Mathematical Physics, Analysis and Geometry - Volume 14

334 Pages·2011·6.04 MB·English

by  A. Boutet de Monvel, V. A. Marchenko, H. McKean (Editors)

Math Phys Anal Geom (2011) 14:1–20
DOI 10.1007/s11040-010-9085-8
Higher Spin Dirac Operators Between Spaces
of Simplicial Monogenics in Two Vector Variables
F. Brackx · D. Eelbode · L. Van de Voorde
Received: 30 November 2009 / Accepted: 20 October 2010 / Published online: 11 November 2010
© Springer Science+Business Media B.V. 2010
Abstract The higher spin Dirac operator Qk,l acting on functions taking values
1
in an irreducible representation space for so(m) with highest weight (k + , l +
2
1 1 1
, , . . . , ), with k, l ∈ N and k ⩾ l, is constructed. The structure of the kernel
2 2 2
space containing homogeneous polynomial solutions is then also studied.
Keywords Clifford analysis · Dirac operators · Higher spin
Mathematics Subject Classifications (2010) 15A66 · 30G35 · 22E46
1 Introduction
Consider an oriented spin manifold, i.e., a Riemannian manifold with a spin
structure which allows the construction of vector bundles whose underlying
symmetry group is Spin(m) rather than SO(m), see e.g., [17]. On such a
Riemannian spin manifold there is a whole system of conformally invariant,
elliptic, first-order differential operators acting on sections of an appropriate
spin bundle, whose existence and construction can be established through
F. Brackx · L. Van de Voorde (B)
Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering,
Ghent University, Ghent, Belgium
e-mail:
2 F. Brackx et al.
geometrical and representation theoretical methods, see e.g., [5, 13, 21, 22].
In Clifford analysis these operators are studied from a function theoretical
point of view, mainly focusing on their rotational invariance with respect to
the spin group Spin(m), or its Lie algebra so(m), and considering functions on
m
R instead of sections. The simplest example is the Dirac operator acting on
spinor-valued functions; we refer to the standard references [1, 11, 15]. Next in
line are the Rarita–Schwinger operator, acting on functions with values in the
3 1 1
irreducible so(m)-representation with highest weight ( , , . . . , ), and its gen-
2 2 2
eralizations to the case of functions taking values in irreducible representation
1 1 1
spaces with highest weight (k + , , . . . , ), see e.g., [7, 8]. Also higher spin
2 2 2
Dirac operators acting on spinor-valued forms have been studied in detail, see
e.g., [6, 20].
Our aim is to combine techniques from Clifford analysis and from repre-
sentation theory, in order to investigate, from the function theoretical point
of view, general higher spin Dirac operators acting between functions taking
values in an arbitrary finite-dimensional half-integer highest-weight represen-
tation. As the case of the Rarita–Schwinger operator (and its generalizations)
does not yet contain the seed from which the most general case can be
derived, we study, in this paper, the particular case of the operator acting on
functions taking values in the irreducible representation with highest weight
1 1 1 1
(k + , l + , , . . . , ), with k, l ∈ N and k ⩾ l. This is done using the elegant
2 2 2 2
framework of Clifford analysis in several vector variables.
2 Clifford Analysis and Definitions
m
Let (e , . . . , e ) be an orthonormal basis for the Euclidean space R . We
1 m
denote by Cm the complex universal Clifford algebra, generated by these
basis elements, its multiplication being governed by the relations e e +
i j
m
e
je i = −2δij, i, j = 1, . . . , m. The space R is embedded in Cm by identifying
∑
m
(x1, . . . , xm) with the real Clifford vector x =
j=1 e jx j. The multiplication of
two vectors x and y is given by xy = −⟨x, y⟩ + x ∧ y with
m
∑ ∑
⟨x, y⟩ = x jy j and x ∧ y = e
ie j(xi y j − x jyi)
j=1 1⩽i< j⩽m
the scalar-valued Euclidean inner product and the bivector-valued wedge
m
product respectively. The wedge product of a finite number of vectors in R
may also be defined using the Clifford product:
Definition 1 The wedge product of N Clifford vectors x ,. . ., x is defined as
1 N
∑
1
x ∧ . . . ∧ x := sgn(σ) x . . . x ,
1 N σ(1) σ (N)
N!
σ∈SN
where SN denotes the symmetric group in N elements.
Higher Spin Dirac Operators in Two Vector Variables 3
For convenience, we will work in odd dimension m = 2n + 1. In this case
there is a unique spinor space S, as opposed to the even-dimensional case
m = 2n where there are two spinor representations (often referred to as
even and odd spinors). However, these cases do not differ from each other
conceptually: in case of even dimension m = 2n, it suffices to take into
account that the vector-valued (higher spin) Dirac operator will change the
parity of the underlying values. The spinor space S should be thought of
as a minimal left ideal in Cm, which can be defined in terms of a primitive
1 1
idempotent; it is characterized by the highest weight ( , . . . , ) under the
2 2
standard multiplicative action of the spin group
⎧ ⎫
⎨ 2k ⎬
∏
m−1
Spin(m) = s j : k ∈ N , s j ∈ S ,
⎩ ⎭
j=1
m−1 m
with S the unit sphere in R . In case one prefers working with its Lie
algebra so(m), which can be identified with the subspace of bivectors in the
algebra Cm, the derived action should be used.
∑
m
The Dirac operator is denoted ∂
x = j=1 e j∂x j . It is an elliptic Spin(m)-
invariant first-order differential operator acting on spinor-valued functions
m 2 m
f (x) on R . It factorizes the Laplace operator: x = −∂
x on R . An S-valued
m
function f is monogenic in an open region ⊂ R if and only if it satisfies
∂ f = 0 in . For a detailed account of the theory of monogenic functions,
x
so called Euclidean Clifford analysis, we refer the reader to e.g., [1, 11, 15].
∑
We also mention the Euler operator Ex =
i xi∂xi , measuring the degree of
homogeneity in the variable x.
Irreducible (finite-dimensional) modules for the spin group can be de-
scribed in terms of spaces of traceless tensors satisfying certain symmetry
conditions expressed in terms of Young diagrams, see e.g., [14, 16], but they
can also be realized in terms of vector spaces of polynomials, see e.g., [10, 15].
We mention the following well-known examples from harmonic and Clifford
analysis: the vector space Hk of C-valued harmonic homogeneous polynomials
of degree k ∈ N corresponds to the irreducible Spin(m)-module with highest
weight (k, 0, . . . , 0), and the vector space Mk of spinor-valued monogenic
homogeneous polynomials of degree k forms an irreducible representation of
1 1 1
Spin(m) with highest weight (k + , , . . . , ).
2 2 2
In what follows, N ∈ N and ∂ is short for the Dirac operator ∂ .
i ui
Remark 1 In the sequel we will often need to refer to the highest weight of
a representation; to that end we introduce the short notation (λ1, . . . , λN)
′
for (λ1, . . . , λN, 0, . . . , 0) and denote by (λ1, . . . , λN) the highest weight (λ1 +
1 1 1 1
, . . . , λN + , , . . . , ).
2 2 2 2
Nm
Definition 2 A function f : R → C, (u , . . . , u ) ↦→ f(u , . . . , u ) is sim-
1 N 1 N
plicial harmonic if the following conditions are satisfied:
⟨∂ , ∂ ⟩ f = 0, i, j = 1, . . . , N
i j
⟨u , ∂ ⟩ f = 0, 1 ⩽ i < j ⩽ N.
i j
4 F. Brackx et al.
The vector space of C-valued simplicial harmonic polynomials, λi-
homogeneous in the variable u
i, will be denoted by Hλ1,...,λN (with λ1 ⩾ . . . ⩾
λN ⩾ 0 from now on).
Nm
Definition 3 A function f : R → S, (u , . . . , u ) ↦→ f(u , . . . , u ) is sim-
1 N 1 N
plicial monogenic if the following conditions are satisfied:
∂ f = 0, i = 1, . . . , N
i
⟨u , ∂ ⟩ f = 0, 1 ⩽ i < j ⩽ N.
i j
The vector space of S-valued simplicial monogenic polynomials, λi-
homogeneous in the variable u
i, will be denoted by Sλ1,...,λN (with λ1 ⩾ . . . ⩾
λN ⩾ 0 from now on).
Remark 2 It is clear that if a function is simplicial monogenic in an open region
Nm
of R , then each of its scalar components is simplicial harmonic in , or in
other words: Sλ
1,...,λN ⊂ Hλ1,...,λN ⊗ S.
Remark 3 The second condition in Definition 2 (respectively 3) implies that
an arbitrary polynomial pλ
1,...,λN ∈ Hλ1,...,λN (respectively Sλ1,...,λN ) can be iden-
tified with a C-valued (resp. S-valued) polynomial f depending only of a
number of specific wedge products of the vector variables:
pλ
1,...,λN (u1, u2, . . . , uN) = f (u1, u1 ∧ u2, u1 ∧ u2 ∧ u3, . . . , u1 ∧ u2 ∧ . . . ∧ uN).
For details we refer to [10], where it is also shown that the spaceHλ
1,...,λN cor-
responds to the irreducible Spin(m)-module with highest weight (λ1, . . . , λN),
with respect to the regular representation H on C-valued simplicial harmonic
polynomials given by
H(s) f (u , u ∧ u , . . . , u ∧ . . . ∧ u ) = f (su s, su ∧ u s, . . . , su ∧ . . . ∧ u s),
1 1 2 1 N 1 1 2 1 N
where s ∈ Spin(m). With respect to the regular representation L on S-valued
simpicial monogenic polynomials, i.e.,
L(s) f (u , u ∧ u , . . . , u ∧ . . . ∧ u ) = sf (su s, su ∧ u s, . . . , su ∧ . . . ∧ u s),
1 1 2 1 N 1 1 2 1 N
the space Sλ
1,...,λN defines a model for the irreducible (finite-dimensional)
′
Spin(m)-module with highest weight (λ1, . . . , λN) .
Remark 4 As opposed to the one-variable case, the extra conditions in
the definition of simplicial monogenic polynomials, involving the operators
⟨u , ∂ ⟩, are needed in order to obtain an irreducible module for Spin(m). For
i j
2m
example, Mk is an irreducible module, while Mλ
1,λ2 := { f : R → S | ∂1 f =
∂ f = 0} can be decomposed into irreducible modules, see e.g., [7], by means
2
of
λ1−λ2
⊕
j
Mλ
1,λ2 = ⟨u2, ∂1⟩ Sλ1+ j,λ2− j.
j=0
Higher Spin Dirac Operators in Two Vector Variables 5
From now on we take N = 2 and (u , u ) = (u, v) in Definitions 2 and 3.
1 2
Our object of interest is the elliptic, Spin(m)-invariant, first-order differential
operator
∞ m ∞ m
Qk,l : C (R , Sk,l) → C (R , Sk,l)
f (x; u, v) ↦→ Qk,l f (x; u, v).
This higher spin Dirac operator Qk,l was already constructed in [12] following
a pragmatic approach. In this paper we will use techniques from representation
theory, which will ease the generalization to the most general case, and
describe its polynomial solutions.
3 Refined Fischer Decomposition for Simplicial Monogenic Polynomials
We proceed as follows for the construction of the higher spin Dirac operator
Qk,l. Let V be a representation of Spin(m) or its Lie algebra so(m). Denote
by Ŵλ the finite-dimensional irreducible representation with highest weight λ.
The multiplicity of Ŵλ in V is denoted mλ(V) and the multiplicity of an arbitrary
weight μ in Ŵλ is denoted nμ(Ŵλ). The following well-known result will be used
(for the proof, we refer to e.g., [16]):
Proposition 1 If ν is a dominant integral weight such that mν(Ŵλ ⊗ Ŵμ) > 0,
′ ′
then there is a weight μ of Ŵμ such that ν = λ + μ and mν(Ŵλ ⊗ Ŵμ) ⩽
nλ−ν(Ŵμ).
One can then also prove the following theorem.
Theorem 1 For any pair of integers k ⩾ l ⩾ 0 with k > 0, one has
′ ′ ′ ′
(k, l) ⊗ (0) = (k, l) ⊕ (1 − δl,0)(k, l − 1) ⊕ (1 − δk,l)(k − 1, l)
′
⊕(1 − δl,0)(k − 1, l − 1) .
′
Proof Take λ = (k, l), μ = (0) the highest weight for S and ν a dominant
integral weight such that mν(Hk,l ⊗ S) > 0. Then, by Proposition 1, there is
a weight s of S such that ν = λ + s and mν(Hk,l ⊗ S) ⩽ ns(S) = 1. The possible
weights ν are given by
( )
1 1 1 1
ν = k ± , l ± , ± , . . . , ± .
2 2 2 2
As ν has to be a dominant integral weight, we only have to deal with the
′ ′ ′
following cases: ν = (k, l) , ν = (k − 1, l) , if k > l, and ν = (k, l − 1) , ν =
′
(k − 1, l − 1) , if k ⩾ l > 0. The representations corresponding to these highest
′
weights appear exactly once in Hk,l ⊗ S. We show this explicitly for ν = (k, l)
1 3 3 1
(the other cases being treated similarly). Let δ = (n − , n − , . . . , , ) be
2 2 2 2
6 F. Brackx et al.
half the sum of the positive roots and W the Weyl group. Using Klimyk’s
formula, see e.g., [14], we find
∑
mν(Hk,l ⊗ S) = sgn(w)nν+δ−w(λ+δ)(S) = nν+δ−1(λ+δ)(S) = n
( 1 ,..., 1 )(S) = 1.
2 2
w∈W
This follows from the fact that w = 1 ∈ W is the only element leading to a non-
trivial result in the summation. Indeed, the action of W changes the sign of the
components λi of the weight (λ1, . . . , λn). In order to satisfy λ1 ⩾ . . . ⩾ λn, only
the trivial action remains. This proves the claim. ⊔⊓
In case l = 0, the previous result encodes the Fischer decomposition for
spinor-valued harmonic polynomials: Hk ⊗ S =Mk ⊕ uMk−1. This result is
well-known in Clifford analysis and states that any S-valued harmonic homo-
geneous polynomial Hk of degree k in the vector variable u can be decomposed
in terms of two monogenic homogeneous polynomials
Hk = Mk + u Mk−1,
with Mλ ∈ Mλ. The factor u in this formula is called an embedding factor: it
realizes an isomorphic copy of the irreducible module Mk−1 inside the tensor
product Hk ⊗ S. Moreover, one can show that these polynomials are explicitly
given by
1
Mk−1 = − ∂
u Hk
m + 2k − 2
( )
u ∂
u
Mk = 1 + Hk.
m + 2k − 2
We will now generalize this result to the present setting k ⩾ l ⩾ 0. Theorem 1
tells us how the space of S-valued simplicial harmonic polynomials Hk,l ⊗ S
decomposes into irreducible summands. It implies the existence of certain
maps which embed each of the spaces Sk,l, Sk−1,l, Sk,l−1 and Sk−1,l−1 (for
appropriate k and l) into the space Hk,l ⊗ S. To ensure that these embedding
maps are indeed morphisms realizing an isomorphic copy of the spaces of
simplicial monogenics inside the space Hk,l ⊗ S, we have to check, next to the
homogeneity conditions, whether the conditions in Definition 2 are satisfied.
Clearly, Sk,l ↪→ Hk,l ⊗ S is the trivial embedding. Also, it is easily verified that
u : Sk−1,l ↪→ Hk,l ⊗ S.
In order to embed the space Sk,l−1 into the tensor product Hk,l ⊗ S, it seems
obvious to start from the basic invariant v, as we need an embedding map of
[ ]
degree (0, 1) in (u, v), but this approach fails since ⟨u, ∂
v⟩ vSk,l−1 = uSk,l−1 ̸=
0. In order to obtain the required embedding map, it suffices to project onto the
kernel of the operator ⟨u, ∂
v⟩, which can be done by fixing c1 in the following
expression:
v − c1 u⟨v, ∂
u⟩ : Sk,l−1 ↪→ Hk,l ⊗ S.
Higher Spin Dirac Operators in Two Vector Variables 7
1
For c1 = all conditions in Definition 2 are indeed satisfied. Similarly,
k−l+1
the last embedding map can be found as a suitable projection of a linear
combination of u v and v u, and is given by
⟨v, u⟩ − c2 v u − c3 ⟨u, u⟩⟨v, ∂
u⟩ : Sk−1,l−1 ↪→ Hk,l ⊗ S
m+k+l−4 1
with c2 = − and c3 = . This can be summarized as follows:
m+2k−4 m+2k−4
Proposition 2 For any pair of integers k ⩾ l ⩾ 0 with k > 0, one has
Hk,l ⊗ S = Sk,l ⊕ (1 − δl,0)νk,l Sk,l−1 ⊕ (1 − δk,l)μk,l Sk−1,l
⊕ (1 − δl,0)κk,l Sk−1,l−1,
with the embedding maps
u⟨v, ∂ ⟩
u
νk,l := v − : Sk,l−1 ↪→ Hk,l ⊗ S
k − l + 1
μk,l := u : Sk−1,l ↪→ Hk,l ⊗ S
m + k + l − 4 ⟨u, u⟩⟨v, ∂ ⟩
u
κk,l := ⟨v, u⟩ + v u − : Sk−1,l−1 ↪→ Hk,l ⊗ S.
m + 2k − 4 m + 2k − 4
Remark 5 The embedding map μk,k clearly does not exist, in view of the
dominant weight condition. If l = 0 the embedding maps νk,l and κk,l do not
exist neither.
Let k > l > 0 and suppose ψ ∈ Hk,l ⊗ S. According to Proposition 2, there
exists ψp,q ∈ Sp,q such that
ψ = ψk,l + νk,l ψk,l−1 + μk,l ψk−1,l + κk,l ψk−1,l−1. (1)
An explicit expression for the projection operators on each of the summands
inside Hk,l ⊗ S can then be obtained as follows. First, the action of ∂
v on (1)
annihilates two summands and leads to
∂
vψ = − (m + 2l − 4) ψk,l−1
( )
m + k + l − 4
+ 1 − (m + 2l − 2) u ψk−1,l−1. (2)
m + 2k − 4
Acting again with ∂ , we find
u
(m + 2k − 4) ∂ ∂ ψ
u v
ψk−1,l−1 = .
(m + 2k − 2)(m + 2l − 4)(m + k + l − 3)
This gives rise to a projection operator πk−1,l−1, defined as
πk−1,l−1 : Hk,l ⊗ S → Sk−1,l−1
(m + 2k − 4) ∂ ∂ ψ
u v
ψ ↦→ . (3)
(m + 2k − 2)(m + 2l − 4)(m + k + l − 3)
8 F. Brackx et al.
Substituting the expression for ψk−1,l−1 in (2), we find
( )
1 u ∂
u
ψk,l−1 = − 1 + ∂
vψ,
m + 2l − 4 m + 2k − 2
which defines a second projection operator πk,l−1:
πk,l−1 : Hk,l ⊗ S → Sk,l−1
( )
1 u ∂
u
ψ ↦→ − 1 + ∂ ψ.
v
m + 2l − 4 m + 2k − 2
Finally, using the previous results, the action of ∂ on (1) leads to
u
[( ( ) )
1 k − l v ∂
v
ψk−1,l = − 1 + ∂
u
m + 2k − 2 k − l + 1 m + 2l − 4
( ) ]
1 u ∂
u
+ 1 + ⟨v, ∂ ⟩∂ ψ,
u v
k − l + 1 m + 2l − 4
which defines the third projection operator:
πk−1,l : Hk,l ⊗ S → Sk−1,l
[( ( ) )
1 k − l v ∂
v
ψ ↦→ − 1 + ∂
u
m + 2k − 2 k − l + 1 m + 2l − 4
( ) ]
1 u ∂
u
+ 1 + ⟨v, ∂ ⟩∂ ψ.
u v
k − l + 1 m + 2l − 4
The last projection operator on the summand Sk,l is then given by
πk,l := 1 − πk−1,l − πk,l−1 − πk−1,l−1.
We gather all this information in the following theorem.
Theorem 2 (Refined Fischer Decomposition for Simplicial Monogenics) Each
S-valued simplicial harmonic polynomial Hk,l in two vector variables can be
uniquely decomposed in terms of simplicial monogenic polynomials:
Hk,l = Mk,l + νk,l Mk,l−1 + μk,l Mk−1,l + κk,l Mk−1,l−1,
with the embedding maps def ined in Proposition 2 and with
Mk,l = πk,l(Hk,l) Mk,l−1 = πk,l−1(Hk,l)
Mk−1,l = πk−1,l(Hk,l) Mk−1,l−1 = πk−1,l−1(Hk,l).
Proof Only the uniqueness has to be addressed, but this can easily be proved.
⊔⊓
4 Construction of the Operator Qk,l
We now use the refined Fischer decomposition of Theorem 2 to construct the
higher spin Dirac operator Qk,l. Since the classical Dirac operator ∂
x can be
Higher Spin Dirac Operators in Two Vector Variables 9
seen as an endomorphism on S-valued functions, the action of ∂
x on a Hk,l ⊗ S-
valued function preserves the values. This gives rise to a collection of invariant
operators defined through the following diagram:
∂x
∞ m / ∞ m
C (R ,Hk,l ⊗ S) C (R ,Hk,l ⊗ S)
= =
Qk,l
∞ m / ∞ m
C (R
⊕,SPkP,lP)XPXPXPPXPXT
PXk
PXk,l,
P−lX
P1XXXXXXXXXXXC+ (R⊕, Sk,l)
C∞(Rm, νk,lSk,l−1) PPPP C∞(Rm, νk,lSk,l−1) .
P
Pk,
Pl
T
k−1,Pl PP
⊕ PP ⊕
P
P
(
∞ m ∞ m
C (R , μk,lSk−1,l) C (R , μk,lSk−1,l)
⊕ ⊕
$
∞ m ∞ m
C (R , κk,lSk−1,l−1) C (R , κk,lSk−1,l−1)
∞ m
The non-existence of an invariant operator from C (R , Sk,l) to
∞ m
C (R , κk,lSk−1,l−1) (i.e., the dotted arrow in the diagram above) can be
proved by means of the construction method of conformally invariant
operators using generalized gradients, see e.g., [13, 22]. It essentially follows
m
from the fact that the tensor product Sk,l ⊗ C does not contain the summand
Sk−1,l−1. The next lemma shows that this can also be verified through direct
calculations in Clifford analysis.
∞ m
Lemma 1 For every f ∈ C (R , Sk,l) one has ∂
u∂v∂x f = 0.
Proof The definition of the Euclidean inner product leads to
∂ ∂ ∂ f = −2∂ ⟨∂ , ∂ ⟩ f − ∂ ∂ ∂ f = 0,
u v x u v x u x v
since ∂ f = ∂ f = 0. ⊔⊓
u v
∞ m
Hence, it follows from (3) that πk−1,l−1(∂
x f ) ≡ 0, for every f ∈ C (R , Sk,l).
k,l k,l
An explicit expression for the operators Qk,l, T
k,l−1 and Tk−1,l in the diagram
above is then obtained using results of the previous section.
Definition 4 For all integers k ⩾ l ⩾ 0 with k > 0, there are (up to a multiplica-
tive constant) unique invariant first-order differential operators Qk,l defined
by
∞ m ∞ m
Qk,l : C (R , Sk,l) → C (R , Sk,l) : f ↦→ πk,l(∂
x f ),
10 F. Brackx et al.
or explicitly,
[ ]
u ∂ v ∂ u⟨v, ∂ ⟩∂
u v u v
Qk,l f = 1 + + − 2 ∂
x f.
m + 2k − 2 m + 2l − 4 (m + 2k − 2)(m + 2l − 4)
In case k = l > 0, the operators reduce to
[ ]
(v − u⟨v, ∂ ⟩)∂
u v
Qk,k f = 1 + ∂
x f.
m + 2k − 4
Remark 6 In case l = 0, we find the Rarita–Schwinger operators Rk, as defined
in [7]:
( )
u ∂
u
Qk,0 = Rk = 1 + ∂
x. (4)
m + 2k − 2
The ellipticity of this operator Qk,l follows e.g., from [5], and the Spin(m)-
invariance can be expressed through the following commutative diagram:
Qk,l
∞ m / ∞ m
C (R , Sk,l) C (R , Sk,l)
L(s) L(s)
9

∞ m / ∞ m
C (R , Sk,l) C (R , Sk,l)
Qk,l
( )
with L(s) f (u, v) = sf (s¯us, s¯vs) the natural action of Spin(m) on higher spin
fields.
Similar calculations lead to the so-called dual twistor operators, which
are visualized as the diagonal arrows in the diagram above. We adopt the
convention that each operator of twistor-type will be denoted by means of
the letter T , together with upper and lower indices. The upper indices denote
the highest weight of the source space, whereas the lower indices denote the
highest weight of the target space (discarding the half-integers).
k,l
Definition 5 For all integers k > l ⩾ 0, the dual twistor operators T are
k−1,l
defined as the unique invariant first-order differential operators
k,l ∞ m ∞ m
T
k−1,l : C (R , Sk,l) → C (R , μk,lSk−1,l) : f ↦→ μk,lπk−1,l(∂x f ),