Abstract
In this paper, we use the representation theory of the group \(\textrm{Spin}(m)\) to develop aspects of the global symbolic calculus of pseudo-differential operators on \(\textrm{Spin}(3)\) and \(\textrm{Spin}(4)\) in the sense of Ruzhansky–Turunen–Wirth. A detailed study of \(\textrm{Spin}(3)\) and \(\textrm{Spin}(4)\)-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 \(\textrm{Spin}(4)\) 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.
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1 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 \(G\times {\widehat{G}}\) 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 \(\textrm{Spin}(m)\). The classic approach to a Fourier symbol calculus over \(\textrm{Spin}(m)\), 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 \(\textrm{Spin}(4)\). This choice is based on two reasons. First of all the structure of group representations of \(\textrm{Spin}(m)\) makes it difficult to obtain explicit formulae in the general case so that the case of \(\textrm{Spin}(4)\) will provide further insight into the general case. Secondly, the case of \(\textrm{Spin}(4)\) is important by itself in applications. For instance, \(\textrm{Spin}(4)\) 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 \(\mathrm {Spin(4)}\) in the same way as in the classic case. Additionally, \(\textrm{Spin}(4)\) also appears in quantum gravity (see, e.g., in \(\textrm{Spin}(4)\) 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 \(\textrm{Spin}(4)\).
After recalling some necessary facts about the abstract case of the full symbol calculus by Ruzhansky, Turunen, and Wirth and \(\textrm{Spin}(3)\)-representations, we are going to study \(\textrm{Spin}(4)\)-representations and their connections with harmonic and spinor-valued 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 \(\textrm{Spin}(4)\), 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.
2 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 \(\xi \) of G is a continuous group homomorphism \(\xi : G \rightarrow U(d_{\xi })\) of G into the group of unitary matrices of a certain dimension \(d_\xi .\) The representation \(\xi \) is irreducible if \(\xi (x)A = A \xi (x)\) for all \(x \in G\) and some \(A \in {\mathbb {C}}^{d_\xi \times d_\xi }\) implies \(A=cI\) is a multiple of the identity. This is equivalent to the statement that \({\mathbb {C}}^{d_\xi }\) does not have non-trivial \(\xi \)-invariant subspaces \(V \subset {\mathbb {C}}^{d_{\xi }}\) with \(\xi (x)V \subset V\) for all \(x \in G.\)
Two representations \(\xi _1\) and \(\xi _2\) are equivalent if there exists an invertible matrix B with \(\xi _1(x)B = B \xi _2(x)\) for all \(x \in G.\) Let \({\widehat{G}}\) 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, \(L_x: \phi \mapsto \phi (x^{-1} \, \cdot )\) and \(R_x: \phi \mapsto \phi (\cdot \, x)\) of functions on the group. These left- and right-translations are unitary on the Hilbert space \(\mathrm L^2(G)\) of square integrable functions and therefore the translations give rise to unitary representations \(x\mapsto L_x\) and \(x\mapsto R_x\) of the group G on the Hilbert space \(\mathrm L^2(G)\). 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 \(\mathrm L^2(G)\) decomposes as the orthogonal direct sum of minimal bi-invariant subspaces parameterised by \({\widehat{G}},\) that is
The Fourier transform of \(f \in \mathrm L^2(G)\) is a matrix valued function on \({\widehat{G}}\) defined by
with inverse given by
Furthermore, the following Parseval identity holds
where \(\Vert A \Vert _{\textrm{HS}}^2 = {{\,\textrm{tr}\,}}(A^*A)\) is the Frobenius or Hilbert–Schmidt norm of a matrix A.
On the group G the convolution of two integrable functions \(\phi , \psi \in \mathrm L^1(G)\) is defined by
The following convolution theorem on G is well-known. Note the change in the order of the factors.
Theorem 2.2
Let \(\phi , \psi \in \mathrm L^1(G).\) Then \(\phi * \psi \in \mathrm L^1(G)\) and \((\widehat{\phi * \psi })(\xi ) = {\widehat{\psi }}(\xi )\, {\widehat{\phi }}(\xi )\).
The Laplace–Beltrami operator \({\mathcal {L}}_G\in \textrm{Diff}^2(G)\) 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 \(\mathrm L^2(G).\) Since \({\mathcal {H}}_\xi \) are minimal bi-invariant subspaces, each of them has to be eigenspace of \({\mathcal {L}}_G\) and we denote the corresponding eigenvalue by \(-\lambda _\xi ^2.\) Hence, we obtain the following decomposition
The notion of Fourier series extends naturally to \(C^\infty (G)\) and the space of distributions \({\mathcal {D}}'(G)\) with convergence in the respective topologies. Now, any operator A on G mapping \(C^\infty (G)\) to \({\mathcal {D}}'(G)\) gives rise to a matrix-valued full symbol \(\sigma _A(x, \xi ) \in {\mathbb {C}}^{d_{\xi }\times d_{\xi }},\) \(x \in G\) defined by
which can be understood either pointwise or distributionally, as the product of a smooth matrix-valued function \(\xi ^*(x)\) with the matrix-valued distribution \(A\xi ,\) i.e. \(\sigma _A(\cdot ,\xi )=\xi ^*A\xi \) as a distribution in the first variable, for all \([\xi ] \in {\widehat{G}}.\) Then it can be shown that
holds as \({\mathcal {D}}'\)-convergent series. If it happens that the operator A maps \(C^\infty (G)\) to itself, then (2.8) holds in the strong topology of \(C^\infty (G).\) For A and \(\sigma _A\) related by (2.8) we write \(A={{\,\textrm{Op}\,}}(\sigma _A).\) 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 \(\sigma _A\) and the right-convolution kernel \(R_A\) are related by \(\sigma _A(x,\xi )= \int _{G} R_A(x,y)\, \xi (y)^* \,\textrm{d}y.\)
The class \(\Psi ^m(G)\) 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 \(\Psi ^m(G)\) we fix some notations.
We say that \(Q_\xi \) is a difference operator of order k if it is given by \(Q_\xi {\widehat{f}}(\xi ) = \widehat{\varphi _Q f}(\xi )\) for a function \(\varphi =\varphi _Q \in C^{\infty }(G)\) vanishing of order k at the identity \(e \in G,\) i.e., \((P_x \, \varphi _Q)(e)=0\) for all left-invariant differential operators \(P_x \in \textrm{Diff}^{k-1}(G)\) of order \(k-1.\) We denote the set of all difference operators of order k as \(\textrm{diff}^k({\widehat{G}}).\) In the sequel, for a function \(\varphi \in C^\infty (G)\) it will be also convenient to denote the associated difference operator, acting on Fourier coefficients, by \(\Delta _\varphi {\widehat{f}}(\xi ):= \widehat{\varphi f}(\xi ).\)
Definition 2.3
(cf. [23]) Let G be a compact Lie group of dimension n with unit element e. A collection of \(k \ge n\) first order difference operators \(\Delta _1, \ldots , \Delta _k \in \textrm{diff}^1({\widehat{G}})\) is called admissible, if the corresponding functions \(\varphi _1,\ldots , \varphi _k \in C^\infty (G)\) satisfy d\(\varphi _j \ne 0, j=1,\ldots ,k,\) and \({{\,\textrm{rank}\,}}(\textrm{d} \varphi _1(e), \ldots , \textrm{d} \varphi _k(e))=n.\) It follows, in particular, that e is an isolated zero of the family \(\{\varphi _j\}_{j=1}^k.\) An admissible collection is called strongly admissible if \(\cap _j \{ x \in G: \varphi _j(x)=0\}=\{e\}.\)
The previous definition was adapted to a collection of \(k\ge n\) 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
Furthermore, there exist corresponding differential operators \(\partial _x^{(\alpha )} \in \textrm{Diff}^{|\alpha |}(G)\) such that the Taylor’s formula
holds true for any smooth function \(f \in C^\infty (G)\) and with h(x) the geodesic distance from x to the identity element e (see [21, Sec. 10.6]). Additionally, we introduce operators \(\partial _x^\alpha \) as follows. Let \(\partial _{x_j} \in \textrm{Diff}^1(G), 1 \le j \le n=\text {dim}\, G,\) 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 \(\partial _x^\alpha = \partial _{x_1}^{\alpha _1}\cdots \partial _{x_n}^{\alpha _n}.\) The following theorem characterises Hörmander’s class of pseudo-differential operators \(\Psi ^m(G)\) by the behaviour of their matrix symbols.
Theorem 2.4
(cf. [23, Thm. 2.2]) Let A be a linear continuous operator from \(C^\infty (G)\) to \({\mathcal {D}}'(G).\) Then the following statements are equivalent:
-
(A)
\(A \in \Psi ^m(G).\)
-
(B)
For every left-invariant differential operator \(P_x \in \) \(\textrm{Diff}^k(G)\) of order k and every difference operator \(Q_\xi \in \textrm{diff}^l({\widehat{G}})\) of order l the symbol estimate
$$\begin{aligned} \Vert Q_\xi \, P_x \, \sigma _A(x,\xi )\Vert _\textrm{op} \le C_{Q_\xi , P_x}\, \langle \xi \rangle ^{m-l} \end{aligned}$$(2.12)is valid, where \(\langle \xi \rangle = (1+ \lambda _\xi ^2)^{1/2}\) and \(-\lambda _\xi ^2\) are the eigenvalues of \({\mathcal {L}}_G\).
-
(C)
For an admissible selection \(\Delta _1,\ldots , \Delta _m \in \) \(\textrm{diff}^1({\widehat{G}})\) we have
$$\begin{aligned} \Vert \Delta _\xi ^\alpha \, \partial _x^\beta \, \sigma _A(x,\xi )\Vert _\textrm{op} \le C_{\alpha ,\beta } \langle \xi \rangle ^{m-|\alpha |} \end{aligned}$$(2.13)for all multi-indices \(\alpha , \beta .\) Moreover, sing supp \(R_A(x,\cdot ) \subseteq \{e\}.\)
-
(D)
For a strongly admissible selection \(\Delta _1, \ldots , \Delta _m \in \) \(\textrm{diff}^1({\widehat{G}})\) we have
$$\begin{aligned} \Vert \Delta _\xi ^\alpha \, \partial _x^\beta \,\sigma _A(x,\xi ) \Vert _\text {op} \le C_{\alpha ,\beta } \,\langle \xi \rangle ^{m-|\alpha |} \end{aligned}$$(2.14)for all multi-indices \(\alpha ,\beta .\)
The set of symbols \(\sigma _A\) satisfying either of equivalent conditions (B)-(D) is also denoted by \(S^m(G)\), such that the operator quantization gives an isomorphism \({{\,\textrm{Op}\,}}: S^m(G) \rightarrow \Psi ^m(G)\). The composition of pseudodifferential operators gives again a pseudo-differential operator with a symbol, which can be expressed as an asymptotic expansion.
Theorem 2.5
(cf. [21, Thm. 10.7.9]) If \(A\in \Psi ^{m_1}(G)\) and \(B\in \Psi ^{m_2}(G)\) then \(A\circ B \in \Psi ^{m_1+m_2}(G)\) satisfies
in the sense that \(\sigma _{A\circ B} - \sum _{|\alpha |<N}\frac{1}{\alpha !} \big ( \Delta _\xi ^\alpha \sigma _A\big )\big (\partial _x^{(\alpha )} \sigma _B\big ) \in S^{m_1+m_2-N}(G)\) holds true.
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].
3 Irreducible Representations of \(\textrm{Spin}(m)\)
3.1 Notation
First we introduce some basic notation about Clifford algebras. We refer to [5] for a more detailed overview. Let \((\textrm{e}_1,\ldots ,\textrm{e}_m)\) be the standard basis of the Euclidean space \({\mathbb {R}}^m\) and \({\mathbb {R}}_{0,m}\) be the real Clifford algebra generated by the vectors \(\textrm{e}_1,\ldots , \textrm{e}_m\) such that \(\textrm{e}_j^2=-1\) for \(j=1,\ldots ,m,\) and \(\textrm{e}_i \textrm{e}_j = -\textrm{e}_j\textrm{e}_i,\) for \(i,j=1,\ldots ,m,\) and \(i \ne j.\) An element \(a \in {\mathbb {R}}_{0,m}\) is of the form \(a= \sum _{A} a_A \, \textrm{e}_A,\) \(a_A \in {\mathbb {R}}\) for ordered subsets \(A\subseteq \{1,\ldots ,m\}\) and with \(\textrm{e}_{\varnothing }=\textrm{e}_0=1\). The k-vector part of a is given by \([a]_k = \sum _{|A|=k}a_A \textrm{e}_A\) and \(a=\sum _{k=0}^{m}[a]_k.\) Vectors \(\underline{{x}} \in {\mathbb {R}}^m\) are identified with 1-vectors \(\underline{{x}} = \sum _{j=1}^{m} x_j \textrm{e}_j\in {\mathbb {R}}_{0,m}\). The Clifford product of two 1-vectors \(\underline{{x}}\) and \(\underline{{y}}\) in \({\mathbb {R}}^m\) splits in a scalar part given by minus the inner product in \({\mathbb {R}}^m\) and the wedge product:
It holds \( - \underline{{x}}\cdot \underline{{y}} = \frac{1}{2}(\underline{{x}}\,\underline{{y}}+\underline{{y}}\,\underline{{x}})\) and \(\underline{{x}} \wedge \underline{{y}} = \frac{1}{2}(\underline{{x}}\,\underline{{y}}-\underline{{y}}\,\underline{{x}}).\) These can be extended to the whole Clifford algebra by setting
The Dirac operator on \({\mathbb {R}}^m\) is given by \(\partial _{\underline{{x}}}=\textrm{e}_1 \partial _{x_1} + \ldots + \textrm{e}_m \partial _{x_m}\) 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 \({\mathbb {C}}_m\) is the complexification of \({\mathbb {R}}_{0,m}\), i.e. \({\mathbb {C}}_m={\mathbb {C}}\otimes {\mathbb {R}}_{0,m}.\) The main anti-involution in \({\mathbb {C}}_m\) is defined by
The complex Clifford algebra is equipped with the Clifford inner product defined by
We will frequently use the Witt basis vectors
for \(M=\lfloor \frac{m}{2}\rfloor \). They satisfy \(T_j^2=0=(T_j^\dagger )^2\) together with \(T_iT_j=-T_jT_i\) and \(T_iT_j^\dagger = -T_j^\dagger T_i\) for \(i\ne j\) and \(T_iT_i^\dagger + T_i^\dagger T_i =1\). For even m they generate all of \(\mathbb C_m\).
Later on we will use spaces of \({\mathbb {C}}_m\)-valued polynomials on \({\mathbb {R}}^m\). 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 \(\partial _{x_i}\) are Fischer-adjoint while the generators \(\textrm{e}_i\) of the Clifford algebra \({\mathbb {C}}_m\) are skew-adjoint
3.2 The Spin Group and \(\mathrm H\)- and \(\mathrm L\)-Representations
The spin group \(\textrm{Spin}(m)\) is realised as the set of even products of unit vectors, that is,
where \({\mathbb {R}}_{0,m}^+={{\,\textrm{span}\,}}_{{\mathbb {R}}} \{ \textrm{e}_A \mid \text {|A| even} \}\) denotes the even subalgebra of \(\mathbb R_{0,m}\). The spin group is a double covering of \(\textrm{SO}(m)\) as seen by the group action \({\mathbb {R}}^m \ni \underline{{x}} \mapsto {\overline{{s}}}\underline{{x}}s\in {\mathbb {R}}^m\) on vectors. There are two distinguished representations of the spin group on \({\mathbb {C}}_m\)-valued functions on \({\mathbb {C}}_m\) defined by
where \(x \in {\mathbb {C}}_m,\) \(s \in \textrm{Spin}(m)\), and \(f: {\mathbb {C}}_m \rightarrow {\mathbb {C}}_m\). The \(\mathrm H\)-representation corresponds to the standard representation of \(\textrm{SO}(m)\) on scalar-valued functions \(f\in \mathrm L^2({\mathbb {S}}^{m-1})\), while the \(\mathrm L\)-representation corresponds to the half-spin representations. Models for all irreducible representations arise from decomposing \(\mathrm H\) and \(\mathrm L\) 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 \(\mathfrak {spin}(m)\) can be realised as the space
of bivectors in \({\mathbb {R}}_{0,m}.\) The Cartan subalgebra \({\mathfrak {h}}\subset \mathfrak {spin}(m)\) is given by
The exponential of \({\mathfrak {h}}\) yields the maximal torus \({\mathbb {T}}\subset \textrm{Spin}(m)\)
from which we can label all the unitary irreducible spin representations. Any representation \(R:\textrm{Spin}(m) \rightarrow \textrm{Aut}(V)\) of \(\textrm{Spin}(m)\) on some vector space V is determined by the restriction to the maximal torus and any representation of the maximal torus invariant under the adjoint action of the group comes from a representation of the group itself. The space V splits into subspaces generated by weight vectors \(v\in V\) satisfying
for the derived representation \(\mathrm d R: \mathfrak {spin}(m) \rightarrow \textrm{Aut}(V)\) and with weights \(l=(l_1,\ldots ,l_M)\) 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 \(l_i \in {\mathbb {Z}}\) or all \(l_i \in {\mathbb {Z}}+\frac{1}{2}\).
3.3 Explicit Models
To construct explicit models for irreducible representations of \(\textrm{Spin}(m)\) we follow [27] and consider \(k\le m\) vector variables \(\underline{{x}}_1,\ldots , \underline{{x}}_k\), where \(\underline{{x}}_i=\sum _{j=1}^{m}x_{ij}\textrm{e}_j\) and \({\mathbb {C}}_m\)-valued polynomials in these k vector variables. A polynomial \(P(\underline{{x}}_1,\underline{{x}}_1 \wedge \underline{{x}}_2,\ldots ,\underline{{x}}_1\wedge \ldots \wedge \underline{{x}}_k)\) depending on the simplicial variables \(\underline{{x}}_1 \wedge \ldots \wedge \underline{{x}}_i\,\) is called a simplicial polynomial. A harmonic simplicial polynomial P is a simplicial polynomial satisfying
The space of these polynomials is denoted by \({\mathcal {H}}(\underline{{x}}_1,\ldots ,\underline{{x}}_k)\). It is invariant under the \(\mathrm H\)-action. A monogenic simplicial polynomial P is characterised by the condition
The space of monogenic simplicial polynomials is denoted by \({\mathcal {M}}(\underline{{x}}_1,\ldots ,\underline{{x}}_k)\) and is invariant under the \(\mathrm L\)-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.Footnote 1
Case 1 For the highest weight \((l_1,\ldots ,l_{M-1},l_M)\), \(l_i \in {\mathbb {N}}_0\), we take the highest weight vector
and let \(\textrm{Spin}(m)\) act by the \(\mathrm H\)-representation on it. We recall that for \(s\in \textrm{Spin}(m)\) the \(\mathrm H\)-representation on simplicial functions is given by
We denote the resulting representation space by
Case 2 For the highest weight \((l_1,\ldots ,l_{M-1},-l_M)\), \(l_i \in {\mathbb {N}}_0\), we have to consider the H-action on the highest weight vector
The resulting representation space will be denoted by \( \mathcal H_{(l_1,\ldots ,- l_M)} ({\mathbb {R}}^m) \).
For half-integer weights we have to realise the representations in the spinor space of the complex Clifford algebra \({\mathbb {C}}_m\). We distinguish between even and odd m. For even \(m=2M\) we use the pairwise commuting idempotents \({\mathcal {I}}_j=T_j T_j^\dagger =\frac{1}{2}(1-{\textbf{i}}\,\textrm{e}_{2j-1}\textrm{e}_{2j})\), \( j=1,\ldots ,M\), together with \({\mathcal {I}}_M' = T_M^\dagger T_M=\frac{1}{2}(1+{\textbf{i}}\,\textrm{e}_{2j-1}\textrm{e}_{2j})\), to construct the primitive idempotents
and define
They are both \(\textrm{Spin}(m)\)-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 \((l_1,\ldots ,l_{M-1},l_M)\), \(l_i \in {\mathbb {N}}_0+\frac{1}{2}\), we take the highest weight vector
and let \(\textrm{Spin}(m)\) act by the \(\mathrm L\)-representation on it. We recall that for \(s \in \textrm{Spin}(m)\) the \(\mathrm L\)-representation on simplicial spinor functions is given by
We denote the resulting representation space by
Case 4 For the highest weight \((l_1,\ldots ,l_{M-1},-l_M)\), \(l_i \in {\mathbb {N}}_0+\frac{1}{2}\), we have to consider the \(\mathrm L\)-action on the highest weight vector
The resulting representation space will be denoted by \( \mathcal M_{(l_1,\ldots ,- l_M)} ({\mathbb {R}}^m) \).
To summarize, in the odd dimensional case \((m=2M+1)\) the irreducible representations of \(\textrm{Spin}(m)\) are obtained considering the H- and L-actions on the weight vectors (3.21) and (3.28), correspondingly. In the even dimensional case \((m=2M)\) all the irreducible representations of \(\textrm{Spin}(m)\) are obtained considering the H-action on the weight vectors (3.18) and (3.21), and the L-action on the weight vectors (3.25) and (3.28).
4 \(\textrm{Spin}(3)\) Representations
In this section we collect results on \(\textrm{Spin}(3)\)-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 \(\textrm{Spin}(4)\)-representations later on.
The group \(\textrm{Spin}(3)\) is the universal cover of \(\textrm{SO}(3)\) and can be realised inside the even Clifford algebra \({\mathbb {R}}_{3}^+\). Moreover, it is isomorphic to the special unitary group \(\textrm{SU}(2)\) and also isomorphic to the unit 3-sphere \({\mathbb {S}}^3\) understood as the group of unit length quaternions, i.e.
We want to make these isomorphisms explicit for later use. An element of \(\textrm{Spin}(3)\) is of the form \(s=a_0+ a_1\textrm{e}_{12}+a_2\textrm{e}_{13}+a_3\textrm{e}_{23}\) such that \(|s|^2=a_0^2+a_1^2+a_2^2+a_3^2=1\). It acts on vectors \(\underline{{x}}=x_1\textrm{e}_1+x_2\textrm{e}_2+x_3\textrm{e}_3 \in {\mathbb {R}}^3\) by \(\underline{{x}} \mapsto {\overline{{s}}}\underline{{x}} s\) and this mapping is represented by the \(\textrm{SO}(3)\) rotation matrix
as a straightforward calculation within \({\mathbb {R}}_{0,3}\) shows. Identifying \(\textrm{e}_{12}={\textbf{i}}\), \(\textrm{e}_{13}={\textbf{j}}\) und \(\textrm{e}_{23}= {\textbf{k}}\) with the quaternion units yields an isomorphism \(\mathbb R_{0,3}^+ \cong {\mathbb {H}}\) and identifies the spin group with the group of unit length quaternions. We identify \({\mathbb {H}} \cong {\mathbb {C}}^2\) by writing \(q=a_0 + a_1 {\textbf{i}}+ a_2 {\textbf{j}} + a_3 {\textbf{k}} = q_1 + q_2 {\textbf{j}}\) with \(q_1 = a_0+{\textbf{i}}a_1\in {\mathbb {C}}\) and \(q_2=a_2+{\textbf{i}}a_3\in {\mathbb {C}}\). Then in particular \({\overline{{q}}} = \overline{q_1} - q_2 {\textbf{j}}\) is the quaternion conjugation and quaternion multiplication corresponds to matrix multiplication for the associated matrices
which belong to \(\textrm{SU}(2)\) whenever \(|q|=1\). This completes the isomorphisms in (4.1). We rewrite the rotation matrix (4.2) in these complex coordinates for \(\textrm{Spin}(3)\). This yields
For later calculations we need the \(\textrm{H}\)-action on the polynomials \(z_1=x_1+{\textbf{i}}x_2\), \({\overline{{z}}}_1=x_1-{\textbf{i}}x_2\) and \(x_3\). Using the rotation matrix (4.4) applied to \(\underline{{x}}=(x_1,x_2,x_3) \in {\mathbb {R}}^3\) we obtain
4.1 Representations of \({\mathbb {S}}^3\subset {\mathbb {H}}\)
Before explicitly giving irreducible \(\textrm{Spin}(3)\)-representations based on the general theory of Sect. 3, we will recall the closely related irreducible \({\mathbb {S}}^3\)-representations from [13, 21] (see also [10, 23, 28]). Since the quaternionic unit sphere \({\mathbb {S}}^3\) can be viewed as a subset of \({\mathbb {C}}^2\) through \({\mathbb {S}}^3=\{(z_1,z_2) \in {\mathbb {C}}^2:|z_1|^2 + |z_2|^2=1\}\) we write the quaternion multiplication and conjugation in \({\mathbb {S}}^3\) as
Let \({\mathcal {P}}\) be the space of all polynomials \(P(z,w)=\sum c_{j,k} z^j w^k\) in two complex variables, and let \({\mathcal {P}}_m \subset {\mathcal {P}}\) be the space of homogeneous polynomials of degree m
An orthonormal basis of \({\mathcal {P}}_m\) with respect to the Fischer inner product (or equivalently, with respect to the normalised \(\mathrm L^2\) inner product) consists of the set of functions
The group \({\mathbb {S}}^3\) naturally acts on \({\mathcal {P}}_m\) by right translation \(R_{(w_1,w_2)}f(z_1,z_2) = f((z_1,z_2)\bullet (w_1,w_2))\). Next, we follow [13] and express the linear map \(R_{(w_1,w_2)}: {\mathcal {P}}_m \rightarrow {\mathcal {P}}_m\) with respect to the orthonormal basis \(P_k^m\), \(j=0,\ldots ,m\). By straightforward computations we have
Hence, for each \(j=0,\ldots ,m,\) we obtain
where
and, therefore, the matrix elements of the unitary representation
are given by
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 \(m \in {\mathbb {N}}_0\) and all \(0\le i,j \le m\) 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.
Definition 4.2
(cf. [13]) For \(m \ge 0\) we define the matrices \(a_-(m), a_+(m) \in {\mathbb {R}}^{(m+1)\times (m+2)}\) by
For \(m \ge 1\) we define the matrices \(b_-(m), b_+(m) \in {\mathbb {R}}^{(m+1)\times m}\) by
and for convenience, we set \(b_-(0)=b_+(0)=0.\)
Theorem 4.3
(cf. [13]) For every \(m \in {\mathbb {N}}_0\) the following recurrence relations hold
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 \(\mathbb S^3\).
Definition 4.4
(cf. [13]) For differentiable functions \({\mathbb {S}}^3\ni (z_1,z_2)\mapsto f(z_1,z_2) \in {\mathbb {C}}\) we define the following differential operators
It is easy to see that the operators \(\partial _{\pm }\) and \(\partial _{\pm }^\dagger \) are linear combination of rotational derivatives. The operators \(\partial _{\pm }\) are left invariant and the operators \(\partial _{\pm }^\dagger \) are right invariant. This means that
holds true for the right translation \(R_{(w_1,w_2)}f(z_1,z_2) = f((z_1,z_2)\bullet (w_1,w_2))\) and the left translation \(L_{(w_1,w_2)}f(z_1,z_2) = f((w_1,w_2)^{-1}\bullet (z_1,z_2))=f(({\overline{{w}}}_1,-w_2)\bullet (z_1,z_2))\) on the group \({\mathbb {S}}^3\).
Theorem 4.5
(cf. [13]) For every \(m \in {\mathbb {N}}_0\) and \(0 \le i,j \le m\) 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 \(\sigma _+(m/2)\) and \(\sigma _-(m/2)\) 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.
Definition 4.6
(cf. [13]) For \(m \ge 0\) we define the matrices \(\sigma _+(m/2), \sigma _-(m/2) \in {\mathbb {R}}^{(m+1)\times (m+1)}\) by
They satisfy \(\sigma _+(m/2) = \sigma _-(m/2)^{\top }\).
Corollary 4.7
(cf. [13]) For every \(m \in {\mathbb {N}}_0\) the following relations hold
where \(\sigma _+^\dagger (m/2) = -\sigma _+(m/2)^\top =-\sigma _-(m/2)\) and \(\sigma _-^\dagger (m/2) = - \sigma _-(m/2)^\top = -\sigma _+(m/2)\).
Remark 4.8
The matrices \(\sigma _\pm (m/2)\) can be obtained as
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 \(\partial _\pm \in \textrm{Diff}^1({\mathbb {S}}^3)\) in the sense of Sect. 2.
4.2 H-Representations: Spherical Harmonics in \({\mathbb {R}}^3\)
We follow the approach of Sect. 3. By (3.21) and (3.5) the weight vector \(\omega _l(\underline{{x}})\) for \(l\in {\mathbb {N}}_0\) is given by
We renormalise the weight vector considering \(2T_1\) instead of \(T_1\) obtaining the polynomial \({\overline{{z}}}_1^l.\) To perform the \(\mathrm H\)-action on \(\omega _l(\underline{{x}})\) we consider (4.5b) and we get
Using multi-index notation \(\alpha =(\alpha _1,\alpha _2,\alpha _3)\in {\mathbb {N}}_0^3\), \(|\alpha |=\alpha _1+\alpha _2+\alpha _3,\) and \(\alpha !=\alpha _1!\alpha _2!\alpha _3!,\) and the multinomial theorem to further expand (4.27) we obtain
where in the last line we put \(\alpha _2+2\alpha _3=j,\) which yields \(2\alpha _1+\alpha _2=2l-j\) from \(|\alpha |=l\). The \(2l+1\) polynomials given by
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)
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 \(z_1 \mapsto 2 \partial _{{\overline{{z}}}_1}\), \({\overline{{z}}}_1 \mapsto 2 \partial _{z_1}\), and \(x_3 \mapsto \partial _{x_3}\).
The following lemma provides a relation between trinomial and binomial coefficients.
Lemma 4.9
For \(\alpha =(\alpha _1,\alpha _2,\alpha _3)\in {\mathbb {N}}_0^3\) it holds
Proof
Multiplication by \(x^j=x^{\alpha _2+2\alpha _3}\) and summing over j yields
based on the multinomial theorem. Comparing powers of x yields the desired result. \(\square \)
Using Lemma 4.9 we can write the normalising constants \(c_{l,j}\) as
and obtain \(2l+1\) orthonormal spherical harmonics (4.29) on \({\mathbb {R}}^3\) spanning the representation space \({\mathcal {H}}_{l}({\mathbb {R}}^3)\) for the given weight \(l \in {\mathbb {N}}\). It remains to express \(\mathrm H(q)\) in this basis in order to obtain the matrix coefficients and to obtain the relation to the representations constructed in Sect. 4.1.
For this, we consider the H-action on the orthogonal polynomials \({P}^l_j\). For each \(j=0,\ldots , 2l\), we use first the actions (4.5a)-(4.5c) and then the multinomial theorem to obtain
Putting \(\beta _\lambda +\gamma _\lambda +\delta _\lambda =\epsilon _\lambda ,\) \(\lambda =1,2,3\) and since \(|\alpha |=l\), \(|\beta |=\alpha _1\), \(|\gamma |=\alpha _2\), \(|\delta |=\alpha _3\) we obtain that \(|\epsilon |=\epsilon _1+\epsilon _2+\epsilon _3=l.\) Moreover, putting \(\epsilon _2+2\epsilon _3=i\) and \(\beta _2+2\beta _3+\gamma _3+\gamma _2-n=k\) and since \(\alpha _2+2\alpha _3=|\gamma |+2|\delta |=j\) we obtain the following identities
Since \(2\,l-j-k \ge 0,\) \(i-k \ge 0,\) and \(j-i+k \ge 0\) we get that \(\max (0,i-j) \le k \le \min (i,2l-j).\) Therefore, (4.35) can be written as
for some coefficients \(t_{i,j,2l}^k.\) Comparing (4.37) with (4.10), we must have the following identifications between the matrix coefficients (4.10) and (4.37): \(m \mapsto 2\,l,\) \(w_1 \mapsto q_1,\) and \(w_2 \mapsto q_2.\) Thus, the matrix coefficients related to the (normalised) basis of harmonic polynomials \({P}^l_j\) associated with the weight \(l \in {\mathbb {N}}_0\) are given by
where \(C_{i,j,2l}^k\) is given by (4.11) with \(m=2l\). Therefore, it holds \(t_{i,j,2l}^k = C_{i,j,2l}^k.\) Consequently, the relationship between these \(\textrm{H}\)-representations and the representations constructed in Sect. 4.1 is given by
Next we give two examples. For \(l=1,\) the module \({\mathcal {H}}_1({\mathbb {R}}^3)\) has dimension 3 and an orthonormal basis is given by \(\{\frac{1}{\sqrt{2}}{\overline{{z}}}_1,\, -x_3,\, -\frac{1}{\sqrt{2}}z_1 \}\). The matrix coefficients associated to this basis are
For \(l=2,\) the module \({\mathcal {H}}_2({\mathbb {R}}^3)\) has dimension 5 and an orthonormal basis is given by
The matrix coefficients \(\xi _2^{\mathrm {Spin(3)}}(q_1,q_2) \) associated to this basis are
4.3 L-Representations: Spinor-Valued Monogenics in \({\mathbb {R}}^3\)
In the 3-dimensional case there is only one basic spinor representation \({\mathcal {S}}\cong {\mathcal {S}}_4^+ ={\mathbb {C}}_4^+{\mathcal {I}}_+ = {{\,\textrm{span}\,}}_{\mathbb {C}}\{1, \textrm{e}_{13} \}{\mathcal {I}}_+\), where the idempotent \({\mathcal {I}}_+\) is given in terms of the Witt basis elements (as introduced in Sect. 3)
by \({\mathcal {I}}_+={\mathcal {I}}_1{\mathcal {I}}_2,\) with \(\mathcal I_1=T_1 T_1^\dagger \) and \({\mathcal {I}}_2=T_2 T_2^\dagger \). For \(z \in {\mathcal {S}}_4^+\) we put \(z=(z^+,z^-)\in {\mathbb {C}}^2\) such that \(z=z^+ {\mathcal {I}}_+- z^- \textrm{e}_{13}{\mathcal {I}}_+.\) Now, considering the left multiplication by \(q=a_1+a_2\textrm{e}_{12} + a_3\textrm{e}_{13} + a_4\textrm{e}_{23} \in \textrm{Spin}(3)\) on the spinor representation \({\mathcal {S}}_4^+\) we obtain
where we have used the multiplication rules of the Clifford algebra and the identities \(\textrm{e}_{12}{\mathcal {I}}_+={\textbf{i}}{\mathcal {I}}_+\) and \(\textrm{e}_{23}{\mathcal {I}}_+=-{\textbf{i}}\textrm{e}_{13}{\mathcal {I}}_+\) following from the basic rules \(\textrm{e}_2{\mathcal {I}}_+=-{\textbf{i}}\textrm{e}_1{\mathcal {I}}_+\) and \(\textrm{e}_4{\mathcal {I}}_+=-{\textbf{i}}\textrm{e}_3{\mathcal {I}}_+\). The linear transformation (4.44) can again be written as an \(\textrm{SU}(2)\)-action given by
where \(q_1=a_0+ {\textbf{i}}a_1\) and \(q_2=a_2+ {\textbf{i}}a_3\) and \(z=z^+\mathcal I_+ - z^-\textrm{e}_{13}{\mathcal {I}}_+\).
Next, we perform the \(\mathrm L\)-action on the weight vector \(\omega _{l}(\underline{{x}})={\overline{{z}}}_1^{l-\frac{1}{2}}\, {\mathcal {I}}_+\) for \(l\in {\mathbb {N}}_0+\frac{1}{2}\) 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 \(P_{-1}^{l-\frac{1}{2}}(z_1,{\overline{{z}}}_1,x_3)=P_{2l}^{l-\frac{1}{2}}(z_1,{\overline{{z}}}_1,x_3)=0\). Therefore, we have obtained the \(2\,l+1\) \({\mathcal {S}}_4^+\)-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 \(\widetilde{P}_0^{l}(z_1,{\overline{{z}}}_1,x_3)={\overline{{z}}}_1^{l-\frac{1}{2}}\,{\mathcal {I}}_+\), i.e. the weight vector, while for \(j=2l\) we get \(\widetilde{P}_{2\,l}^{l}(z_1,{\overline{{z}}}_1,x_3)=-(-z_1)^{l-\frac{1}{2}} \textrm{e}_{13}\mathcal I_+\).
To obtain an orthonormal basis for \({\mathcal {M}}_{l}({\mathbb {R}}^3)\), 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 \(j=0,\ldots ,2l\). We will endow the representation space \({\mathcal {M}}_{l}({\mathbb {R}}^3)\) with the basis obtained by normalising \({\widetilde{P}}^{l}_j\).
To obtain the matrix coefficients associated to (4.47), we calculate the \(\mathrm L\)-action on these orthogonal polynomials \({\widetilde{P}}^{l}_j\). Using (4.45) we obtain for each \(j=0,\ldots , 2\,l\)
Using (4.37) we see that the maximal exponents of \(q_1\) and \(q_2\) are \(q_1^{2\,l-j-k}\) and \(q_2^{j-i+k}\). Therefore, we can write (4.50) using the renormalised basis (4.47) in the form
for some still to be determined coefficients \(t_{i,j,2l}^k\). Comparing (4.51) with (4.10), we must have the identifications \(m \mapsto 2\,l\), \(w_1 \mapsto q_1\), and \(w_2 \mapsto q_2\) between the matrix coefficients (4.10) and (4.51). Therefore, the matrix coefficients related to the basis of monogenic polynomials \({{\widetilde{P}}}^{l}_j\) associated with the weight \( l \in \mathbb N_0+\frac{1}{2}\) are given by
for \(i,j=0,\ldots ,2l\) and with \(C_{i,j,2l}^k\) given by (4.11) for \(m=2l\). Thus, \(t_{i,j,2\,l}^k = C_{i,j,2\,l}^k\) and consequently, the relationship between the spinor-valued representations and the representations constructed in Sect. 4.1 is
Next we give two examples. For \(l=\frac{1}{2}\), the representation space \({\mathcal {M}}_{\frac{1}{2}}({\mathbb {R}}^3)\) has dimension 2 and an orthonormal basis is given by \(\{{\mathcal {I}}_+,\, -\textrm{e}_{13}\mathcal I_+\}\). The matrix coefficients associated to this basis are
For \(l=\frac{3}{2}\), the module \({\mathcal {M}}_{\frac{3}{2}}({\mathbb {R}}^3)\) has dimension 4 and an orthonormal basis is given by
The associated matrix coefficients are
5 \(\textrm{Spin}(4)\) Representations
5.1 Prerequisites
We start by recalling the notation from Sect. 3. We use an orthonormal basis \(\{\textrm{e}_1,\textrm{e}_2,\textrm{e}_3,\textrm{e}_4\}\) of \({\mathbb {R}}^4\) and denote by \({\mathbb {R}}_{0,4}\) the real \(2^4\)-dimensional Clifford algebra over \({\mathbb {R}}^4\) generated by the relations \(\textrm{e}_i^2=-1\), \(i=1,\ldots ,4\) and \(\textrm{e}_i\textrm{e}_j =-\textrm{e}_j\textrm{e}_i\), \(i \ne j\). The group \(\textrm{Spin}(4)\) consists of even products of unit vectors
Later on we will make use of the isomorphism \(\textrm{Spin}(4)\cong \textrm{Spin}(3) \times \textrm{Spin}(3)\) following from the identifications \({\mathbb {R}}_{0,4}^+ \cong {\mathbb {R}}_{0,3} \cong {\mathbb {H}} \oplus {\mathbb {H}}\), which we recall next. Due to [5, Section 0.5.4] we can write \(a \in {\mathbb {R}}_{0,4}^+\) in the form \(a = \omega _+ a_+ + \omega _- a_-\) with \(a_{\pm } \in {\mathbb {R}}_{0,3}^+ = {{\,\textrm{span}\,}}\{1,\textrm{e}_{12},\textrm{e}_{13},\textrm{e}_{23}\} \cong {\mathbb {H}}\) and \(\omega _{\pm } = \frac{1}{2}(1 \pm \textrm{e}_{1234})\). These elements \(\omega _{\pm }\) are mutually annihilating idempotents satisfying
As \(\omega _{\pm }\) commute with \(a_{+}\) and \(a_-\) we can also write \(a = a_+\omega _+ + a_- \omega _-.\) Applying this decomposition to elements \({\textbf{s}} \in \textrm{Spin}(4)\) we obtain
together with
making the isomorphism explicit. For an arbitrary element \(\underline{{x}} \in {\mathbb {R}}^4\) putting \(\underline{{x}}=\sum _{i=1}^{4}x_i\textrm{e}_i\) and observing that for each \(i=1,\ldots ,4\) it holds \(\omega _+ \textrm{e}_i = \textrm{e}_i \omega _-\) and \(\omega _- \textrm{e}_i = \textrm{e}_i \omega _+\) we obtain
which describes a rotation in \({\mathbb {R}}^4\) induced by \((q,s) \in \textrm{Spin}(3) \times \textrm{Spin}(3)\). Due to the isomorphism \(\textrm{Spin}(3)\cong {\mathbb {S}}^3\) and the identification of \({\mathbb {R}}^4\) with \({\mathbb {C}}^2\) we can write the action \({\overline{{{\textbf{s}}}}}\, \underline{{x}}\, {{\textbf{s}}}\) inside \({\mathbb {C}}^2\). Putting \(q=(q_1,q_2),s=(s_1,s_2)\in {\mathbb {S}}^3\), and \(\underline{{x}}=(z_1,z_2) \in {\mathbb {C}}^2\) yields after a lengthy calculation
identifying \({\mathbb {C}}^2\) with \({\mathbb {H}}\). Next, we describe the \(\textrm{Spin}(4)\) action on the spinor spaces \({\mathcal {S}}_4^\pm \). A realisation of \({\mathcal {S}}_4^+\) was already described in Sect. 4.3. Considering \({\textbf{s}} = s \omega _+ + q \omega _-\) with \(q,s \in \textrm{Spin}(3)\) we obtain for \({\mathcal {S}}_4^+\) the half-spin action
as \(\omega _+{\mathcal {I}}_+=0\), \(\omega _-{\mathcal {I}}_+=\mathcal I_+\), and \(\omega _+\), \(\omega _-\) commute with \(\textrm{e}_{13}\). Thus on \({\mathcal {S}}_4^+\) the action corresponds to the left multiplication by \(q \in \textrm{Spin}(3) \cong {\mathbb {S}}^3\) and hence by (4.45) also to the matrix multiplication
For \({\mathcal {S}}_4^-\), we get the half-spin action
Thus, this action corresponds to the left multiplication by \(s \in \textrm{Spin}(3) \cong {\mathbb {S}}^3\) and as in (4.45) we see that it corresponds to the matrix-multiplication
5.2 H-Representations: Spherical Harmonics in \({\mathbb {R}}^4\)
Following Sect. 3, we construct explicitly the representations associated to the weights \((l_1,l_2),\) with \(l_1,l_2 \in {\mathbb {N}}_0,\) and \(0 \le l_2 \le l_1\). Considering \(\underline{{x}}=\sum _{i=1}^{4}x_i\textrm{e}_i\), \(\underline{{y}}=\sum _{i=1}^{4}y_i\textrm{e}_i\) by (3.18) the (normalised) weight vector is given by
where \(z_1=x_1+ {\textbf{i}}x_2\), \(z_2=x_3+ {\textbf{i}}x_4\), \(w_1=y_1+{\textbf{i}}y_2\) and \(w_2=y_3+ {\textbf{i}}y_4\). Representing an element \({\textbf{s}} = s\omega _++q\omega _-\in \textrm{Spin}(4)\) by two unit quaternions \(q=(q_1,q_2), s =(s_1,s_2) \in {\mathbb {S}}^3\) and using complex coordinates for \(\underline{{x}}=(z_1,z_2),\underline{{y}}=(w_1,w_2) \in {\mathbb {C}}^2\) equation (5.6) yields
and
Thus, we obtain for the \(\textrm{H}\)-action on the weight vector \(\omega _{(l_1,l_2)}(\underline{{x}},\underline{{y}})\) and by using shorthand notations \(Q_j\) for the terms marked above
Collecting the powers of q this can be written as
The terms \(Q_3\), \(Q_4\) and \(Q_5\) are independent of the spinor \({\textbf{s}}=s\omega _++q\omega _-\). For \(Q_1\) and \(Q_2\) we extract the dependence on s and obtain (with \(l=l_1-l_2\))
Thus, defining the following polynomials
where \(\underline{{z}}=z_1T_1-{\overline{{z}}}_1T_1^\dagger + z_2T_2 - {\overline{{z}}}_2T_2^\dagger \) and \(\underline{{w}}=w_1T_1-{\overline{{w}}}_1T_1^\dagger + w_2T_2 - {\overline{{w}}}_2T_2^\dagger \) are the Hermitian variables constructed from \(\underline{{x}}\) and \(\underline{{y}},\) we can write (5.15) as
From (5.19) the tensor product structure of the \(\textrm{Spin}(4)\)-representations of weight \((l_1,l_2)\), \(l_i \in {\mathbb {N}}_0\), \(l_1\ge l_2\) can be seen.
Lemma 5.1
Let \(l_1,l_2\in {\mathbb {N}}_0\) with \(l_1\ge l_2\). Then the polynomials
for \(k=0,\ldots , l_1+l_2\) and \(m=0,\ldots , l_1-l_2\), form an orthogonal basis of \(\mathcal H_{(l_1,l_2)}({\mathbb {R}}^4)\). 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 \({\mathcal {H}}_{(l_1,l_2)} ({\mathbb {R}}^4)\). 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 \({\overline{{z}}}_1,{\overline{{z}}}_2,{\overline{{w}}}_1,{\overline{{w}}}_2\). Then due to (5.18) the polynomial \(S^{l_1-l_2}_{m,i}(\underline{{z}})\) is of holomorphic degree i and of antiholomorphic degree \(l_1-l_2-i\), while \(Q^{l_2}_j(\underline{{z}}\wedge \underline{{w}})\) 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 \(z_1,{\overline{{z}}}_2, w_1, {\overline{{w}}}_2\) the mixed degree of the polynomial \(S^{l_1-l_2}_{m,i}(\underline{{z}})\) is of degree m and \(Q^{l_2}_j(\underline{{z}}\wedge \underline{{w}})\) 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. \(\square \)
Next, we construct the representations associated to the weights \((l_1,l_2)\) with \(l_1,l_2 \in {\mathbb {Z}}\) and \(l_1\ge -l_2 >0\). By (3.21) the corresponding (normalised) weight vector is given by
Using again \(q=(q_1,q_2), s =(s_1,s_2) \in {\mathbb {S}}^3\) to describe elements of the spin group and \(\underline{{x}}=(z_1,z_2)\), \(\underline{{y}}=(w_1,w_2) \in {\mathbb {C}}^2\) as complex coordinates, equation (5.6) yields again
and now
Using these short-hand notations we compute the \(\textrm{H}\)-action on the weight vector \(\omega _{(l_1,l_2)}(\underline{{x}},\underline{{y}})\) following the lines of (5.14) as
with
and obtained from \(S_{k,i}^l(\underline{{z}})\) by complex conjugating \(z_2\) and
obtained from \(Q^{|l_2|}_j(\underline{{z}} \wedge \underline{{w}})\) by complex conjugating \(z_2\) and \(w_2\).
Lemma 5.2
Let \(l_1,l_2\in {\mathbb {Z}}\) with \(l_1\ge -l_2>0\). Then the polynomials
for \(k=0,\ldots , l_1+l_2\) and \(m=0,\ldots , l_1-l_2\), form an orthogonal basis of \({\mathcal {H}}_{(l_1,l_2)}({\mathbb {R}}^4)\).
Proof
The proof follows again by considering holomorphic, antiholomorphic and mixed degrees of the polynomials. \(\square \)
5.3 L-Representations: Spinor-Valued Monogenics in \({\mathbb {R}}^4\)
In this section we construct the representations associated to the weights \(\big (l_1,l_2\big )\) with \(l_1,l_2 \in {\mathbb {Z}}+\frac{1}{2}\), and \(l_1\ge |l_2|.\) First we consider the case \(l_2\ge 0\) and the corresponding \(\mathrm L\)-representation on the space of \({\mathcal {S}}_4^+\)-valued simplicial monogenic polynomials. From (5.7), (5.8), and (5.19) we obtain
where we used \(\sum _{i+j=-1} S_{m,i}^{l_1-l_2}(\underline{{z}})\,Q_j^{l_2-\frac{1}{2}}(\underline{{z}}\wedge \underline{{w}})=\sum _{i+j=l_1+l_2} S_{m,i}^{l_1-l_2}(\underline{{z}})\,Q_j^{l_2-\frac{1}{2}}(\underline{{z}}\wedge \underline{{w}})=0\). Due to the orthogonality between our basis functions of \({\mathcal {H}}_{(l_1,l_2)}({\mathbb {R}}^4)\) we conclude:
Corollary 5.3
Let \(l_1,l_2\in {\mathbb {N}}_0+\frac{1}{2}\) with \(l_1\ge l_2\). Then the monogenic polynomials
for \(k=0,\ldots ,l_1+l_2\) and \(m=0,\ldots ,l_1-l_2\) form an orthogonal basis of \({\mathcal {M}}_{(l_1,l_2)}({\mathbb {R}}^4)\).
The calculations for the representations of weight \(\big (l_1,l_2\big )\) with \(l_1,l_2 \in {\mathbb {Z}}+\frac{1}{2}\) and \(l_1 \ge - l_2 >0\) on the space of \({\mathcal {S}}_4^-\)-valued polynomials are similar. Indeed,
where we used \(\sum _{i+j=-1} \widetilde{S}_{k,i}^{l_1+l_2}(\underline{{z}})\,{\widetilde{Q}}_j^{l_2+\frac{1}{2}}(\underline{{z}}\wedge \underline{{w}})=\sum _{i+j=l_1-l_2} \widetilde{S}_{k,i}^{l_1+l_2}(\underline{{z}})\,{\widetilde{Q}}_j^{l_2+\frac{1}{2}}(\underline{{z}}\wedge \underline{{w}})=0\). Again we conclude:
Corollary 5.4
Let \(l_1,l_2\in {\mathbb {Z}}+\frac{1}{2}\) with \(l_1\ge -l_2>0\). Then the monogenic polynomials
for \(k=0,\ldots , l_1+l_2\) and \(m=0,\ldots , l_1-l_2\) form an orthogonal basis of \({\mathcal {M}}_{(l_1,l_2)}({\mathbb {R}}^4)\).
5.4 Matrix Coefficients
To describe all \(\textrm{Spin}(4)\) representations \(\xi _{(l_1,l_2)}\), 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 \(\textrm{Spin}(4)\) representations \(\xi _{(l_1,l_2)}\) for all \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\). This choice of bases allows to make direct use of the tensor product structure of these representations. By construction, we have
where \({\textbf{s}} = s \omega _+ + q \omega _-\) and \(\otimes \) denotes the Kronecker product of the matrices. Each \(\xi _{(l_1,l_2)}({\textbf{s}})\) is a unitary matrix of dimension
Figure 1 shows the infinite triangle of \(\textrm{Spin}(4)\) representations acting on harmonic modules \({\mathcal {H}}\) and monogenic modules \({\mathcal {M}}\). At the borders of the infinite triangle, we can observe the \(\textrm{Spin}(3)\) representations \(\xi ^{\textrm{Spin}(3)}_{\frac{l_1+l_2}{2}}\otimes 1\) in red and the \(\textrm{Spin}(3)\) representations \(1\otimes \xi ^{\textrm{Spin}(3)}_{\frac{l_1-l_2}{2}}\) in blue.
Harmonic and monogenic modules of \(\textrm{Spin}(4)\) representations \(\xi _{(l_1,l_2)}\)
5.5 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.
Definition 5.5
For all \((l_1,l_2) \in \Gamma _{\textrm{Spin}(4)} \) we define the following matrices:
where \(a_{\pm }(l_1\pm l_2)\) and \(b_{\pm }(l_1 \pm l_2)\) are the matrices given in Definition 4.2.
Using the matrices just defined we can transfer the recurrence relations given in Theorem 4.3 to the \(\textrm{Spin}(4)\) case. To simplify the expressions we use \(A_{\pm }\) instead of \(A_{\pm }(l_1,l_2)\) 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 \((l_1,l_2) \in \Gamma _{\textrm{Spin}(4)} \) the three term recurrence relations
hold true, where we used the notation
for the neighbouring weights in the weight lattice.
Proof
Theorem 4.3 together with the mixed product property \((A \otimes B)(C\otimes D)=(AC)\otimes (BD)\) of the Kronecker product yields
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 \(\pm 1\). \(\square \)
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 \(a_\pm \) and \(b_\pm \) 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 \((l_1,l_2) \in \Gamma _{\textrm{Spin}(4)} \) the following matrix recurrence relations hold
where every expression out of domain is interpreted as zero.
5.6 Differential Relations for Matrix Coefficients
The matrix coefficients of the \(\textrm{Spin}(4)\) 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 \(q=(q_1,q_2) \in {\mathbb {S}}^3\) and \(s=(s_1,s_2) \in {\mathbb {S}}^3\) of unit quaternions as coordinates for \({\textbf{s}}= s\omega _++q\omega _- \in \textrm{Spin}(4)\), cf. Sect. 5.
Definition 5.9
On \(\textrm{Spin}(4)\) 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 \({\textbf{s}},{\textbf{s}}_{0} \in \textrm{Spin}(4)\) be given by \({\textbf{s}}=s \omega _+ + q\omega _-\) and \({\textbf{s}}_{0}=s_0 \omega _+ + q_0\omega _-\) with \(q,s,q_0,s_0 \in {\mathbb {S}}^3\). Then by the properties of the idempotents \(\omega _{\pm }\) it is easy to see that the left translation on \(\textrm{Spin}(4)\) is given by \({\textbf{s}}_0^{-1} {\textbf{s}} = {\overline{{s}}}_0 s \,\omega _+ + {\overline{{q}}}_0q\,\omega _-\), while the right translation is given by \( {\textbf{s}} {\textbf{s}}_0 = s {s}_0 \,\omega _+ + q {q}_0 \,\omega _-\). 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).
Theorem 5.10
The operators \(\partial _{q\pm }, \partial _{s\pm }\) are left invariant and the operators \(\partial _{q\pm }^\dagger , \partial _{s\pm }^\dagger \) are right invariant on \(\textrm{Spin}(4)\), that is,
together with
We now determine derivatives of our representations. Using the matrix relations (4.24a)–(4.24d) we can obtain the following differential relations.
Theorem 5.11
For all \((l_1,l_2) \in \Gamma _{\textrm{Spin}(4)}\) the following identities
hold true with
and for \(\sigma _\pm \) given by (4.23).
Proof
We prove only the identity for \(\partial _{q+}^\dagger \), the remaining ones are obtained analogously. By using (5.33), (4.24c), and the mixed-product property for the Kronecker product, we obtain
\(\square \)
Example 5.12
On the level of matrix coefficients this allows to switch around between different indices. We only give some formulas
where \(r=l_1+l_2+1\), \(p=l_1-l_2+1\), \(\delta ^+=1\), \(\delta ^-=0\), and every expression out of domain is interpreted as zero.
Corollary 5.13
For all \((l_1,l_2) \in \Gamma _{\textrm{Spin}(4)}\) the following relations hold
for \(\sigma _\pm \) and \(\sigma _\pm ^\dagger \) given by (4.23).
6 Calculus
6.1 Left Invariant Differential Operators
The differential operators \(\partial _{q\pm }\), \(\partial _{s\pm }\) 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 \(\textrm{Spin}(4)\), i.e. for the Lie algebra \(\mathfrak {spin}(4)\).
Note, that the differential operators \(\partial _{q\pm }\) and \(\partial _{s\pm }\) are complex derivatives and that on a formal level \(\partial _{q+}^* = -\partial _{q-}\). To obtain elements of the Lie algebra, we form combinations of them.
Definition 6.1
We define the left invariant differential operators
Remark 6.2
The operators \(\partial _{q+}\) and \(\partial _{s+}\) are sometimes called creation operators, while the operators \(\partial _{q-}\) and \(\partial _{s-}\) are called annihilator operators (cf. [23, Remark 12.2.3.] where the operators are denoted by \(\partial _+\) and \(\partial _-,\) and [28, p. 140] where the operators are denoted by \({\hat{H}}_+\) and \({\hat{H}}_-\)). It is also customary to define
and denote this as the neutral operator. The operator \(\mathrm D_{3q}\) can thus be written as \(\mathrm D_{3q}=-{\textbf{i}}\partial _{q0}\).
Proposition 6.3
The commutator relations hold
together with \([\mathrm D_{\nu q},\mathrm D_{\mu s}]\) for all \(\mu ,\nu \in \{1,2,3\}\). Furthermore, the set \(\{D_{\mu q},\mathrm D_{\nu s} \mid \mu ,\nu =1,2,3\}\) forms an (orthonormal) basis of the Lie algebra \(\mathfrak {spin}(4)\).
Proof
The last identity follows from the direct product decomposition of \(\textrm{Spin}(4)\). 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
such that
Therefore, we obtain
together with
\(\square \)
The Laplace–Beltrami operator \({\mathcal {L}}\) on \(\textrm{Spin}(4)\) is given by
If we denote by \({\mathcal {H}}^{(l_1,l_2)}\) the complex linear span of the matrix coefficients of \(\xi _{(l_1,l_2)}\),
the following theorem holds true.
Theorem 6.4
The space \({\mathcal {H}}^{(l_1,l_2)}\) is an eigenspace of \({\mathcal {L}}\) with eigenvalue
Proof
This follows directly from (6.8) in combination with the formulas of Theorem 4.5 for the matrix coefficients of the \(\textrm{Spin}(3)\)-representation. \(\square \)
For later use we define
6.2 The Group Fourier Transform on \(\textrm{Spin}(4)\)
First, we introduce notation and recall some basic facts. Then the characterisations of function spaces on \(\textrm{Spin}(4)\) 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 \(\textrm{Spin}(4)\) is given in terms of all equivalence classes of irreducible representations
For an integrable function \(f\in \mathrm L^1(\textrm{Spin}(4))\) we define
where we integrate with respect to the normalised Haar measure on the group \(\textrm{Spin}(4)\). Note that by uniqueness of the Haar measure and by the direct product structure \(\textrm{Spin}(4)\simeq \textrm{Spin}(3)\times \textrm{Spin}(3)\) the Haar measure on \(\textrm{Spin}(4)\) is also the tensor product of the normalised Haar measures on both factors.
The Fourier transform maps \(\mathrm L^2(\textrm{Spin}(4))\) unitarily onto a sequence space. For this we define \(\ell ^2:=\ell ^2(\widehat{\textrm{Spin}(4)})\) to be the space of all sequences
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 \(\mathrm L^2(\textrm{Spin}(4))\) to \(\ell ^2(\widehat{\textrm{Spin}(4)})\) with inverse
and Plancherel identity
Remark 6.6
In the particular case \(f(s\omega _+ + q \omega _-)=f_1(q)f_2(s)\) with \(f_i\in \mathrm L^2(\textrm{Spin}(3))\), the Kronecker product representation of the Fourier coefficients
in terms of the \(\textrm{Spin}(3)\) Fourier transforms imply
Here we made use of \({{\,\textrm{tr}\,}}(A\otimes B) = {{\,\textrm{tr}\,}}(A){{\,\textrm{tr}\,}}(B)\) together with (5.34). This also allows to split the Plancherel formula into a double sum
based on the product formula \(\Vert A\otimes B\Vert _{\textrm{HS}} = \Vert A\Vert _{\textrm{HS}} \Vert B\Vert _{\textrm{HS}}\) for Hilbert-Schmidt norms.
The group Fourier transform extends naturally to distributions. The space \({\mathcal {D}}'(\textrm{Spin}(4))\) of distributions is the topological dual space of smooth functions \(C^\infty (\textrm{Spin}(4))\). As usual for a function \(f \in L^1(\textrm{Spin}(4))\) and \(\varphi \in C^\infty (\textrm{Spin}(4))\) we define \(T_f \in {\mathcal {D}}'(\textrm{Spin}(4))\) by
and use the same notation for the dual pairing between distributions and functions. For a distribution \(T \in {\mathcal {D}}'(\textrm{Spin}(4))\) its Fourier transform \({\widehat{T}}\) is defined by \({\widehat{T}}(l_1,l_2)=\langle T, \xi ^*_{(l_1,l_2)} \rangle \).
6.3 Function Spaces
Sobolev spaces are characterised in terms of the Laplacian. Thus, for \(r\in {\mathbb {R}}\) the space \(\mathrm H^r(\textrm{Spin}(4))\) has the familiar characterisation
in terms of the group Fourier transform. This allows to characterise spaces of smooth functions and of distributions. In the following, we denote by \({\mathfrak {s}}:=\mathfrak s(\widehat{\textrm{Spin}(4)})\) the space of rapidly decaying matrix sequences
such that the dimension of \(\rho (l_1,l_2)\) 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 \(\langle \xi _{(l_1,l_2)}\rangle \) in the dimension \(d_{(l_1,l_2)}\). We also denote by \({\mathfrak {s}}':=\mathfrak s'(\widehat{\textrm{Spin}(4)})\) the space of all such matrix sequences with
for one number N. Both are equipped with their natural locally convex topology arising from (6.24) and (6.25), respectively. Using this topology the spaces \({\mathfrak {s}}\) and \({\mathfrak {s}}'\) are dual to each other.
Theorem 6.7
The Fourier transform provides isomorphisms
-
1.
\({\mathcal {F}}: C^\infty (\textrm{Spin}(4)) \rightarrow {\mathfrak {s}}\)
-
2.
\({\mathcal {F}}: {\mathcal {D}}' (\textrm{Spin}(4)) \rightarrow {\mathfrak {s}}'\)
and thus characterises smooth functions in terms of rapidly decaying sequences of Fourier coefficients and distributions in terms of moderately growing sequences of Fourier coefficients.
6.4 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 \(\textrm{Spin}(4)\).
6.4.1 Symbolic Calculus of Invariant Operators
Now, we discuss the symbolic calculus for operators on \(\textrm{Spin}(4)\). We recall from Sect. 2 that an operator \(A: C^\infty \rightarrow {\mathcal {D}}'\) is said to be left-invariant if it commutes with left translations, i.e. if \(A\circ L_{{\textbf{s}}_0} = L_{{\textbf{s}}_0} \circ A\) for any \({\textbf{s}}_0 = q_0\omega _+ +s_0\omega _-\) with L given by (5.40a). Any such operator A can be expressed in terms of a left-symbol \(\sigma _A\) in the form of a Fourier multiplier
Similarly, right invariant operators \(B: C^\infty \rightarrow {\mathcal {D}}'\) with \(B\circ R_{{\textbf{s}}_0} = R_{{\textbf{s}}_0} \circ B\) correspond to right multiplication of the Fourier coefficients by a right-symbol \(\sigma _B^\dagger \)
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 \(\textrm{Spin}(4)\). The formulas for symbols are a consequence of Theorem 5.11 and Corollary 5.13 and given in terms of \(\sigma _\pm \) from (4.23) and with \(\sigma _0=[\sigma _+,\sigma _-]\), i.e. with \(\sigma _0(m/2)_{ij} = \frac{1}{2} (m-2j)\delta _{ij}\) for every \(m \in {\mathbb {N}}_0\) and \(0 \le i,j \le m\).
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 \(\Delta : \mathfrak s' \rightarrow {\mathfrak {s}}'\) acting on moderate matrix sequences is defined in terms of a function \(\varphi \in C^\infty (\textrm{Spin}(4))\) via \(\Delta {\widehat{f}}= \widehat{\varphi f}\) using the group Fourier transform \({{\widehat{f}}}\) of distributions \(f\in \mathcal D'(\textrm{Spin}(4))\). If \(\varphi \) vanishes to first order at the identity, we call \(\Delta \) 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 \(\xi _{(\frac{1}{2},\pm \frac{1}{2})}\). 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.
We recall the following result from [25]. We will use multi-index notation for difference operators and write \(\Delta ^\alpha = (\Delta _{q}^{--})^{\alpha _1}\cdots (\Delta _{s}^{++})^{\alpha _8}\) for \(\alpha \in {\mathbb {N}}_0^{8}\) and formulate the multiplier theorem [25, Theorem 3.5] for the particular case of \(\textrm{Spin}(4)\).
Theorem 6.8
(Multiplier theorem, [25]) Let \(A: C^\infty (\textrm{Spin}(4)) \rightarrow \mathcal D'(\textrm{Spin}(4))\) be a left-invariant operator on \(\textrm{Spin}(4)\) with left-symbol \(\sigma _A\) satisfying
with the particular second order difference \(\triangle \!\!\!\!* = \Delta _q^{++} + \Delta _q^{--} + \Delta _s^{++} + \Delta _s^{--} \in \textrm{diff}^2(\widehat{\textrm{Spin}(4)})\). Then A is bounded on \(\mathrm L^p(\textrm{Spin}(4))\) for \(1<p<\infty \) and of weak type (1, 1).
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
Proof
We consider the first difference operator \(\Delta _q^{--}\). 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 \(\Gamma _{\textrm{Spin}(4)}\) the matrices \(A_\pm \) and \(B_\pm \) vanish, the last sums can be rewritten as sums over \(\Gamma _{\textrm{Spin}(4)}\). Making use of the identities
and subtracting by \(\sigma (l_1,l_2)\) we obtain the expression for \(\Delta _q^{--}\). The remaining formulas are obtained similarly from Theorem 5.6. \(\square \)
Corollary 6.10
When \(\sigma (l_1,l_2)=\rho (\frac{l_1+l_2}{2})\otimes \tau (\frac{l_1-l_2}{2})\) is of tensor product form, the difference operators of order 1 given in Table 3 are explicitly given by
where
are the difference operators on \(\textrm{Spin}(3)\) given in terms of \(a_\pm \) and \(b_\pm \) from Definition 4.2.
Corollary 6.11
A matrix sequence is of the form \(\sigma (l_1,l_2) = \rho (\frac{l_1+l_2}{2}) \otimes \mathrm I_{l_1-l_2+1}\) if and only if
Similarly, it is of the form \(\sigma (l_1,l_2) = \mathrm I_{l_1+l_2+1} \otimes \tau (\frac{l_1+l_2}{2})\) if and only if
In Table 4 we show the difference operators applied to the symbols of the elementary first order differential operators \(\partial _{q\nu },\partial _{s\mu },~ \nu ,\mu \in \{0,+,-\}\) and the Laplacian \({\mathcal {L}}\) 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
and similarly for \(\Delta _s^{ij}\). We show how to apply this to compute the difference operators acting on the symbol of the partial Laplacians
and thus the Laplacian \({\mathcal {L}}\). As \(\sigma _{{\mathcal {L}}_q} = (-\frac{1}{2} ( \sigma _+ \sigma _- + \sigma _-\sigma _+) - \sigma _0^2)\otimes \mathrm I\) it follows that
where we made use of \(\Delta ^{++}\sigma _+=0\) and \(\Delta ^{+-}\sigma _0=0\) to simplify the expression. The calculation for the remaining differences \(\Delta ^{\pm \pm }_q \sigma _{ \mathcal L_q}\) and \(\Delta ^{\pm \pm }_s \sigma _{ {\mathcal {L}}_s}\) is similar and due to \(\Delta _s^{\pm \pm }\sigma _{{\mathcal {L}}_q}= \Delta _q^{\pm \pm }\sigma _{{\mathcal {L}}_s} = 0\) the formulas for differences applied to \(\sigma _{{\mathcal {L}}}\) follow.
6.4.2 Symbolic Calculus of Pseudodifferential Operators
A continuous linear operator A mapping \(C^\infty (\textrm{Spin}(4))\) to \({\mathcal {D}}'(\textrm{Spin}(4))\) can be characterised by its matrix-valued full left-symbol
defined by
By definition
holds true as \({\mathcal {D}}'\)-convergent series. For A and \(\sigma _A\) related by (6.41) we write \(A={{\,\textrm{Op}\,}}(\sigma _A).\)
In [21, 23] the Hörmander class \(\Psi ^k(G)\) of pseudo-differential operators of order k on a compact Lie group G was characterised in terms of these full symbols, also \((\rho ,\delta )\)-classes \(\Psi ^k_{\rho ,\delta }\) have been introduced there. We recall this and the resulting characterisations of ellipticity and hypoellipticity of operators for the particular case of \(\textrm{Spin}(4)\).
Adapted to the difference operators \(\Delta ^\alpha \), \(\alpha \in {\mathbb {N}}_0^8\), we find left-invariant differential operators \(\partial ^{(\alpha )}\) of order \(|\alpha |\) 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
and point out that \(\partial ^\alpha \ne \partial ^{(\alpha )}\) are different operators. However, \(\partial ^{(\alpha )}\) is a complex linear combination of \(\partial ^\beta \) with \(|\beta |\le |\alpha |\).
Theorem 6.12
([23, Theorems 2.2 and 2.6])] Let A be a linear continuous operator from \(C^\infty (\textrm{Spin}(4))\) to \({\mathcal {D}}'(\textrm{Spin}(4))\) with matrix-valued full symbol \(\sigma _A({\textbf{s}},l_1,l_2) \in {\mathbb {C}}^{d_{(l_1,l_2)}\times d_{(l_1,l_2)}}\). Then \(A \in \Psi ^k(\textrm{Spin}(4))\) if and only if
for all multi-indices \(\alpha \) and \(\beta \) uniformly in \(\textbf{s}\in \textrm{Spin}(4)\) and \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\).
Moreover, the rapid off-diagonal decay property of the symbol
holds true uniformly in \({\textbf{s}} \in \textrm{Spin}(4)\), \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\) and \(0\le i,j < d_{(l_1,l_2)}\).
In order to obtain symbolic estimates for the (multiplicative) inverse of a symbol \(\sigma =\sigma ({\textbf{s}}, l_1, l_2)\) we will use the following result taking from [23]. It implies ellipticity and also hypoellipticity of certain operators within the calculus.
Lemma 6.13
([23, Lem. 4.5]) Let \(k \ge k_0\) and \(0 \le \delta < \rho \le 1\). Let the matrix symbol \(\sigma ({\textbf{s}}, l_1, l_2)\) satisfy the \((\rho ,\delta )\)-estimates of order k
for all multi-indices \(\alpha \) and \(\beta \) and uniformly in \(\textbf{s}\in \textrm{Spin}(4)\) and \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\). Assume further that \(\sigma ({\textbf{s}}, l_1, l_2)\) is invertible for all \({\textbf{s}} \in \textrm{Spin}(4)\) and \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\) and satisfies
for all \({\textbf{s}}\in \textrm{Spin}(4)\) and \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\) and if \(k_0 < k\) in addition that
for all \({\textbf{s}}\in \textrm{Spin}(4)\) and \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\).
Then the symbol \(\sigma ^{-1}({\textbf{s}},l_1,l_2)=\sigma (\textbf{s}, l_1, l_2)^{-1}\) satisfies the \((\rho ,\delta )\)-estimates of order \(-k_0\)
for all multi-indices \(\alpha \) and \(\beta \) and uniformly in \(\textbf{s}\in \textrm{Spin}(4)\) and \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\).
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 \(\textrm{Spin}(4)\) later on. First we give a characterisation of the elliptic operators in \(\Psi ^{k}(\textrm{Spin}(4))\) in terms of their global symbols.
Theorem 6.14
[23, Thm. 4.1] An operator \(A \in \Psi ^{k}(\textrm{Spin}(4))\) is elliptic if and only if its matrix valued symbol \(\sigma _A({\textbf{s}}, l_1, l_2)\) is invertible for all but finitely many \((l_1,l_2) \in \Gamma _{\textrm{Spin}(4)}\) and for all such \((l_1,l_2)\) satisfies
uniformly in \({\textbf{s}} \in \textrm{Spin}(4)\).
We say that a symbol \(\sigma _A\) belongs to the symbol class \(S_{\rho ,\delta }^k(\textrm{Spin}(4))\) if estimate (6.45) holds true for all multi-indices \(\alpha \), \(\beta \) and uniformly in the remaining variables. We recall the existence of parametrices for operators with such symbols under suitable conditions on the multiplicative inverse of the symbol. It is again taken from [23] and provides an analogue of the well-known hypoellipticity result of Hörmander [15], requiring conditions on lower order terms of the symbol.
Theorem 6.15
([23, Thm. 5.1.]) Let \(k \ge k_0\) and \(1 \ge \rho > \delta \ge 0\). Let \(A \in {{\,\textrm{Op}\,}}(S_{\rho ,\delta }^k(\textrm{Spin}(4)))\) be a pseudo-differential operator with the matrix-valued symbol \(\sigma _A=\sigma _A({\textbf{s}}, l_1, l_2) \in S_{\rho ,\delta }^k(\textrm{Spin}(4))\) which is invertible for all but finitely many \((l_1,l_2) \in \Gamma _{\textrm{Spin}(4)}\) and satisfies for all such \((l_1,l_2)\)
uniformly in \({\textbf{s}} \in \textrm{Spin}(4)\). Assume further that
for all multi-indices \(\alpha \) and \(\beta \), all \({\textbf{s}} \in \textrm{Spin}(4)\), and all but finitely many \((l_1,l_2)\). Then there exists an operator \(B \in {{\,\textrm{Op}\,}}(S_{\rho ,\delta }^{-k_0}(\textrm{Spin}(4))\) such that \(AB-I\) and \(BA-I\) map \({\mathcal {D}}'(\textrm{Spin}(4))\) to \(C^\infty (\textrm{Spin}(4))\).
Consequently, A is locally hypoelliptic and
Note, that the parametrix B provided by this theorem satisfies the subelliptic estimates \(\Vert Bf\Vert _{H^r} \lesssim \Vert f\Vert _{{H}^{r+k_0}}\) with \(k_0\) independent of \(r \in {\mathbb {R}}\).
7 Examples
To conclude this paper we provide examples of operators on \(\textrm{Spin}(4)\) having interesting ellipticity and hypoellipticity properties.
Example 7.1
(See [23, Example 5.2]) Consider the operator \(\mathrm D_{3q}+c,\) with \(c \in {\mathbb {C}}\). Using the identity \(\mathrm D_{3q}= -\frac{{\textbf{i}}}{2} \partial _{q0}\) and the symbols given in Table 1, we have
and this symbol is invertible for all \((l_1,l_2)\)-representations if and only if \({\textbf{i}}c \notin \frac{1}{2}{\mathbb {Z}}\). The inverse satisfies \((\mathrm D_{3q}+c )^{-1}\in {{\,\textrm{Op}\,}}(S_{0,0}^0(\textrm{Spin}(4)))\) since
together with
by using Corollary 6.10. Therefore, the operators \(\mathrm D_{3q}+c\) satisfy subelliptic estimates with loss of one derivative and are thus globally hypoelliptic.
Example 7.2
Any left-invariant vector field on \(\textrm{Spin}(4)\) can be conjugated by an inner automorphism to a constant multiple of
with parameter \(\varkappa \in {\mathbb {R}}\). We investigate this in dependence of \(\varkappa \). Considering the symbol of the operator we obtain
where \(\sigma _0(\frac{m}{2}) = \frac{1}{2}(m-2j) \delta _{ij} = \textrm{diag}(-\frac{m}{2}, -\frac{m}{2}+1,\ldots ,\frac{m}{2}--1,\frac{m}{2})\). This matrix is diagonal with entries given as sums
for \(0\le j \le l_1+l_2\) and \(0\le k \le l_1-l_2\). Both terms vanish for some admissible k and l if both \(l_1+l_2\) and \(l_1-l_2\) are even. Thus, there are always infinitely many \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\) for which \(\sigma _{\mathrm X}(l_1,l_2)\) is not invertible and none of the left-invariant vector fields on \(\textrm{Spin}(4)\) can be globally hypoelliptic.
This is in contrast to tori \({\mathbb {T}}^n\), where hypoelliptic left-invariant vector fields are known to exist, see [12] and [6]. In [17] a new notion of hypoellipticity was introduced. We say that \(\mathrm X\) is hypoelliptic modulo \(\ker \mathrm X\), if for \(f\in {\mathcal {D}}'\) with \(\mathrm Xf\in C^\infty \) we find an element \(g\in \ker \mathrm X\) such that \(f-g\in C^\infty \).
The vector field \(\mathrm X\) is globally hypoelliptic modulo \(\ker \mathrm X\) if (and only if) \(\varkappa \) is irrational and non-Liouville. Indeed, such a number \(\varkappa \) can only be approximated to finite order by rationals (cf. [14, §11.7]) and hence
with constants c and \(\mu \) for all \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\) and \(0\le j \le l_1+l_2\) and \(0\le k \le l_1-l_2\) such that \((\frac{l_1+l_2}{2} - j) + \varkappa ( \frac{l_1-l_2}{2} - k)\ne 0\). The term in parantheses on the right hand side is clearly bounded by \(\langle \xi _{(l_1,l_2)}\rangle \) and thus we obtain a lower bound on the symbol by a multiple of \(\langle \xi _{(l_1,l_2)}\rangle ^{-\mu }\). This implies the subelliptic estimate
where the Fourier coefficients of g equal those of f on the zeros of \(\sigma _{\mathrm X}\) and in turn the global hypoellipticity modulo \(\ker \mathrm X\) follows.
Example 7.3
Next, we consider the (partial) Laplacians
and
together with the Laplacian \({\mathcal {L}} = {\mathcal {L}}_s + {\mathcal {L}}_q\). Their symbols are given by
and
(see also Theorem 6.4). While the latter one is elliptic, the symbols of the two partial Laplacians \({\mathcal {L}}_q\) and \({\mathcal {L}}_s\) vanish for \(l_1=-l_2\) and for \(l_1=l_2\), respectively.
Clearly any combination \({\mathcal {L}}_q + \varkappa {\mathcal {L}}_s\) with \(\varkappa >0\) is elliptic. For \(\varkappa <0\) the behaviour of this operator depends on number theoretic properties of \(\varkappa \) as the next example shows.
Example 7.4
The ultra-hyperbolic operator
with \(\varkappa >0\) is hypoelliptic for irrational non-Liouville numbers \(\varkappa \). To see this, we consider its symbol
This is clearly invertible for irrational \(\varkappa \) and \((l_1,l_2)\ne (0,0)\). Furthermore, if \(\varkappa \) is non-Liouville we find a maximal approximation order and hence there is a number \(\mu \) such that \(\varkappa \) can not be rationally approximated to this order (see e.g. [14, Chapter §11.4 und §11.7]) and hence there is a constant \(C>0\) such that there are at most finitely many couples \((l_1,l_2)\) with \(l_1\ne l_2\) and
Thus, apart from those \((l_1,l_2)\), it follows that
together with the estimate
Thus, the operator satisfies the subelliptic estimate
between Sobolev spaces with loss of \(2\mu \) derivatives. We leave it open whether the parametrix to this operator is pseudodifferential in our calculus.
Example 7.5
We consider the following sub-Laplacian on \(\textrm{Spin}(4)\)
with symbol given by the diagonal matrix
As all the entries are non-zero, we conclude that the operator \({\mathcal {L}}_{\textrm{sub}}\) has trivial null space. As symbol of a second order differential operator it belongs to the class \(S^2(\textrm{Spin}(4))\) and thus satisfies the norm estimate
for all multi-indices \(\alpha \). The operator is not elliptic, but it is subelliptic with loss of one derivative. We are going to show this next by appealing to Theorem 6.14. The pointwise inverse of the symbol satisfies the norm estimate
for all admissible \((l_1,l_2)\) with \(C= \sqrt{7}\) being the sharp constant. In particular, we obtain the subelliptic estimate
for all Sobolev regularities \(r \in {\mathbb {R}}\).
The condition (6.51) of Theorem 6.14 is satisfied with \(\rho =\frac{1}{2}\) and for \({\alpha }=0\) or \(|{\alpha }|\ge 2\). Hence, it remains to check the case of first order differences. For this we use \(\Delta _q^{--} \sigma _{{\mathcal {L}}_{\textrm{sub}}} = \Delta _q^{++} \sigma _{{\mathcal {L}}_{\textrm{sub}}} =-\frac{1}{2} \mathrm I_d\) together with
and an analogous statement for \(\Delta _q^{-+}\). Concerning the differences in s we can make use of the relations \(\Delta _s^{--} \sigma _{{\mathcal {L}}_{\textrm{sub}}} = -\sigma _{\partial _{s0}} - \frac{3}{4}\mathrm I_d\), \(\Delta _s^{++} \sigma _{{\mathcal {L}}_{\textrm{sub}}} = \sigma _{\partial _{s0}} - \frac{3}{4}\mathrm I_d\), \(\Delta _s^{+-} \sigma _{{\mathcal {L}}_{\textrm{sub}}} = -\sigma _{\partial _{s+}}\), and \(\Delta _s^{--} \sigma _{{\mathcal {L}}_{\textrm{sub}}} = -\sigma _{\partial _{s-}}\) directly arising from the partial Laplacian.
Hence, \({\mathcal {L}}_{\textrm{sub}}\) has a pseudo-differential parametrix \({\mathcal {L}}_\textrm{sub}^{\natural } \in {{\,\textrm{Op}\,}}(S_{\frac{1}{2},0}^{-1}(\textrm{Spin}(4)))\).
Further examples can be constructed along the lines of [23, Section 5]. As proofs are similar, we only provide the results.
Example 7.6
The following analogue of the heat operator on \(\textrm{Spin}(4)\)
has a parametrix \(\mathrm H^\natural \in {{\,\textrm{Op}\,}}(S_{\frac{1}{2},0}^{-1}(\textrm{Spin}(4)))\) and, consequently, it satisfies the sub-elliptic estimate
for all Sobolev regularities \(r \in {\mathbb {R}}\).
Example 7.7
The operators
are analogues of the Schrödinger operator on \(\textrm{Spin}(4)\). These operators have non-trivial distributions in their null-spaces and can thus be not hypoelliptic.
However, the operators \(\mathrm S_\pm + c\) with \(c \in {\mathbb {C}}{\setminus }{\mathbb {R}}\) are globally hypoelliptic.
Our last example shows a differential operator with non-constant coefficients which cannot be written as tensor product of \(\textrm{Spin}(3)\) operators.
Example 7.8
Let us consider the operator
with \(a,b \in C^\infty (\textrm{Spin}(4))\) such that \(\Re (\theta a({\textbf{s}}))\ge c_a>0\) and \(\Re (\theta b({\textbf{s}}))\ge c_b>0\) for all \({\textbf{s}} \in \textrm{Spin}(4)\) and for a fixed complex number \(\theta \in {\mathbb {C}}\setminus \{0\}\). The full symbol of the operator \(\mathrm A\) can be written as the Kronecker sum
with
Since \(\Re (\theta a)\) and \(\Re (\theta b)\) are both positive and \({(l-2j)^2} - {l(l+2)} \le -2\,l\) and \(0 \le j \le l\) and every \(l\in {\mathbb {N}}_0\), we conclude that the symbol is invertible for all \({\textbf{s}}\in \textrm{Spin}(4)\) and all \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\setminus \{(0,0)\}\) with eigenvalues satisfying
where \(\textrm{Spec}\,\sigma _A\) denotes the spectrum of the matrix \(\sigma _A\). As symbol of a second order differential operator we also know that \(\sigma _{\mathrm A} \in S^2(\textrm{Spin}(4))\) and it remains to ask for symbolic properties of \(\sigma _{\mathrm A}^{-1}({\textbf{s}}, l_1, l_2)\) for \((l_1,l_2)\ne (0,0)\). As the absolutely smallest eigenvalue corresponds to the operator norm of the inverse, we observe
for all \((l_1,l_2)\in \Gamma _{\textrm{Spin}(4)}\setminus \{(0,0)\}\). Hence, condition (6.51) of Theorem 6.15 holds with \(\rho =\frac{1}{2}\) and \(\delta =0\). Thus, the operator \(\mathrm A\) has a pseudo-differential parametrix \(\mathrm A^{\natural } \in {{\,\textrm{Op}\,}}(S_{\frac{1}{2},0}^{-1}(\textrm{Spin}(4)))\) and therefore satisfies the subelliptic estimate
for all \(s \in {\mathbb {R}}\).
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Acknowledgements
The work of P. Cerejeiras, M. Ferreira, and U. Kähler was supported by Portuguese funds through CIDMA-Center for Research and Development in Mathematics and Applications, and FCT–“Fundação para a Ciência e a Tecnologia”, within project UIDB/04106/2020 and UIDP/04106/2020. The present paper was supported by the project “Global operator calculi on compact and non-compact Lie groups”, Ações Integradas Luso-Alemãs – Acção No. A-42/16.
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Cerejeiras, P., Ferreira, M., Kähler, U. et al. Global Operator Calculus on Spin Groups. J Fourier Anal Appl 29, 32 (2023). https://doi.org/10.1007/s00041-023-10015-5
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DOI: https://doi.org/10.1007/s00041-023-10015-5
Keywords
- Spin group
- Spin representations
- Difference operators
- Pseudo-differential operators
- Fourier transform
- Microlocal analysis
- Elliptic operators
- Global hypoellipticity