Global Operator Calculus on Spin Groups

In this paper, we use the representation theory of the group Spin(m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Spin}(m)$$\end{document} to develop aspects of the global symbolic calculus of pseudo-differential operators on Spin(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Spin}(3)$$\end{document} and Spin(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Spin}(4)$$\end{document} in the sense of Ruzhansky–Turunen–Wirth. A detailed study of Spin(3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Spin}(3)$$\end{document} and Spin(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Spin}(4)$$\end{document}-representations is made including recurrence relations and natural differential operators acting on matrix coefficients. We establish the calculus of left-invariant differential operators and of difference operators on the group Spin(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Spin}(4)$$\end{document} and apply this to give criteria for the subellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of first and second order globally hypoelliptic differential operators are given, including some that are locally neither invertible nor hypoelliptic. The paper presents a particular case study for higher dimensional spin groups.


Introduction
The classic principal calculus of Hörmander over manifolds, which is based on the notion of the symbol via localizations, has several limitations such as in the characterisation of global and local hypoellipticity. This is due to the fact that one uses local Euclidean Fourier analysis on manifolds which makes only the principal part of a symbol to be coordinate-invariant. But in the case of Lie groups one has another approach based on harmonic analysis over these groups which allows for a global approach. For instance, Zelditch [29] used the non-Euclidean harmonic analysis of Helgason to replace the local Euclidean Fourier analysis to obtain a pseudo-differential calculus on hyperbolic surfaces in the plane. Hereby, Helgason's non-Euclidean harmonic analysis is based on a Fourier transform given by eigenfunctions of the invariant Laplacian over a suitable homogeneous space which has its own drawbacks. For a detailed description on the historic development of calculi of pseudo-differential operators we refer to [21,26].
During the last decade, a new and full symbol calculus over compact groups was developed by Ruzhansky, Turunen, and Wirth which represents a non-commutative extension of the classical Kohn-Nirenberg quantization. This calculus has several advantages over the classic principal calculus of Hörmander. Given a Lie group G one makes full use of its representation theory and the corresponding harmonic analysis to create a global Fourier transform which allows the study of matrix-valued symbols defined on G × G and their characterisation using results from harmonic analysis on phase space. This full symbol calculus has been extended to the case of type-1 groups [18] and, recently, to a subelliptic pseudo-differential calculus on compact Lie group [2] as well as to the case of nilpotent groups [9,20].
As with all these abstract approaches there appears always the question of its realization in concrete cases. The full symbol calculus over compact groups has been explicitly worked out in the case of the n-dimensional torus and the case of SU (2). But it also raises the question of how this calculus would look like in one of the most important cases of compact groups, the case of Spin(m). The classic approach to a Fourier symbol calculus over Spin(m), i.e., the case of spinor-valued functions, consists in constructing Gelfand pairs and employing the spherical Dirac or Laplace operator, see the classic work by Dieudonné [7]. Naturally, these approaches restrict also the class of pseudo-differential operator symbols which can be considered and to overcome this problem a full symbol calculus in the sense of Ruzhansky, Turunen, and Wirth becomes even more important. In this paper, we are going to establish this calculus for the group Spin (4). This choice is based on two reasons. First of all the structure of group representations of Spin(m) makes it difficult to obtain explicit formulae in the general case so that the case of Spin(4) will provide further insight into the general case. Secondly, the case of Spin(4) is important by itself in applications. For instance, Spin(4) is the translation group on the three-sphere which appears in the study of diffraction tomography and the construction of wavelet and Gabor frames over the three-sphere [3]. While classically diffraction tomography is used to establish the so-called orientation density function of a fixed specimen, recent advances have also created new interests in time-dependent versions of diffraction tomography in structural analysis. Furthermore, discussion of perturbations of wavelet and Gabor frames over the three-sphere will require the study of pseudo-differential operators over Spin (4) in the same way as in the classic case. Additionally, Spin(4) also appears in quantum gravity (see, e.g., in Spin(4) BF models [19]). Among other things, these investigations require tools for the study of symbols of pseudo-differential operators and global hypoellipticity of differential operators defined over function spaces on Spin (4).
After recalling some necessary facts about the abstract case of the full symbol calculus by Ruzhansky, Turunen, and Wirth and Spin(3)-representations, we are going to study Spin(4)-representations and their connections with harmonic and spinorvalued monogenic polynomials. Hereby, we establish the necessary tools for a full symbol calculus like matrix-coefficients, recurrence relations and difference operators acting on them. This will allow us to work out details of the Fourier transform on Spin (4), which in turn gives rise to the full symbol calculus. Furthermore, we are going to obtain conditions on ellipticity and hypoellipticity.
In the end we provide some examples of differential operators to show the symbol calculus in action.

Preliminaries on the Harmonic Analysis for Compact Groups
We start with some basic notations and results about harmonic analysis of a compact Lie group. Let G be a compact Lie group of real dimension n with unit element e. A finite-dimensional unitary representation ξ of G is a continuous group homomorphism ξ : G → U (d ξ ) of G into the group of unitary matrices of a certain dimension d ξ . The representation ξ is irreducible if ξ(x)A = Aξ(x) for all x ∈ G and some A ∈ C d ξ ×d ξ implies A = cI is a multiple of the identity. This is equivalent to the statement that Two representations ξ 1 and ξ 2 are equivalent if there exists an invertible matrix B with ξ 1 (x)B = Bξ 2 (x) for all x ∈ G. Let G denote the set of all equivalence classes of irreducible representations.
We further equip G by its normalized Haar measure. The group structure gives rise to left and right translations, L x : φ → φ(x −1 ·) and R x : φ → φ(· x) of functions on the group. These left-and right-translations are unitary on the Hilbert space L 2 (G) of square integrable functions and therefore the translations give rise to unitary representations x → L x and x → R x of the group G on the Hilbert space L 2 (G). These representations split into irreducibles and give rise to the Peter-Weyl Theorem in the following form, see [10,21], or [28].

Theorem 2.1 (Peter-Weyl) The space L 2 (G) decomposes as the orthogonal direct sum of minimal bi-invariant subspaces parameterised by G, that is
The Fourier transform of f ∈ L 2 (G) is a matrix valued function on G defined by with inverse given by Furthermore, the following Parseval identity holds

4) where A 2 HS = tr(A * A) is the Frobenius or Hilbert-Schmidt norm of a matrix A.
On the group G the convolution of two integrable functions φ, ψ ∈ L 1 (G) is defined by The following convolution theorem on G is well-known. Note the change in the order of the factors.
The Laplace-Beltrami operator L G ∈ Diff 2 (G) on the group G is bi-invariant, i.e., it commutes with all L x and R x . Therefore, all of its eigenspaces are bi-invariant subspaces of L 2 (G). Since H ξ are minimal bi-invariant subspaces, each of them has to be eigenspace of L G and we denote the corresponding eigenvalue by −λ 2 ξ . Hence, we obtain the following decomposition The notion of Fourier series extends naturally to C ∞ (G) and the space of distributions D (G) with convergence in the respective topologies. Now, any operator A on G mapping C ∞ (G) to D (G) gives rise to a matrix-valued full symbol σ A (x, ξ) ∈ C d ξ ×d ξ , x ∈ G defined by σ A (x, ξ) := ξ(x) * (Aξ)(x) (2.7) which can be understood either pointwise or distributionally, as the product of a smooth matrix-valued function ξ * (x) with the matrix-valued distribution Aξ, i.e. σ A (·, ξ) = ξ * Aξ as a distribution in the first variable, for all [ξ ] ∈ G. Then it can be shown that holds as D -convergent series. If it happens that the operator A maps C ∞ (G) to itself, then (2.8) holds in the strong topology of C ∞ (G). For A and σ A related by (2.8) we write A = Op(σ A ). For a comprehensive treatment of this quantization we refer to [21] and [24]. We denote the right-convolution kernel of A by R A , so that The symbol σ A and the right-convolution kernel R A are related by The class m (G) of Hörmander's pseudo-differential operators on G was fully characterised in [21] and [22] using commutator properties with the vector fields in Sobolev spaces, and in [23] by the behaviour of their matrix symbols. Before we give a characterisation of the class m (G) we fix some notations.
We say that Q ξ is a difference operator of order k if it is given by Q ξ f (ξ ) = ϕ Q f (ξ ) for a function ϕ = ϕ Q ∈ C ∞ (G) vanishing of order k at the identity e ∈ G, i.e., (P x ϕ Q )(e) = 0 for all left-invariant differential operators P x ∈ Diff k−1 (G) of order k − 1. We denote the set of all difference operators of order k as diff k ( G). In the sequel, for a function ϕ ∈ C ∞ (G) it will be also convenient to denote the associated difference operator, acting on Fourier coefficients, by ϕ f (ξ ) := ϕ f (ξ ). Definition 2.3 (cf. [23]) Let G be a compact Lie group of dimension n with unit element e. A collection of k ≥ n first order difference operators 1 , . . . , k ∈ diff 1 ( G) is called admissible, if the corresponding functions ϕ 1 , . . . , ϕ k ∈ C ∞ (G) satisfy dϕ j = 0, j = 1, . . . , k, and rank(dϕ 1 (e), . . . , dϕ k (e)) = n. It follows, in particular, that e is an isolated zero of the family {ϕ j } k j=1 . An admissible collection is called The previous definition was adapted to a collection of k ≥ n first order difference operators since this happens in our case. For a given admissible selection of difference operators on a compact Lie Group G we use the multi-index notation α ξ := α 1 1 · · · α k k and ϕ α (x) := ϕ 1 (x) α 1 · · · ϕ k (x) α k (2.10) Furthermore, there exist corresponding differential operators ∂ (α) x ∈ Diff |α| (G) such that the Taylor's formula holds true for any smooth function f ∈ C ∞ (G) and with h(x) the geodesic distance from x to the identity element e (see [21,Sec. 10.6]). Additionally, we introduce operators ∂ α x as follows. Let ∂ x j ∈ Diff 1 (G), 1 ≤ j ≤ n = dim G, be a collection of left-invariant first order differential operators corresponding to some linearly independent family of the left-invariant vector fields on G. We denote ∂ α x = ∂ α 1 x 1 · · · ∂ α n x n . The following theorem characterises Hörmander's class of pseudo-differential operators m (G) by the behaviour of their matrix symbols.
For every left-invariant differential operator P x ∈ Diff k (G) of order k and every difference operator Q ξ ∈ diff l ( G) of order l the symbol estimate for all multi-indices α, β. Moreover, sing supp R A (x, ·) ⊆ {e}. (D) For a strongly admissible selection 1 for all multi-indices α, β.
The set of symbols σ A satisfying either of equivalent conditions (B)-(D) is also denoted by S m (G), such that the operator quantization gives an isomorphism Op : The composition of pseudodifferential operators gives again a pseudo-differential operator with a symbol, which can be expressed as an asymptotic expansion.
Note that the proof in [21] omits the crucial remainder estimates for the underlying Taylor expansion. For a complete proof including the remainder estimates see e.g. [4, Sections 9.5 and 9.7].

Notation
First we introduce some basic notation about Clifford algebras. We refer to [5] for a more detailed overview. Let (e 1 , . . . , e m ) be the standard basis of the Euclidean space R m and R 0,m be the real Clifford algebra generated by the vectors e 1 , . . . , e m such that e 2 j = −1 for j = 1, . . . , m, and e i e j = −e j e i , for i, j = 1, . . . , m, and i = j. An element a ∈ R 0,m is of the form a = A a A e A , a A ∈ R for ordered subsets A ⊆ {1, . . . , m} and with e ∅ = e 0 = 1. The k-vector part of a is given by x j e j ∈ R 0,m . The Clifford product of two 1-vectors x and y in R m splits in a scalar part given by minus the inner product in R m and the wedge product: It holds −x · y = 1 2 (x y + y x) and x ∧ y = 1 2 (x y − y x). These can be extended to the whole Clifford algebra by setting The Dirac operator on R m is given by ∂ x = e 1 ∂ x 1 + . . . + e m ∂ x m and its null solutions are called (left) monogenic functions. Right monogenic functions can also be defined considering the multiplication of the partial derivatives by the basis elements on the right.
The complex Clifford algebra is equipped with the Clifford inner product defined by We will frequently use the Witt basis vectors For even m they generate all of C m . Later on we will use spaces of C m -valued polynomials on R m . For this, we recall the Fischer inner product defined for two such polynomials P and Q. This definition implies immediately that homogeneous polynomials of different degree are Fischer orthogonal. The multiplication by the variable x i and the derivative ∂ x i are Fischer-adjoint while the generators e i of the Clifford algebra C m are skew-adjoint (3.8)

The Spin Group and H-and L-Representations
The spin group Spin(m) is realised as the set of even products of unit vectors, that is, where R + 0,m = span R {e A | |A| even} denotes the even subalgebra of R 0,m . The spin group is a double covering of SO(m) as seen by the group action R m x → sxs ∈ R m on vectors. There are two distinguished representations of the spin group on C m -valued functions on C m defined by where x ∈ C m , s ∈ Spin(m), and f : C m → C m . The H-representation corresponds to the standard representation of SO(m) on scalar-valued functions f ∈ L 2 (S m−1 ), while the L-representation corresponds to the half-spin representations. Models for all irreducible representations arise from decomposing H and L into irreducibles. Although spin representations are an old topic (see [1]), here we follow [27] which construct representation models based on simplicial harmonic and monogenic polynomials, i.e., harmonic and monogenic polynomials of simplicial variables. This is an extension of the work in [11], where the authors consider simplicial harmonic polynomials which provide only models for irreducible representations with integer weight of the SO(m) group.
The Lie algebra spin(m) can be realised as the space spin(m) ∼ = R The exponential of h yields the maximal torus T ⊂ Spin(m) for the derived representation d R : spin(m) → Aut(V ) and with weights l = (l 1 , . . . , l M ) consisting entirely of either integer or half integer numbers. Factoring out the action of the Weyl group, we obtain the highest weights given by the ordering where all l i ∈ Z or all l i ∈ Z + 1 2 .

Explicit Models
To construct explicit models for irreducible representations of Spin(m) we follow [27] and consider k ≤ m vector variables x 1 , . . . , x k , where x i = m j=1 x i j e j and C mvalued polynomials in these k vector variables. A polynomial P(x 1 , x 1 ∧ x 2 , . . . , x 1 ∧ . . . ∧ x k ) depending on the simplicial variables x 1 ∧ . . . ∧ x i is called a simplicial polynomial. A harmonic simplicial polynomial P is a simplicial polynomial satisfying (3.16) The space of these polynomials is denoted by H(x 1 , . . . , x k ). It is invariant under the H-action. A monogenic simplicial polynomial P is characterised by the condition The space of monogenic simplicial polynomials is denoted by M(x 1 , . . . , x k ) and is invariant under the L-action. Different to the notation from [27] we will parameterise representations and representation spaces by their weights and not by degrees of homogeneity of the polynomials in the representation spaces. 1 Case 1 For the highest weight (l 1 , . . . , l M−1 , l M ), l i ∈ N 0 , we take the highest weight vector and let Spin(m) act by the H-representation on it. We recall that for s ∈ Spin(m) the H-representation on simplicial functions is given by We denote the resulting representation space by (3.21) The resulting representation space will be denoted by For half-integer weights we have to realise the representations in the spinor space of the complex Clifford algebra C m . We distinguish between even and odd m. For even m = 2M we use the pairwise commuting idempotents , to construct the primitive idempotents I + = I 1 · · · I M and I − = I 1 · · · I M−1 I M . (3.22) and define They are both Spin(m)-invariant, minimal and inequivalent.
In the odd dimensional case m = 2M + 1 there is up to equivalence only one spinor space and we use Case 3 For the highest weight (l 1 , . . . , l M−1 , l M ), l i ∈ N 0 + 1 2 , we take the highest weight vector and let Spin(m) act by the L-representation on it. We recall that for s ∈ Spin(m) the L-representation on simplicial spinor functions is given by

Spin(3) Representations
In this section we collect results on Spin(3)-representations, in particular constructing the irreducible modules from the theory explained in Sect. 3. These turn out to be also important in the construction of Spin(4)-representations later on.
The group Spin (3) is the universal cover of SO(3) and can be realised inside the even Clifford algebra R + 3 . Moreover, it is isomorphic to the special unitary group SU (2) and also isomorphic to the unit 3-sphere S 3 understood as the group of unit length quaternions, i.e.
We want to make these isomorphisms explicit for later use. An element of Spin (3) is of the form s = a 0 + a 1 e 12 + a 2 e 13 + a 3 e 23 such that |s| 2 = a 2 0 + a 2 1 + a 2 2 + a 2 3 = 1. It acts on vectors x = x 1 e 1 + x 2 e 2 + x 3 e 3 ∈ R 3 by x → sxs and this mapping is represented by the SO(3) rotation matrix ⎛ as a straightforward calculation within R 0,3 shows. Identifying e 12 = i, e 13 = j und e 23 = k with the quaternion units yields an isomorphism R + 0,3 ∼ = H and identifies the spin group with the group of unit length quaternions. We identify H ∼ = C 2 by writing q = a 0 + a 1 i + a 2 j + a 3 k = q 1 + q 2 j with q 1 = a 0 + ia 1 ∈ C and q 2 = a 2 + ia 3 ∈ C. Then in particular q = q 1 − q 2 j is the quaternion conjugation and quaternion multiplication corresponds to matrix multiplication for the associated matrices which belong to SU(2) whenever |q| = 1. This completes the isomorphisms in (4.1). We rewrite the rotation matrix (4.2) in these complex coordinates for Spin(3). This yields For later calculations we need the H-action on the polynomials z 1 = x 1 + ix 2 , z 1 = x 1 − ix 2 and x 3 . Using the rotation matrix (4.4) applied to

Representations of S 3 ⊂ H
Before explicitly giving irreducible Spin(3)-representations based on the general theory of Sect. 3, we will recall the closely related irreducible S 3 -representations from [13,21] (see also [10,23,28]). Since the quaternionic unit sphere S 3 can be viewed as a subset of C 2 through S 3 = {(z 1 , z 2 ) ∈ C 2 : |z 1 | 2 + |z 2 | 2 = 1} we write the quaternion multiplication and conjugation in S 3 as Let P be the space of all polynomials P(z, w) = c j,k z j w k in two complex variables, and let P m ⊂ P be the space of homogeneous polynomials of degree m An orthonormal basis of P m with respect to the Fischer inner product (or equivalently, with respect to the normalised L 2 inner product) consists of the set of functions The group S 3 naturally acts on P m by right translation ). Next, we follow [13] and express the linear map R (w 1 ,w 2 ) : P m → P m with respect to the orthonormal basis P m k , j = 0, . . . , m. By straightforward computations we have Hence, for each j = 0, . . . , m, we obtain and, therefore, the matrix elements of the unitary representation are given by (4.13) In particular, the first row contains the holomorphic polynomials, i.e.
while the first column contains polynomials of the form The following recurrence relations are taken from [13] and follow easily by direct computation.
Theorem 4.1 (cf. [13]) For every m ∈ N 0 and all 0 ≤ i, j ≤ m the following recurrence relations hold where every expression out of domain is interpreted as zero.
The previous recurrence relations can be written in matrix form using suitable matrices filled up with zeros and suitable weights in their entries.
and for convenience, It is possible to define shift operators acting on the matrix coefficients of a given representation. They are related to left-respectively right-invariant differential operators on S 3 .

Definition 4.4 (cf. [13]) For differentiable functions
we define the following differential operators It is easy to see that the operators ∂ ± and ∂ † ± are linear combination of rotational derivatives. The operators ∂ ± are left invariant and the operators ∂ † ± are right invariant. This means that holds true for the right translation Theorem 4.5 (cf. [13]) For every m ∈ N 0 and 0 ≤ i, j ≤ m the following relations hold where every matrix coefficient outside of the matrix is understood as zero.
The previous relations can be written in matrix form using two special matrices σ + (m/2) and σ − (m/2) defined as follows. The use of m/2 instead of m as argument is related to the parametrisation of representations by weights instead of by homogeneity used later on.

Remark 4.8
The matrices σ ± (m/2) can be obtained as (1,0) and where we apply the operators entrywise and use the point evaluation at (1, 0). Thus, they are the matrix-valued symbols of the left-invariant differental operators ∂ ± ∈ Diff 1 (S 3 ) in the sense of Sect. 2.

H-Representations: Spherical Harmonics in R 3
We follow the approach of Sect. 3. By (3.21) and (3.5) the weight vector ω l (x) for l ∈ N 0 is given by We renormalise the weight vector considering 2T 1 instead of T 1 obtaining the polynomial z l 1 . To perform the H-action on ω l (x) we consider (4.5b) and we get Using multi-index notation α = (α 1 , α 2 , α 3 ) ∈ N 3 0 , |α| = α 1 + α 2 + α 3 , and α! = α 1 !α 2 !α 3 !, and the multinomial theorem to further expand (4.27) we obtain where in the last line we put α 2 + 2α 3 = j, which yields 2α 1 + α 2 = 2l − j from |α| = l. The 2l + 1 polynomials given by (4.29) suffice to build the representation space from the weight vector. As the dimension of the representation to weight l is 2l + 1, the polynomials must also belong to the representation space and form a basis. The polynomials are orthogonal with respect to the Fischer inner product (3.7) (4.30) due to the non-matching orders of the monomials and in order to calculate their norm, we use The inner product is calculated using the Fischer duality z 1 → 2∂ z 1 , z 1 → 2∂ z 1 , and The following lemma provides a relation between trinomial and binomial coefficients.
Proof Multiplication by x j = x α 2 +2α 3 and summing over j yields 2l j=0 |α|=l based on the multinomial theorem. Comparing powers of x yields the desired result.
Using Lemma 4.9 we can write the normalising constants c l, j as and obtain 2l + 1 orthonormal spherical harmonics (4.29) on R 3 spanning the representation space H l (R 3 ) for the given weight l ∈ N. It remains to express H(q) in this basis in order to obtain the matrix coefficients and to obtain the relation to the representations constructed in Sect. 4.1.

L-Representations: Spinor-Valued Monogenics in R 3
In the 3-dimensional case there is only one basic spinor representation S ∼ = S + 4 = C + 4 I + = span C {1, e 13 }I + , where the idempotent I + is given in terms of the Witt basis elements (as introduced in Sect. 3) by I + = I 1 I 2 , with I 1 = T 1 T † 1 and I 2 = T 2 T † 2 . For z ∈ S + 4 we put z = (z + , z − ) ∈ C 2 such that z = z + I + − z − e 13 I + . Now, considering the left multiplication by q = a 1 + a 2 e 12 + a 3 e 13 + a 4 e 23 ∈ Spin(3) on the spinor representation S + 4 we obtain where we have used the multiplication rules of the Clifford algebra and the identities e 12 I + = iI + and e 23 I + = −ie 13 I + following from the basic rules e 2 I + = −ie 1 I + and e 4 I + = −ie 3 I + . The linear transformation (4.44) can again be written as an SU(2)-action given by where q 1 = a 0 + ia 1 and q 2 = a 2 + ia 3 and z = z + I + − z − e 13 I + .
Next, we perform the L-action on the weight vector ω l (x) = z l− 1 to obtain the representation space for the half-integer weight. Based on (4.5b), (4.45), and (4.28), we get where we set for convenience P Therefore, we have obtained the 2 l + 1 S + 4 -valued monogenic polynomials As this is the dimension of the representation space, we know that they are indeed linearly independent and thus must form a basis. Note that for j = 0 we obtain i.e. the weight vector, while for j = 2l we get To obtain an orthonormal basis for M l (R 3 ), we still need to compute the Fischer inner products (3.7) of the polynomials. By (4.30) follows and the normalizing constants are obtained for j = 0, . . . , 2l. We will endow the representation space M l (R 3 ) with the basis obtained by normalising P l j . To obtain the matrix coefficients associated to (4.47), we calculate the L-action on these orthogonal polynomials P l j . Using (4.45) we obtain for each j = 0, . . . , 2 l (4.50) Using (4.37) we see that the maximal exponents of q 1 and q 2 are q 2 l− j−k 1 and q j−i+k 2 . Therefore, we can write (4.50) using the renormalised basis (4.47) in the form for some still to be determined coefficients t k i, j,2l . Comparing (4.51) with (4.10), we must have the identifications m → 2 l, w 1 → q 1 , and w 2 → q 2 between the matrix coefficients (4.10) and (4.51). Therefore, the matrix coefficients related to the basis of monogenic polynomials P l j associated with the weight l ∈ N 0 + 1 2 are given by has dimension 4 and an orthonormal basis is given by (4.55) The associated matrix coefficients are

Prerequisites
We start by recalling the notation from Sect. 3. We use an orthonormal basis {e 1 , e 2 , e 3 , e 4 } of R 4 and denote by R 0,4 the real 2 4 -dimensional Clifford algebra over R 4 generated by the relations e 2 i = −1, i = 1, . . . , 4 and e i e j = −e j e i , i = j. The group Spin(4) consists of even products of unit vectors Later on we will make use of the isomorphism Spin(4) ∼ = Spin (3)×Spin (3) following from the identifications R + 0,4 ∼ = R 0,3 ∼ = H ⊕ H, which we recall next. Due to [5, Section 0.5.4] we can write a ∈ R + 0,4 in the form a = ω + a + + ω − a − with a ± ∈ R + 0,3 = span{1, e 12 , e 13 , e 23 } ∼ = H and ω ± = 1 2 (1 ± e 1234 ). These elements ω ± are mutually annihilating idempotents satisfying As ω ± commute with a + and a − we can also write a = a + ω + + a − ω − . Applying this decomposition to elements s ∈ Spin(4) we obtain together with (3) which describes a rotation in R 4 induced by (q, s) ∈ Spin(3) × Spin(3). Due to the isomorphism Spin(3) ∼ = S 3 and the identification of R 4 with C 2 we can write the action s x s inside C 2 . Putting q = (q 1 , q 2 ), s = (s 1 , s 2 ) ∈ S 3 , and x = (z 1 , z 2 ) ∈ C 2 yields after a lengthy calculation identifying C 2 with H. Next, we describe the Spin(4) action on the spinor spaces S ± 4 . A realisation of S + 4 was already described in Sect. 4.3. Considering s = sω + + qω − with q, s ∈ Spin(3) we obtain for S + 4 the half-spin action as ω + I + = 0, ω − I + = I + , and ω + , ω − commute with e 13 . Thus on S + 4 the action corresponds to the left multiplication by q ∈ Spin(3) ∼ = S 3 and hence by (4.45) also to the matrix multiplication For S − 4 , we get the half-spin action Thus, this action corresponds to the left multiplication by s ∈ Spin(3) ∼ = S 3 and as in (4.45) we see that it corresponds to the matrix-multiplication

H-Representations: Spherical Harmonics in R 4
Following Sect. 3, we construct explicitly the representations associated to the weights (l 1 , l 2 ), with l 1 , l 2 ∈ N 0 , and 0 ≤ l 2 ≤ l 1 . Considering (3.18) the (normalised) weight vector is given by 4 , w 1 = y 1 + iy 2 and w 2 = y 3 + iy 4 . Representing an element s = sω + + qω − ∈ Spin(4) by two unit quaternions q = (q 1 , q 2 ), s = (s 1 , s 2 ) ∈ S 3 and using complex coordinates for and Thus, we obtain for the H-action on the weight vector ω (l 1 ,l 2 ) (x, y) and by using shorthand notations Q j for the terms marked above (5.14) Collecting the powers of q this can be written as The terms Q 3 , Q 4 and Q 5 are independent of the spinor s = sω + + qω − . For Q 1 and Q 2 we extract the dependence on s and obtain (with l = l 1 − l 2 ) where z = z 1 T 1 − z 1 T † 1 + z 2 T 2 − z 2 T † 2 and w = w 1 T 1 − w 1 T † 1 + w 2 T 2 − w 2 T † 2 are the Hermitian variables constructed from x and y, we can write (5.15) as From (5.19) the tensor product structure of the Spin(4)-representations of weight (l 1 , l 2 ), l i ∈ N 0 , l 1 ≥ l 2 can be seen. In particular the dimension of the (l 1 , l 2 )-representation is (l 1 + l 2 + 1)(l 1 − l 2 + 1).
Proof By construction, these polynomials generate the representation space H (l 1 ,l 2 ) (R 4 ). To obtain orthogonality with respect to the Fischer inner product we consider their degrees of homogeneity in certain sets of variables. First we distinguish holomorphic variables z 1 , z 2 , w 1 , w 2 from their antiholomorphic counterparts z 1 , z 2 , w 1 , w 2 . Then due to (5.18) the polynomial S l 1 −l 2 m,i (z) is of holomorphic degree i and of antiholomorphic degree l 1 − l 2 − i, while Q l 2 j (z ∧ w) is of holomorphic degree j and antiholomorphic degree 2l 2 − j. Thus, the polynomials (5.20) are holomorphic of degree k and antiholomorphic of degree l 1 + l 2 − k.
Next we consider a mixed form of degree, considering the variables z 1 , z 2 , w 1 , w 2 the mixed degree of the polynomial S l 1 −l 2 m,i (z) is of degree m and Q l 2 j (z ∧ w) of mixed degree l 2 and thus the above polynomials (5.20) are of mixed degree m + l 2 .
It follows that for given l 1 and l 2 and different k and m the polynomials (5.20) have different homogeneities and are thus Fischer orthogonal.
Next, we construct the representations associated to the weights (l 1 , l 2 ) with l 1 , l 2 ∈ Z and l 1 ≥ −l 2 > 0. By (3.21) the corresponding (normalised) weight vector is given by (5.21) Using again q = (q 1 , q 2 ), s = (s 1 , s 2 ) ∈ S 3 to describe elements of the spin group and x = (z 1 , z 2 ), y = (w 1 , w 2 ) ∈ C 2 as complex coordinates, equation (5.6) yields again and now (5.23) Using these short-hand notations we compute the H-action on the weight vector ω (l 1 ,l 2 ) (x, y) following the lines of (5.14) as (5.25) and obtained from S l k,i (z) by complex conjugating z 2 and Proof The proof follows again by considering holomorphic, antiholomorphic and mixed degrees of the polynomials.

L-Representations: Spinor-Valued Monogenics in R 4
In this section we construct the representations associated to the weights l 1 , l 2 with l 1 , l 2 ∈ Z + 1 2 , and l 1 ≥ |l 2 |. First we consider the case l 2 ≥ 0 and the corresponding L-representation on the space of S + 4 -valued simplicial monogenic polynomials. From (5.7), (5.8), and (5.19) we obtain

Matrix Coefficients
To describe all Spin(4) representations ξ (l 1 ,l 2 ) , we consider the lattice consisting of pairs of integers and pairs of half-integers. In Sects. 5.2 and 5.3 we constructed orthogonal bases for the representation spaces of the Spin(4) representations ξ (l 1 ,l 2 ) for all (l 1 , l 2 ) ∈ Spin(4) . This choice of bases allows to make direct use of the tensor product structure of these representations. By construction, we have

Recurrence Relations
In the following two sections we discuss the general structure of matrix coefficients of the representations we have constructed. First, we prove recurrence relations for the matrix coefficients and then we define differential operators corresponding (up to factors) to shifts in the matrix coefficients of a given representation.
Using the matrices just defined we can transfer the recurrence relations given in Theorem 4.3 to the Spin(4) case. To simplify the expressions we use A ± instead of A ± (l 1 , l 2 ) and the same for the other matrices. The notation introduced will also be used later on when dealing with difference operators and the group Fourier transform.

Theorem 5.6
For all (l 1 , l 2 ) ∈ Spin(4) the three term recurrence relations hold true, where we used the notation and thus the first identity follows. The remaining identities are proved analogously. Note that for the last four formulas the sum of the weights stays the same while the difference is altered by ±1.

Example 5.7
The matrix recurrence relations yield three term relations for coefficients. We provide one example obtained by plugging in the definition of the matrices a ± and b ± from (4.2). It follows that with r = l 1 + l 2 + 1 and p = l 1 − l 2 + 1.
The next theorem provides second order recurrence relations following from Theorem 5.6.

Corollary 5.8
For all (l 1 , l 2 ) ∈ Spin(4) the following matrix recurrence relations hold where every expression out of domain is interpreted as zero.

Differential Relations for Matrix Coefficients
The matrix coefficients of the Spin(4) representations can be generated using particular first order differential operators. This is a consequence of Theorem 4.5, for which we provide the details now. We use again a pair q = (q 1 , q 2 ) ∈ S 3 and s = (s 1 , s 2 ) ∈ S 3 of unit quaternions as coordinates for s = sω + + qω − ∈ Spin(4), cf. Sect. 5.

Definition 5.9
On Spin(4) we define the following first order differential operators These operators are complex linear combinations of rotational derivatives and, moreover, are left or right invariant operators.
Let s, s 0 ∈ Spin(4) be given by s = sω + + qω − and s 0 = s 0 ω + + q 0 ω − with q, s, q 0 , s 0 ∈ S 3 . Then by the properties of the idempotents ω ± it is easy to see that the left translation on Spin(4) is given by s −1 0 s = s 0 s ω + + q 0 q ω − , while the right translation is given by ss 0 = ss 0 ω + +qq 0 ω − . Consequently, left and right translations for functions f (q, s) are given by in these coordinates. The following statement follows by direct calculation similar to (4.21).
We now determine derivatives of our representations. Using the matrix relations (4.24a)-(4.24d) we can obtain the following differential relations.
Proof We prove only the identity for ∂ † q+ , the remaining ones are obtained analogously. By using (5.33), (4.24c), and the mixed-product property for the Kronecker product, we obtain

Example 5.12
On the level of matrix coefficients this allows to switch around between different indices. We only give some formulas where r = l 1 + l 2 + 1, p = l 1 − l 2 + 1, δ + = 1, δ − = 0, and every expression out of domain is interpreted as zero.

Left Invariant Differential Operators
The differential operators ∂ q± , ∂ s± appearing in the recurrence relations for matrix coefficients allow to construct all left-invariant differential operators. To see this, we first provide a basis for the space of left-invariant vector fields on Spin(4), i.e. for the Lie algebra spin(4). Note, that the differential operators ∂ q± and ∂ s± are complex derivatives and that on a formal level ∂ * q+ = −∂ q− . To obtain elements of the Lie algebra, we form combinations of them.

Remark 6.2
The operators ∂ q+ and ∂ s+ are sometimes called creation operators, while the operators ∂ q− and ∂ s− are called annihilator operators (cf. [23,Remark 12.2.3.] where the operators are denoted by ∂ + and ∂ − , and [28, p. 140] where the operators are denoted byĤ + andĤ − ). It is also customary to define and denote this as the neutral operator. The operator D 3q can thus be written as D 3q = −i∂ q0 .

Proof
The last identity follows from the direct product decomposition of Spin (4). For the remaining identities we are concentrating on derivatives with respect to the component q, the remaining ones are obtained similarly. The first commutator relation holds by definition. For the second commutator relation we observe that by straightforward calculation Therefore, we obtain The Laplace-Beltrami operator L on Spin(4) is given by If we denote by H (l 1 ,l 2 ) the complex linear span of the matrix coefficients of ξ (l 1 ,l 2 ) , the following theorem holds true. For later use we define ξ l 1 ,l 2 = 1 − λ l 1 ,l 2 = 1 + l 1 + l 2 1 + l 2 2 2 . (6.11)

The Group Fourier Transform on Spin(4)
First, we introduce notation and recall some basic facts. Then the characterisations of function spaces on Spin(4) follows from abstract arguments, as presented in [21] and [23]. See also [16] for the relation between certain function spaces on direct products of groups. The group Fourier transform on Spin(4) is given in terms of all equivalence classes of irreducible representations For an integrable function f ∈ L 1 (Spin (4)) we define where we integrate with respect to the normalised Haar measure on the group Spin(4). Note that by uniqueness of the Haar measure and by the direct product structure Spin(4) Spin(3) × Spin(3) the Haar measure on Spin(4) is also the tensor product of the normalised Haar measures on both factors. The Fourier transform maps L 2 (Spin(4)) unitarily onto a sequence space. For this we define 2 := 2 ( Spin(4)) to be the space of all sequences σ : Spin(4) (l 1 , l 2 ) → σ (l 1 , l 2 ) ∈ C d (l 1 ,l 2 ) ×d (l 1 ,l 2 ) (6.14) such that This space is clearly a Hilbert space and we endow it with its natural inner product. Now, Peter-Weyl Theorem 2.1 implies Theorem 6.5 The Fourier transform is unitary from L 2 (Spin(4)) to 2 ( Spin(4)) with inverse f (s) = (l 1 ,l 2 )∈ Spin(4) d (l 1 ,l 2 ) tr( f (l 1 , l 2 )ξ (l 1 ,l 2 ) (s)) (6. 16) and Plancherel identity Remark 6. 6 In the particular case f (sω (Spin(3)), the Kronecker product representation of the Fourier coefficients in terms of the Spin (3) Fourier transforms imply Here we made use of tr(A ⊗ B) = tr(A) tr(B) together with (5.34). This also allows to split the Plancherel formula into a double sum based on the product formula A⊗ B HS = A HS B HS for Hilbert-Schmidt norms.
The group Fourier transform extends naturally to distributions. The space D (Spin(4)) of distributions is the topological dual space of smooth functions C ∞ (Spin(4)). As usual for a function f ∈ L 1 (Spin(4)) and ϕ ∈ C ∞ (Spin (4)) we define T f ∈ D (Spin(4)) by (6.21) and use the same notation for the dual pairing between distributions and functions. For a distribution T ∈ D (Spin(4)) its Fourier transform T is defined by T (l 1 , l 2 ) = T , ξ * (l 1 ,l 2 ) .

Function Spaces
Sobolev spaces are characterised in terms of the Laplacian. Thus, for r ∈ R the space H r (Spin(4)) has the familiar characterisation (4)) (6.22) in terms of the group Fourier transform. This allows to characterise spaces of smooth functions and of distributions. In the following, we denote by s := s( Spin(4)) the space of rapidly decaying matrix sequences ρ : Spin(4) → d C d×d (6.23) such that the dimension of ρ(l 1 , l 2 ) equals d (l 1 ,l 2 ) = (l 1 + l 2 + 1)(l 1 − l 2 + 1) and for any number N . The particular choice of matrix norm does not matter due to the polynomial growth of ξ (l 1 ,l 2 ) in the dimension d (l 1 ,l 2 ) . We also denote by s := s ( Spin (4)

Differential and Pseudo-Differential Operators on Spin(4)
In a next step we provide details on the differential and pseudo-differential calculus on the group Spin(4).

Symbolic Calculus of Invariant Operators
Now, we discuss the symbolic calculus for operators on Spin(4). We recall from Sect.
It is important to distinguish between left and right-symbols here, right-invariant operators also posses (variable coefficient) left-symbols.
In Tables 1 and 2 we present symbols for some left-invariant respectively right-invariant differential operators on Spin(4). The formulas for symbols are a consequence of Theorem 5.11 and Corollary 5.13 and given in terms of σ ± from (4.23) and with σ 0 = [σ + , σ − ], i.e. with σ 0 (m/2) i j = 1 2 (m − 2 j)δ i j for every m ∈ N 0 and 0 ≤ i, j ≤ m.
Mapping properties of left-invariant operators are characterised in terms of difference operators acting on their symbols. We recall the definition first before providing properties of the difference operators of our choice. A difference operator : s → s acting on moderate matrix sequences is defined in terms of a function ϕ ∈ C ∞ (Spin(4)) via f = ϕ f using the group Fourier transform f of distributions f ∈ D (Spin(4)). If ϕ vanishes to first order at the identity, we call a first order difference operator.
There are different ways to construct first-order difference operators. At first glance the concept of difference operators introduced in [8] seems to be a natural choice for difference operators defined over tensor products of compact Lie groups, but it has a major drawback. In general, arbitrary tensor products of representations are not irreducible and require another decomposition making the construction of an admissible collection rather difficult. In the present case this approach leads to the same difference operators which we introduce here in a more direct way.
We use particular difference operators related to the matrix entries of the representations ξ ( 1 2 ,± 1 2 ) . As pointed out in [23] this construction leads to difference operators satisfying a finite Leibniz rule. To fix notation, we collect them in Table 3.
Explicit formulas for the difference operators from Table 3 follow from the recurrence relations given in Theorem 5.6. Theorem 6.9 The difference operators of order 1 given in Table 3 are explicitly given by ±± q σ (l 1 , l 2 ) = Proof We consider the first difference operator −− q . By using the first matrix recurrence relation from Theorem 5.6 and the cyclic property of the trace, we obtain As for indices just outside Spin(4) the matrices A ± and B ± vanish, the last sums can be rewritten as sums over Spin (4) . Making use of the identities d (l 1 ,l 2 ) = l 1 + l 2 l 1 + l 2 + 1 (6.31) and subtracting by σ (l 1 , l 2 ) we obtain the expression for −− q . The remaining formulas are obtained similarly from Theorem 5.6. Corollary 6.10 When σ (l 1 , l 2 ) = ρ( l 1 +l 2 2 ) ⊗ τ ( l 1 −l 2 2 ) is of tensor product form, the difference operators of order 1 given in Table 3 are explicitly given by are the difference operators on Spin(3) given in terms of a ± and b ± from Definition 4.2. In Table 4 we show the difference operators applied to the symbols of the elementary first order differential operators ∂ qν , ∂ sμ , ν, μ ∈ {0, +, −} and the Laplacian L on the group. The table can be computed using Corollary 6.10 in combination with the first columns of [23, Table 1], the symbol of the Laplacian is corrected here.
By construction, difference operators are mutually commuting operators. Differences acting on symbols of differential operators are best calculated using the Leibniz rule for difference operators combined with Table 4. From [23] it follows that Table 4 Difference operators acting on some symbols and similarly for i j s . We show how to apply this to compute the difference operators acting on the symbol of the partial Laplacians and thus the Laplacian L.
where we made use of ++ σ + = 0 and +− σ 0 = 0 to simplify the expression. The calculation for the remaining differences ±± q σ L q and ±± s σ L s is similar and due to ±± s σ L q = ±± q σ L s = 0 the formulas for differences applied to σ L follow.

Symbolic Calculus of Pseudodifferential Operators
A continuous linear operator A mapping C ∞ (Spin(4)) to D (Spin(4)) can be characterised by its matrix-valued full left-symbol By definition holds true as D -convergent series. For A and σ A related by (6.41) we write A = Op(σ A ).
In [21,23] the Hörmander class k (G) of pseudo-differential operators of order k on a compact Lie group G was characterised in terms of these full symbols, also (ρ, δ)-classes k ρ,δ have been introduced there. We recall this and the resulting characterisations of ellipticity and hypoellipticity of operators for the particular case of Spin (4).
Adapted to the difference operators α , α ∈ N 8 0 , we find left-invariant differential operators ∂ (α) of order |α| such that Taylor's formula (2.11) and Theorem 2.5 hold true. Although these differential operators play a crucial role for the calculus, we will use a different set of differential operators for our purposes. We use the multi-index notation  (4)) to D (Spin (4)) with matrix-valued full symbol σ A (s, l 1 , (4)) if and only if for all multi-indices α and β uniformly in s ∈ Spin(4) and (l 1 , l 2 ) ∈ Spin (4) . Moreover, the rapid off-diagonal decay property of the symbol k−|α| (6.44) holds true uniformly in s ∈ Spin(4), (l 1 , l 2 ) ∈ Spin(4) and 0 ≤ i, j < d (l 1 ,l 2 ) .
As consequence, one obtains combined with Theorem 2.5 and the standard construction of parametrices within the calculus characterisations of ellipticity and local hypoellipticity. We recall these theorems before giving examples on Spin(4) later on. First we give a characterisation of the elliptic operators in k (Spin (4)) in terms of their global symbols.  (4)) is elliptic if and only if its matrix valued symbol σ A (s, l 1 , l 2 ) is invertible for all but finitely many (l 1 , l 2 ) ∈ Spin (4) and for all such (l 1 , l 2 ) satisfies uniformly in s ∈ Spin(4).
Consequently, A is locally hypoelliptic and sing supp Au = sing supp u for all u ∈ D (Spin (4)). (6.52) Note, that the parametrix B provided by this theorem satisfies the subelliptic estimates B f H r f H r +k 0 with k 0 independent of r ∈ R.

Examples
To conclude this paper we provide examples of operators on Spin(4) having interesting ellipticity and hypoellipticity properties. Using the identity D 3q = − i 2 ∂ q0 and the symbols given in Table 1, we have  (4)) ≤ C X f H r +μ−1 (Spin(4)) , (7.8) where the Fourier coefficients of g equal those of f on the zeros of σ X and in turn the global hypoellipticity modulo ker X follows. (see also Theorem 6.4). While the latter one is elliptic, the symbols of the two partial Laplacians L q and L s vanish for l 1 = −l 2 and for l 1 = l 2 , respectively. Clearly any combination L q + κL s with κ > 0 is elliptic. For κ < 0 the behaviour of this operator depends on number theoretic properties of κ as the next example shows.

Example 7.7
The operators S ± = ±iD 3q − L sub . (7.27) are analogues of the Schrödinger operator on Spin(4). These operators have non-trivial distributions in their null-spaces and can thus be not hypoelliptic. However, the operators S ± + c with c ∈ C\R are globally hypoelliptic.
Our last example shows a differential operator with non-constant coefficients which cannot be written as tensor product of Spin(3) operators.
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