l ≤m = i + j i. Thus,≥ substituting− j = l m i in ( 11.12 ) and letting≥ i vary in− the range≥ max 0, n m i min l −n, l − m , we obtain formula ( 11.11 ). This completes the{ proof− of Theorem} ≤ ≤ 11.7.1{ .− − } Exercise 11.7.4. Prove that ( 1) l+m( 1) l+n = ( 1) m−n (11.13) − − − if l 1 N = k : 2 k N and m, n l, l + 1 , , +l 1, +l . ∈ 2 0 { ∈ 0} ∈ {− − · · · − } Corollary 11.7.5. For each θ, we have the identities l l P−m, −n(cos( θ)) = Pmn (cos( θ)) , (11.14) l l Pnm(cos( θ)) = Pmn (cos( θ)) . (11.15) 11.8. Multiplication formulae for representations of SU(2) 635

l l Proof. By Theorem 11.7.1 , notice that Pmn (cos( θ)) = tmn (u) if

a b a = cos( θ/ 2) , u = , where b a b = i sin( θ/ 2) . ! , Then Theorem 11.7.1 yields

l−m l+m l−n l+n l d d (z1a + z2b) (z1b + z2a) tmn (u) = , dz1 dz2 (l m)!( l + m)!( l n)!( l + n)! ! ! − − and the ( m, n)-symmetry in this formula implies ( 11.14 ). Notice also that ± ± l−m l+m l−n l+n l ∗ d d (z1a z2b) ( z1b + z2a) tmn (u ) = − − dz1 dz2 (l m)!( l + m)!( l n)!( l + n)! ! ! − − l−m l+m l−n l+n d d (x1a + x2b) ( x1b x2a) = − − − dx1 dx2 (l m)!( l + m)!( l n)!( l + n)! ! ! − − = ( 1) l+m( 1) l+n tl (u) − − mn (11 .13 ) = ( 1) m−n tl (u). − mn This leads to

l l Pnm(cos( θ)) = tnm(u) l ∗ = tmn (u ) = ( 1) m−n tl (u) − mn = ( 1) m−n P l (cos( θ)) − mn Theorem 11 .7.1 l = Pmn (cos( θ)) , proving ( 11.15 ).

11.8 Multiplication formulae for representations of SU(2)

On a compact group G, a function f : G C is called a trigonometric polynomial if its translates span the ﬁnite-dimensional→ vector space

span (x f(y−1x)) : G C y G , &→ → | ∈ see Section 7.6 for details.- Trigonometric polynomials can be. expressed as a linear combination of matrix elements of irreducible unitary representations. Thus a trigonometric polynomial is continuous, and on a Lie group even C∞-smooth. 636 Chapter 11. Fourier analysis on SU (2)

Moreover, trigonometric polynomials form an algebra when endowed with the usual multiplication. On SU(2), actually,

l+l′ ′ ′ l′ l ll (l+k) ll (l+k) l+k tm′n′ tmn = Cm′m(m′+m) Cn′n(n′+n) t(m′+m)( n′+n), ′ k=%|l−l | ll ′(l+k) where Cm′m(m′+m) are so-called Clebsch-Gordan coeﬃcients, for which there are explicit formulae [ 145 ]. We are going to derive basic multiplication formulae on l trigonometric polynomials tmn : SU(2) C; for general multiplication of trigonometric polynomials, one can use this computation→ iteratively. Theorem 11.8.1. Let 1/2 1/2 t−− t−+ 1/2 t−1/2,−1/2 t−1/2,+1 /2 := t = 1/2 1/2 t+− t++ t t ! / +1 /2,−1/2 +1 /2,+1 /20 and x± := x 1/2 for x R. Then ± ∈

l (l m + 1)( l n + 1) l+ (l + m)( l + n) l− tmn t = − − tm−n− + tm−n− , −− 2l + 1 2l + 1 l (l + m + 1)( l + n + 1) l+ (l m)( l n) l− tmn t++ = tm+n+ + − − tm+n+ , 2l + 1 2l + 1 l (l m + 1)( l + n + 1) l+ (l + m)( l n) l− tmn t + = − tm−n+ − tm−n+ , − 2l + 1 − 2l + 1 l (l + m + 1)( l n + 1) l+ (l m)( l + n) l− tmn t+ = − tm+n− − tm+n− . − 2l + 1 − 2l + 1 Remark 11.8.2 . Notice the pattern of the signs above! ± l Proof. It is enough to consider multiplication formulae for functions Pmn , because by Theorem 11.7.1

l i( φm+ψn ) l tmn (φ, θ, ψ ) = e − Pmn (cos( θ)) . l l Moreover, by Corollary 11.7.5 , the tmn t formula implies the tmn t++ formula, l l −− and the tmn t + formula implies the tmn t+ formula. Indeed, evaluating P below at x = cos θ,− we have −

l 1/2 Pmn P1/2,1/2 l 1/2 = P m, n P 1/2, 1/2 − − − − (l + m + 1)( l + n + 1) l+ (l m)( l n) l− = P m+, n+ + − − P m+, n+ 2l + 1 − − 2l + 1 − − (l + m + 1)( l + n + 1) l+ (l m)( l n) l− = Pm+,n + + − − Pm+,n + 2l + 1 2l + 1 11.8. Multiplication formulae for representations of SU(2) 637 and

l 1/2 Pmn P1/2, 1/2 − l 1/2 = P m, n P 1/2,1/2 − − − (l + m + 1)( l n + 1) l+ (l m)( l + n) l− = − P m+, n− + − P m+, n− 2l + 1 − − 2l + 1 − − (l + m + 1)( l n + 1) l+ (l m)( l + n) l− = − Pm+,n − + − Pm+,n − . 2l + 1 2l + 1 l Let us now prove the tmn t formula. By Theorem 11.7.1 , we have −−

l+ Pm−n− (x) (n m)/2 l m+1 l+ (1 x) − d − l n+1 l+n = c − − − (1 x) − (1 + x) , m n (1 + x)(m+n 1) /2 dx − − ! (n m)/2 l m " # l+ (1 x) − d − = c − − − m n (1 + x)(m+n 1) /2 dx − ! d l n+1 l+n (1 x) − (1 + x) , dx − " # where

d l n+1 l+n (1 x) − (1 + x) dx − l n l+n 1 = (1 " x) − (1 + x) − [ (l# n + 1)(1 + x) + ( l + n)(1 x)] − l n l+n 1 − − − = (1 x) − (1 + x) − [ (2 l + 1)(1 + x) + 2( l + n)] − l n − l+n = (2 l + 1) (1 x) − (1 + x) − − l n l+n 1 +2( l + n) (1 x) − (1 + x) − , − yielding

1/2 l P (x) l+ l+ Pmn (x) 1/2, 1/2 P − − (x) = (2 l + 1) c − − − − m n m n cl 1/2 mn c 1/2, 1/2 − − l− l+ Pm−n− (x) +2( l + n) cm−n− l− cm−n− l 1/2 Pmn (x) P 1/2, 1/2(x) = (2 l + 1) − − (l m + 1)( l n + 1) − − l− P − − (x) (l + m)( l + n) m n . − (l m + 1)( l n + 1) − − 638 Chapter 11. Fourier analysis on SU (2)

Thereby

l 1/2 (2 l + 1) Pmn (x) P 1/2, 1/2(x) − − l+ l− = (l m + 1)( l n + 1) P − − (x) + (l + m)( l + n)P − − (x), − − m n m n l l proving the tmn t formula. The tmn t + case is similar. Indeed, by Theorem 11.7.1 , we have −− −

l+ Pm−n+ (x) (n m+1) /2 l m+1 l+ (1 x) − d − l n l+n+1 = c − + − (1 x) − (1 + x) , m n (1 + x)(m+n)/2 dx − ! (n m+1) /2 l m " # l+ (1 x) − d − = c − + − m n (1 + x)(m+n)/2 dx ! d l n l+n+1 (1 x) − (1 + x) , dx − where " #

d l n l+n+1 (1 x) − (1 + x) dx − l n 1 l+n = (1 " x) − − (1 + x) [ (l# n)(1 + x) + ( l + n + 1)(1 x)] − l n 1 l+n − − − = (1 x) − − (1 + x) [(2 l + 1)(1 x) 2( l n)] − l n l+n − − − = (2 l + 1) (1 x) − (1 + x) − l n 1 l+n 2( l n) (1 x) − − (1 + x) , − − − yielding

1/2 l P (x) l+ l+ Pmn (x) 1/2,+1 /2 P − + (x) = (2 l + 1) c − + − m n m n cl 1/2 mn c 1/2,+1 /2 − l− l+ Pm−n+ (x) 2( l n) cm−n+ l− − − cm−n+ l 1/2 Pmn (x) P 1/2,+1 /2(x) = (2 l + 1) − (l m + 1)( l + n + 1) − l− P − + (x) + (l + m)( l n) m n . − (l m + 1)( l + n + 1) − Thereby

l 1/2 (2 l + 1) Pmn (x) P 1/2,+1 /2(x) − l+ l− = (l m + 1)( l + n + 1) P − + (x) (l + m)( l n)P − + (x), − m n − − m n l proving the tmn t + formula. − 11.9. Laplacian and derivatives of representations on SU(2) 639

l l Exercise 11.8.3. Calculate formulae for tmn t++ and tmn t+ in Theorem 11.8.1 l − directly from deﬁnitions of tmn , t ++ and t+ . − Exercise 11.8.4. There are other forms of multiplication formulae that can be de- rived. For example, by using symmetries as in Corollary 11.7.5 derive the following formulae

l (l m + 1)( l n + 1) l+ (l + m)( l + n) l− tmn t++ = − − tm−n− + tm−n− , 2l + 1 2l + 1 l (l m + 1)( l + n + 1) l+ (l + m)( l n) l− tmn t+ = − tm−n+ − tm−n+ . − 2l + 1 − 2l + 1 11.9 Laplacian and derivatives of representations on SU(2)

Our aim now is to derive the formula for the Laplacian in terms of Euler’s angles and determine its eigenvalues and eigenfunctions. We also want to ﬁnd formulae for derivatives of representations. First, recall the invariant diﬀerential operators D1, D 2, D 3 and their expressions in Euler angles in view of Proposition 11.5.9 :

∂ sin( ψ) ∂ cos( θ) ∂ D := cos( ψ) + sin( ψ) , 1 ∂θ sin( θ) ∂φ − sin( θ) ∂ψ

∂ cos( ψ) ∂ cos( θ) ∂ D := sin( ψ) + cos( ψ) , 2 − ∂θ sin( θ) ∂φ − sin( θ) ∂ψ ∂ D := . 3 ∂ψ 2 2 2 The Laplacian is given by = D1 + D2 + D3 and we have [ , D j] = 0 , see L L 2 2 2 L Remark 11.5.11 . Therefore, we see that D3 = ∂ /∂ψ , as well as calculate

∂2 cos θ sin ψ ∂ sin ψ ∂2 D2 = cos ψ cos ψ + − + 1 ∂θ 2 sin( θ)2 ∂φ sin θ ∂θ ∂φ

sin( θ)2 cos( θ)2 ∂ cos θ ∂2 − − sin ψ sin ψ − sin( θ)2 ∂ψ − sin θ ∂θ ∂ψ ! sin ψ ∂2 sin ψ ∂2 cos θ ∂2 + cos ψ + sin ψ sin θ ∂θ ∂φ sin θ ∂φ 2 − sin θ ∂φ ∂ψ ! cos θ ∂ ∂2 sin ψ sin ψ + cos ψ − sin θ − ∂θ ∂θ ∂ψ

cos ψ ∂ sin ψ ∂2 + + sin θ ∂φ sin θ ∂φ ∂ψ cos θ ∂ cos θ ∂2 cos ψ sin ψ , − sin θ ∂ψ − sin θ ∂ψ 2 ! 640 Chapter 11. Fourier analysis on SU (2) so that after cancellations we get ∂2 cos ψ cos θ sin ψ ∂ 2 cos ψ sin ψ ∂2 D2 = cos( ψ)2 2 + 1 ∂θ 2 − sin( θ)2 ∂φ sin θ ∂θ ∂φ (1 + cos( θ)2) cos ψ sin ψ ∂ cos ψ cos θ ∂2 + 2 sin ψ sin( θ)2 ∂ψ − sin θ ∂θ ∂ψ sin( ψ)2 ∂2 cos θ ∂2 + 2 sin( ψ)2 sin( θ)2 ∂φ 2 − sin( θ)2 ∂φ ∂ψ cos( θ) ∂ + sin( ψ)2 sin( θ) ∂θ cos( θ)2 ∂2 + sin( ψ)2 . sin( θ)2 ∂ψ 2 Similarly, we have ∂2 D2 = sin ψ sin ψ 2 − − ∂θ 2

cos θ cos ψ ∂ cos ψ ∂2 +− + sin( θ)2 ∂φ sin θ ∂θ ∂φ sin( θ)2 cos( θ)2 ∂ cos θ ∂2 − − cos ψ cos ψ − sin( θ)2 ∂ψ − sin θ ∂θ ∂ψ ! cos ψ ∂2 cos ψ ∂2 cos θ ∂2 + sin ψ + cos ψ sin θ − ∂θ ∂φ sin θ ∂φ 2 − sin θ ∂φ ∂ψ ! cos θ ∂ ∂2 cos ψ cos ψ sin ψ − sin θ − ∂θ − ∂θ ∂ψ

sin ψ ∂ cos ψ ∂2 +− + sin θ ∂φ sin θ ∂φ ∂ψ cos θ ∂ cos θ ∂2 + sin ψ cos ψ , sin θ ∂ψ − sin θ ∂ψ 2 ! and after cancellations we get ∂2 cos θ cos ψ sin ψ ∂ cos ψ sin ψ ∂2 D2 = sin( ψ)2 + 2 2 2 ∂θ 2 sin( θ)2 ∂φ − sin θ ∂θ ∂φ 1 + cos( θ)2 ∂ cos θ ∂2 cos ψ sin ψ + 2 cos ψ sin ψ − sin( θ)2 ∂ψ sin θ ∂θ ∂ψ cos( ψ)2 ∂2 cos θ ∂2 + 2 cos( ψ)2 sin( θ)2 ∂φ 2 − sin( θ)2 ∂φ ∂ψ cos θ ∂ + cos( ψ)2 sin θ ∂θ cos( θ)2 ∂2 + cos( ψ)2 . sin( θ)2 ∂ψ 2 11.9. Laplacian and derivatives of representations on SU(2) 641

Hence we obtain Proposition 11.9.1 (Laplacian on SU(2) ). In terms of Euler angles, the Laplacian on SU(2) is given by

= D2 + D2 + D2 L 1 2 3 ∂2 1 ∂2 cos θ ∂2 1 ∂2 = + 2 + ∂θ 2 sin( θ)2 ∂φ 2 − sin( θ)2 ∂φ ∂ψ sin( θ)2 ∂ψ 2 cos θ ∂ + . sin θ ∂θ We now determine derivatives of representations.

l Proposition 11.9.2 (Derivatives D1, D 2, D 3 of tmn ). We have

l (l n)( l + n + 1) l (l + n)( l n + 1) l D1tmn = − tm,n +1 + − tm,n 1, 2i 2i − − − l (l n)( l + n + 1) l (l + n)( l n + 1) l D2tmn = − tm,n +1 − tm,n 1, 2 − 2 − D tl = in t l . 3 mn − mn Proof. Recall that d D f(u) = f(u ω (t)) , j dt j |t=0 l where f : SU(2) C. Recall also that tmn is a matrix element of the irreducible unitary representation→ T : SU(2) End( V ), acting by l → l

Tl(u)v(z) = v(zu ),

2 where v Vl is a homogeneous polynomial v : C C of order 2 l N0. Matrix elements ∈tl : SU(2) C of T with respect to the→ basis ∈ mn → l q k l, l + 1 , , +l 1, +l { lk | ∈ {− − · · · − }} satisfy l qln (zu ) = Tl(u)qln (z) = tmn (u) qlm(z), m % where l m l+m z1− z2 qlm(z) = . (l m)! ( l + m)! − Especially,

D q (zu ) := D (u q (zu )) = D tl (u) q (z) j ln j &→ ln j mn lm m % * + 642 Chapter 11. Fourier analysis on SU (2)

First, d D q (zu ) = q (z u ω (t)) 3 ln dt ln 3 't=0 it/ 2 l 'n it/ 2 l+n d (zu )1 e −' (zu )2 e− = ' dt (l n)! ( l + n)! ' * +− * + 't=0 l n l+n ' d itn (zu )1− (zu )2 ' = e− ' dt t=0 (l n)! ( l + n)! ' − = in q ln ('zu ). − Thereby D tl (u) = in t l (u). 3 mn − mn Next,

D2qln (zu ) d = q (z u ω (t)) dt ln 2 't=0 t ' t l n t t l+n d (zu )1 cos +' ( zu )2 sin − (zu )1 sin + ( zu )2 cos = 2 ' 2 − 2 2 dt (l n)! ( l + n)! ' * +− * + 't=0 l n 1 l+n l n l+n 1 ' (l n)( zu )2(zu ) − − (zu ) (zu ) − (l + n)( zu )1(zu ) − ' = − 1 2 − 1 2 ' 2 (l n)! ( l + n)! − (l n)( l + n + 1) ql,n +1 (zu ) (l + n)( l n + 1) ql,n 1(zu ) = − − − − . 2 l Thus D2tmn (u) equals to

l l (l n)( l + n + 1) tm,n +1 (u) (l + n)( l n + 1) tm,n 1(u) − − − − . 2 Finally,

D1qln (zu ) d = q (z u ω (t)) dt ln 1 't=0 t ' t l n t t l+n d (zu )1 cos +' ( zu )2i sin − (zu )1i sin + ( zu )2 cos = 2 ' 2 2 2 dt (l n)! ( l + n)! ' * +− * + 't=0 l n 1 l+n l n l+n 1 ' (l n)( zu )2(zu ) − − (zu ) + ( zu ) − (l + n)( zu )1(zu ) − ' = − 1 2 1 2 ' 2i (l n)! ( l + n)! − − (l n)( l + n + 1) ql,n +1(zu ) + (l + n)( l n + 1) ql,n 1(zu ) = − − − . 2i − 11.9. Laplacian and derivatives of representations on SU(2) 643

l Thus D1tmn (u) equals to

l l (l n)( l + n + 1) tm,n +1 (u) + (l + n)( l n + 1) tm,n 1(u) − − − , 2i − completing the proof.

In the sequel, we will work with operators ∂+, ∂ , ∂ 0 rather than with D1, D 2, D 2, and the relation between them was given in Deﬁnition− 11.5.10 , which we recall:

i ∂+ := i D1 D2 D1 = − (∂ + ∂+) 2 − − 1 ∂ := i D1 + D2 i.e. D2 = (∂ ∂+) (11.16)  −  2 − − ∂0 := i D3 D3 = i∂0. −   l Theorem 11.9.3 (Derivatives ∂+, ∂ , ∂ 0 and Laplacian of tmn ). We have − L ∂ tl = (l n)( l + n + 1) tl , + mn − − m,n +1 l l ∂ tmn = (l + n)( l n + 1) tm,n 1, − − − − l l ∂0tmn = n t mn , tl = l(l + 1) tl . L mn − mn

Proof. Formulae for ∂+, ∂ , ∂ 0 follow from formulae in Proposition 11.9.2 and formulae ( 11.16 ). Since by− Remark 11.5.11 we have

2 = ∂0 (∂+∂ + ∂ ∂+)/2, L − − − − we get

tl L mn 2 l 1 l = n tmn + (l + n)( l n + 1) ∂+tm,n 1 − 2 − − 5 l + (l n)( l + n + 1) ∂ tm,n +1 − − 1 6 = − 2n2 + (l + n)( l n + 1) (l (n 1))( l + ( n 1) + 1) 2 − − − − 5 + (l n)( l + n + 1) (l + ( n + 1))( l (n + 1) + 1) tl − − mn 1 6 = − n2 + ( l + n)( l n + 1) + ( l n)( l + n + 1) tl 2 − − mn 1 * + = − 2n2 + 2( l2 n2) + ( l + n) + ( l n) tl 2 − − mn = l(l*+ 1) tl , + − mn completing the proof. 644 Chapter 11. Fourier analysis on SU (2)

Remark 11.9.4 . In Proposition 11.9.2 we saw that

l l l (l n)( l + n + 1) tm,n +1 + (l + n)( l n + 1) tm,n 1 D1tmn = − − − , 2i l − l l (l n)( l + n + 1) tm,n +1 (l + n)( l n + 1) tm,n 1 D2tmn = − − − − , 2 D tl = in t l , 3 mn − mn so that (l n)( l + n + 1) tl + (l + n)( l n + 1) tl l m,n +1 m,n 1 D1tmn = − − − , +2i (l n)( l + n + 1) tl (l + n)( l n + 1) tl l m,n +1 m,n 1 D2tmn = − − − − , 2 l l D3tmn = +i n t mn , implying

l l ∂+tmn = + (l + n)( l n + 1) tm,n 1, − − l l ∂ tmn = + (l n)( l + n + 1) tm,n +1 , − − ∂ tl = n t l . 0 mn − mn 11.10 Fourier series on SU(2) and on SO(3)

By the Peter–Weyl Theorem (Theorem 7.5.14 ), an orthogonal basis for L2(SU(2)) l consists of functions tnm for l N0/2 and l m, n l, where l m, l n Z and ∈ − ≤ ≤ − − ∈ l i( nφ +mψ ) l tnm(ω(φ, θ, ψ )) = e − Pnm(cos( θ)) , and where

l m m n l l ( 1) − i − (l + n)! Pnm(z) = 2 − − (l m)!( l + m)! (l n)! $ − −m−n 2 l n (1 z) d − l m l+m − n+m l n (1 z) − (1 + z) . × (1 + z) 2 dz − − " # Note that here we changed the order of indices ( m, n ) into ( n, m ) compared with the formulation of the Peter–Weyl theorem in order to have ( m, n ) entries for the Fourier coeﬃcients in ( 11.17 ) and in the sequel. Notice that for SO(3), these same formulae are valid with appropriate Eu- l ler’s angles, but then l N0 (not l N0/2). Nevertheless, t (u) U(2 l + 1) (2 l+1) (2 l+1) ∈ ∈ ∈ ⊂ C × . For instance,

T0(u(φ, θ, ψ )) = 1 , * + 11.10. Fourier series on SU(2) and on SO(3) 645

ei( φ+ψ)/2 cos θ ei( φ ψ)/2i sin θ T (u(φ, θ, ψ )) = 2 − 2 1/2 e i( φ ψ)/2i sin θ e i( φ+ψ)/2 cos θ − − 2 − 2 ! a b = u(φ, θ, ψ ) = , c d !

ei( φ+ψ) cos 2 θ eiφ i sin θ ei( φ ψ) sin 2 θ 2 √2 − 2 iψ i sin θ − iψ i sin θ T (u(φ, θ, ψ )) = e cos θ e− 1  √2 √2  i( φ ψ) 2 θ iφ i sin θ i( φ+ψ) 2 θ e− − sin e− e− cos − 2 √2 2   a2 √2ab b 2  = √2ac ad + bc √2bd .   c2 √2cd d 2   Consequently, the collection 1 √2l + 1 tl : l N , l m, n l, l m, l n Z { nm ∈ 2 0 − ≤ ≤ − − ∈ } is an orthonormal basis for L2(SU(2)), and thus by the Peter–Weyl Theorem (Theorem 7.5.14 ) any function f C∞(SU(2)) has a Fourier series representation ∈ l f(x) = (2 l + 1) f(l)mn tnm(x), (11.17) 1 l N0 m n ∈%2 % % = where the Fourier coeﬃcients are computed by

l l f(l) := f(x) t (x) d x = f, t 2 , mn nm ) nm*L (SU(2)) >SU(2) = (2 l+1) (2 l+1) so that f(l) C × . The series ( 11.17 ) converges almost everywhere on SU(2) as well∈ as in L2(SU(2)). = Deﬁnition 11.10.1 (Quantum numbers, notation f(l), and summation on SU(2) ). In the case of SU(2), we simplify the notation writing f(l) instead of f(tl), etc.

In sums ( 11.17 ), our convention will be that summations= m n are over m, n such that l m, n l and l m, l n Z. The index= l is called the =quantum number . − ≤ ≤ − − ∈ & & On SO(3), the collection √2l + 1 tl : l N , m, n Z, l m, n l { nm ∈ 0 ∈ − ≤ ≤ } 2 is an orthonormal basis for L (SO(3)), and thus f C∞(SO(3)) has a Fourier series representation ∈

l l ∞ l f(x) = (2 l + 1) f(l)mn tnm(x), l=0 m= l n= l % %− %− = 646 Chapter 11. Fourier analysis on SU (2) where the Fourier coeﬃcients are computed by

l l f(l) := f, t 2 = f(x) t (x) d µ (x). mn ) nm*L (SO(3)) nm SO(3) >SO(3) Notice that= by Remark 10.2.3 we hav e

l l l 1 tnm(x) = ( t (x)∗)mn = tmn (x− ).

2 Evidently, the values of f C∞(S ) C∞(SO(3)) do not depend on the Euler ∈ ⊂ angle ψ, so that in this case f(l) = 0 whenever n = 0. nm + = Chapter 12

Pseudo-diﬀerential operators on SU (2)

In this chapter we carry out the analysis of operators on SU(2) with an application to the operators on the 3-dimensional sphere S3. In particular, we derive a much simpler symbolic characterisation of pseudo-differential operators on SU(2) than the one given in Definition 10.9.5 . In turn, this will also yield a characterisation of full symbols of pseudo-differential operators on the 3-sphere S3. We note that this approach works globally on the whole sphere, since the version of the Fourier analysis that we use is different from the one in e.g. [ 107 , 119 , 108 ] which covers only a hemisphere, with singularities at the equator. For a general introduction and motivation for the analysis on SU(2) we refer the reader to the introduction in Part IV where the cases of SU(2) and S3 were put in a perspective. On SU(2), the conventional abbreviations in summation indices are

= , = , 1 1 l l N0 l m,n l N0 m l, l +m Z n l, l +n Z % ∈%2 % % ∈%2 | |≤ % ∈ | |≤ % ∈ where N0 = 0 N = 0, 1, 2, . As before, the space of all linear mappings from a ﬁnite{ dimensional} ∪ { (inner-product)· · · } vector space to itself is denoted by H 1 End( ), a mapping U ( ) is called unitary if U ∗ = U − and the space of all unitaryH linear mappings∈ on L aH ﬁnite dimensional inner product space is denoted by ( ). H U H 12.1 Symbols of operators on SU(2)

First we summarise the approach to symbols from Section 10.4 in the case of SU(2), also simplifying the notation in this case. We recall that in the case of SU(2), we simplify the notation writing f(l) instead of f(tl), etc.

= = 648 Chapter 12. Pseudo-diﬀerential operators on SU (2)

By the Peter-Weyl theorem (Theorem 7.5.14 ) the collection

1 √2l + 1 tl : l N , l m, n l, l m, l n Z (12.1) { nm ∈ 2 0 − ≤ ≤ − − ∈ }

2 is an orthonormal basis for L (SU(2)), and thus f C∞(SU(2)) has a Fourier series representation ∈

l f(x) = (2 l + 1) f(l)mn tnm(x) 1 l N0 m n ∈%2 % % = = (2 l + 1) Tr f(l) tl(x) , 1 l N0 ∈%2 5 6 = where the Fourier coeﬃcients are computed by

l l f(l) := f(x) t (x) d x = f, t 2 , mn nm ) nm*L (SU(2)) >SU(2)

= (2 l+1) (2 l+1) l (2 l+1) (2 l+1) so that f(l) C × . We recall that t C × by Deﬁnition ∈ l ∈ 11.10.1 is a matrix with components tnm, with the convention that indices m and n vary as= in ( 12.1 ). Let A : C∞(SU(2)) C∞(SU(2)) be a continuous linear operator and let → R ′(SU(2) SU(2)) be its right-convolution kernel, i.e. A ∈ D ×

1 Af (x) = f(y) R (x, y − x) d y = ( f R (x, ))( x) A ∗ A · >SU(2) in the sense of distributions. According to Deﬁnition 10.4.3 , by the symbol of A we mean the sequence of matrix-valued mappings

(2 l+1) (2 l+1) (x σ (x, l )) : SU(2) C × , &→ A → where 2 l N , obtained from ∈ 0

l σA(x, l )mn = RA(x, y ) tnm(y) d y. (12.2) >SU(2) th That is, σA(x, l ) is the l Fourier coeﬃcient of the function y RA(x, y ). Then by Theorem 10.4.4 we have &→

l Af (x) = (2 l + 1) Tr t (x) σA(x, l ) f(l) %l 5 6 = l = (2 l + 1) t (x)nm σA(x, l )mk f(l)kn . m,n / 0 %l % %k = 12.1. Symbols of operators on SU(2) 649

Alternatively, by Theorem 10.4.6 we have

l l σA(x, l ) = t (x)∗ At (x), (12.3) that is * + l l σA(x, l )mn = tkm(x)( At kn )( x), (12.4) %k by formula ( 10.21 ). Formula ( 10.20 ) expressing the right-convolution kernel in terms of the symbol becomes

l RA(x, y ) = (2 l + 1) tnm(y) σA(x, l )mn , (12.5) m,n %l % with a similar distributional interpretation for the series. In the case of SU(2), the quantity tl in ( 10.9 ) for the representation ξ = tl can be calculated as

? @ l 1/2 1/2 t = (1 + λ[tl]) = (1 + l(l + 1)) , in view of Theorem 11.9.3? @ . Consequently, Deﬁnition 10.9.5 of the symbol class Σm(SU(2)) becomes: Deﬁnition 12.1.1. We write that σ Σm(SU(2)) if A ∈ 0 sing supp ( y R (x, y )) e (12.6) &→ A ⊂ { } and if α β m α l ∂x σA(x, l ) C2l+1 C2l+1 CAαβm (1 + l) −| | (12.7) △ → ≤ for all x G, allA multi-indices Aα, β , and l 1 N . Here ∈ A A ∈ 2 0

β β1 β2 β3 α α1 α2 α3 ∂x = ∂0 ∂+ ∂ and l = 0 + − △ △ △ △− are defined in the general situation in Definition 10.7.1 , but in Section 12.3 we dis- m cuss the simplification of these difference operators. Moreover, σA Σk+1 (SU(2)) if and only if ∈

σ Σm(SU(2)) , (12.8) A ∈ k [σ , σ ] = σ σ σ σ Σm(SU(2)) , (12.9) ∂j A ∂j A − A ∂j ∈ k γ m+1 γ ( σ ) σ Σ −| |(SU(2)) , (12.10) △l A ∂j ∈ k for all γ > 0 and j 0, +, . Let | | ∈ { −}

m ∞ m Σ (SU(2)) := Σk (SU(2)) , kB=0 so that by Theorem 10.9.6 we have OpΣ m(SU(2)) = Ψ m(SU(2)). 650 Chapter 12. Pseudo-diﬀerential operators on SU (2)

β β1 β2 β3 Remark 12.1.2 . The ordering of operators ∂0, ∂ +, ∂ in ∂x = ∂0 ∂+ ∂ may seem to be of importance because they do not commute− (see Exercise 11.5.12− ). However, this is not really an issue in view of Remark 10.5.4 : indeed applying their commutator relations from Exercise 11.5.12 iteratively, we see that we can take β any ordering in ∂x in ( 12.7 ) to obtain the same class of symbols. Remark 12.1.3 . We would like to provide a more direct definition for the sym- m m bol class Σ (SU(2)), without resorting to classes Σ k (SU(2)). Condition ( 12.7 ) is just an analogy of the usual symbol inequalities. Conditions ( 12.6 ) and ( 12.8 ) are α β straightforward. We may have difficulties with differences l , but derivatives ∂x do not cause problems; if we want, we may assume that the symbols△ are constant in x. In Section 12.4 we present such a simplification, thus providing a more straightforward characterisation of operators from Ψ m(SU(2)) in terms of quantizations and full symbols developed here.

α β Exercise 12.1.4. From the deﬁnition of operators △l and ∂x verify the following properties: α β β α △l ∂x σA(x, l ) = ∂x △l σA(x, l ),

∂j (σA(x, l ) σB(x, l )) = ( ∂jσA(x, l )) σB(x, l ) + σA(x, l ) ∂jσA(x, l ),

β β ∂y (σA(x, l ) σB(y, l ) σC (z, l )) = σA(x, l ) ∂y σB(y, l ) σC (z, l ). 12.2 Symbols of ∂+, ∂ −, ∂ 0 and Laplacian L

In this section we calculate symbols of the creation, annihilation and neutral operators ∂+, ∂ −, ∂ 0, and of the Laplacian L. We will use the fact that the symbol of the operator A is obtained by

l l σA(x, l ) = t (x)∗ (At )( x),

l l that is, σA(x, l )mn = k tkm(x) ( At kn )( x), see ( 12.3 ) and ( 12.4 ). Theorem 12.2.1. We! have

σ∂+ (x, l )mn = − (l − n)( l + n + 1) δm,n +1

= −"(l − m + 1)( l + m) δm 1,n , − σ∂− (x, l )mn = −"(l + n)( l − n + 1) δm,n 1 − = −"(l + m + 1)( l − m) δm+1 ,n ,

σ∂0 (x, l )mn = n" δ mn = m δ mn , σ (x, l )mn = −l(l + 1) δmn , L where δmn is the Kronecker delta: δmn = 1 for m = n and, δmn = 0 otherwise. 12.2. Symbols of ∂+, ∂ , ∂ 0 and Laplacian L 651 −

Proof. Let e ∈ SU(2) be the neutral element of SU(2) and let tl be a unitary matrix representation of SU(2). First we note that

l l 1 δmn = t (e)mn = t (x− x)mn = l 1 l l l = t (x− )mk t (x)kn = t (x)km t (x)kn , #k #k see also Remark 10.2.11 for the general form of this identity. Similarly, we note the identity l l δmn = t (x)mk t (x)nk . #k From this, formula ( 12.4 ), and Theorem 11.9.3 we get

l l σ∂+ (x, l )mn = tkm(x) ∂+tkn (x) k # l l = − (l − n)( l + n + 1) tkm(x) tk,n +1 (x) k " # = − (l − n)( l + n + 1) δm,n +1 , " and the case of σ∂− (x, l ) is analogous:

l l σ∂− (x, l )mn = t (x) ∂ tkn (x) km − k # l l = − (l + n)( l − n + 1) tkm(x) tk,n 1(x) − k " # = − (l + n)( l − n + 1) δm,n 1, − Finally, "

l l l l σ∂0 (x, l )mn = tkm(x) ∂0tkn (x) = n tkm(x) tkn (x) = n δ m,n , k k # # and similarly for L. Exercise 12.2.2. Complete the proof of Theorem 12.2.1 for the symbol σ of the Laplacian L. L

Remark 12.2.3 . Notice that σ∂0 (x, l ) and σ (x, l ) are diagonal matrices. The non- L zero elements reside just above the diagonal of σ∂+ (x, l ), and just below the diagonal of σ∂− (x, l ). Because of this the operators ∂0, ∂+ and ∂ may be called neutral , creation and annihilation operators, respectively, and this− explains our preference to work with them rather than with Dj’s, which have more non-zero entries.

Finally we note that vector fields D1, D 2, D 3 related to ∂+, ∂ , ∂ 0 by Defini- tion 11.5.10 can be conjugated as follows (recall the definition of conjugations− and their properties, e.g. in Proposition 10.4.18 ): 652 Chapter 12. Pseudo-differential operators on SU (2)

Proposition 12.2.4 (Conjugation of D1, D 2, D 3 and their symbols). We have

(D3)(w1)R = D2, (D1)(w2)R = D3, (D2)(w3)R = D1. (12.11)

The symbols of the operators D1, D 2 can be turned to that of D3 by taking suitable conjugations:

l l σD1 (x, l ) = t (w2) σD3 (x, l ) t (w2)∗, (12.12) l l σD2 (x, l ) = t (w1)∗ σD3 (x, l ) t (w1). (12.13) Moreover, if D ∈ su (2) there exists u ∈ SU(2) such that

l l σD(l) = t (u)∗ σD3 (l) t (u). Proof. Combining Lemma 10.4.19 with Proposition 11.5.7 , we see that formulae

(12.11 ) hold. Since D1, D 2, D 3 are left-invariant operators, their symbols σDj (x, l ) do not depend on x ∈ G, and by Proposition 10.4.18 we obtain ( 12.12 ) and ( 12.13 ). The last statement follows from Proposition 10.4.18 since D is a rotation of D3.

12.3 Diﬀerence operators for symbols

We are now going to introduce “difference operators” △+, △ , △0 acting on symbols of operators on SU(2) that resemble first order forward− and backward dif- α n ferences △ξ acting on symbols σA(x, ξ ) on a torus, where we had ξ ∈ Z (see Definition 3.3.1 ). Then we will apply these differences to first order differential operators, as well as to products of special type.

12.3.1 Diﬀerence operators on SU(2) From Theorem 11.7.1 and Theorem 11.8.1 we recall the notation

1/2 t t + t = −− − t+ t++ $ − % 1/2 1/2 t 1/2, 1/2 t 1/2,+1 /2 = −1/2 − −1/2 &t+1 /2, 1/2 t+1 /2,+1 /2' − cos( θ/ 2) e i(+ φ+ψ)/2 i sin( θ/ 2) e i(+ φ ψ)/2 = − . i sin( θ/ 2) e i( φ+ψ)/2 cos( θ/ 2) e i( φ ψ)/2 $ − − − %

Definition 12.3.1 (Differences △q for Fourier coefficients). For q ∈ C∞(SU(2)) and f ∈ D ′(SU(2)), let △qf(l) := qf (l). We shall use abbreviations ( )

△+ = △q+ , △ = △q− and △0 = △q0 , − 12.3. Diﬀerence operators for symbols 653 where

1/2 q := t + = t 1/2,+1 /2, − − − 1/2 q+ := t+ = t+1 /2, 1/2, − − 1/2 1/2 q0 := t − t++ = t 1/2, 1/2 − t+1 /2,+1 /2. −− − −

Thus each trigonometric polynomial q+, q , q 0 ∈ C∞(SU(2)) vanishes at the − neutral element e ∈ SU(2). In this sense trigonometric polynomials q + q+, q − − − 3 q+, q 0 on SU(2) are analogues of polynomials x1, x 2, x 3 in the Euclidean space R . Now our aim is to let these diﬀerence operators act on symbols. For this purpose we may only look at symbols independent of x corresponding to left-invariant (right-convolution) operators since the following construction is independent of x:

Definition 12.3.2 (Differences △+, △ , △0 acting on symbols). Let a = a(ξ) be a symbol as in Definition 10.4.3 . It follows− that a = s for some right-convolution kernel s ∈ D ′(SU(2)) so that the operator Op( a) is given by ( Op( a)f = f ∗ s.

We deﬁne “ diﬀerence operators ” △+, △ , △0 acting on the symbol a by −

△+a := q+ s, (12.14) △ a := q s, (12.15) − *− △0a := q0 s. (12.16) * Obviously, Definitions 12.3.1 and 12.3.2 are consistent. We note once more * that this construction is analogous to the one producing usual derivatives in Rn or difference operators on the torus Tn. To analyse the structure of difference operators on SU(2), we first need to l know how to multiply functions tmn by q+, q , q 0, and the necessary formulae are given in Theorem 11.8.1 . We recall the notation−

1/2 t t + t q t = −− − = −− − , q 0 = t − t++ , t+ t++ q+ t++ −− $ − % $ % and summarise the multiplication formulae as follows:

Corollary 12.3.3. For x ∈ R, let x± := x ± 1/2. Then

− l l+ l (2 l + 1) q+tmn = + (l + m + 1)( l − n + 1) tm+n− − (l − m)( l + n)tm+n− ,

− l " l+ " l (2 l + 1) q tmn = + (l − m + 1)( l + n + 1) tm−n+ − (l + m)( l − n)tm−n+ , − − l l+ l " − − " − − (2 l + 1) q0tmn = + (l − m + 1)( l − n + 1) tm n + (l + m)( l + n)tm n − " l+ " l − (l + m + 1)( l + n + 1) tm+n+ − (l − m)( l − n)tm+n+ . " " 654 Chapter 12. Pseudo-diﬀerential operators on SU (2)

We also recall the relation between symbols and kernels which follows from (12.2 ) and ( 12.5 ), and where we switch the order of m and n to adopt it to the proof of Theorem 12.3.5 :

Corollary 12.3.4 (Kernel and symbol). Symbol a(x, l ) = a(l) and kernel s(x) of an operator are related by

l l a(x, l )nm = anm = s(l)nm = s(y) tmn (y) d y, +SU(2) ( and

l l l s(x) = (2 l + 1) Tr a(x, l ) t (x) = (2 l + 1) anm tmn . (12.17) l l m,n # # # Let us now derive explicit expressions for the first order difference operators △+, △ , △0 defined in ( 12.14 )-( 12.16 ). To abbreviate the notation, we will also −l write anm = a(x, l )nm, even if symbol a(x, l ) depends on x, keeping in mind that the following theorem holds pointwise in x.

Theorem 12.3.5 (Formulae for diﬀerence operators △+, △ , △0). The diﬀerence operators are given by −

− l (l − m)( l + n) l (l + m + 1)( l − n + 1) l+ (△ a)nm = an−m+ − an−m+ , − " 2l + 1 " 2l + 1 − l (l + m)( l − n) l (l − m + 1)( l + n + 1) l+ (△+a)nm = an+m− − an+m− , " 2l + 1 " 2l + 1 − l (l − m)( l − n) l (l + m + 1)( l + n + 1) l+ (△0a)nm = an+m+ + an+m+ − " 2l + 1 " 2l + 1 − (l + m)( l + n) l (l − m + 1)( l − n + 1) l+ − an−m− − an−m− , " 2l + 1 " 2l + 1

1 where k± = k ± 2 , and satisfy commutator relations

[△0, △+] = [ △0, △ ] = [ △ , △+] = 0 . (12.18) − −

Proof. Identities ( 12.18 ) follow immediately from ( 12.14 )–( 12.16 ). As discussed before, we can abbreviate a(x, l ) by a(l) since none of the arguments in the proof will act on the variable x. In the proof we will heavily rely on the relation between kernels and symbols in Proposition 12.3.4 . Also, in the calculation below we will not worry about boundaries of summations keeping in mind that we can always view finite matrices as infinite ones simply by extending them be zeros. Recalling 12.3. Difference operators for symbols 655

that q = t + and using Theorem 11.8.1 , we can calculate − −

(12 .17 ) l l q s = (2 l + 1) anm q tmn − − m,n #l # Cor 12 .3.3 l l+ = anm tm−n+ (l − m + 1)( l + n + 1) l m,n , # # − " l −tm−n+ (l + m)( l − n)

− l l " - = tmn an−m+ (l − m)( l + n) l m,n # # , " l+ −an−m+ (l + m + 1)( l − n + 1) . " -

Since △ a = q s, we obtain the desired formula for △ : − − −

* − l (l − m)( l + n) l (l + m + 1)( l − n + 1) l+ (△ a)nm = an−m+ − an−m+ . − " 2l + 1 " 2l + 1

Similarly, for △+, we calculate

l l q+ s = (2 l + 1) anm q+ tmn m,n #l # l l+ = anm tm+n− (l + m + 1)( l − n + 1) l m,n , # # − " l −tm+n− (l − m)( l + n)

− l l " - = tmn an+m− (l + m)( l − n) l m,n # # , " l+ −an+m− (l − m + 1)( l + n + 1) . " -

From this we obtain the desired formula for △+:

− l (l + m)( l − n) l (l − m + 1)( l + n + 1) l+ (△+a)nm = an+m− − an+m− . " 2l + 1 " 2l + 1 656 Chapter 12. Pseudo-diﬀerential operators on SU (2)

Finally, for △0, we calculate

q0 s = l l = (2 l + 1) anm q0 tmn m,n #l # l = anm × l m,n # # − l+ l × tm−n− (l − m + 1)( l − n + 1) + tm−n− (l + m)( l + n)

− , l+" l " − tm+n+ (l + m + 1)( l + n + 1) − tm+n+ (l − m)( l − n) l " " - = tmn × l m,n # #− l l+ × an+m+ (l − m)( l − n) + an+m+ (l + m + 1)( l + n + 1)

− , l " l+ " −an−m− (l + m)( l + n) − an−m− (l − m + 1)( l − n + 1) . " " - From this we obtain the desired formula for △0:

− l (l − m)( l − n) l (l + m + 1)( l + n + 1) l+ (△0a)nm = an+m+ + an+m+ − " 2l + 1 " 2l + 1 − (l + m)( l + n) l (l − m + 1)( l − n + 1) l+ − an−m− − an−m− , " 2l + 1 " 2l + 1 and the proof of Theorem 12.3.5 is complete.

12.3.2 Differences for symbols of ∂+, ∂ −, ∂ 0 and Laplacian L Remark 10.7.3 and Proposition 10.7.4 said that application of difference operators reduce the order of the symbol of differential operators. However, for operators ∂+, ∂ , ∂ 0 and for the Laplacian L we calculate this now more explicitly: − Theorem 12.3.6. We have

σI = △+σ∂+ = △ σ∂− = △0σ∂0 . (12.19) − If µ, ν ∈ { +, −, 0} are such that µ %= ν, then

△µσ∂ν = 0 , (12.20) and for every ν ∈ { +, −, 0}, we have

△ν σI (x) = 0 . (12.21) Moreover, if L is the bi-invariant Laplacian, then

△+σ = −σ∂− , △ σ = −σ∂+ , △0σ = −2σ∂0 . (12.22) L − L L 12.3. Diﬀerence operators for symbols 657

The proof of this theorem will depend on explicit calculations. In trying to simplify the presentation, we prove this theorem in the form of several propositions dealing with different parts of the statement. We recall that the symbols of the first order partial differential operators ∂+, ∂ , ∂ 0 have many zero elements, and altogether they are as in Theorem 12.2.1 : −

− (l − n)( l + n + 1) , if m = n + 1 , σ∂+ (l)mn = (12.23) .0," otherwise .

− (l + n)( l − n + 1) , if m = n − 1, σ∂− (l)mn = .0," otherwise .

n, if m = n, σ∂0 (l)mn = .0, otherwise .

Proposition 12.3.7. We have identities ( 12.19 ), i.e.

σI = △+σ∂+ = △ σ∂− = △0σ∂0 . −

l Proof. From Corollary 12.3.3 we get an expression for q+tmn , which is used in the following calculation together with ( 12.17 ) in Corollary 12.3.4 and Theorem 12.2.1 :

(12 .17 ) l q+ s∂+ = q+ (2 l + 1) σ∂+ (l)mn tnm m,n #l # (12 .23 ) l = σ∂+ (l)n+1 ,n (2 l + 1) q+ tn,n +1 n #l # 2 − Cor 12 .3.3 l+ l = − (l − n)( l + n + 1) tn+n+ − tn+n+ l n # # /" 0 / 0 (12 .24 ) l = (2 l + 1) tkk #l #k (12 .17 ) = δe

= sI , where we made a change in indices and used the identity

+ + (l− − n)( l− + n + 1) − (l − n)( l + n + 1) = −2l − 1. (12.24) 658 Chapter 12. Pseudo-diﬀerential operators on SU (2)

Hence △+σ∂+ = σI . Similarly, △ σ∂− = σI , because − l q s∂− = q (2 l + 1) σ∂− (l)mn tnm − − m,n #l # l = σ∂− (l)n 1,n (2 l + 1) q tn,n 1 − − − n #l # 2 − l+ l = − (l − n + 1)( l + n) tn−n− − tn−n− l n # # /" 0 / 0 l = (2 l + 1) tkk #l #k = δe

= sI .

Moreover,

l q0 s∂0 = q0 (2 l + 1) σ∂0 (l)mn tnm m,n #l # l = σ∂0 (l)nn (2 l + 1) q0 tnn l n # # − l+ l = n +( l − n + 1) tn−n− + ( l + n) tn−n− l n / # # − l+ l −(l + n + 1) tn+n+ − (l − n) tn+n+

l+ 0 = tn+n+ (( n + 1)( l − n) − n(l + n + 1)) l n / # # − l +tn+n+ (( n + 1)( l + n + 1) − n(l − n))

l+ 2 0 = tn+n+ (l − 2n − 2n) l n / # # − l 2 +tn+n+ (l + 1 + 2 n + 2 n)

l+ 2 0 2 = tn+n+ (l − 2n − 2n) + ( l + 2 + 2 n + 2 n) l n # # + l = (2 l + 2) tn+n+ n #l # l = (2 l + 1) tnn n #l # = δe

= sI . 12.3. Diﬀerence operators for symbols 659

Proposition 12.3.8. We have identities ( 12.20 ), i.e.

△µσ∂ν = 0 , where µ, ν ∈ { +, −, 0} such that µ %= ν. Proof. We can calculate

l q+ s∂− = q+ (2 l + 1) σ∂− (l)mn tnm m,n #l # l = σ∂− (l)n 1,n (2 l + 1) q+ tn,n 1 − − n #l # = − (l + n)( l − n + 1) l n # # " l+1 /2 (l + n + 1)( l − n + 2) tn+1 /2,n 3/2 − / l 1/2 " − − (l − n)( l + n − 1) tn+1 /2,n 3/2 − l"+1 /2 0 = tn+1 /2,n 3/2 n − #l # − (l + n + 1)( l − n + 2) (l − n + 1)( l + n) / + "(l + n)( l − n + 1) (l +"n + 1)( l − n + 2) = 0 . " " 0

Analogously,

l q s∂+ = q (2 l + 1) σ∂+ (l)mn tnm − − m,n #l # l = σ∂+ (l)n+1 ,n (2 l + 1) q tn,n +1 − n #l # = − (l − n)( l + n + 1) l n # # " l+1 /2 (l − n + 1)( l + n + 2) tn 1/2,n +3 /2 − / l 1/2 " − − (l + n)( l − n − 1) tn 1/2,n +3 /2 − l"+1 /2 0 = tn 1/2,n +3 /2 n − #l # − (l − n)( l + n + 1) (l − n + 1)( l + n + 2) / + "(l − n + 1)( l + n +" 2) (l + n + 1)( l − n) = 0 , " " 0 660 Chapter 12. Pseudo-diﬀerential operators on SU (2) and

l q+ s∂0 = q+ (2 l + 1) σ∂0 (l)mn tnm m,n #l # l = σ∂0 (l)nn (2 l + 1) q+ tnn n #l # l+ = n (l + n + 1)( l − n + 1) tn−,n + l n / # # " − l − (l − n)( l + n) tn−,n +

l+" 0 = n t n−,n + (l + n + 1)( l − n + 1) l n # # /" − (l − n + 1)( l + n + 1) = 0 . " 0

Analogously,

l q s∂0 = q (2 l + 1) σ∂0 (l)mn tnm − − m,n #l # l = σ∂0 (l)nn (2 l + 1) q tnn − n #l # l+ = n (l + n + 1)( l − n + 1) tn+,n − l n / # # " − l − (l − n)( l + n) tn+,n −

l+" 0 = n t n+,n − (l + n + 1)( l − n + 1) l n # # /" − (l − n + 1)( l + n + 1) = 0 . " 0 12.3. Diﬀerence operators for symbols 661

We also have

l q0 s∂− = q0 (2 l + 1) σ∂− (l)mn tnm m,n #l # l = σ∂− (l)n 1,n (2 l + 1) q0 tn,n 1 − − n #l # = − (l + n)( l − n + 1) l n # # " l+1 /2 + (l − n + 1)( l − n + 2) tn 1/2,n 3/2+ − − / l 1/2 " − + (l + n)( l + n − 1) tn 1/2,n 3/2 − − − " l+1 /2 − (l + n + 1)( l + n)tn+1 /2,n 1/2 − − l 1/2 " − − (l − n)( l − n + 1) tn+1 /2,n 1/2 − " l 0 = tn,n 1 − n #l # + (l + n)( l − n + 1) ( l − n)+

/+( l"+ n + 1) (l − n + 1)( l + n) − −(l + n − 1) " (l − n + 1)( l + n) − − (l + n)( l"− n + 1) ( l − n + 2) = 0 . " 0 662 Chapter 12. Pseudo-diﬀerential operators on SU (2)

Analogously,

l q0 s∂+ = q0 (2 l + 1) σ∂+ (l)mn tnm m,n #l # l = σ∂+ (l)n+1 ,n (2 l + 1) q0 tn,n +1 n #l # = − (l − n)( l + n + 1) l n # # " l+1 /2 + (l − n + 1)( l − n)tn 1/2,n +1 /2+ − / l 1/2 " − + (l + n)( l + n + 1) tn 1/2,n +1 /2 − − " l+1 /2 − (l + n + 1)( l + n + 2) tn+1 /2,n +3 /2 − l 1/2 " − − (l − n)( l − n − 1) tn+1 /2,n +3 /2 " l 0 = tn,n +1 n #l # +( l − n − 1) (l + n + 1)( l − n) +

+/ (l − n)( l +"n + 1) ( l + n + 2) − −"(l − n)( l + n + 1) ( l + n) − −"(l − n + 1) (l + n + 1)( l − n) = 0 . " 0

Proposition 12.3.9. We have identities ( 12.21 ), i.e.

△ν σI (x) = 0 , for every ν ∈ { +, −, 0}. Proof. Similarly to the propositions before, we calculate

l q+ sI = q+ (2 l + 1) δmn tnm m,n #l # l = (2 l + 1) q+ tnn l n # # − l+ l = (l + n + 1)( l − n + 1) tn+,n − − (l − n)( l + n) tn+,n − l n / 0 # # "− " l = tn+,n − (l + n)( l − n) − (l − n)( l + n) l n # # /" " 0 = 0 . 12.3. Diﬀerence operators for symbols 663

Analogously,

l q sI = q (2 l + 1) δmn tnm − − m,n #l # l = (2 l + 1) q tnn − l n # # − l+ l = (l − n + 1)( l + n + 1) tn−,n + − (l + n)( l − n) tn−,n + l n / 0 # # "− " l = tn−,n + (l − n)( l + n) − (l + n)( l − n) l n # # /" " 0 = 0 .

Moreover,

l q0 sI = q0 (2 l + 1) δmn tnm m,n #l # l = (2 l + 1) q0 tnn l n # # − l+ l = +( l − n + 1) tn−n− + ( l + n) tn−n− l n / # # − l+ l −(l + n + 1) tn+n+ − (l − n) tn+n+

l 0 = tnn (+( l − n) + ( l + n + 1) − (l + n) − (l − n + 1)) n #l # = 0 .

Proposition 12.3.10. We have identities ( 12.22 ), i.e.

△+σ = −σ∂− , △ σ = −σ∂+ , △0σ = −2σ∂0 . L − L L

Proof. Since

σ (x, l )mn = −l(l + 1) δmn , L 664 Chapter 12. Pseudo-diﬀerential operators on SU (2) we get

l q+ s = q+ (2 l + 1) σ (x, l )mn tnm −L −L m,n #l # l = (2 l + 1) l(l + 1) q+ tnn n #l # l+ = l(l + 1) + (l + n + 1)( l − n + 1) tn−,n + l n / # # " − l − (l − n)( l + n) tn−,n +

l+ " 0 = tn−,n + +l(l + 1) (l + n + 1)( l − n + 1) l n # # / " −(l + 1)( l + 2) (l − n + 1)( l + n + 1)

" l+ 0 = −2( l + 1) (l + n + 1)( l − n + 1) tn−,n + l n # # " l = (2 l + 1) − (l + n)( l − n + 1) tn 1,n − l n # # " = s∂− .

Analogously,

l q s = q (2 l + 1) σ (x, l )mn tnm − −L − −L m,n #l # l = (2 l + 1) l(l + 1) q tnn − n #l # l+ = l(l + 1) + (l − n + 1)( l + n + 1) tn−,n + l n / # # " − l − (l + n)( l − n) tn−,n +

l+ " 0 = tn−,n + +l(l + 1) (l − n + 1)( l + n + 1) l n # # / " −(l + 1)( l + 2) (l + n + 1)( l − n + 1)

" l+ 0 = −2( l + 1) (l + n + 1)( l − n + 1) tn−,n + l n # # " l = (2 l + 1) − (l + n)( l − n + 1) tn 1,n − l n # # " = s∂+ . 12.3. Diﬀerence operators for symbols 665

Moreover,

l q0 s = q0 (2 l + 1) σ (x, l )mn tnm −L −L m,n #l # l = (2 l + 1) l(l + 1) q0 tnn l n # # − l+ l = l(l + 1) +( l − n + 1) tn−n− + ( l + n) tn−n− l n / # # − l+ l −(l + n + 1) tn+n+ − (l − n) tn+n+ 0 = l(l + 1) l n # # − l+ l tn+,n + (( l − n) − (l + n + 1)) + tn+,n + (( l + n + 1) − (l − n))

− / l+ l 0 = l(l + 1) −(2 n + 1) tn+,n + − tn+,n + n #l # / 0 l+ = − tn+,n + (2 n + 1) ( l(l + 1) − (l + 1)( l + 2)) n #l # l+ = tn+,n + (2 n + 1) (2 l + 2) n #l # l = (2 l + 1) 2n t nn n #l # = 2 s∂0 , completing the proof.

Remark 12.3.11 . The proof of Theorem 12.3.6 relied on explicit calculations in representations and we decided to include them in detail for didactic purposes as well as in preparation for the proof of Theorem 12.3.12 . However, we may also argue using formula ( 10.28 ) in Proposition 10.7.4 . There, if q(e) = 0, we get

△qσ∂+ = −(∂+q)( e) σI ,

△qσ∂− (x, ξ ) = −(∂ q)( e) σI , −

△qσ∂0 (x, ξ ) = −(∂0q)( e) σI .

a b If we write u ∈ SU(2) by u = then c d $ %

q+ = c, q = b, q 0 = a − d, − 666 Chapter 12. Pseudo-diﬀerential operators on SU (2) and by Exercise 11.5.14 we have

−b 0 0 0 ∂ u| = = , + u=e −d 0 −1 0 $ %u=e $ % 0 −a 0 −1 ∂ u| = = , − u=e 0 −c 0 0 $ %u=e $ % −a/ 2 0 −1/2 0 ∂ u| = = . 0 u=e 0 d/ 2 0 1 /2 $ %u=e $ % This implies that

σI = △+σ∂+ = △ σ∂− = △0σ∂0 , −

0 = △+σ∂− = △+σ∂0

= △ σ∂+ = △ σ∂0 − −

= △0σ∂+ = △0σ∂− .

Let us now apply the diﬀerence operators to the symbol of the Laplacian L. First, by Remark 11.5.11 we write

2 L = −∂0 − (∂+∂ + ∂ ∂+)/2. − − Now

1 −a b 1 a b u=e 1 1 0 ∂2u = ∂ = = , 0 0 2 −c d 4 c d 4 0 1 $ % $ % $ % 0 −a 0 b u=e 0 0 ∂+∂ u = ∂+ = = , − 0 −c 0 d 0 1 $ % $ % $ % −b 0 a 0 u=e 1 0 ∂ ∂+u = ∂ = = , − − −d 0 c 0 0 0 $ % $ % $ % so that

2 △ σ 2 = q (e) σ 2 − 2( ∂ q )( e) σ + ( ∂ q )( e) σ + ∂0 + ∂0 0 + ∂0 0 + I = 0 ,

△+σ∂+∂− = q+(e) σ∂+∂− − [( ∂+q+)( e) σ∂− +

+( ∂ q )( e) σ∂+ ] + ( ∂+∂ q+)( e) σI − − − = σ∂− ,

△+σ∂−∂+ = q+(e) σ∂−∂+ − [( ∂+q+)( e) σ∂− +

+( ∂ q )( e) σ∂+ ] + ( ∂ ∂+q+)( e) σI − − − = σ∂− .

Therefore

△+σ = −σ∂− . L 12.3. Diﬀerence operators for symbols 667

Analogously,

△ σ = −σ∂+ . − L Finally,

△0σ = −2σ∂0 , L because

2 △ σ 2 = q (e) σ 2 − 2( ∂ q )( e) σ + ( ∂ q )( e) σ 0 ∂0 0 ∂0 0 0 ∂0 0 0 I

= 2 σ∂0 ,

△0σ∂+∂− = q0(e) σ∂+∂− − [( ∂+q0)( e) σ∂− +

+( ∂ q )( e) σ∂+ ] + ( ∂+∂ q0)( e) σI − − − = −σI ,

△0σ∂−∂+ = q0(e) σ∂−∂+ − [( ∂+q0)( e) σ∂− +

+( ∂ q )( e) σ∂+ ] + ( ∂ ∂+q0)( e) σI − − − = + σI .

12.3.3 Diﬀerences for aσ ∂0

Let us now calculate higher order diﬀerences of the symbol aσ ∂0 which will be needed in the sequel.

3 Theorem 12.3.12. For any α ∈ N0, we have the formula

α1 α2 α3 l △+ △ △0 (aσ ∂0 ) = − nm α1 α2 α3 l α1 α2 α3 1 l 1 = ( m − α1/2 + 2α2/2) △+ △ △0 a + α3 △0△+ △ △0 − a , − nm − nm 1 2 1 2 wher e △0 is given by

l (△0a)nm =

− 1 (l − m)( l − n) l (l + m + 1)( l + n + 1) l+ = an+m+ + an+m+ + 2 3" 2l + 1 " 2l + 1

− (l + m)( l + n) l (l − m + 1)( l − n + 1) l+ + an−m− + an−m− , " 2l + 1 " 2l + 1 4 and satisﬁes [△0, △0] = 0 . Proof. First we observe that we have

l l l (a σ ∂0 )nm = ank k δ km = m a nm. #k 668 Chapter 12. Pseudo-diﬀerential operators on SU (2)

Then using Theorem 12.3.5 , we get

l △ (aσ ∂0 )nm − − (l − m)( l + n) + l (l + m + 1)( l − n + 1) + l+ = m an−m+ − m an−m+ " 2l + 1 " 2l + 1 + l = ( m △ a)nm, −

+ and we can abbreviate this by writing △ (aσ ∂0 ) = m △ a. Further, we have − − l △ (△ (aσ ∂0 )) nm = − − − (l − m)( l + n) l = [△ (aσ ∂0 )] n−m+ " 2l + 1 − (l + m + 1)( l − n + 1) l+ − [△ (aσ ∂0 )] n−m+ " 2l + 1 − − (l − m)( l + n) l = (m + 1)( △ a)n−m+ " 2l + 1 − (l + m + 1)( l − n + 1) l+ − (m + 1)( △ a)n−m+ " 2l + 1 − 2 l = ( m + 1)( △ a)nm. − Continuing this calculation we can obtain

k l k l △ (aσ ∂0 ) = ( m + k/ 2)( △ a)nm. (12.25) − nm − 1 2 By Theorem 12.3.5 we also have

l [△+(△ (aσ ∂0 ))] = − nm + l = △+ m △ a − nm − 1 (l + m)( l − n2 ) + l = m △ a n+m− " 2l + 1 − + (l − m + 1)( l + n + 1) + l − m △ a n+m− " 2l + 1 − l = m(△+△ a)nm. − By induction, and using ( 12.25 ), we then get

l k1 k2 k1 k2 l △+ △ (aσ ∂0 ) = ( m − k1/2 + k2/2)( △+ △ a)nm. (12.26) − nm − , - The situation with △0 is more complicated because there are more terms. Using 12.3. Diﬀerence operators for symbols 669

Theorem 12.3.5 we have

l △0(aσ ∂0 )nm =

− (l − m)( l − n) l (l + m + 1)( l + n + 1) l+ = (ma )n+m+ + (ma )n+m+ − " 2l + 1 " 2l + 1 − (l + m)( l + n) l (l − m + 1)( l − n + 1) l+ − (ma )n−m− − (ma )n−m− " 2l + 1 " 2l + 1 − (l − m)( l − n) + l (l + m + 1)( l + n + 1) + l+ = m an+m+ + m an+m+ − " 2l + 1 " 2l + 1 − (l + m)( l + n) l (l − m + 1)( l − n + 1) l+ − m−an−m− − m−an−m− " 2l + 1 " 2l + 1 − l 1 (l − m)( l − n) l = m(△0a)nm + an+m+ + 2 3" 2l + 1

(l + m + 1)( l + n + 1) l+ + an+m+ + " 2l + 1 − (l + m)( l + n) l (l − m + 1)( l − n + 1) l+ + an−m− + an−m− " 2l + 1 " 2l + 1 4 l l = m(△0a)nm + (△0a)nm,

i.e.

l l l △0(aσ ∂0 )nm = m(△0a)nm + (△0a)nm, (12.27)

where △0 is a weighted averaging operator given by

l (△0a)nm

− 1 (l − m)( l − n) l (l + m + 1)( l + n + 1) l+ = an+m+ + an+m+ + 2 3" 2l + 1 " 2l + 1

− (l + m)( l + n) l (l − m + 1)( l − n + 1) l+ + an−m− + an−m− . " 2l + 1 " 2l + 1 4 670 Chapter 12. Pseudo-diﬀerential operators on SU (2)

k We want to ﬁnd a formula for △0 , and for this we ﬁrst calculate

l △0(△0a) nm =

− 1 (l − m)(2l − n) l (l + m + 1)( l + n + 1) l+ = (△0a)n+m+ + (△0a)n+m+ − " 2l + 1 " 2l + 1 − (l + m)( l + n) l (l − m + 1)( l − n + 1) l+ − (△0a)n−m− − (△0a)n−m− " 2l + 1 " 2l + 1 + + −− (l − m)( l − n) 1 (l− − m )( l− − n ) l = an++m++ + " 2l + 1 2 3" 2l− + 1

+ + − (l− + m + 1)( l− + n + 1) l + + an++m++ + " 2l− + 1 + + −− (l− + m )( l− + n ) l + an+−m+− + " 2l− + 1 + + − (l− − m + 1)( l− − n + 1) l + + an+−m+− + " 2l− + 1 4

+− (l + m + 1)( l + n + 1) 1 1 + + + + l + (l − m )( l − n )a ++ ++ + 2l + 1 2 2l+ + 1 n m " ++ + + + + l ," + (l + m + 1)( l + n + 1) an++m++ + +− + + + + l +"(l + m )( l + n )an+−m+− + ++ + + + + l +" (l − m + 1)( l − n + 1) an+−m+− −

" - −− (l + m)( l + n) 1 1 l − (l− − m−)( l− − n−)an−+m−+ + 2l + 1 2 2l− + 1 " − ," l + + (l− + m− + 1)( l− + n− + 1) an−+m−+ + −− l +"(l− + m−)( l− + n−)an−−m−− + − l + "(l− − m− + 1)( l− − n− + 1) an−−m−− −

" - +− (l − m + 1)( l − n + 1) 1 1 + + l − (l − m )( l − n )a −+ −+ + 2l + 1 2 2l+ + 1 − − n m " ++ + + l ," + (l + m− + 1)( l + n− + 1) an−+m−+ + +− + + l +"(l + m−)( l + n−)an−−m−− + ++ + + l "(l − m− + 1)( l − n− + 1) an−−m−− . " - 12.3. Diﬀerence operators for symbols 671

From this we get l △0(△0a) nm =

−− 1 (l − m)(2l − n) 1 1 l = (l − m − 1)( l − n − 1) an++m++ + " 2l + 1 2 2l ," l + (l + m + 1)( l + n + 1) an++m++ + −− l l +" (l + m)( l + n)anm + (l − m)( l − n)anm + " " - (l + m + 1)( l + n + 1) 1 1 l + (l − m)( l − n)an++m++ + " 2l + 1 2 2l + 2 l++ ," + (l + m + 2)( l + n + 2) an++m++ + l l++ +" (l + m + 1)( l + n + 1) anm + (l − m + 1)( l − n + 1) anm −

" " −− - (l + m)( l + n) 1 1 l − (l − m)( l − n)anm + " 2l + 1 2 2l l ," + (l + m)( l + n)anm + −− l +"(l + m − 1)( l + n − 1) an−−m−− + l "(l − m + 1)( l − n + 1) an−−m−− − " - (l − m + 1)( l − n + 1) 1 1 l − (l − m + 1)( l − n + 1) anm+ " 2l + 1 2 2l + 2 l++ ," + (l + m + 1)( l + n + 1) anm + l l++ +" (l + m)( l + n)an−−m−− + (l − m + 2)( l − n + 2) an−−m−− ,

" " −− - (l − m)( l − n) 1 1 l = (l − m − 1)( l − n − 1) an++m++ + " 2l + 1 2 2l ," l + (l + m + 1)( l + n + 1) an++m++ ] +

"(l + m + 1)( l + n + 1) 1 1 l + (l − m)( l − n)an++m++ + " 2l + 1 2 2l + 2 l++ ," + (l + m + 2)( l + n + 2) an++m++ + ] −

−− "(l + m)( l + n) 1 1 l − (l + m − 1)( l + n − 1) an−−m−− + " 2l + 1 2 2l ," l + (l − m + 1)( l − n + 1) an−−m−− − "(l − m + 1)( l − n + 1) 1 1 - − [ " 2l + 1 2 2l + 2 l l++ (l + m)( l + n)an−−m−− + (l − m + 2)( l − n + 2) an−−m−− ,

−− " " l l++ - where we used that pairs of terms with anm , anm canceled, and also four terms 672 Chapter 12. Pseudo-diﬀerential operators on SU (2)

l with anm canceled in view of the identity

(l − m)( l − n) (l + m + 1)( l + n + 1) + − (2 l + 1)(2 l) (2 l + 1)(2 l + 2) (l + m)( l + n) (l − m + 1)( l − n + 1) − − (2 l + 1)(2 l) (2 l + 1)(2 l + 2) −2l(m + n) (2 l + 2)( m + n) = + (2 l + 1)(2 l) (2 l + 1)(2 l + 2) = 0 . 12.3. Diﬀerence operators for symbols 673

Calculating in the other direction, we get

l △0(△0a) nm =

− 11 (l − m2)( l − n) l 1 (l + m + 1)( l + n + 1) l+ = (△0a)n+m+ + (△0a)n+m+ + 2 " 2l + 1 2 " 2l + 1 − 1 (l + m)( l + n) l 1 (l − m + 1)( l − n + 1) l+ + (△0a)n−m− + (△0a)n−m− 2 " 2l + 1 2 " 2l + 1 −− (l − m)( l − n) 1 1 + + l = (l− − m )( l− − n )an++m++ + 2l + 1 2 2l− + 1 " −+ + +," l + (l− + m + 1)( l− + n + 1) an++m++ − −− + + l −"(l− + m )( l− + n )an+−m+− − −+ + + l −"(l− − m + 1)( l− − n + 1) an+−m+− +

" - +− (l + m + 1)( l + n + 1) 1 1 + + + + l + (l − m )( l − n )a ++ ++ + 2l + 1 2 2l+ + 1 n m " ++ + + + + l ," + (l + m + 1)( l + n + 1) an++m++ − +− + + + + l −"(l + m )( l + n )an+−m+− − ++ + + + + l −" (l − m + 1)( l − n + 1) an+−m+− +

" - −− (l + m)( l + n) 1 1 l + (l− − m−)( l− − n−)an−+m−+ + 2l + 1 2 2l− + 1 " − ," l + + (l− + m− + 1)( l− + n− + 1) an−+m−+ − −− l −"(l− + m−)( l− + n−)an−−m−− − − l + −" (l− − m− + 1)( l− − n− + 1) an−−m−− +

" - +− (l − m + 1)( l − n + 1) 1 1 + + l + (l − m )( l − n )a −+ −+ + 2l + 1 2 2l+ + 1 − − n m " ++ + + l ," + (l + m− + 1)( l + n− + 1) an−+m−+ − +− + + l −"(l + m−)( l + n−)an−−m−− − ++ + + l −" (l − m− + 1)( l − n− + 1) an−−m−− . " - 674 Chapter 12. Pseudo-diﬀerential operators on SU (2)

From this we get l △0(△0a) nm =

−− 1 (l − m)(2l − n) 1 1 l = (l − m − 1)( l − n − 1) an++m++ + " 2l + 1 2 2l ," l + (l + m + 1)( l + n + 1) an++m++ − −− l l −" (l + m)( l + n)anm − (l − m)( l − n)anm + " " - (l + m + 1)( l + n + 1) 1 1 l + (l − m)( l − n)an++m++ + " 2l + 1 2 2l + 2 l++ ," + (l + m + 2)( l + n + 2) an++m++ − l l++ −" (l + m + 1)( l + n + 1) anm − (l − m + 1)( l − n + 1) anm +

" " −− - (l + m)( l + n) 1 1 l + (l − m)( l − n)anm + " 2l + 1 2 2l l ," + (l + m)( l + n)anm − −− l −"(l + m − 1)( l + n − 1) an−−m−− − l −" (l − m + 1)( l − n + 1) an−−m−− + " - (l − m + 1)( l − n + 1) 1 1 l + (l − m + 1)( l − n + 1) anm+ " 2l + 1 2 2l + 2 l++ ," + (l + m + 1)( l + n + 1) anm − l l++ −" (l + m)( l + n)an−−m−− − (l − m + 2)( l − n + 2) an−−m−−

" " −− - (l − m)( l − n) 1 1 l = (l − m − 1)( l − n − 1) an++m++ + " 2l + 1 2 2l ," l + (l + m + 1)( l + n + 1) an++m++ ] +

"(l + m + 1)( l + n + 1) 1 1 l + (l − m)( l − n)an++m++ + " 2l + 1 2 2l + 2 l++ ," + (l + m + 2)( l + n + 2) an++m++ ] +

−− "(l + m)( l + n) 1 1 l + (l + m − 1)( l + n − 1) an−−m−− − " 2l + 1 2 2l ," l − (l − m + 1)( l − n + 1) an−−m−− + "(l − m + 1)( l − n + 1) 1 1 - + [ " 2l + 1 2 2l + 2 l l++ − (l + m)( l + n)an−−m−− − (l − m + 2)( l − n + 2) an−−m−− ,

−− " "l l l++ - where we used again the fact that terms anm, anm and anm canceled. Comparing 12.4. Symbol classes on SU(2) 675 these calculations we get the commutativity propert y

△0△0a = △0△0a.

From this and ( 12.27 ) we obtain

2 2 △0(ma ) = △0(m△0a + △0a) = m△0a + 2△0△0a, and, consequently, by induction we get

k k k 1 △0 (ma ) = m△0 a + k△0△0− a.

Let us now apply this to ( 12.26 ). Using commutativity of △0, △+ and △ from Theorem 12.3.5 , we get −

l k1 k2 k3 △+ △ △0 (aσ ∂0 ) = − nm , -l k3 k1 k2 = △0 △+ △ (aσ ∂0 ) − nm , - l k3 k1 k2 = △0 (m − k1/2 + k2/2) △+ △ a − nm , / l 0- l k3 k1 k2 k3 k1 k2 = △0 m△+ △ a − △0 (k1/2 − k2/2) △+ △ a − nm − nm , / l0- , / l 0- k3 k1 k2 k3 1 k1 k2 = m △0 △+ △ a + k3 △0△0 − △+ △ a − − nm − nm , - , l - k3 k1 k2 −(k1/2 − k2/2) △0 △+ △ a − nm , - l l k1 k2 k3 k1 k2 k3 1 = ( m − k1/2 + k2/2) △+ △ △0 a + k3 △0△+ △ △0 − a , − nm − nm , - , - completing the proof.

12.4 Symbol classes on SU(2)

The goal of this section is to simplify the symbol class Σ m(SU(2)) from Definition 12.1.1 , yielding Hörmander’s class Ψ m(SU(2)) of pseudo-differential operators on SU(2). For this purpose, we introduce and investigate the symbol class Sm(SU(2)). Definition 12.4.1 (Symbol class Sm(SU(2)) ). For u ∈ SU(2), denote

1 Auf := A(f ◦ φ) ◦ φ− , where φ(x) = xu ; note that by Proposition 10.4.18 we have

1 1 RAu (x, y ) = RA(xu − , uyu − ), l 1 l σAu (x, l ) = t (u)∗ σA(xu − , l ) t (u). 676 Chapter 12. Pseudo-diﬀerential operators on SU (2)

m The symbol class S (SU(2)) consists of the symbols σA of those operators A ∈ L(C∞(SU(2))) for which

sing supp ( y '→ RA(x, y )) ⊂ { e}, and for which α β N m α △l ∂x σAu (x, l )ij ≤ CAαβmN +i − j,− (1 + l) −| | (12.28) uniformly in x,5 u ∈ SU(2), for every5 N ≥ 0, all l ∈ 1 N , all multi-indices α, β ∈ N3, 5 5 2 0 0 and for all matrix column/row numbers i, j . Thus, the constant in ( 12.28 ) may depend on A, α, β, m and N, but not on x, u, l, i, j . Remark 12.4.2 ( Rapid off-diagonal decay) . We note that inequality ( 12.28 ) con- tains the rapid off-diagonal decay property since we can take N as large as we want. We now formulate the main theorem of this section: Theorem 12.4.3 (Equality of classes Op Sm(SU(2)) = Ψ m(SU(2)) ). We have A ∈ m m Ψ (SU(2)) if and only if σA ∈ S (SU(2)) . Moreover, we have the equality of symbol classes Sm(SU(2)) = Σ m(SU(2)) . (12.29) In fact, we need to prove only the equality of symbol classes ( 12.29 ), from which the first part of the theorem would follow by Theorem 10.9.6 . In the process of proving equality ( 12.29 ), we establish a number of auxiliary results. m γ δ m γ Remark 12.4.4 . By Corollary 10.9.8 , if σA ∈ Σ (SU(2)) then △l ∂xσA ∈ Σ −| |(SU(2)). We show the analogous result for Sm(SU(2)): m γ δ m γ Lemma 12.4.5. If σA ∈ S (SU(2)) then σB = △l ∂xσA ∈ S −| |(SU(2)) . γ Proof. First, let |γ| = 1. Then △l f(l) = qf (l) for some 1/2 q ∈ P ol 1(SU(2)) := span( tij) : i, j ∈ {− 1/2, +1 /2}

61 7 for which q(e) = 0. Let r(y) := q(uyu − ). Then r ∈ P ol 1(SU(2)), because

1/2 1 1/2 1/2 1/2 1 tij (uyu − ) = tik (u) tkm (y) tmj (u− ). #k,m Moreover, we have r(e) = 0. Hence f(l) '→ rf (l) is a linear combination of dif- 1 ference operators △0, △+, △ because f ∈ P ol (SU(2)) : f(e) = 0 is a three- − 3 γ δ dimensional vector space spanned by q(0, q +, q ). Now let γ ∈ N0 and σB = △ ∂xσA. 8 − 9 l We have α β α β l 1 l △l ∂x σBu (x, l ) = △l ∂x t (u)∗ σB(xu − , l ) t (u) α β l γ δ 1 l = △l ∂x t (u)∗ △l ∂xσA(xu − , l ) t (u) ′ α+γ ′ β+δ = λu,γ △l ∂x σAu (x, l ), γ′ = γ | #| | | 12.4. Symbol classes on SU(2) 677

for some scalars λu,γ ′ ∈ C depending only on u ∈ SU(2) and multi-indices γ′ ∈ 3 N0. Remark 12.4.6 . Let D be a left-invariant vector field on SU(2). From the very m m definition of the symbol classes Σ (SU(2)) = k∞=0 Σk (SU(2)), it is evident that m m [σD, σ A] ∈ Σ (SU(2)) if σA ∈ Σ (SU(2)). We now prove the similar invariance for Sm(SU(2)). : m Lemma 12.4.7. Let D be a left-invariant vector field on SU(2) . Let σA ∈ S (SU(2)) . m m+1 Then [σD, σ A] ∈ S (SU(2)) and σA σD ∈ S (SU(2)) . Proof. For D ∈ su (2) we write D = i E, so that E ∈ i su (2). By Proposition 12.2.4 l l there is some u ∈ SU(2) such that σE(l) = t (u)∗ σ∂0 (l) t (u). Now, we have l l l l [σE, σ A]( l) = t (u)∗ σ∂0 (l), t (u) σA(x, l ) t (u)∗ t (u)

= σ , σ − (l). ∂0 A1 u 1 u 2 m Next, notice that S (SU(2))1 is invariant2 under the mappings σB '→ σBu and

σB '→ [σ∂0 , σ B]; here [ σ∂0 , σ B]( l)ij = ( i − j) σB(l)ij . Finally, l l l l σA(x, l ) σE(l) = t (u)∗ t (u) σA(x, l ) t (u)∗ σ∂0 (l) t (u)

= σ − (x, l ) σ (l) . Au 1 ∂0 u m+1 Just like in the first part of the proof, we see that σ A σD belongs to S (SU(2)) m+1 m since σB σ∂0 ∈ S (SU(2)) if σB ∈ S (SU(2)), by Theorem 12.3.12 . Proof of Theorem 12.4.3 . We have to show that Sm(SU(2)) = Σ m(SU(2)), so that the theorem would follow from Theorem 10.9.6 . Note that both classes Sm(SU(2)) m and Σ (SU(2)) require the singular support condition ( y '→ RA(x, y )) ⊂ { e}, so we do not have to consider this; moreover, the x-dependence of the symbol is not essential here, and therefore we abbreviate σA(l) := σA(x, l ). First, let us show that m m m m Σ (SU(2)) ⊂ S (SU(2)). Take σA ∈ Σ (SU(2)). Then also σAu ∈ Σ (SU(2)) (either by the well-known properties of pseudodifferential operators and Theorem m 10.9.6 , or by checking directly that the definition of the classes Σ k (SU(2)) is conjugation-invariant). Let us define cN (B) by N σcN (B)(l)ij := ( i − j) σB(l)ij .

m + m Now σcN (Au) ∈ Σ (SU(2)) for every N ∈ Z , because σAu ∈ Σ (SU(2)) and

[σ∂0 , σ B]( l)ij = ( i − j) σB(l)ij .

This implies the “rapid oﬀ-diagonal decay” of σAu : N m |σAu (x, l )ij | ≤ CAmN +i − j,− (1 + l) , implying the norm comparability

.· · · σAu (l).op ∼ sup |· · · σAu (l)ij | (12.30) i,j 678 Chapter 12. Pseudo-diﬀerential operators on SU (2)

α β m α in view of Lemma 12.6.5 in Section 12.6 . Moreover, △l ∂x σAu ∈ Σ −| |(SU(2)) by Corollary 10.9.8 , so that we obtain the symbol inequalities ( 12.28 ) from ( 10.37 ). Thereby Σ m(SU(2)) ⊂ Sm(SU(2)). Now we have to show that Sm(SU(2)) ⊂ Σm(SU(2)). Again, we may exploit m m the norm comparabilities ( 12.30 ): thus clearly S (SU(2)) ⊂ Σ0 (SU(2)). Conse- m m + quently, S (SU(2)) ⊂ Σk (SU(2)) for all k ∈ Z , due to Lemmas 12.4.5 and 12.4.7 .

Remark 12.4.8 . Notice that in Deﬁnition 12.4.1 , we demanded inequalities ( 12.28 ) uniformly in u ∈ SU(2). However, it suﬃces to assume this for only u ∈ { e, ω 1(π/ 2) }, 1 0 1 i where e = and ω (π/ 2) = √2 . 0 1 1 2 i 1 $ % $ % Exercise 12.4.9. Prove the claim of Remark 12.4.8 . Hint: Notice that

u(φ, θ, ψ ) = ω3(φ) ω2(θ) ω3(ψ)

1 √2 1 i and that ω (θ) = w− ω (θ)w , where w = ω (π/ 2) = . Recall also 2 1 3 1 1 1 2 i 1 l $ % Proposition 10.4.18 . Conjugating a symbol σA(x, l ) with t (ω3(t)) does not affect the “rapid off-diagonal decay” property. Let q ∈ Q1 := span {q+, q , q 0} \ { 0}, and − let △q be the corresponding first-order difference operator, i.e.

△qf(l) = qf (l).

1 Defining q1 ∈ Q1 by q1(z) := q(u− (zu ) and)σB := △q1 σA, we have △qσAu (l) = σBu (l). This shows us that difference operators can essentially be moved through the tl(u)-conjugations; also, such conjugations do not affect the x-derivatives; moreover, these conjugations behave well with respect to taking commutators of symbols. Remark 12.4.10 ( Topology on Sm(SU(2)) ). It is natural to defined the topology on Sm(SU(2)) by seminorms

α β N △l ∂x σAu (x, l )ij pα,β,m,i,j,N,u (σA) := sup +i − j, m α . (12.31) 1 N (1 + l) x SU(2) ,l 2 0 . 5 −| | 5; ∈ ∈ 5 5 Notice that by Exercise 12.4.9 , it is sufficient to consider only the cases u ∈ {e, ω 2(π/ 2) }. Compared to Corollary 10.9.9 , we can replace the convergence in the Hilbert– Schmidt norm by the pointwise ℓ∞ converges to relate the convergence to symbols to the convergence of operators: Corollary 12.4.11 (Convergence of symbols and operators). Let σ ∈ S0(SU(2)) , 0 and assume that a sequence σk ∈ S (SU(2)) satisfies inequalities ( 12.28 ) uniformly 12.5. Pseudo-differential operators on S3 679 in k (i.e. with constants independent of k). Assume that for all |β 2 we have the convergence | ≤ β β ∂ σk(x, l ) ∂ σ(x, l ) as k (12.32) x → x → ∞ 1 in the ℓ∞ norm, uniformly over all x G and all l 2 N0. Then Op σk Op σ strongly on L2(SU(2)) . ∈ ∈ → Moreover, if the convergence ( 12.32 ) holds for all β, then Op σk Op σ strongly on Hs(SU(2)) for any s R. → ∈ Proof. We observe that Theorem 10.5.5 implies that

Op σk Op σ (L2(SU(2))) $ − $L ≤ β β C1 sup ∂x σk(x, l ) ∂x σ(x, l ) op ≤ 1 N $ − $ x∈SU(2) ,l ∈ 2 0,|β|≤ 2 β β C2 sup ∂x σk(x, l ) ∂x σ(x, l ) ℓ∞ , ≤ 1 N $ − $ x∈SU(2) ,l ∈ 2 0,|β|≤ 2 where the last estimate follows from Lemma 12.6.5 in Section 12.6 , with constant 2 C2 independent of k. The strong convergence on L (SU(2)) now follows directly from ( 12.32 ). The strong convergence on Hs(SU(2)) follows from Theorem 10.8.1 by the same argument.

12.5 Pseudo-diﬀerential operators on S3

In this section we discuss how the construction of Section 10.10 yields full symbols of pseudo-differential operators on the 3-sphere S3. For this, we will use a global S3 isomorphism ∼= SU(2) from Proposition 11.4.2 . First we recall the quaternion space H from Section 11.4 which is the associative R-algebra with a vector space basis 1, i, j, k , where 1 H is the unit and { } ∈ i2 = j2 = k2 = 1 = ijk . − 3 4 The mapping x = ( xm)m=0 x01 + x1i + x2j + x3k identifies R with H. In S3'→ R4 H particular, the unit sphere ∼= is a multiplicative group. A bijective homomorphism Φ −1 : S3 SU(2)⊂ in ( 11.6 ) is defined by → x + i x x + i x x Φ−1(x) = 0 3 1 2 , '→ x1 + i x2 x0 ix3 − − and its inverse Φ : SU(2) S3 gives rise to the global quantisation of pseudo- differential operators on S3→induced by that on SU(2), as shown in Section 10.10 . The diffeomorphism Φ induces the Fourier analysis on S3 in terms of the representations of SU(2). To fix the notation for this in terms of S3, let

tl : S3 U(2 l + 1) C(2 l+1) ×(2 l+1) , → ⊂ 680 Chapter 12. Pseudo-diﬀerential operators on SU (2)

1 l 2 N0, be a family of group homomorphisms, which are the irreducible continuous∈ (and hence smooth) unitary representations of S3 when it is endowed with the SU(2) structure via the quaternionic product, see Section 12.5 for details. The Fourier coefficient f(l) of f C∞(S3) is defined by ∈ ! f(l) = f(x) tl(x)∗ dx, S3 " where the integration is performed! with respect to the Haar measure, and f(l) C(2 l+1) ×(2 l+1) . The corresponding Fourier series is given by ∈ ! f(x) = (2 l + 1) Tr f(l) tl(x) . 1 N l∈#2 0 $ % ! Now, if A : C∞(S3) C∞(S3) is a continuous linear operator, we define its full symbol as a mapping →

(x, l ) σ (x, l ), σ (x, l ) = tl(x)∗(At l)( x) C(2 l+1) ×(2 l+1) . '→ A A ∈ Then we have the representation the of operator A in the form

l Af (x) = (2 l + 1) Tr t (x) σA(x, l ) f(l) , 1 N l∈#2 0 $ % ! see Theorem 10.4.4 . We also note that if

−1 Af (x) = KA(x, y ) f(y) d y = f(y) RA(x, y x) d y, S3 S3 " " where RA is the right-convolution kernel of A, then

l ∗ σA(x, l ) = RA(x, y ) t (y) dy S3 " by Theorem 10.4.6 , where, as usual, the integration is performed with respect to the Haar measure with a standard distributional interpretation. We now introduce symbol classes Sm(S3) which allow us to characterise operators from H¨ormander’s class Ψ m(S3). Deﬁnition 12.5.1 (Symbol class Sm(S3)). We write σ Sm(S3) if the corre- A ∈ sponding kernel KA(x, y ) is smooth outside the diagonal x = y and if we have the estimate

α∂βσ (x, l ) C (1 + i j )−N (1 + l)m−| α|, (12.33) △l x Au ij ≤ AαβmN | − | for every & N 0, every u & S3, and all multi-indices α, β , where the symbol σ & & Au is the symbol≥ of the operator∈

A f = A(f ϕ ) ϕ−1, u ◦ u ◦ u 12.6. Appendix: inﬁnite matrices 681

α α1 α2 α3 with ϕu(x) = xu the quaternionic product. We write l = + − 0 , where the operators , , are discrete difference operators△ △ acting△ on△ matrices △+ △− △0 σA(x, l ) in the variable l, and explicit formulae for them and their properties are given in Definition 12.3.2 and Theorem 12.3.5 , with polynomials q+, q− and q0 defined as in Remark 10.6.2 . Constants CAαβmN in ( 12.33 ) may depend on A, α, β, m, N but not on i, j, l .

Remark 12.5.2 . As in the case of SU(2), the symbols of Au and A are related by

l ∗ −1 l σAu (x, l ) = t (u) σA(xu , l ) t (u), see Proposition 10.4.18 . We notice that imposing the same conditions on all sym- m 3 bols σAu in ( 12.33 ) simply refers to the well-known fact that the class Ψ (S ) should be in particular “translation”-invariant (i.e. invariant under the changes of m 3 variables induced by quaternionic products ϕu), namely that A Ψ (S ) if and m 3 3 ∈ only if Au Ψ (S ), for all u S . Condition ( 12.33 ) is the growth condition with respect∈ to the quantum number∈ l combined with the condition that matrices σA(x, l ) must have a rapid off-diagonal decay. With Definition 12.5.1 , we have the following characterisation, which follows immediately from Theorem 12.4.3 : Theorem 12.5.3 (Equality Op Sm(S3) = Ψ m(S3)). We have A Ψm(S3) if and only if σ Sm(S3). ∈ A ∈ 12.6 Appendix: infinite matrices

In this section we discuss infinite matrices. The main conclusion that we need is that the operator-norm and the l∞-norm are equivalent for matrices arising as full symbols of pseudo-differential operators in Ψ m(SU(2)), used in ( 12.30 ) in the proof of Theorem 12.4.3 . The reader should already know basic linear algebra, but for the sake of completeness, we review necessary matrix operations. Let Cm×n denote the complex vector space of matrices with m rows and n columns; the rows are numbered 1, . . . , m downwards, the columns 1 ,...,n from left to right. Let Aij C denote the element of matrix A Cm×n on row i and column k. Let λ ∈ C and A, B Cm×n; let matrices λA,∈ A + B Cm×n and the adjoint A∗ C∈n×m be defined∈ by ∈ ∈ (λA )ij := λA ij ,

(A + B)ij := Aij + Bij , ∗ (A )ij := Aji . The product of A Cm×p and B Cp×n is AB Cm×n deﬁned by ∈ ∈ ∈ p

(AB )ij := Aik Bkj . k#=1 682 Chapter 12. Pseudo-diﬀerential operators on SU (2)

The trace of A Cn×n is ∈ n Tr( A) := Ajj . j=1 # We may naturally identify vector space Cn with Cn×1, and a mapping ( x Ax ) : Cm×1 Cn×1 can be seen as a linear mapping Cm Cn. '→ → → The Euclidean inner product (or Hilbert–Schmidt inner product ) of A, B Cm×n is a special case of that in Subsection B.5.1 , and is given by ∈

m n A, B := Tr( B∗A)1/2 = B A , -HS ij ij i=1 j=1 # # 1/2 and the corresponding norm of A is A HS := A, A HS . The operator norm of A Cm×n is $ $ , - ∈ n×1 A = A := sup Ax 2 : x C , x 2 1 , $ $ $ $op $ $ℓ ∈ $ $ℓ ≤ n 2 1/2 ' ( where x ℓ2 = ( j=1 xj ) is the usual Euclidean norm. Of course, due to the ﬁnite dimensionality$ $ here,| | supremum could be replaced by maximum. ) Theorem 12.6.1. Let A, B Cn×n. Then ∈ AB A B . $ $HS ≤ $ $ $ $HS Moreover, A = sup AX : X Cn×n, X 1 . $ $ {$ $HS ∈ $ $HS ≤ } n×1 th Proof. Let bj C denote the j column vector of the matrix B. Then Ab j Cn×1 is the jth∈ column vector of the matrix AB , so that ∈

n n AB 2 = Ab 2 A 2 b 2 = A 2 B 2 . $ $HS $ j$HS ≤ $ $ $ j$HS $ $ $ $HS j=1 j=1 # # n×1 Let x C be such that x HS = 1 and Ax HS = A . Let each column vector∈ of X Cn×n be x/ √n$. Then$ X =$ 1 and$ $ $ ∈ $ $HS n AX 2 = Ax 2 /n = A 2. $ $HS $ $HS $ $ j=1 # Taking square roots completes the proof.

Definition 12.6.2 (Operator as an infinite matrix). Let CZ denote the space of complex sequences x : Z C. We will write x = ( x ) Z, where x = x(j). Let → j j∈ j V = x CZ : j Z : x = 0 finite . ! ∈ { ∈ ! j / } ! ' ( ! 12.6. Appendix: infinite matrices 683

A matrix A CZ×Z is a function A : Z Z C, sometimes presented as an inﬁnite table ∈ × →

......  A−1,−1 A−1,0 A−1,1 A−1,2 A−1,3  · · · A A A A A · · · · · · 0,−1 00 01 02 03 · · ·  A = A =  A A A A A  , ij i,j ∈Z · · · 1,−1 10 11 12 13 · · ·   A A A A A  * + · · · 2,−1 20 21 22 23 · · ·   A A A A A  · · · 3,−1 30 31 32 33 · · ·   ......   ......      Z where Aij := A(i, j ). The ( standard ) matrix of a linear operator A : V C is the matrix A where the numbers A C are obtained from → ij i,j ∈Z ij ∈ * + (Ax)i = Aij xj; Z #j∈ ! there should be no confusion in denoting the linear operator and the corresponding standard matrix with the same letter.

2 2 Remark 12.6.3 . Let δi = ( δij )j∈Z ℓ = ℓ (Z), where δii = 1 and δij = 0 if i = j. Then for a linear operator A : ℓ2 ∈ ℓ2, the standard matrix is / →

A = Aδ , δ 2 , (12.34) ij , j i-ℓ i.e. the standard matrix is the matrix of the operator A with respect to the basis δ : i Z . { i ∈ } Deﬁnition 12.6.4. Let A : V CZ be a linear operator. For each k Z, let us deﬁne a linear operator A(k) :→V CZ by ∈ →

Aij , if i j = k, A(k)ij = − . 0, if i j = k 2 − /

Notice that now A = k∈Z A(k) and ) A(k) ℓ2→ℓ2 = sup A(k)j+k,j . (12.35) $ $ j | |

Let

A ℓ∞ := sup Aij , $ $ i,j ∈Z | | and recall the notation i j = (1 + i j 2)1/2. , − - | − | 684 Chapter 12. Pseudo-diﬀerential operators on SU (2)

Lemma 12.6.5. Let A : V CZ be a linear operator. Then →

A ∞ A 2 2 . $ $ℓ ≤ $ $ℓ →ℓ −r Moreover, if A c i j A ∞ for a constant c < where r > 1, then | ij | ≤ r, − - || || ℓ r ∞ ′ A 2 2 c A ∞ $ $ℓ →ℓ ≤ r $ $ℓ ′ −r for the constant c = c k < ; hence in this case the norms 2 2 r r k∈Z, - ∞ $ · $ ℓ →ℓ and ℓ∞ are equivalent. $ · $ ) Proof. The ﬁrst claim follows from the Cauchy–Schwarz inequality:

(12 .34 ) A = (Aδ , δ ) 2 A 2 2 . | ij | j i ℓ ≤ $ $ℓ →ℓ & & Next, we can assume that A ℓ∞ &< since& otherwise there is nothing to prove. || || ∞ Since A = k∈Z A(k), we get ) A ℓ2→ℓ2 A(k) ℓ2→ℓ2 $ $ ≤ Z $ $ k#∈ (12 .35 ) = sup A(k)j+k,j Z j | | k#∈ −r ′ cr k A ℓ∞ =: cr A ℓ∞ , ≤ Z , - || || || || k#∈ this last sum converging since r > 1. This concludes the proof.

Definition 12.6.6 (Matrices with rapid off-diagonal decay). A matrix A CZ×Z is said to decay (rapidly ) off-diagonal if ∈

A c i j −r | ij | ≤ Ar , − - for every i, j Z and r N, where constants cAr < depend on r, A , but not on i, j . The set of∈ oﬀ-diagonally∈ decaying matrices is denoted∞ by . D Proposition 12.6.7. Let A, B . Then AB . ∈ D ∈ D Proof. In principle, we must be cautious here, since linear operators A, B : V CZ in general cannot be composed to get AB : V CZ. Here, however, there→ is no problem as A, B , so that → ∈ D

(AB ) x = ( AB )x = A(Bx) = A B x , ik k  ij jk k 3 k 4 Z j k # i∈ # # i∈Z ! !   12.6. Appendix: inﬁnite matrices 685 where

(AB ) = A B | ik | & ij jk & & j & &# & & & & A B & ≤ & | ij | | jk &| j # c c i j −r j k s ≤ Ar Bs , − - , − - j # Peetre 3.3.31 2|r|c c i k −r k j |r| j k s ≤ Ar Bs , − - , − - , − - j # = 2 |r|c c i k −r j |r|+s, Ar Bs , − - , - j # which converges if r + s < 1. This shows that AB . | | − ∈ D Remark 12.6.8 . Proposition 12.6.7 dealt with matrix multiplication in . For matrices A, B CZ×Z in general, notice that D ∈

(A + B)ij = Aij + Bij , 2(λA )ij = λA ij .

Moreover, we may deﬁne involution A A∗ by '→ ∗ (A )ij := Aji .

Of course, on the algebra ( ) this corresponds to the usual adjoint operation ∗ ∗ L H A A , where A x, y H = x, Ay H for every x, y . We may collect these observations:'→ , - , - ∈ H Theorem 12.6.9. (ℓ2) is a unital involutive algebra. Moreover, for A , D ⊂ L ∈ D the norms A and A ∞ are equivalent. $ $op $ $ℓ 686 Chapter 12. Pseudo-diﬀerential operators on SU (2) Chapter 13

Pseudo-diﬀerential operators on homogeneous spaces

In this section we discuss pseudo-differential operators on homogeneous spaces. The main question addressed here is how operators on such a space are related to pseudo-differential operators on the group that acts on the space. Once such a correspondence is established, one can use it to map the whole construction developed earlier from the group to the homogeneous space. We also note that among other things, this chapter provides an application to the characterisation of pseudo- differential operators in terms of Σ m-classes in Theorem 10.9.6 . An important Sn class of examples to keep in mind here are the spheres ∼= SO( n) SO( n + 1) ∼= SO( n + 1) /SO( n). \

13.1 Analysis on closed manifolds

We start with closed manifolds. Let M be a C∞-smooth, closed (i.e. compact, without a boundary) orientable manifold. We refer to Section 5.2 for the basic constructions on M, so we now review only a few of them. The test function space (M) is the space of C∞(M) endowed with the usual Fr´echet space topology. Its Ddual ′(M) = ( (M), C) is the space of distributions, endowed with the weak- - topology,D see RemarkL D 5.2.15 . The duality is expressed by the bracket f, ϕ = f(ϕ∗) ϕ (M), f ′(M)). The embedding (M) ֒ ′(M) is interpreted, - by) ∈ D ∈ D D → D ψ, ϕ := ψ(x) ϕ(x) d x. , - "M The Schwartz kernel theorem states that ( (M)) is isomorphic to (M) ′(M); the isomorphism is given by L D D ⊗D

Aϕ, f = K , ϕ f , , - , A ⊗ - 688 Chapter 13. Pseudo-diﬀerential operators on homogeneous spaces

′ where A ( (M)), ϕ (M), f (M), and distribution KA (M) ′(M) is∈ called L D the Schwartz∈ D kernel of∈A D . Then A can be uniquely extended∈ D (by⊗ duality)D to A ( ′(M)), and it is customary to write informally ∈ L D

(Af )( x) = KA(x, y ) f(y) d y "M instead of ϕ Af, ϕ (ϕ (M)). Recall that L2(M) = H0(M), ′(M) = s '→ , - ∈ D s s 2 D s∈RH (M) and (M) = s∈RH (M), where H (M) is the ( L -type) Sobolev ∪space of order s DR, see Definition∩ 5.2.16 . There are different spaces of distributions available∈ more specifically on homogeneous spaces, see e.g. [ 90 ] for spaces ′ L1 (M) of summable distributions. D An operator A ( (M)) is a pseudo-differential operator of order m R m ∈ L D m dim( M) ∈ on M, A Ψ (M), if ( MφAM ψ)κ Ψ (R ) for every chart ( U, κ ) of M ∈ ∞ ∈ and for every φ, ψ C0 (U), where Mφ is the multiplication operator f φf , and ∈ '→ (M AM ) f := ( M AM (f κ)) κ−1 (f C∞(κU )) . φ ψ κ φ ψ ◦ ◦ ∈ m dim( M) We sometimes write MφAM ψ Ψ (R ), thus omitting the subscript κ and leaving he chart mapping implicit.∈ Equivalently, pseudo-differential operators can be characterised by commutators (see Theorem 5.3.1 ): A ( (M)) belongs to m ∞ m 0 ∈ L D Ψ (M) if and only if ( Ak)k=0 (H (M), H (M)) for every sequence of smooth ∞ ⊂ L vector fields ( Dk)k=1 on M, where A0 := A and Ak+1 := [ Dk+1 , A k]. Definition 13.1.1 (Right transformation group). A smooth right transformation group is (G, M, m ), where G is a Lie group, M is a C∞-manifold and m : M G M is a C∞-mapping called a right action , satisfying m(p, e ) = p and m(m(×p, y )→, x ) = m(p, yx ) for all x, y G and p M, where e G is the neutral element of the group. The action is free ,∈ if m(p, x ) =∈ p implies x =∈e. It is evident how one defines a left transformation group ( G, M, m ) with a left action m : G M M. × → Definition 13.1.2 (Fiber bundles). A smooth fiber bundle is

(E, B, F, p E→B), where E, B, F are C∞-manifolds and p C∞(E, B ) is a surjective mapping E→B ∈ such that there exists an open cover = Uj j J of B and diffeomor- −1 U { | ∈ } phisms φj : p (Uj) Uj F satisfying φj(x) = ( pE→B(x), ψ j(x)) for every −1 → × x pE→B(Uj). The spaces E, B, F are called the total space , the base space , and the∈ fiber of the bundle, respectively. The cover is called a locally trivialising U cover of the bundle. Sometimes only the mapping pE→B is called the fiber bundle. Definition 13.1.3 (Principal fiber bundles). A principal fiber bundle is

(E, B, F, p E→B, m ), 13.2. Analysis on compact homogeneous spaces 689

where ( E, B, F, p E→B) is a smooth ﬁber bundle with cover and mappings φj, ψ j as above and ( F, E, m ) is a smooth right transformation groupU with a free action satisfying pE→B(m(x, y )) = pE→B(x) for every ( x, y ) E F and ψj(m(x, y )) = ψ (x)y for every ( x, y ) p−1 (U ) F . ∈ × j ∈ E→B j × 13.2 Analysis on compact homogeneous spaces

Here we review some elements of the analysis on homogeneous spaces. The group will be acting on the right to adopt the construction to the previously constructed symbolic calculus on groups. Definition 13.2.1 (Homogeneous spaces I). Let ( G, M, m ) be a smooth right transformation group. The manifold M is called a homogeneous space if the action m : M G M is transitive , i.e. if for every p, q M there exists x G such that m(×p, x )→ = q. ∈ ∈ For this line of thought we can refer to e.g. [ 146 ]. However, let us also give another, equivalent definition for a homogeneous space: Definition 13.2.2 (Homogeneous spaces II). Let G be a Lie group with a closed subgroup K. The homogeneous space K G is the set of classes Kx = kx k K (x G) endowed with the topology co-induced\ by x Kx , and equipped{ | ∈ with} the∈ unique C∞-manifold structure such that the mapping'→ ( x, Ky ) Kyx belongs to C∞(G (K G), K G), and such that there is a neighbourhood'→ U K G of Ke K G× and\ a mapping\ ψ C∞(U, G ) satisfying Kψ (Kx ) = Kx . The⊂ group\ G acts∈ smoothly\ from the right∈ on the manifold K G by ( Ky, x ) Kyx . \ '→ Exercise 13.2.3. Actually a smooth homogeneous space M is diffeomorphic to Gp G, where Gp = x G m(p, x ) = p is the isotropy subgroup (see Definition 6.3.3\ and Theorem 6.3.4{ ∈). Thus,| show that} two definitions are equivalent. Exercise 13.2.4. Show that ( G, K G, K, x Kx, (x, k ) kx ) has a structure of a principal fiber bundle. For a further\ development'→ of this'→ point of view see e.g. [20 ]. Remark 13.2.5 ( Homogeneous spaces K G vs G/K ). Clearly one can consider homogeneous spaces G/K with the action\ of the left transformation group. As it turns out, once we have chosen to identify the Lie algebra of a Lie group with the left invariant vector fields, the further analysis is fixed from the point of view of “right”/“left”, see Remark 10.4.2 for the starting point of this choice. Moreover, since on the group we wanted to have the composition formulae for pseudo-differential operators in the usual form σ = σ σ + and not in the A◦B A B · · · form σA◦B = σBσA + , also the choice of the definition of the Fourier transform was fixed, see Remark· · · 10.4.13 . However, the right/left constructions are very symmetric, and since the notation G/K reminding the division of numbers may be more familiar, we chose to introduce homogeneous spaces in this setting in the 690 Chapter 13. Pseudo-differential operators on homogeneous spaces definition of the right quotient in Definition 6.2.12 . Consequently, this led to the corresponding definition of the quotient topology in Definition 7.1.8 , as well as the corresponding discussion of the group actions in Section 6.3 and the invariant integration in Section 7.4.1 . However, as we pointed out in Remark 6.2.14 the choice between “right” and “left” is completely symmetric, so the reader should have no difficulty in translating those results to the setting of the right actions considered here. From now on we assume the Lie group G to be compact, and we observe that by Remark 7.3.2 , (8), and by Proposition 7.1.10 the space K G is a compact Hausdorff space. \ We can regard functions (or distributions) constant on the cosets Kx (x G) as functions (or distributions) on K G; it is obvious how one embeds the spaces∈ C∞(K G) and ′(K G) into the spaces\ C∞(G) and ′(G), respectively. Let us define P\ D(C∞(\G)) by D K\G ∈ L

(PK\Gf)( x) := f(kx ) d µK (k), (13.1) "K where d µK is the Haar measure on the compact Lie group K. Hence PK\Gf ∞ 2 ∈ C (K G), and PK\G extends uniquely to the orthogonal projection of L (G) onto the\ subspace L2(K G). Let us consider an operator A (C∞(G)) with the symbol satisfying \ ∈ L σ (kx, ξ ) = σ (x, ξ ) ( x G, k K, ξ Rep( G)); (13.2) A A ∈ ∈ ∈ this condition is equivalent to

RA(kx, y ) = RA(x, y ) (13.3) in the sense of distributions for the right-convolution kernels, in view of ( 10.18 ) in Section 10.4.1 . Consequently, the Schwartz integral kernel KA of A satisﬁes −1 −1 KA(kx, kxy ) = KA(x, xy ) in view of Proposition 10.4.1 . Replacing xy −1 by y we have

KA(kx, ky ) = KA(x, y ). This means that A maps the space C∞(K G) into itself. Of course, for a general A (C∞(G)) the equality ( 13.2 ) does not\ have to be true, but then we can define∈ L an operator A ( (G)) by the right convolution kernel K\G ∈ L D R := ( P id) R , AK\G K\G ⊗ A m with PK\G as in ( 13.1 ). We note that for A Ψ (G) its right operator-valued sym- ∈ ∞ m 0 bol rA in Definition 10.11.9 satisfies the property that rA C (G, (H (G), H (G))), so that the right operator-valued symbol ∈ L

rAK\G (x) = rA(kx ) d µK (k) "K 13.2. Analysis on compact homogeneous spaces 691

of AK\G exists as a weak integral (Pettis integral), with the interpretation as in Remark 7.9.3 . Consequently, by ( 10.60 ) in Theorem 10.11.16 , or directly by Deﬁnition 10.4.3 , the symbol of AK\G satisﬁes

σAK\G (x, ξ ) = σA(kx, ξ ) d µK (k) "K for all x G and ξ Rep( G). ∈ ∈ Remark 13.2.6 ( Calculus of K-invariant operators) . Suppose we are given symbols of pseudodifferential operators A1, A 2 on G satisfying the K-invariance ( 13.2 ). If we look at the asymptotic expansion formulae for σ , σ ∗ and σ t in Section A1A2 A1 A1 10.7.3 , we see that all the terms there are K-invariant in the same sense. Moreover, for an elliptic K-invariant symbol the terms in the asymptotic expansion for a parametrix in Theorem 10.9.10 are also K-invariant. In this way the calculus of the K-invariant operators is immediately obtained from the corresponding calculus of operators on the group G. Theorem 13.2.7 and Corollary 13.2.8 below show how to “project” pseudo- differential operators on G to pseudo-differential operators on K G. The history of such averaging processes for pseudo-differential operators can be\ traced at least back to the work of M. F. Atiyah and I. M. Singer in the 1960s, and H. Stetkær studied related topics for classical pseudo-differential operators in [ 116 ]. Theorem 13.2.7 (Averaging of operators). Let G be a compact Lie group with a closed Lie subgroup K. If A Ψm(G), then A Ψm(G). ∈ K\G ∈ Proof. We will use Theorem 10.9.6 characterising symbols of operators from Ψ m(G). First, notice that PK\G is right-invariant, and hence

(∂β α)( P id) σ = ( P id)( ∂β α)σ x ⊗ △ ξ K\G ⊗ A K\G ⊗ x ⊗ △ ξ A for a left-invariant partial diﬀerential operator ∂β and a diﬀerence operator α, x △ξ for all α, β Ndim( G). Therefore ∈ 0 Op( α∂βσ ) = Op( α∂βσ ) . △ξ x AK\G △ξ x A K\G Since A Ψm(G), by Theorem 10.9.6 we* have + ∈ α∂βσ (x, ξ ) C ξ m−| α|, $△ ξ x A $op ≤ Aαβm, - and hence we can estimate

α∂βσ (x, ξ ) = α∂βσ (kx, ξ ) d µ (k) $△ ξ x AK\G $op △ξ x A K 5"K 5op 5 5 5 α β 5 5 ∂ σA(kx, ξ ) op dµK (5k) ≤ $△ ξ x $ "K C ξ m−| α|. ≤ Aαβm, - 692 Chapter 13. Pseudo-diﬀerential operators on homogeneous spaces

At the same time, formula ( 13.3 ) implies that the right-convolution kernel of m Op( σAK\G ) has singularities only at y = e. This proves that σAK\G Σ0 (G). Let now B (C∞(G)) be a left-invariant (right-convolution) pseudo-diﬀerential∈ operator. Then ∈σ L (x, ξ ) = σ (ξ) is independent of x G in view of Remark B B ∈ 10.4.10 , and hence B = BK\G. Consequently, we have

(Op( σAσB)) K\G = Op( σAK\G σB) and

(Op( σBσA)) K\G = Op( σBσAK\G ).

To argue by induction, assume now that for some k ∈ N0 we have proven that r r R σCK\G ∈ Σk(G) for every C ∈ Ψ (G), for every r ∈ . By Remark 13.2.6 we hence get m [σ∂j , σ AK\G ] = [ σ∂j , σ A]K\G ∈ Σk (G),

γ γ m+1 −| γ| (△ξ σ∂j )σAK\G = (( △ξ σ∂j )σA)K\G ∈ Σk (G) and γ γ m+1 −| γ| (△ξ σAK\G )σ∂j = (( △ξ σA)σ∂j )K\G ∈ Σk (G); m m this means that σAK\G ∈ Σk+1 (G), so that by induction we get σAK\G ∈ Σ (G) = m k∞=0 Σk (G).

Once we get a pseudo-diﬀerential operator of the form AK G, it can be pro- jected to the homogeneous space K\G: \

Corollary 13.2.8 (Projection of operators). Let K\G be orientable. Then AK G|C∞(K G) ∈ Ψm(K\G) for every A ∈ Ψm(G). \ \ Proof. Let us denote

m m Ψ (G)K G := {AK G | A ∈ Ψ (G)} \ \ and m m Ψ (G)K G|C∞(K G) = {AK G|C∞(K G) : A ∈ Ψ (G)}. \ \ \ \ m m By Theorem 13.2.7 we know that Ψ (G)K G ⊂ Ψ (G). Let D be a smooth vector field on K\G. Since by Exercise 13.2.4 , ( G,\ K \G, K, x "→ Kx, (x, k ) "→ kx ) is a principal fiber bundle, there exists a smooth vector field X = XK G on G such \ that X|C∞(K G) = D (see [ 112 ]). Then we have \

m [D, Ψ (G)K G|C∞(K G)] = \ \ m m = [ X, Ψ (G)K G]|C∞(K G) ⊂ Ψ (G)K G|C∞(K G), \ \ \ \ m m 0 and this combined with Ψ (G)K G|C∞(K G) ⊂ L (H (K\G), H (K\G)) yields the conclusion due to the commutator\ characterisation\ of pseudo-diﬀerential operators on closed manifolds in Theorem 5.3.1 . 13.3. Analysis on K\G, K a torus 693

Definition 13.2.9 (Lifting of operators). We will say that the operator A ∈ Ψm(G) m is a lifting of the operator B ∈ Ψ (K\G) if A = AK G and if A|C∞(K G) = B. \ \ Remark 13.2.10 ( Calculus of liftings) . It already follows from Corollary 13.2.8 that at least sometimes a pseudo-differential operator on K\G can be (possibly non- mj uniquely) lifted to a pseudo-differential operator on G. If Bj ∈ Ψ (K\G) can be mj ∗ mj lifted to Cj = ( Cj)K G ∈ Ψ (G) (i.e. Cj|C∞(K G) = Bj), then Cj ∈ Ψ (G) \ ∗ mj \ m1+m2 is a lifting of the adjoint operator Bj ∈ Ψ (K\G), and B1B2 ∈ Ψ (K\G) m1+m2 is lifted to C1C2 ∈ Ψ (G). Moreover, if C1 is elliptic with a parametrix −m1 m1 D ∈ Ψ (G) as in Theorem 10.9.10 , then D = DK\G and B1 ∈ Ψ (K\G) is −m1 elliptic with a parametrix D|C∞(K\G) ∈ Ψ (K\G).

13.3 Analysis on K\G, K a torus

In this section we assume that the subgroup K of G is a torus, K =∼ Tq. For example, K may be the maximal torus which has an additional importance in the representation theory of G in view of Cartan’s maximal torus theorem (Theorem 7.8.8 ). However, it may me a lower dimensional torus as well. Remark 13.3.1 ( Sphere S2). Let Bn be the unit ball of the Euclidean space Rn, and Sn−1 its boundary, the ( n−1)-sphere. The two-sphere S2 can be considered as the base space of the Hopf fibration S3 → S2, where the fibers are diffeomorphic to the unit circle S1 ⊂ R2. In the context of harmonic analysis, S3 is diffeomorphic 1 to the compact non-commutative Lie group G = SU(2), having a maximal torus K =∼ S1 =∼ T1. Then the homogeneous space K\G is diffeomorphic to S2, so that the canonical projection pG→K\G : x "→ Kx is interpreted as the Hopf fiber bundle G → K\G; in the sequel we treat the two-sphere S2 always as the homogeneous space K\G. Notice that also S2 =∼ T1\SO(3). For a sketch of operators there see [137 ]. Remark 13.3.2 ( Spherical symbols) . In [ 122 ] a subalgebra of Ψ m(S2) was described in terms of the so-called spherical symbols. Functions f ∈ C∞(S2) can be expanded in series ∞ l m f(φ, θ ) = f(l)m Yl (φ, θ ), l=0 m =−l ! m where ( φ, θ ) ∈ [0 , 2π] × [0 , π ] are the spherical coordinates, and the functions Yl are the spherical harmonics with “spherical” Fourier coefficients π 2π m f(l)m := f(φ, θ ) Yl (φ, θ ) sin( θ) d φ dθ. "0 "0 Let us define ! ∞ l m (Af )( φ, θ ) := a(l) f(l)m Yl (φ, θ ), l=0 m =−l 1See Proposition 11.4.2 . ! 694 Chapter 13. Pseudo-differential operators on homogeneous spaces

m 2 where a : N0 → C is a rational function; in [ 122 ], Svensson states that A ∈ Ψ (S ) if and only if m |a(l)| ≤ CA,m(l + 1) . Let us now present another proof for a special case of Theorem 13.2.7 and Corollary 13.2.8 for the torus subgroup K; this method of proof turns out to be useful when we develop an analogous method for showing that the mapping m m (A "→ AK\G|C∞(K\G)) : Ψ (G) → Ψ (K\G) is surjective if K is a torus subgroup (see the proof of Theorem 13.3.5 ). Theorem 13.3.3. Let G be a compact Lie group with a torus subgroup K. If A ∈ m m m Ψ (G), then AK\G ∈ Ψ (G) and the restriction AK\G|C∞(K\G) ∈ Ψ (K\G). q Proof. Let dim( G) = p + q, where K =∼ T . Let V = {Vi | i ∈ I} be a locally trivialising open cover of the base space K\G for the principal ﬁber bundle (G, K \G, K, x "→ Kx, (x, k ) "→ kx ). Let U = {Uj | 1 ≤ j ≤ N} be an open cover of K\G such that for every j1, j 2 ∈ { 1, . . . , N } there exists Vi ∈ V containing

Uj1 ∪ Uj2 whenever Uj1 ∩ Uj2 *= ∅; notice that we can always refine any open cover on a finite-dimensional manifold to get a new cover satisfying this additional re- quirement. Then each Ui∪Uj (1 ≤ i, j ≤ N) is a chart neighbourhood on K\G, and −1 furthermore there exist diffeomorphisms φij : ( Ui ∪ Uj) × K → pG→K\G(Ui ∪ Uj) such that pG→K\G(φij (x, k )) = x for every x ∈ Ui ∪ Uj and k ∈ K. To simplify p the notation, we treat the neighbourhood Ui ∪ Uj ⊂ K\G as a set Ui ∪ Uj ⊂ R , −1 q p q and pG→G/K (Ui ∪ Uj) ⊂ G as a set ( Ui ∪ Uj) × T ⊂ R × T . Let {(Uj, ψ j) | 1 ≤ j ≤ N} be a partition of unity subordinate to U, and m let Aij = Mψi AM ψj ∈ Ψ (G). With the localised notation we consider Aij ∈ m p q p q m p q p q Ψ (R × T ; R × Z ), so that it has the symbol σAij ∈ S (R × T ; R × Z ). We note that the notation that we use for symbols here is slightly different from before: Rp × Tq stands for the space variables, and Rp × Zq is for dual frequencies. Then

σ(AK\G)ij (x, ξ )

= σ(Aij )K\G (x, ξ )

= σAij (x1,...,x p, x p+1 + z1,...,x p+q + zq; ξ) d z1 · · · dzq, Tq " m Rp Tq Rp Zq and it is now easy to check that σ(AK\G)ij ∈ S ( × ; × ). This yields m (AK\G)ij ∈ Ψ (G), and hence

m AK\G = (AK\G)ij ∈ Ψ (G), i,j completing the proof. Theorem 13.3.3 has the inverse which will be given in Theorem 13.3.5 . But ﬁrst, we prepare a lemma on the extension of symbols in the Euclidean space. 13.3. Analysis on K\G, K a torus 695

Because of the commutator characterisations in Chapter 5 (especially the equality (5.4 ) in Theorem 5.4.1 ), and in view of Corollary 4.6.13 , all of the symbol classes on Tn in both the Euclidean and toroidal quantizations coincide. That is why, to simplify the notation, we will skip writing the space for the frequency variable and will only write the space which will usually be Rp × Tq. Thus, the class Ψm(Rp×Tq) will stand for either Ψ m(Rp×Tq; Rp×Zq) or for Ψ m(Rp×Tq; Rp×Rq), which we know to be equal, with the correspondence between the Euclidean and toroidal symbols given in Theorem 4.5.3 . The same will apply for symbols, with the quantization clear from the context. Lemma 13.3.4 (Extension of symbols). Let χ ∈ C∞(Rp+q) be homogeneous of order 0 in Rp+q \B(0 , 1) , i.e. 2 χ(ξ) = χ(ξ/ ,ξ,) when ,ξ, ≥ 1. Furthermore, assume that p q χ satisﬁes χ|(U×Rq )\B(0 ,1) ≡ 0 and χ|Rp×V ≡ 1, where U ⊂ R and V ⊂ R are m p neighbourhoods of zeros. Let σB ∈ S (R ) and denote

σA(x, ξ ) := χ(ξ) σB(P x, P ξ ),

p+q p where P : R → R is deﬁned by P (x1,...,x p+q) = ( x1,...,x p). Then σA ∈ m p+q m p q S (R ) and σA|(Rp×Rq )×(Rp×Zq ) ∈ S (R × T ). Proof. We shall ﬁrst prove that

γ −r r−| γ| |(∂ξ χ)( ξ)| ≤ Cγr /P ξ 0 /ξ0 (13.4)

p+q for every r ∈ R and for every γ ∈ N0 . It is trivial that ( x, ξ ) "→ χ(ξ) belongs to S0(Rp+q). If r ≥ 0 then obviously ( 13.4 ) is true. Since we are not interested in the behaviour of the symbols when ,ξ, is small, we assume that ,ξ, > 1 from here on. There exists r0 ∈ (0 , 1) such that χ(ξ) = 0 when ,P ξ , < r 0. Let r < 0 and ξ ∈ supp( χ). Then ,P ξ , ≥ r0,ξ,, and thus γ −| γ| |(∂ξ χ)( ξ)| ≤ Cγ /ξ0 −r r −| γ| = Cγ /P ξ 0 /P ξ 0 /ξ0 −r r −| γ| ≤ Cγ /P ξ 0 /r0ξ0 /ξ0 r −r r−| γ| ≤ Cγ r0 /P ξ 0 /ξ0 . Hence the inequality ( 13.4 ) is proven. Now

α β |∂ξ ∂x σA(x, ξ )| α ≤ |(∂γ χ)( ξ)| | (∂α−γ ∂βσ )( P x, P ξ )| γ ξ ξ x B γ ≤α # $ α ≤ C /P ξ 0−rγ /ξ0rγ −| γ| C /P ξ 0m−| α−γ| γ γr γ B(α−γ)βm γ ≤α # $ m−| α| ≤ CBαβmχ /ξ0 ,

2Here B(0 , 1) stands for the unit ball in Rp+q centred at the origin and of radius 1. 696 Chapter 13. Pseudo-diﬀerential operators on homogeneous spaces

m p+q if we choose rγ = m − | α − γ|. Thereby σA ∈ S (R ). Clearly we can regard p q p q this symbol as a function σA : ( R × T ) × (R × R ) → C and study its restriction m p q σA|(Rp×Tq )×(Rp×Zq ). We claim that this restriction belongs to S (R ×T ). Indeed, the Taylor expansion of a function σ ∈ C∞(Rq) yields

γ △γ σ(ξ) = (−1) |γ−δ| σ(ξ + δ) ξ δ δ ≤γ # $ γ = (−1) |γ−δ| δ δ ≤γ # $ 1 1 × δρ (∂ρσ)( ξ) + δρ (∂ρσ)( ξ + θ δ)  ρ! ξ ρ! ξ δ  |ρ |<|γ| |ρ |=|γ|  1 γ  = (∂ρσ)( ξ) (−1) |γ−δ|δρ ρ! ξ δ |ρ |<|γ| δ ≤γ # $ 1 + δρ (∂ρσ)( ξ + θ δ) ρ! ξ δ δ ≤γ |ρ |=|γ| 1 = δρ (∂ρσ)( ξ + θ δ), ρ! ξ δ δ ≤γ |ρ |=|γ| because γ (−1) |γ−δ|δρ = △γ ξρ| = 0 δ ξ ξ=0 δ ≤γ # $ whenever |ρ| < |γ|. Therefore 1 |△ γ σ(ξ)| ≤ δρ |(∂ρσ)( ξ + θ δ)| ξ ρ! ξ δ δ ≤γ |ρ |=|γ| ρ ≤ cγ sup |(∂ξ σ)( ξ + η)|, η∈Sγ ,|ρ|=|γ|

q ′ ′′ ′ where Sγ is the rectangle j=1 [0 , γ j]. Let α = ( P α, 0,..., 0) and let α = α − α ; then ) α′ α′′ β α′+ρ β |∂ξ △ξ ∂x σA(x, ξ )| ≤ Cα sup |∂ξ ∂x σA(x, ξ + η)| ′′ η∈Sα′′ ,|ρ|=|α | m−| α| ≤ Cα CAαβm sup /ξ + η0 η∈Sα |m−| α|| |m−| α|| m−| α| ≤ Cα CAαβm 2 sup /η0 /ξ0 η∈Sα |m−| α|| |m−| α|| m−| α| ≤ Cα CAαβm 2 /α0 /ξ0 ′ m−| α| = CAαβm /ξ0 ; 13.3. Analysis on K\G, K a torus 697 notice the application of the Peetre inequality (Proposition 3.3.31 ):

/ξ + η0s ≤ 2|s| /ξ0s /η0|s|.

m p q Hence σA|(Rp×Tq )×(Rp×Zq ) ∈ S (R × T ). Now we are ready to give the converse to Theorem 13.3.3 : Theorem 13.3.5 (Lifting of operators). Let G be a compact Lie group with a torus m subgroup K. Let B ∈ Ψ (K\G). Then there exists an operator A = AK\G ∈ m Ψ (G) such that A|C∞(K\G) = B. q Proof. Let K =∼ T , dim( G) = p + q, and let {(Uj, ψ j) | 1 ≤ j ≤ N} be the same partition of unity as in the proof of Theorem 13.3.3 . Let Bij = Mψi BM ψj ∈ m m p p Ψ (K\G). With the localised notation we consider Bij ∈ Ψ (R ; R ), so that p p it has the symbol σBij : R × R → C, and the mapping ( x, ξ ) "→ σBij (x, ξ ) is p zero when x ∈ R \ (Ui ∪ Uj). By Lemma 13.3.4 there exists a pseudo-diﬀerential m p q p q p q p q operator Aij ∈ Ψ (R × T ; R × Z ) such that σAij : ( R × T ) × (R × Z ) → C satisﬁes

σAij (x; P ξ, 0,..., 0) = σBij (P x ; P ξ ), p+q where P y = ( y1,...,y p) ( y ∈ R ). Because Aij are independent of the K- m m variables, we have A = AK\G = i,j Aij ∈ Ψ (G) and A|C∞(K\G) ∈ Ψ (K\G). ∞ ∞ Let f = fk ∈ C (K\G) ⊂ C (G), fk = fψ k. Then we have k * *(Af )( x) =

= (Aij fk)( x) i,j,k i2 πx ·ξ = σAij (x, ξ ) fk(ξ) e dξ1 · · · dξp Rp Z i,j,k " ξp+1 ,...,ξ p+q ∈ ! i2 π(P x )·(P ξ ) = σAij (x; P ξ, 0,..., 0) fk(P ξ, 0,..., 0) e dξ1 · · · dξp Rp i,j,k " ! i2 π(P x )·(P ξ ) = σBij (P x ; P ξ ) fk(P ξ, 0,..., 0) e dξ1 · · · dξp Rp i,j,k " ! = (Bij fk)( P x ) i,j,k = ( Bf )( Kx ), completing the proof. Remark 13.3.6 . Theorem 13.3.5 provides just one way of lifting operators in Ψm(K\G) to operators in Ψ m(G), unfortunately destroying ellipticity: this is due to the apparent non-ellipticity of the symbol χ in Lemma 13.3.4 . We now discuss this problem and a possibility of other liftings. 698 Chapter 13. Pseudo-diﬀerential operators on homogeneous spaces

Remark 13.3.7 ( Lifting the identity) . Let us lift the identity operator I ∈ Ψ0(Rp) using the process suggested by Lemma 13.3.4 . Of course, it would be desirable if 0 p 0 p+q I ∈ Ψ (R ) could be extended to the identity in Ψ (R ), but now σI (x, ξ ) ≡ 1, and thereby its lifting A ∈ Ψ0(Rp+q) has the non-elliptic homogeneous symbol 0 p+q σA = χ ∈ S (R ). m p Given an elliptic symbol σB ∈ S (R ) one can occasionally modify the construction in Lemma 13.3.4 to get an extended elliptic symbol in Sm(Rp+q). m p+q Sometimes the following trick helps: Let σA1 ∈ S (R ) be an extension of σB1 as in Lemma 13.3.4 ,

σA1 (x, ξ ) = χ1(ξ) σB1 (x1,...,x p; ξ1,...,ξ p),

0 p+q where χ1 ∈ S (R ) is a homogeneous symbol such that χ1|(U×Rq )\B(0 ,1) ≡ 0, p q χ1|Rp×V ≡ 1, where U ⊂ R and V ⊂ R are neighbourhoods of zeros. Take any m q elliptic symbol σB2 ∈ S (R ), and modify Lemma 13.3.4 to construct an extension m p+q σA2 ∈ S (R ) such that

σA2 (x, ξ ) = χ2(ξ) σB2 (xp+1 ,...,x p+q; ξp+1 ,...,ξ p+q)

0 p+q for a homogeneous symbol χ2 ∈ S (R ) satisfying χ2|(U×Rq )\B(0 ,1) ≡ 1 and m p+q χ2|(Rp×V )\B(0 ,1) ≡ 0. Then σA1 +σA2 ∈ S (R ) is an extension for σB1 (modulo 0 p inﬁnitely smoothing operators). For instance, if B1 = I ∈ Ψ (R ), let B2 = I ∈ 0 q 0 p+q Ψ (R ) and χ2(ξ) = 1 −χ1(ξ) (for |ξ| > 1), then A1 +A2 = I ∈ Ψ (R ) (modulo inﬁnitely smoothing operators). Remark 13.3.8 ( No elliptic liftings) . It may happen that any lifting process for an m p m p+q elliptic symbol σB ∈ S (R ) yields a non-elliptic symbol in S (R ). Consider, for instance, a case where B ∈ Ψm(R2) is an elliptic convolution operator and 2 ξ "→ f(ξ) ≡ σB(x, ξ ) is homogeneous outside the unit ball B(0 , 1) ⊂ R . If the 1 restricted mapping f|S1 : S → C \ { 0} is not homotopic to a constant mapping m 3 (i.e. f|S1 has a non-zero winding number) then no lifting σA ∈ S (R ) of σB can be elliptic. Multiplications on K\G have already been lifted to multiplications on G via x "→ Kx , and A = AK\G for any right-convolution operator (multiplier) ∞ A ∈ L (C (G)) (in fact, then σA(x, ξ ) = σA(ξ) for every x ∈ G). Sometimes on K\G we have operators that resemble convolution operators. Suppose we are given a right-convolution operator A ∈ Ψm(SU(2)). Then the restriction B = m 2 A|C∞(S2) ∈ Ψ (S ) is of the form

∞ l l m (Bf )( φ, θ ) = a(l)mn f(l)n Yl (φ, θ ), (13.5) + , l=0 m =−l n =−l ! where the coeﬃcients a(l)mn ∈ C can be calculated from the data

m {BY l | l ∈ N0, m ∈ {− l, −l + 1 ,...,l − 1, l }} . 13.4. Lifting of operators 699

It is even true that the original operator A can be retrieved from the coeﬃcients ∞ 2 a(l)mn . In fact, any operator B ∈ L (C (S )) of the form ( 13.5 ) can be lifted to a unique right-convolution operator belonging to L(C∞(SU(2))). An interesting special case is (Bf )( x) = κ(x · y) f(y) d y, S2 " where κ ∈ D ′(S2), ( x, y ) "→ x · y is the scalar product of R3, and the integration is with respect to the angular part of the Lebesgue measure of R3. Then

∞ l m (Bf )( φ, θ ) = cl κ(l)0 f(l)m Yl (φ, θ ) l=0 m =−l ! ! for some normalising constants cl depending only on l ∈ N0.

13.4 Lifting of operators

We now describe the lifting of operators from K\G to G for general closed subgroup K, which can be done similar to the proof of Theorem 13.3.5 :

Theorem 13.4.1 (Lifting of operators). Let G be a compact Lie group with a closed m subgroup K. Let B ∈ Ψ (K\G). Then there exists an operator A = AK\G ∈ m Ψ (G) such that A|C∞(K\G) = B. The rest of this section is devoted to the proof of this theorem. Since we proved Theorem 13.3.5 and Lemma 13.3.4 in detail, we sketch the construction here. We start with the following

Lemma 13.4.2 (Some properties of representations). Let φ0 ∈ Rep( G) be the trivial one-dimensional representation given by φ0(x) = 1 for all x ∈ G. Then for every φ ∈ Rep( G) we have

1, if φ = φ , 1( φ) = δ I = 0 φ,φ 0 dim φ 0, if φ *= φ , - 0 ! where 0 on the right hand side is the zero operator 0 ∈ L (Hφ). Moreover, for every non-trivial φ ∈ Rep( G), i.e. φ *∈ [φ0], we have G φ(x) d x = 0 . . Proof. If φ ∈ Rep( G) is such that φ *∈ [φ0], then G φ(x) d x = 0 follows from the orthogonality of φij , 1 ≤ i, j ≤ dim φ, and φ0, given in Lemma 7.5.12 . Consequently, ∗ . ∗ 1( φ) = G φ(x) dx = 0 ∈ L (Hφ) if φ *∈ [φ0], and 1( φ0) = G φ0(x) dx = 1 . . . !Exercise 13.4.3. Show that for every φ ∈ Rep( !G) we have δ(φ) = Idim φ, the identity operator on L(Hφ). ! 700 Chapter 13. Pseudo-diﬀerential operators on homogeneous spaces

By the argument similar to that in the proof of Theorem 13.3.5 we can use the partition of unity to reduce the question to the extension of symbols from Rp to Rp × K. Let x = ( x′, x ′′ ) ∈ Rp × K, let ξ ∈ Rp and φ ∈ Rep( K). Assume that ′ m p p σB = σ(x , ξ ) ∈ S (R ) has an extension to R ×K, i.e. that there exists a symbol ′ ′′ m p m σA = σA(x , x , ξ, φ ) ∈ S (R ) ⊗ Σ (K) such that ′ ′ σA(x , 0, ξ, φ 0) = σB(x , ξ ), (13.6) where φ0 ∈ Rep( K) is the trivial representation. Then by the argument of Theorem 13.3.5 , and using Lemma 13.4.2 , we have (Af )( x) =

= (Aij fk)( x) i,j,k ′ ′′ i2 πx ′·ξ ′′ = σAij (x , x , ξ, φ ) fk ⊗ 1( ξ, φ ) e φ(x ) d ξ Rp i,j,k " [φ ]∈K ′ i2 πx ′·ξ = σAij (x , 0, ξ, φ 0) fk(ξ) e dξ Rp i,j,k " ′ ! = (Bij fk)( x ) i,j,k = ( Bf )( Kx ).

p p Thus, we need to construct an extension σA of σB from R to R × K satisfying property ( 13.6 ). We note that it is enough to do it for symbols σB with compact support in x′. First, let us deﬁne operator C ∈ Ψ0(Rp × K) by −1 C := (1 − 2LK )(1 − L Rp×K ) , where LK is the Laplace operator on K and LRp×K = LRp + LK is the Laplace p operator on R × K. For each φ ∈ Rep( K), φ : K → U (Hφ), let λφ ≥ 0 be the 2 eigenvalue of −L K corresponding to the eigenspace Hφ ⊂ L (K). For the details of this construction on compact groups we refer to Theorem 8.3.47 . Consequently, 2 we have σ1−2LK (ξ, φ ) = (1+2 λφ)Idim φ and σ1−L Rp×K (ξ, φ ) = (1+ |ξ| +λφ)Idim φ, so that ′ ′′ 2 −1 σC (x , x , ξ, φ ) = σC (ξ, φ ) = (1 + 2 λφ)(1 + |ξ| + λφ) Idim φ. ∞ R Now, let χ ∈ C0 ( ) be such that χ(t) = 1 for |t| ≤ 1/2 and χ(t) = 0 for |t| ≥ 1. Let us denote −1 A0 := χ(C) = χ (1 − 2LK )(1 − L Rp×K ) . 2πiτt 2πiτC 0 Rp By writing χ(t) = R e χ(τ) d /τ, we have A0 = R e 0χ(τ) d τ ∈ Ψ ( × K). This follows from the fact that the operator u(τ) := e 2πiτC is the solution of . . the Cauchy problem ! !

i∂τ u(τ) + 2 πCu (τ) = 0 , u (0) = I, 13.4. Lifting of operators 701 and hence can be constructed by solving the transport equations. We note that such an argument is a special case of solving the hyperbolic Cauchy problems in terms of Fourier integral operators but the situation at hand is simpler because 0 p operator C is of order zero. Thus, A0 ∈ Ψ (R ×K) follows e.g. from [ 127 , Section XII.1]. For the representation of χ(C) on the Fourier transform side one can also see that ′ ′′ 2 −1 σA0 (x , x , ξ, φ ) = σA0 (ξ, φ ) = χ (1 + 2 λφ)(1 + |ξ| + λφ) Idim φ is the diagonal symbol. We claim that/ the operator Op( σA) with 0σA := σA0 σB satisfies the required properties. Let us first show that A ∈ Ψm(G). For this we can apply Theorem 10.9.6 and use the characterisation given in Definition 10.9.5 . On one hand we have α α −| α| m ||△ φσA|| op = || (△φσA0 )σB|| op ≤ C(/φ0 + /ξ0) /ξ0

α m−| α| which implies ||△ φσA|| op ≤ C(/φ0 + /ξ0) , if we use Theorem 10.9.6 for the 0 2 operator A0 ∈ Ψ (G) as well as the fact that λφ ≤ | ξ| , and hence /φ0 = (1 + 1/2 λφ) ≤ / ξ0 on the support of σA0 (ξ, φ ). On the other hand we have

|| ∂ξj σA|| op ≤ || ∂ξj σA0 || op |σB| 2 −1 + |χ((1 + 2 λφ)(1 + |ξ| + λφ) )| | ∂ξj σB| ≤ C(/φ0 + /ξ0)−1/ξ0m + C/ξ0m−1,

m−1 and again this implies || ∂ξj σA|| op ≤ C(/φ0+/ξ0) if we use that /φ0 ≤ / ξ0 on the support of σA0 (ξ, φ ). Similarly, one can extend this to the higher order derivatives α ∂ξ . Let us show ( 10.36 ). Using the usual Euclidean formula together with ( 10.18 ), the right-convolution kernel of A can be written as

′ ′′ ′ ′′ RA(x , x , y , y ) =

2πiy′·ξ ′ ′ = e dim( φ) Tr( φ(y ) σA0 (ξ, φ )) σB(x , ξ ) d ξ. Rp " [φ ]∈K Now, if y′ *= 0 we can integrate by parts in ξ any number of times. At the same time, viewing σA0 as a ξ-dependent symbol on K implies that RA is smooth for ′′ y *= e because the same property holds for σA0 . Hence the singular support of y "→ RA(x, y ) is contained in (0 , e ). The other properties in the Deﬁnition 10.9.5 follow from the diagonality of the symbol σA0 . It remains to show ( 13.6 ). For this we note that φ0 ≡ 1 is the eigenfunction of −L K with the eigenvalue λφ0 = 0. Consequently, we have ′ 2 −1 σA0 (x , 0, ξ, φ 0) = χ (1 + |ξ| ) = 1 ′ ′′ for |ξ| ≥ 1. Finally we note that we can/ modify σA(0x , x , ξ, φ 0) arbitrarily for small ξ, in particular setting it to be zero for |ξ| ≤ 1, similarly to Lemma 13.3.4 , thus completing the proof. 702 Chapter 13. Pseudo-diﬀerential operators on homogeneous spaces Bibliography

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Notation Γ( f), 101 +x, y ,, 105 Part I x⊥y, M⊥N, 105 P , P (x), 107 , x A, x !∈ A, A ⊂ B, 12 M M M ⊥, M ⊕ M ⊥, 108 N∅ , N∈, Z, Q, R, C, Z+, R+, 12 0 M ⊕ M , H , 108 A ∪ B, A ∩ B, A = B, 12 1 2 j∈J j S , Tr( A), 113 P(X), 12 1 S , +A, B , , 114 Ac, 13 2 S2 *A* , 114 f| , 15 HS A a ⊗ b, X ⊗ · · · ⊗ X , 85 ∼, 16 1 r A ⊗ B, 86 min, max, inf, sup, 18 X ⊗π Y , X⊗πY , 92 lim inf, lim sup, 18 ! X ⊗ε Y , X⊗εY , 93 Xj, 20 j∈J a , 117! card( A), 22 "j∈J j m∗, 118 |A|, 22 M(ψ), 120 d(x, y ), 29 Σ( τ), 121 B (x), B (x, r ), 29 r d µ-a.e., 139 d , d , d 29 p 2 ∞ f ∼ g, 139 C([ a, b ]), B([ a, b ]), 30 µ f +, f −, 143 d(A, B ), 30 ∞ # f dµ, 145 (xk)k=1 , 31 p ∞ d L (µ), L (µ), *f*Lp(µ), 154 lim xk = p, xk → p, xk −−−−→ p 31 + − k→∞ k→∞ ν , ν , |ν|, 161 τ, τ ∗, 33 µ⊥λ, 163 τd, τ(d), 34 1A, 193 τA, 35 [A, B ], 194 τ1 ⊗ τ2, 36 σA(x), 194 int d(A), ext d(A), ∂d(A), 37 A =∼ B, Hom( A, B), 197 Vτ (x), V(x), 39 Spec( A), 210 NEF IS (X), 75 ρ(x), 207 K, 81 R(λ), 206 Kn, 82 Gel, Gel −1, 218 V X , 82 H(Ω), 219 span( S), 82 L(V, W ), L(V ), 83 Part II Ker( A), Im( A), 84 σ(A), 84 Spaces, sets + x )→ * x*X , *x*, 94 Z , N0, Z, R, C, 304 C(K), 95 U| W , 336 BX (x, r ), B(x, r ), 96 τX , ( X, τ X ), 336 *A*op , 96 (uj)j∈Z+ , 336 L(X, Y ), L(X), 96 L(X, Y ), L(X, Y ), 336 V ′, L(V, K), 96 supp, 244 , 336 LC (X, Y ), LC (X), 97 Ker, Im, 389 Bibliography 713

n n R /Z , 306 δj,k , 329 ∞ Cm, C (Tn), 306 k (k) (−k) 1 ∂x , ∂x , ∂x , 333 S1 , 305 σ, A $→ σA, 341 n n n T = R /Z , 305 Op, σ $→ Op( σ), 341 2 2 n L , L (T ), 308 a $→ a, 349 , 352 Tn TrigPol( ), 311 a $→ a1, a $→ a2, 352 s n H (T ), 313 f $→ f, FRn , 226 s,t Tn Tn −1 H ( × ), 315 FRn , 230 ∞ Tn Tn −1 C ( × ), 315 FTn , FTn , 307 ∞ n n C (T × Z ), 344 exp, 375 ∞ m n n m n n n ∗ (∗B) (∗H) S (T × Z ), Sρ,δ (T × Z ), Sρ,δ (T × a , a , a , 377 −∞ Zn), S (Tn), 344 [·, ·], 378 m n n S (R × R ), 266 Lj, Rk, 431 m n n Sρ,δ (R × R ), 267 [·, ·]θ, 392 m n n m n n Op( S (T × Z )), Op( Sρ,δ (T × Z )), σA, 268 α 344 Dx , 333 ∞ n −∞ n (α) Op( S (T )), Op( S (T )), 344 (−Dy) , 401 m n m n A (R ), Aρ,δ (R ), 282 m n m n −∞ n A (T ), Aρ,δ (T ), A (T ), 347 Other notations Op( Am(Tn)), Op( Am (Tn)), 347 1 1 ρ,δ p.v. , ± , 242 ∞ n −∞ n x x i0 Op( A (T )), Op( A (T )), 347 A|W , 336 Rn S( ), 228 A(x)|x=x0 , 336 τ Ker s, 389 uj −→ u, 336 sing supp, 390 Ind, 389 s s H (x), H (U), 390 dim, codim, 389 sing supp t, 390 z, 308 !ξ", 225 , 306 ξ(j), 319 Operators, etc. k I, 336 S(j), , 327 k j! cl X (·), cl (·), U )→ U, 336 K, 346 , 357 * · * X , * · * L(X,Y ), 336 m m,ρ,δ u )→ u, 308 ∼, ∼, ∼ , 348 ! [·], 348 τx, R, 248 ∞ 2 0 σ ∼ j=0 σj, 359 (·, ·)L (Tn) = ( ·, ·)H (Tn), 308 , 315 ∞ Op (σ") ∼ Op (σ ), 359 · Hs(Tn), 313 j=0 j ϕs, 314 pj → ∞, "393 α!, 227 (·, ·)Hs(Tn), 314 !· , ·" , 314 α ≤ β, 227 A∗, A(∗B), A(∗H), 315 |α|, 227 α · s,t , 315 D , 228 △, △ξ, 316 L, 229 △, △ξ, 316 · ℓp , 325 Part III 714 Bibliography

K, 442 δmn , 548 Aut( V ), 443 LG, L, 548 Aﬀ( V ), 444 D′(G), 548 0 −1 n −n xA , Ax , AB , A , A , A , A , 444 !· , ·" G, 549 H < G , H ⊳ G, 444 Hs(G), 549 Z(G), 445 Hξ, 550 GL (n, R), O(n), SO (n), 445 dim( ξ), 550 GL (n, C), U(n), SU (n), 445 λξ, 551 G/H , 446 M(G), 551 H\G, 446 L2(G), 551 G , 450 q (·, ·) 2 , 551 L (G) U(H), 451 !ξ", λ , 553 π , π , 452 [ξ] L R S(G), p , 553 φ ∼ ψ, 454 k S′(G) 557 τG/H , 459 !· , ·" , 557 HOM( G1, G 2), 460 G 2 − 1 p p p( p 2 ) j∈J φ|Hj , 462 L (G), ℓ G, dim , 561 Haar( f), 471 p $ % # L (G), 564 P , 474 k G/H K (x, y ), L (x, y ), R (x, y ), 565 Haar , 475 A A A G/H l(f), r(f), 565 , 594 G, 480 ∂α, 548 , 574 TrigPol( G), 486 σA(x, ξ ), 566 2 L (G), 489 fφ, Aφ, 570 Res G ψ, 495 H uL, uR, 570 C (G, H), 497 φ !A, B "HS , || A|| HS , || A|| op , 573 G α Ind φ H, 497 △ξ , 578 exp( X), 504 △q, 579 m Lie K(A), 511 Ak (M), 580 m m Lie (G), g, 511 Σ (G), Σ k (G), 590 gl (n, K), o(n), so (n), u(n), su (n), 512 πL, πR, 594 SL( n, K), sl (n, K), 513 RA(x), LA(x), 597 Ad( A)X, 518 la, la(x), rA, rA(x), 597 ad( X)Y , 518 , 521 D(G), 605 U(g), 519 SU(2), 613 H, 617 Sp( n), 620 Part IV Sp( n, C), 621 w , w , w , 621 Re G, 544 1 2 3 Y , Y , Y , 622 H , ξ : G → U (H ), 545 1 2 3 ξ ξ D , D , D , 624 f(ξ), 545 1 2 3 ∂+, ∂ −, ∂ 0, 625 φu, 545 Vl, Tl, 627 DY f, 546 l l l t , tmn , Pmn , 632 f(ξ)mn , 547 Bibliography 715

t−− , t−+, t+−, t++ , 636 f(l)mn , 648 σ (x, l ), σ (x, l ) , 648 A A mn σ∂+ , σ∂− , σ∂0 , 650 Sm(SU(2)), 675 Sm(S3), 680 ′ DL1 (M), 688 pE→B, 688 K\G, 689 Index

∗-algebra, 215 involution, 215 Ψ( M), 431 isomorphism, 197 Diff( M), 427 Lie, 510 of pseudo-differential operators on quotient, 196 , 201 Tn, 387 radical of, 195 , 213 ∗-homomorphism, 215 semisimple, 195 !ξ" on a group, 553 tensor product, 198 tl on SU(2), 649 topological, 198 µ& -almost' everywhere, 139 unit, inverse, 193 µ-integrable, 145 unital, 193 ∂α on groups, 574 universal enveloping, 519 σ-algebra, 121 algebra of periodic ΨDOs, 375 , 387 algebra reformulation, 530 Abel–Dini theorem, 393 associativity diagram of, 530 action co-algebra, 531 free, 688 co-associativity diagram, 531 left, right, 449 multiplication mapping, 530 linear, 449 tensor product, 530 of a group, 449 unit mapping, 530 transitive, 449 , 689 algebraic adjoint operator, 109 basis, 82 Banach, 103 dimension, 83 on a group, 583 , 606 number, 26 adjoints (Banach, Hilbert), 315 tensor product, 86 Ado–Iwasawa theorem, 520 almost orthogonality lemma, 97 algebra, 193 amplitude ∗-algebra, involutive, 215 of adjoints, 377 , 378 Spec( A), spectrum of, 210 of periodic integral operator, 395 Banach, 202 operator, 282 , 347 C∗-algebra, 215 toroidal, 347 m n m n character of, 210 amplitudes A (R ), Aρ,δ (R ), 282 m n m n commutative, 193 amplitudes A (T ), Aρ,δ (T ), 347 derivations of, 511 Arzelà–Ascoli theorem, 59 homomorphism, 197 asymptotic equivalence, 348 Hopf, 532 asymptotic expansion, 359 Index 717

of adjoint, 285 , 377 Calderón–Zygmund covering lemma, of parametrix, on a group, 591 262 of parametrix, toroidal, 388 canonical mapping of a Lie algebra, of product, 378 519 of transpose, 286 , 376 Carathéodory condition, 127 asymptotic sums, 358 Carathéodory–Hahn extension, 126 atlas, 424 cardinality, 22 automorphism, 442 Cartan’s maximal torus theorem, 494 inner, 545 Cartesian product, 14 , 20 , 73 space Aut( V ), 443 Casimir element, 522 axiom of choice, 20 , 28 , 75 Cauchy’s inequality, 233 for Cartesian products, 21 Cauchy–Schwarz inequality, 105 , 234 chain, 18 Baire’s theorem, 97 character balls Br(x), Bd(x, r ), 29 of a representation, 492 Banach characterisation of S−∞(Tn), 354 adjoint, 103 characteristic function, 15 , 137 algebra, 202 characters, 596 duality, 314 Chebyshev’s inequality, 150 fixed point theorem, 45 choice function, 20 injective tensor product, 93 closed graph theorem, 101 projective tensor product, 93 closure operator, interior operator, 39 Banach space, 96 co-algebra, 531 dual of, 103 monoid, 533 reflexive, 104 co-induced Banach–Alaoglu theorem, 101 family, 16 in Hilbert spaces, 111 topology, 71 in topological vector spaces, 91 commutant, 200 Banach–Steinhaus theorem, 99 commutator, 194 , 378 barrel, 92 commutator characterisation basis Euclidean, 422 algebraic, 82 on a group, 580 orthonormal, 112 on closed manifolds, 429 Bernstein’s theorem, 311 toroidal, 432 bijection, 15 complemented subspace, 103 bilinear mapping, 85 complete topological vector space, 88 Borel completion, 46 σ-algebra, 121 of a topological vector space, 88 measurable function, 137 component, 460 sets, 121 composition, 15 Borel–Cantelli lemma, 124 composition formula bounded inverse theorem, 101 Euclidean, 277 for Fourier series operators, 402 , C∗-algebra, 215 404 718 Index

on a group, 581 , 582 forward, backward, 316 toroidal, 378 on SU(2), △q, △+, △−, △0, 652 continuity on SU(2), formulae for, 654 metric, 32 on a group, 578 topological, 48 Dirac uniform on a group, 464 delta, 243 , 248 continuum hypothesis, 28 delta comb, 312 , 372 generalised, 28 direct sum, 103 , 108 contraction, 45 algebraic, 453 convergence of representations, 462 almost everywhere, 140 discrete almost uniform, 140 cone, 397 in Lp(µ), 157 fundamental theorem of calculus, in measure, 140 320 in metric spaces, 31 integration, 320 (α) in topological spaces, 34 partial derivatives Dx , 333 metric uniform, 44 polynomials, 319 of a net, 79 Taylor expansion, 321 pointwise, 44 , 140 disjoint family, 121 uniform, 44 , 140 distance, 29 convex hull, 91 between sets, 30 convolution, 232 distribution function, 260 left, right, l(f), r(f), 594 distributions associativity of, 232 D′(Ω), 246 non-associativity of, 250 E′(Ω), 247 n of distributions, 248 D′(T ), periodic, 310 of linear operators, 532 on manifolds, 427 of sampling measures, 468 periodic, 313 , 314

on a group, 491 , 546 summable, DL′ 1 (M), 688 translations of, 251 dual convolution kernel, left, right, 597 algebraic, 86 convolution operators, 565 , 594 Banach, Hilbert, 314 Cotlar’s lemma, 413 of Lp(µ), 172 cover, 51 of a Banach space, 103 locally trivialising, 688 second, 104 cyclic vector, representation, 462 space, 87 unitary, 480 de Morgan’s rules, 14 duality density, 39 · , ·! G, 549 derivations of operator-valued sym- · , ·! G, 557 bols, 601 derivatives and differences, 331 Egorov’s theorem, 141 diameter, 30 ellipticity, 384 difference operators on a group, 592 Index 719 embedding, 315 of Gaussians, 230 embedding theorem, 300 of tempered distributions, 238 endomorphism, 442 on D′(G), 560 equicontinuous family, 59 , 87 on L1(G), 562 equivalence relation, 16 on Lp(G), 563 n Euler’s angles on D′(T ), 311 on S3, 618 on a group, 488 on SO(3), 611 on group G, 545 on SU(2), 615 toroidal, periodic, 307 Euler’s identity, 290 Fréchet space, 89 exponential coordinates, 512 Fredholm exponential of a matrix, 504 integral equations, 46 extreme set, 91 operator, 389 freezing principle, 294 family, 11 Frobenius reciprocity theorem, 501 family induced, co-induced, 16 , 136 Fubini theorem, 189 Fatou’s lemma, 148 Fubini–Tonelli theorem, 187 reverse, 149 function, 15 Fatou–Lebesgue theorem, 153 M-measurable, Borel measurable, fiber, 688 Lebesgue measurable, 137 fiber bundle, 688 Hölder continuous, 311 principal, 688 harmonic, 299 finite intersection property, 52 , 75 holomorphic, 219 Fourier coefficients on a group, 488 negative part of, 143 Fourier coefficients, series, 308 periodic, 306 Fourier inversion positive part of, 143 global, 595 simple, 142 Fourier inversion formula test, 90 Euclidean, 230 weakly holomorphic, 206 n functional on S′(R ), 243 n n Haar, 466 –472 on S(Z ), C∞(T ), 307 Fourier series linear, 83 on L2(Tn), 308 positive, 177 , 465 ∗ on a group, 488 positive, in C -algebra, 218 Fourier series operator, 400 , 415 functional calculus at the normal el- Fourier transform ement, 218 f(l)mn , on SU(2), 648 and rotations, 232 Gelfand Euclidean, 226 theorem, 1939, 205 inverse, on S′(G), 560 theorems, 1940, 210 , 212 p inverse, on L (G ), 563 theory, 209 matrix, 547 topology, 211 , 529 multiplication formula, 230 transform, 211 , 529 720 Index

Gelfand–Beurling spectral radius for- functional, 466 –472 mula, 207 integral, 467 Gelfand–Mazur theorem, 207 measure, 466 Gelfand–Naimark theorem, 215 Haar integral commutative, 217 on SO(3), 613 graph, 101 on SU(2), 619 group, 443 Hadamard’s principal value, 242 SU(2), 613 Hahn decomposition, 163 SO(2), 610 Hahn–Banach theorem, 98 SO(3), 610 in locally convex spaces, 90 Sp( n), 620 Hamel basis, 82 Sp( n, C), 621 Hausdorff U(1), 609 maximal principle, 21 , 75 unitary, U(H), 451 space, 55 action, 449 total boundedness theorem, 88 centre of, 445 Hausdorff–Young inequality, 241 , 310 commutative, Abelian, 443 , 503 on G and G, 563 Heaviside function, 243 compact, 463 finite, 444 Heine–Borel property, 92 general linear GL( n, R), GL( n, C), Heine–Borel theorem, 61 445 Hilbert homomorphism, 447 duality, 314 infinitesimal, 510 space, 105 isomorphism, 447 Hilbert–Schmidt, 573 , 682 Lie, 503 operators, 114 linear Lie group, 503 spectral theorem, 111 locally compact, 463 homeomorphism, 50 orthogonal O( n), 445 homomorphism, 197 , 442 permutation, 444 continuous, 460 representation of, 451 differential, 514 special linear SL (n, K), 513 space HOM( G1, G 2), 460 special orthogonal SO( n), 445 Hopf algebra, 532 special unitary SU( n), 445 “everyone with the antipode” di- symmetric, 443 agram, 533 ∗ topological, 457 and C -algebra, 536 transformation, right, left, 688 antipode, 532 unitary U( n), 445 , 451 co-multiplication and unit diagram, 533 Hölder’s inequality, 155 , 234 for compact group, 535 converse of, 175 for finite group, 534 discrete, 325 multiplication and co-multiplication for Schatten classes, 115 diagram, 533 general, 235 multiplication and co-unit dia- Haar gram, 533 Index 721

tensor product, 532 Killing form, 521 Hopf fibration, 693 Krein–Milman theorem, 91 hyperbolic equations, 418 Krull’s theorem, 195 hypoellipticity, 392 Kuratowski’s closure axioms, 39 ideal Laplace operator, 229 spanned by a set, 195 on SU(2), 626 , 641 two-sided, maximal, proper, 195 on a group, 525 , 548 index, 396 on a group, symbol of, 568 of Fredholm operator, 389 law of trichotomy, 24 index sets, 13 Lebesgue induced Lp(µ)-norm, 154 family, 16 Lp(µ)-spaces, 156 G induced representation space Ind φ H, –B.Levi monotone convergence the- 497 orem, 146 injection, 15 conjugate, 155 inner product, 105 covering lemma, 63 on V ⊗ W , 86 decomposition of measures, 170 integral, 145 differentiation theorem, 257 , 258 Haar, 467 dominated convergence theorem, Lebesgue, 145 152 , 226 Pettis, weak, 92 , 495 integral, 145 Riemann, 153 measurable function, 137 integration measure, 130 discrete, 320 measure, translation and rotation interpolation theorems, 392 invariance, 131 invariant measurelet, 119 vector fields, 512 non-measurable sets, 135 isometry, 456 outer measure, 119 p isomorphism, 197 , 442 space L (µG) on a group, 472 canonical, 314 left quotient H\G, 446 intertwining, 545 Leibnitz formula isometric, 50 on an algebra, 511 isotropy subgroup Gq, 450 Leibniz formula asymptotic, 585 Jacobi identity, 510 discrete, 317 for Gaussians, 368 Euclidean, 254 Jacobi’s identity, 368 LF-space, 90 Jordan decomposition, 161 , 162 Lie algebra, 510 kernel algebra sl (n, K), 513 of a linear operator, 84 algebra homomorphism, 510 kernel, null space, 442 , 447 algebra of a Lie group, 511 kernels la, la(x), rA, rA(x), 597 algebra, canonical mapping, 519 722 Index

algebra, semisimple, 521 Lebesgue, 130 algebras gl (n, K), o(n), so (n), u(n), Lebesgue decomposition of, 170 su (n), 512 outer, 118 group, 503 probability, 124 group, dimension of, 512 product of, 183 group, exponential coordinates, Radon–Nikodym derivative of, 164 512 sampling, 467 group, linear, 503 semifinite, 160 group, semisimple, 521 signed, 160 subalgebra, 510 variations of, positive, negative, lifting of operators, 693 total, 161 linear operator measure space, 124 bounded, 96 σ-finite, 166 compact, 97 Borel, 124 norm of, 96 complete, 124 Liouville’s theorem finite, 124 for harmonic functions, 299 measurelet, 118 for holomorphic functions, 219 measures Littlewood’s principles, 144 mutually singular, 163 locality, 272 , 390 metric, 29 logarithm of a matrix, 507 discrete, 29 Luzin’s theorem, 143 Euclidean, 29 interior, closure, boundary, 37 manifold, 67 , 425 subspace, 30 closed, 427 sup-metric, d , 30 differentiable, 68 metric space ∞ orientable, 426 complete, 42 paracompact, 431 sequentially compact, 60 mapping, 15 totally bounded, 64 continuous, uniformly continuous, metrics Lipschitz continuous, 51 equivalent, 51 measurable, 136 Lipschitz equivalent, 35 Marcinkiewicz’ interpolation theorem, Minkowski’s functional, 89 261 Minkowski’s inequality, 155 , 235 maximum, minimum, supremum, in- for integrals, 190 fimum, 18 mollifier, 256 measure, 124 monoid, 468 absolutely continuous, 165 Montel space, 92 action-invariant on G/H , 475 multi-indices, 227 Carathéodory–Hahn extension of, multiplication of distributions, 242 126 Haar, 466 Napier’s constant, 43 Hahn decomposition of, 163 neighbourhood, 32 Jordan decomposition of, 161 net, 79 Index 723

Cauchy, 88 Borel regular, 127 neutral element, inverse, 443 metric, 127 norm, 94 product of, 183 equivalent, 95 operator, 96 parallelogram law, 107 trace, 113 parametrix, 294 , 385 , 388 normal on a group, 591 divisors, 445 Parseval’s identity element, 218 on Rn, 240 subgroup, 444 on a group, 488 , 489 , 552 normal element, 529 partition of unity, 58 nuclear space, 94 path, 69 numbers, 12 Pauli matrices, 622 algebraic, 26 Peetre’s Stirling, 327 inequalities, 327 theorem, 272 open mapping, 100 periodic open mapping theorem, 100 Schwartz kernel, 342 operator Taylor expansion, 334 (α) (−Dy) , 401 periodic integral operator, 395 Dα, 228 periodicity, 306 α (α) Dx , Dx , 333 periodisation, 367 adjoint, 109 compactly supported perturba- classical, 362 tions, 374 compact, 195 of operators, 370 intertwining, 454 of symbols, 373 left-invariant, right-invariant, 566 Peter–Weyl theorem, 483 linear, 83 for Tn, 483 order of, 422 , 428 left, 483 properly supported, 286 Pettis integral, 92 , 495 self-adjoint, 109 Plancherel’s identity, 308 operator norm, 96 in Hilbert space, 112 operators on Rn, 240 Hilbert-Schmidt, 114 on a group, 488 , 489 trace class, 113 point operators ∂+, ∂ −, ∂ 0, 625 accumulation, 39 , 54 applied to tl, 643 fixed, 45 symbols of, 650 isolated, 39 in Euler’s angles, 626 Poisson summation formula, 367 order polynomial partial, 17 discrete, 319 total, linear, 18 trigonometric, TrigPol( G), 486 orthogonal projection, 107 trigonometric, on Tn, 311 outer measure, 118 Pontryagin duality, 481 724 Index power set P(X), 12 tl on SU(2), 632 preimage, 32 regular, left, right, 566 , 594 preimage, image, 15 adjoint, of a Lie algebra, 518 principal symbol, 359 adjoint, of a Lie group, 518 principle cyclic, 462 Littlewood, 144 decomposition of, 480 product dimension dim( ξ), 545 of measures, outer measures, 183 dimension of, 451 topology, 36 , 42 , 74 direct sum of, 462 projection PG/H , 474 equivalent, 454 pseudo-differential operator induced, 497 local, 422 irreducible, 452 on a manifold, 426 matrix, 481 periodic, continuity of, 349 multiplicity of, 501 toroidal, 344 of a group, 451 pseudo-differential operators regular, left, right, πL, πR, 452 , Ψm(G), 587 482 Ψm(M), 426 restricted, 452 m m G Ψ (SU(2)), Ψ (SU(2)), 542 space Ind φ H, 497 on a group, 567 space Rep( G), 544 pseudolocality, 272 , 390 strongly continuous, 461 pushforwards, φ–, 570 topologically irreducible, 462 Pythagoras’ theorem, 105 unitary, 451 unitary matrix, 451 quantization resolvent mapping, 206 on Rn, 282 restriction, 15 on a group, 567 Riemann integral, sums, 153 operator-valued, 598 Riemann–Lebesgue lemma, 227 toroidal, 343 Riesz quantum numbers, 645 , 681 almost orthogonality lemma, 97 quaternions, 617 , 679 representation theorem, 109 quotient topological representation theo- algebra, 196 , 201 rem, 177 , 179 left H\G, 446 Riesz–Thorin interpolation theorem, right G/H , 446 160 topology, 71 , 201 right quotient G/H , 446 topology on G/H , 459 right transformation group, 688 vector space, 84 Russell’s paradox, 13

Radon–Nikodym derivative, theorem, scaling, 245 164 , 166 Schatten class, 114 relation, 15 Schr¨oder–Bernstein theorem, 23 Rellich’s theorem, 301 Schr¨odinger equation, 419 representation Schur’s lemma, 290 , 455 Index 725

p Schwartz kernel, 94 , 346 , 357 , 565 , localisation, Lk(Ω) loc , 253 606 on manifolds, Hs(M), 427 periodic, 342 toroidal, Hs(Tn), 313 Schwartz kernel theorem, 94 space n n Schwartz space, 89 , 228 C∞(T × Z ), 344 n k S(R ), 228 C (M), C∞(M), 425 n S(Z ), 306 C0∞(Ω), 244 Schwartz’ impossibility result, 242 L2(Tn), 308 semigroup, 468 L2(G), 489 seminorm, 94 p L (G), 561 separating points, 57 , 77 Lp(Rn), 225 sequence, 31 Lp(Rn), interpolation, 237 Cauchy, 42 Lp(Tn), 310 p n generalised, 79 Lloc (R ), 246 sequential density of functions, 244 P ol 1(SU(2)), 676 set K-vector, 81 directed, 79 D′(G), 548 , 605 sets, 11 , 12 D′(M), 427 ψ-measurable, 120 D(G), 605 balanced, 82 D(M), 427 Borel, 121 Ψm(M), 426 bounded, 30 S′(G), 557 compact, 52 M(G), 551 convex, 82 S(G), 553 elementary, 118 Diﬀ( M), 427 extreme, 91 barreled, 92 Lebesgue non-measurable, 135 base, 688 linearly independent, 82 homogeneous, 689 open, 32 measure, 124 open, closed, 33 metric, 29 well-ordered, 19 quotient, 16 singular support, 248 , 249 , 390 simply-connected, 516 small sets property, 87 topological, 33 small subgroups, 509 total, 688 smooth mapping, 425 span, 82 smoothing operators, 272 spectral radius formula, 207 smoothing periodic ΨDOs, 354 spectrum Sobolev spaces, 300 of an algebra element, σ(x), 194 Hs(G), 549 , 556 of an algebra, Spec( A), 529 p Lk(G), 564 of an operator, σ(A), 84 p Lk(Ω), 252 Stein–Weiss interpolation, 563 p n Ls(T ), 383 Stirling numbers, 327 biperiodic, 315 recursion formulae, 328 726 Index

Stone–Weierstrass theorem, 66 on a group, 575 subalgebra periodic, 333 , 334 involutive, 66 tempered distributions n subcover, 51 S′(R ), 237 n subgroup, 444 S′(Z ), 306 trivial, 445 tensor product subnet, 79 algebra, 198 , 530 subspace algebraic, 86 compact, 53 Banach, injective, 93 invariant, 452 Banach, projective, 93 metric, 30 , 36 Hopf algebra, 532 trivial, 82 , 452 injective, 93 vector, 82 of operators, 86 subspaces of spaces, 85 orthogonal, 105 projective, 92 sum, inﬁnite sum, 117 spaces, dual of, 86 summation by parts, 319 test function, 90 summation on SU(2), 645 test functions, 306 support, 57 , 247 , 336 Tietze’s extension theorem, 73 surjection, 15 Tihonov’s theorem, 75 Sweedler’s example, 538 topological symbol algebra, 198 classical, 290 , 362 equivalence, 50 elliptic, 296 group, 457 Euclidean, σA, 268 interior, closure, boundary, 38 homogeneous, 289 property, 50 of periodic ΨDO, 364 vector space, 87 on a group, 566 zero divisor, 205 operator-valued, left, right, 597 topological approximation principal, 359 of Lebesgue measurable sets, 133 toroidal, 341 , 342 of measurable sets, 128 symbol class topological space Sm(S3), 680 compact, 52 Sm(SU(2)), 675 complete, 88 m n n Sρ,δ (R × R ), 267 completion of, 88 m n n Sρ,δ (T × R ), 345 connected, disconnected, 68 m n n Sρ,δ (T × Z ), 344 Hausdorﬀ, 55 Σm(G), 590 locally compact, 52 Σm(SU(2)), 649 locally convex, 89 m m Σ0 (SU(2)), Σ k (SU(2)), 649 paracompact, 431 path-connected, 69 Taylor expansion totally bounded, 88 biperiodic, 334 topology, 33 discrete, 321 F-induced, 73 Index 727

base of, 41 as Hopf algebra, 537 co-induced, 71 universality discrete, 77 of enveloping algebra, 520 induced, 50 of permutation groups, 449 injective tensor, 93 of unitary groups, 503 metric, 38 , 42 Urysohn’s lemma, 57 metric, canonical, 34 smooth, 259 metric, comparison, 35 metrisable, 77 vector space, 81 norm, 96 Banach, 96 on R2, 37 dimension of, 83 product, 36 , 42 , 73 Fréchet, 89 projective tensor product, π-topology, Hilbert, 105 92 inner-product, 105 quotient, 71 , 201 LF-, 90 quotient on G/H , 459 locally convex, 89 relative, 35 Montel, 92 second countable, 41 normed, 94 strong operator, 461 nuclear, 94 subbase, subbasis, 41 quotient, 84 weak, 90 , 111 topological, 87 weak ∗, 90 , 101 vectors toroidal orthogonal, orthonormal, 105 amplitude, 347 Vitali’s convergence theorem, 158 torus, 305 inflated, 369 wave front set, 397 tower, 21 weak derivative, 251 trace, trace class, trace norm, 113 weak topology, 90 , 111 weak type ( p, p ), 261 transfinite induction, 19 ∗ mathematical induction, 19 weak -topology, 90 , 101 Weierstrass theorem, 65 transpose, 285 well-ordering principle, 19 , 27 transposed operator Whitney’s embedding theorem, 426 on a group, 584 , 606 triangle inequality, 29 , 106 Young’s inequality, 189 trigonometric polynomials discrete, 326 TrigPol( G), 486 for convolutions, 236 Tn TrigPol( ), 311 for convolutions, general, 236 on Rn, 234 uncertainty principle, 245 uniform boundedness principle, 99 Zermelo–Fraenkel axioms, 28 unit, 193 Zorn’s lemma, 22 unital algebra, 193 unitary dual G, 480 , 544 universal enveloping algebra, 519