FROM HERMITEAN CLIFFORD ANALYSIS TO SUBELLEPITIC DIRAC OPERATORS ON ODD DIMENSIONAL SPHERES AND OTHER CR MANIFOLDS

. We show that the two Dirac operators arising in Hermitian Cliﬀord analysis are identical to standard diﬀerential operators arising in several complex variables. We also show that the maximal subgroup that preserves these operators are generated by translations, dilations and actions of the unitary n-group. So the operators are not invariant under Kelvin inversion. We also show that the Dirac operators constructed via two by two matrices in Hermitian Cliﬀord analysis correspond to standard Dirac operators in euclidean space. In order to develop Hermitian Cliﬀord analysis in a diﬀerent direction we introduce a sub elliptic Dirac operator acting on sections of a bundle over odd dimensional spheres. The particular case of the three sphere is examined in detail. We conclude by indicating how this construction could extend to other CR manifolds.


Introduction
Clifford analysis started as an attempt to generalize one variable complex analysis to n-dimensional euclidean space. It has since evolved into a study of the analyst, geometry and applications of Dirac operators over euclidean space, spheres, real projective spaces, conformally flat spin manifolds and spin manifolds with applications to representation theory arising from mathematical physics, classical harmonic analysis and many other topics.
In recent years a topic referred to as Hermitian Clifford analysis has attracted some attention. See for instance [2,3,4]. It is developed initially over a complexification of even dimensional Euclidean space. An almost complex structure is introduced and two associated projection operators are applied to this complex vector space. This splits this space into two n-dimensional complex spaces. When these operators are applied to the euclidean Dirac operator it splits into two differential operators. Here we show that these operators are respectively the d-bar operator and its dual as arising in several complex variables. See for instance [5,6].
The Euclidean Dirac operator is a conformally invariant operator. In particular it is invariant under Kelvin inversion. We show that these Hermitian Dirac operators have a narrower range of invariance. We show they are no longer invariant under Kelvin inversion. Their maximal invariance group is generated by translation, dilation and a subgroup of SO(2n) isomorphic to U (n).
We also show via lemma given in [1] that the Dirac operator constructed via two by two matrices is in fact a standard Dirac operator over Euclidean space. In order to try and develop fresh ideas for this topic we transfer to odd dimensional spheres and make use of their CR structure to introduce a subelliptic Dirac operator acting on sections of a bundle defined over the sphere. Each fiber is isomorphic to a Clifford algebra generated from the CR structure of the sphere. The square of this Dirac operator gives the Kohn Laplacian on the sphere. The Hopf vibration is used to illustrate this scenario in more detail over S 3 . We conclude by illustrating how this construction might carry over to other CR manifolds.

Preliminaries on Hermitean Clifford Analysis
In Hermitian Clifford analysis one starts with the standard Dirac operator under Clifford algebra multiplication.
So we consider the real 2 m dimensional Clifford algebra C m,− with R m ⊂ C m,− and One now introduces an almost complex structure on R m . This is a matrix J ∈ SO(m) with J 2 = −I. As J ∈ SO(m) this forces m to be even. So m = 2n. For instances, J = 0 −I I 0 . This is the choice of J that we will use here. We now complexify R 2n to obtain C 2n and we complexify C 2n,− to obtain the complex Clifford algebra C 2n (C). Following [5,6] we now have the projection operators 1 2 (I ± iJ). They act on C 2n and split it into two complex spaces W + ⊕ W − each of complex dimension n. So In particular, 1 2 (I ± iJ)e j = 1 2 (e j ± ie j+n ), for j = 1, . . . , n, and it equals 1 2 (e j ∓ ie j−n ), for j = n + 1, . . . , 2n. Note 1 2 (e j ∓ ie j−n = 1 2 i(e k ± ie k+n ) for j = n + 1, · · · , 2n and k = j − n. We denote 1 2 (e k ± ie k+n ) by f ± k respectively, with k = 1, · · · , n. The elements f ± k , k = 1, . . . , n, is known as a Witt basis for W ± , respectively. Note that (f ± k ) 2 = 0, These relations correspond to the relations of differential forms dz j , dz k , and their duals on the alternating algebra ∧(C n ). See, for instance [5,6]. In fact, e j = f + j + f − j for j = 1, . . . , n and e j = −i(f + j − f − j ) for j = n + 1, . . . , 2n. So our complex Clifford algebra and the alternating algebra are the same.
Via these identifications it can be seen that D + corresponds to the operator ∂ = n j=1 dz j ∂ ∂zj and D − corresponds to its dual ∂ * from several complex variables. See for instance [5,6]. In this way the Dirac operator corresponds to the operator ∂ + ∂ * .

Hermitean Clifford Analysis and matrix differential operators
In [13] the pair of matrix differential operators where ∆ 2n is the Laplacian on R 2n . The first matrx operator can be written as and the second as Now, the space R(2) of 2 × 2 matrices is a representation of the Clifford algebra C 2,+ . See for instances [1].
To see this, consider the basis 1, g 1 , g 2 , g 1 g 2 of C 2,+ . So g 2 i = 1 for i = 1, 2. Now, make the identification This defines an algebra isomorphism between C 2,+ and R(2).The matrix differential operators now become of Dirac operators defined over a copy of R 2n lying in C 2n+2,+ .
The fundamental solution to where x = x 1 e 1 + · · · + x 2n e 2n and x ± = 1 2 (I ± iJ)x. In matrix form this corresponds to 1 |x| 2n while the fundamental solution to the operator D − ⊗ 1 + D + ⊗ g 2 is, up to a constant, In matrix form this gives 1 |x| 2n They correspond to the fundamental solutions to the matrix operator given in [3].
The following Borel-Pompeiu and Cauchy integral formulas correspond to the integral formulas appearing in [3].
There are similar integral formulas obtained by replacing

Conformal transformations in the Hermitean case
In [14] and elsewhere it is shown that the Euclidean Dirac operator is invariant under conformal transfor- As for orthogonal transformations, such a transformation must preserve the spaces W ± . So we need to restrict to transformations in SO(2n) that commute with J. It is known that this is a subgroup of SO(2n) isomorphic to U(n).
To proceed further we need the spin group. First, consider This set is a group under Clifford algebra multiplication. It is the pin group and it is denoted by P in(2n).
If we restrict so that p is even we obtain a subgroup called the spin group. It is denoted by Spin(2n).
It is well known, [12] for instance, that for x ∈ R 2n , axã defines a special orthogonal transformation on R 2n . In fact, [12], there is a surjective group homomorphism where θ a x := axã. The kernel of this homomorphism is {±1}, [12].
We have previously noted that the operators D ± are invariant under actions of a copy of U(n) ⊂ SO(2n).
It follows that this copy of U(n) has a double covering, U (n) ⊂ Spin(2n), also isomorphic to U(n). Further, following [8] then if Ω is a domain in R 2n and y = axã ∈ Ω, with a ∈ U (n) then if where D ± now acts with respect to the variable x.
This situation differs from what is considered in several variables for the operators ∂ and ∂ * in C n . There these operators are considered as acting strictly on (p, q) forms, see for instance [5]. The action ofã on the function f (axã) is a spherical action that does not preserve (p, q) forms. It preserves the spinor subspaces of the algebra C 2n,− or C 2n (C), see [12].
We now turn to consider the case of the Kelvin inversion. Given a hypersurface S in R 2n then under Kelvin inversion the surface element n(y)dσ(y) is transformed to G(x)n(x)G(x)dσ(x), where y ∈ S, n(y) is the outer pointing unit vector perpendicular to the tangent space T S y , and σ is the Lebesgue measure on S.
We want to know if the spaces W ± are preserved under the Kelvin inversion and in particular if the operators D ± are also preserved. As G(x) = x |x| 2n , this boils down to determine whether or not J commutes with the action xn(x)x of x on n(x). Note that xn(x)x describes (up to the sign) a reflection of n(x) in the direction of x. So xn(x)x is a vector in R 2n .
As J is a matrix in SO(2n) and the vector xn(x)x is defined in terms of Clifford multiplication it is better to make the notation uniform. The action of J on R 2n gives a counterclockwise rotation of π/2 in each plane spanned by e i , e i+n for i = 1, · · · n. So a lifting of J to Spin(2n) gives (1 + e n e 2n ) = ± 1 2 n/2 (1 + e 1 e 1+n ) · · · (1 + e n e 2n ).
It should be noted that each pair (1 + e i e i+n )(1 + e s e s+n ) commutes with each other. It follows that it is enough to compare the terms (1 + e 1 e 1+n )xn(x)x(1 + e 1+n e 1 ) and x(1 + e 1 e 1+n )n(x)(1 + e 1+n e 1 )x.
In fact, it is enough to compare the subterms (1+e 1 e 1+n )xn(x)x and x(1+e 1 e 1+n )n(x)x. As the hypersurface S is arbitrary we can place x = e 1 . In this case the terms

Kohn Dirac and Laplacian operators in S 2n−1
In the previous sections we have shown that with the exception of the action of U (n) on D ± , (∂, ∂ * ), much of the Hermitian Clifford analysis already exists in several complex variables over C n . This leads to the question of trying to find ways in which to develop Hermitian Clifford analysis that are more meaningful and original.
One starting point is to look at its analogue over odd dimensional spheres.
Consider S 2n−1 ⊂ C n . Consider also x ∈ S 2n−1 and T S 2n−1 x ⊂ C n . In several complex variables [5,6] one Here, n ≥ 2. This gives rise to a situation that does not occur in one complex variable. We now have a complex bundle L ⊂ T S 2n−1 where each fiber is the complex space L x . Consider now the canonical projection This gives rise to the projection P : Now consider a hypersurface S bounding a domain Ω in S 2n−1 . If we consider S to be sufficiently smooth we can consider smooth C 2n (C)−valued functions defined in a neighbourhood of Ω within S 2n−1 and following [10] we have the following version of the Stokes' Theorem.
where n(x) is the outward pointing unit vector in T S x . Further D s is the spherical Dirac operator We may also consider the modified integral Applying the Stokes' Theorem to this integral gives the integral For each x ∈ S 2n−1 , we have ix ∈ T S 2n−1 x , so ix / ∈ L x . Consequently, P x (ix) = 0. Therefore, P x (D s ) = P x (xΓ x ). It follows that this operator is an isomorphic copy of the sub-elliptic Dirac operator ∂ b + ∂ * b over S 2n−1 . This Dirac operator is also known as the Kohn Dirac operator (see [11]). Therefore, (P x (xΓ x )) 2 = b , the Kohn Laplacian. The operators ∂ b , ∂ * b and b are all defined in [5,6] and elsewhere.
The group of conformal transformations that preserves the unit b all B(0, 1) ⊂ R 2n , and its boundary, is SO(2n, 1). However, we have identified R 2n with C n . We need to restrict to the subgroup of SO(2n, 1) that preserves the Hermitian structure of C n . At this point we need to reference or show that this group is U(n, 1).
We also need to show that the invariant group for P x (D s ) is the double cover of U(n, 1) within Spin(2n). A start is the following: note that if (f (x)(D s P x )) = (P x D s )g(x) = 0 then we have the following version of the Cauchy's Theorem: By allowing S to vary it follows that the group of diffeomorphisms that preserves the Kohn-Dirac operator P x (D s ) is a subgroup of Spin(2n, 1) whose projection preserves the bundle L.

Realization on S 3
To make the above points more clear we consider the three-dimensional sphere as a special example.
Furthermore, we will consider for the moment quaternionic valued functions.
Let us introduce the following first-order differential operators These differential operators are skew-symmetric with respect to the Riemannian surface form dS on S 3 given where dx j is generated from the oriented volume form dx with dx j being omitted.
The sub-Laplacian b is given by while the Laplacian is given by

Moreover, we have the sub-Dirac operator
Keep in mind that in general its square is not the sub-Laplacian since X j = [X i , X k ].
We can identify R 4 ∼ H with C 2 in the usual way via z = z 1 + iz 2 , with z 1 , z 2 ∈ C j := {a + bj, a, b ∈ R}.
The one-parameter transformation group generated by X j corresponds to the complex multiplication with λ = (a + bj) from the right. The resulting orbits on S 3 span the complex projective space P 1 C and we get the Hopf bundle π R : S 3 → P 1 C.
The complex line bundle L l on P 1 C is associated to the character χ l (λ) = λ l .
As usual we can now consider the space Π N of homogeneous polynomials of degree N in the variables x 0 , x 1 , x 2 , x 3 . We denote by H N the subspace of harmonic polynomials in Π N . For this subspace we have the classic Fischer decomposition The space H N restricted to the sphere S 3 is the eigenspace H N of the Laplacian ∆ with respect to the eigenvalue λ N = N (N + 2) with dimension (N + 1) 2 .
Let us now consider the subspace Π n,m , n + m = N, of Π N defined by Π n,m = p ∈ Π N : p(z 0 e it , z 1 e it , z 0 e it , z 1 e it ) = e i(n−m)t p(z 0 , z 1 , z 0 , z 1 ) and the subspace H n,m = H N ∩ Π n,m . In this subspace we have X j p = (n − m)p. This leads to the decomposition H N = n+m=N,n,m≥0 ⊕H n,m Furthermore, we have harmonic Fischer decomposition Π n,m = H n,m + |w 0 | 2 + |w 1 | 2 Π n−1,m−1 .
Moreover, each of the eigenspaces H n,m can be decomposed in terms of eigenspaces of the Dirac operator, where M k,s = {p ∈ Π k,s : Dp = 0}.
Furthermore, for any p ∈ H n,m we have as well as and we denote the restriction of H n,m to S 3 by H n,m as well as the restriction of M n,m to S 3 by M n,m .
This leads to the following lemma. i.e. F l ∼ Γ(L l ) and The sub-Laplacian can be identified with the horizontal Laplacian This identification allows us to decompose F l as The sub-Dirac operator can be identified with the horizontal Dirac operator This identification allows us to decompose F l as

Subelliptic Dirac operator
Following [7,9] it is desirable to ask what manifolds besides S 2n−1 admit a subelliptic Dirac operator.
Clearly it should be a CR manifold with some sort of spin structure.
In C n we saw an invariance under a subgroup of Spin(2n) isomorphic to U(n). This suggests that we are locking for a CR manifold with principle bundle whose fibers are isomorphic to U(n) and with a global double cover also with fiber isomorphic to U(n).

Irreducible representations in U (n)
Within the Clifford algebras C m the irreducible representation spaces for Spin(m) are the spinor spaces.
Here though we are dealing with the subgroup U (n) of Spin(2n).
As point out in [15] the spinor spaces are no longer irreducible subspaces of U (n). Furthermore, in several complex variables one is interested specifically in (p, q) sections. If z = aωã and f (z) is the isomorphic equivalent of a (p, q) form in the context we consider here then in generalãf (aωã) is not the equivalent of a (p, q) form.
It follows that if f (z) is the equivalent of a (p, q) form in the context described here then so isãf (aωã)a.