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n-Regular Functions in Quaternionic Analysis

In this paper we study left and right n-regular functions that originally were introduced in [FL4]. When n=1, these functions are the usual quaternionic left and right regular functions. We show that n-regular functions satisfy most of the properties of the usual regular functions, including the conformal invariance under the fractional linear transformations by the conformal group and the Cauchy-Fueter type reproducing formulas. Arguably, these Cauchy-Fueter type reproducing formulas for n-regular functions are quaternionic analogues of Cauchy's integral formula for the n-th order pole expressing the (n-1)-st derivative of a holomorphic function. We also find two expansions of the Cauchy-Fueter kernel for n-regular functions in terms of certain basis functions, we give an analogue of Laurent series expansion for n-regular functions, we construct an invariant pairing between left and right n-regular functions and we describe the irreducible representations associated to the spaces of left and right n-regular functions of the conformal group and its Lie algebra.


Introduction
The foundational result in quaternionic analysis is the Cauchy-Fueter integral formulas for left and right regular functions. Thus it is natural to ask about quaternionic analogue of Cauchy's integral formula for the n-th order pole for all positive integers n f (n−1) (w) = (n − 1)! 2πi For n = 2 we suggested an answer to this question in [FL1] with the first derivative replaced by the Maxwell equations in the quaternionic case. In a recent paper [FL4] we proposed a different quaternionic counterpart of (1), for general n, introducing left and right n-regular functions. For n = 1 we get the usual left and right regular functions; when n = 2 we call them doubly left and right regular functions, and in the doubly regular case the first derivative in (1) is replaced by the degree operator plus two. In this paper we study left and right n-regular functions in more detail. Let n be a positive integer and V n 2 the irreducible representation of SU (2) of dimension n + 1. Then n-regular functions are functions on the space of quaternions H or H × = H \ {0} with values in V n 2 and satisfying n regularity conditions. The spaces of left and right n-regular functions form the most degenerate series of unitary representations of SU (2, 2) that are often called the spin n 2 representations of positive and negative helicities and play important role in physics. In the context of quaternionic analysis, n-regular functions first appeared briefly in [FL4]. In the present paper we study these spaces in more detail as natural generalizations of quaternionic regular functions.
The spaces of n-regular functions provide a class of irreducible representations of the conformal group that were considered before, for example, by H. P. Jakobsen and M. Vergne in [JV] and in a more general case by S. T. Lee [Le].
We show that n-regular functions satisfy most of the properties of the usual regular functions and doubly regular functions, including the conformal invariance under the fractional linear transformations by the conformal group SL(2, H C ) ≃ SL(4, C) and the Cauchy-Fueter type reproducing formulas. Arguably, these Cauchy-Fueter type reproducing formulas for n-regular functions are quaternionic analogues of Cauchy's integral formula for the n-th order pole (1). We also study in detail the K-type bases of the spaces of n-regular functions, the duality between left and right regular functions and the u(2, 2)-invariant inner products on these spaces.
In [FL4] we constructed an algebra of quaternionic functions using the product of spaces of left and right regular functions. We expect this construction to have a straightforward generalization to n-regular functions, thus yielding an explicit realization of an infinite family of certain non-highest, non-lowest weight representations of the conformal group, parametrized by positive integers n with an intrinsic algebra structure.
The conformal groups of the quaternions, Minkowski space and split quaternions are locally isomorphic to SO(5, 1), SO(4, 2) and SO(3, 3) respectively. Thus our constructions complement a thorough study of minimal representations of the indefinite orthogonal groups O(p, q) by T. Kobayashi and B. Ørsted [KØ] (see references therein for the previous work on this subject). Their work also uses the space of solutions of the ultrahyperbolic wave equation on R p−1,q−1 to give concrete realizations of these minimal representations. Many results of quaternionic analysis extend to higher dimensions in the form of Clifford analysis (see, for example, [BDS, DSS, R]). It is interesting to see how results of this paper are generalized to Clifford analysis.
The paper is organized as follows. In Section 2 we define left and right n-regular functions (Definitions 3, 4) and prove conformal invariance under the fractional linear transformations by group GL(2, H C ) (Theorem 5). In Section 3 we derive the Cauchy-Fueter type formulas for n-regular functions (Theorem 9). In Section 4 we find two expansions of the Cauchy-Fueter kernel k n/2 (Z −W ) for n-regular functions in terms of certain basis functions F (n) l,µ,ν (Z), F ′(n) l,µ,ν (Z), G (n) l,µ,ν (Z) and G ′(n) l,µ,ν (Z) (Proposition 12). In Section 5 we prove a technical result that a certain differential operator D n that enters the Cauchy-Fueter formulas for n-regular functions can be inverted for (left or right) n-regular functions defined on all of H × (Proposition 15). We also give an analogue of Laurent series expansion for n-regular functions on H × (Corollary 16). In Section 6 we use the technical result from the previous section to construct a gl(2, H C )-invariant pairing (14) between left and right n-regular functions on H × . We also prove orthogonality relations between the basis functions F l,µ,ν (Z) and G ′(n) l,µ,ν (Z) (Proposition 20). In Section 7 we consider gl(2, H C )-modules associated to left and right n-regular functions. We prove that these gl(2, H C )-modules are irreducible (Theorem 26) and identify their K-types, where K is the maximal compact subgroup U (2) × U (2) of U (2, 2) (Proposition 24). In Section 8 we give explicit descriptions of the u(2, 2)invariant inner products on the gl(2, H C )-modules F ± n and G ± n (Theorem 27). Since this paper is a continuation of [FL1,FL3,FL4], we follow the same notations and instead of introducing those notations again we direct the reader to Section 2 of [FL3].

Definitions and Conformal Invariance
We continue to use notations established in [FL1]. In particular, e 0 , e 1 , e 2 , e 3 denote the units of the classical quaternions H corresponding to the more familiar 1, i, j, k (we reserve the symbol i for √ −1 ∈ C). Thus H is an algebra over R generated by e 0 , e 1 , e 2 , e 3 , and the multiplicative structure is determined by the rules e 0 e i = e i e 0 = e i , (e i ) 2 = e 1 e 2 e 3 = −e 0 , e i e j = −e j e i , 1 ≤ i < j ≤ 3, and the fact that H is a division ring. Next we consider the algebra of complexified quaternions (also known as biquaternions) H C = C ⊗ R H and write elements of H C as Z = z 0 e 0 + z 1 e 1 + z 2 e 2 + z 3 e 3 , z 0 , z 1 , z 2 , z 3 ∈ C, so that Z ∈ H if and only if z 0 , z 1 , z 2 , z 3 ∈ R: For Z = z 0 e 0 + z 1 e 1 + z 2 e 2 + z 3 e 3 ∈ H C , we use to denote the quaternionic conjugation and to denote the quadratic norm. Let be the sets of invertible elements in H and H C respectively. The algebra H C can be naturally identified with the algebra of 2 × 2 matrices with complex entries. Recall that we denote by S (respectively S ′ ) the irreducible 2-dimensional left (respectively right) H C -module, as described in Subsection 2.3 of [FL1]. The spaces S and S ′ can be realized as respectively columns and rows of complex numbers. Then Note that S ⊗ S and S ′ ⊗ S ′ are respectively left and right modules over H C ⊗ H C . Fix a positive integer n. The n-fold tensor product and contains the n-fold symmetric product which we denote S ⊙ · · · ⊙ S n times as a subspace. Similarly, the n-fold tensor product S ′ ⊗ · · · ⊗ S ′ n times is a right module over (3) and contains the subspace of n-fold symmetric tensors We have a natural bilinear pairing between S ′ and S: This pairing extends to a multilinear pairing by taking the product of pairings of the respective components, and then to by multilinearity. For convenience, we restate Lemma 5 from [FL4]: Lemma 1. Let t ∈ S ⊙ · · · ⊙ S n times and t ′ ∈ S ′ ⊙ · · · ⊙ S ′ n times , then, for any 1 ≤ j, k ≤ n, j = k, We consider spaces of functionŝ F n = f : H C → S ⊗ · · · ⊗ S n times andĜ n = g : H C → S ′ ⊗ · · · ⊗ S ′ n times (possibly with singularities), and let the group GL(2, H C ) act on these spaces as follows: Clearly, these two actions preserve the subspaces of functions with values in n-fold symmetric products S ⊙ · · · ⊙ S and S ′ ⊙ · · · ⊙ S ′ respectively.
Differentiating π nl and π nr , we obtain actions of the Lie algebra gl(2, H C ), which we still denote by π nl and π nr respectively. Using notations we can describe these actions of the Lie algebra (cf. Lemma 4 in [FL4]).
We introduce 2n first order differential operators (1 ⊗ · · · ⊗e 3 ⊗ k-th place · · · ⊗ 1), k = 1, . . . , n, which can be applied to functions with values in n-fold tensor products S ⊗ · · · ⊗ S or S ′ ⊗ · · · ⊗ S ′ as follows. If U is an open subset of H or H C and f : U → S ⊗ · · · ⊗ S is a differentiable function, then these operators can be applied to f on the left. For example, Similarly, these operators can be applied on the right to differentiable functions g : U → S ′ ⊗ · · · ⊗ S ′ ; we often indicate this with an arrow above the operator. For example, Definition 3. Let U be an open subset of H. A C 1 -function f : U → S ⊙ · · · ⊙ S is left n-regular if it satisfies n differential equations for all points in U . Similarly, a C 1 -function g : U → S ′ ⊙ · · · ⊙ S ′ is right n-regular if for all points in U . Since left and right n-regular functions are harmonic.
One way to construct n-regular functions is to start with a harmonic function ϕ : H → S ⊙ · · · ⊙ S, then (∇ ⊗ · · · ⊗ ∇)ϕ is left n-regular. Similarly, if ϕ : We also can talk about n-regular functions defined on open subsets of H C . In this case we require such functions to be holomorphic.
for all points in U . Similarly, a holomorphic function g : for all points in U .
Let R n and R ′ n denote respectively the spaces of (holomorphic) left and right n-regular functions on H C , possibly with singularities.
1. The space R n of left n-regular functions H C → S ⊙ · · · ⊙ S (possibly with singularities) is invariant under the π nl action (5) of GL(2, H C ).

The space R ′
n of right n-regular functions H C → S ′ ⊙ · · · ⊙ S ′ (possibly with singularities) is invariant under the π nr action (6) of GL(2, H C ).
Proof. Since the Lie group GL(2, for example, the case of π nl 0 0 C 0 f , the other cases are similar. For k = 1, . . . , n, we have: the first summand is zero essentially because the space of left regular functions is invariant under the action π l (equation (22) in [FL1]), the second summand is zero because f satisfies ∇ + k f = 0, and the third summand is zero by Lemma 1.

Cauchy-Fueter Formulas for n-Regular Functions
In this section we derive the Cauchy-Fueter type formulas for n-regular functions from the classical Cauchy-Fueter formulas for left and right regular functions.
Lemma 6. Let f (Z) be a left n-regular function, then the S ⊗ · · · ⊗ S-valued functions are "left regular in k-th place" in the sense that they satisfy Similarly, if g(Z) is a right n-regular function, then the S ′ ⊗ · · · ⊗ S ′ -valued functions are "right regular in k-th place" in the sense that they satisfy Proof. We have: the first term is zero because f satisfies ∇ + k f = 0 and the second term is zero by Lemma 1.
Proof of the second part of the lemma is similar.
Recall the quaternionic valued holomorphic 3-form Dz on H C : We also consider holomorphic 3-forms on H C with values in H C ⊗ · · · ⊗ H C . We have an analogue of Cauchy's integral theorem for n-regular functions.
Lemma 7. Let U ⊂ H be an open bounded subset with piecewise C 1 boundary ∂U . Suppose that f (Z) is left n-regular and g(Z) is right n-regular on a neighborhood of the closure U . Then, for each k = 1, . . . , n, Note that the expression inside the integral is a C-valued function obtained by applying the pairing (4).
Proof. Essentially by the definition of Dz, By Lemma 6, both summands are zero, and the result follows.
Recall the degree operator deg acting on functions on H: and let (deg +m), m ∈ Z, denote the degree operator plus m times the identity: Similarly, we can define operators deg and (deg +m) acting on functions on H C . For convenience we recall Lemma 8 from [FL2] (it applies to both cases).
We introduce an operator

Define a function of Z and W taking values in
.
We have the following analogue of the Cauchy-Fueter formulas for n-regular functions.
Theorem 9. Let U ⊂ H be an open bounded subset with piecewise C 1 boundary ∂U . Suppose that f (Z) is left n-regular on a neighborhood of the closure U , then, for each k = 1, . . . , n, If g(Z) is right n-regular on a neighborhood of the closure U , then, for each k = 1, . . . , n, Proof. By Lemma 6, the S ⊗ · · · ⊗ S-valued function is "left regular in the k-th place" in the sense that it is annihilated by ∇ + k . From the classical Cauchy-Fueter formula for left regular functions, we obtain: wherẽ .
The case of right n-regular function is similar.
We have an analogue of Liouville's theorem for n-regular functions: Corollary 10. Let f : H → S ⊙ · · · ⊙ S be a function that is left n-regular and bounded on H, then f is constant. Similarly, if g : H → S ′ ⊙ · · · ⊙ S ′ is a function that is right n-regular and bounded on H, then g is constant.
Proof. The proof is essentially the same as for the (classical) left and right regular functions on H, so we only give a sketch of the first part. From Theorem 9 we have: If f is bounded, one easily shows that the integral on the right hand side tends to zero as R → ∞. Thus ∂ ∂x 0 D n f = 0. Similarly, the other partial derivatives It follows that D n f and hence f are constant.

Expansion of the Cauchy-Fueter Kernel for n-Regular Functions
We often identify H C with 2×2 matrices with complex entries. Similarly, it will be convenient to identify H C ⊗ · · · ⊗ H C n times with complex 2 n × 2 n matrices using the Kronecker product. Let C k×k denote the algebra of k × k complex matrices. For example, if A = a 11 a 12 a 21 a 22 , B = b 11 b 12 b 21 b 22 ∈ C 2×2 , then their Kronecker product is It is easy to see that the Kronecker product satisfies Next we recall the matrix coefficients t l ν µ (Z)'s of SU (2) described by equation (27) Z = z 11 z 12 z 21 z 22 ∈ H C , the integral is taken over a loop in C going once around the origin in the counterclockwise direction. We regard these functions as polynomials on H C .
In order to construct the dual basis, we consider left n-regular functions using Lemmas 22 and 23 in [FL1], one can see that these are columns with entries being some scalar multiples of Shift the indexμ so thatμ − n/2 = µ and call the resulting function F ′(n) l,µ,ν (Z). Note that this is not always the same as differentiating N (Z) −1 · t l ν µ+n/2 (Z −1 ), since µ + n/2 may be bigger than l, which is outside of allowed range. Similarly, consider right n-regular functions using Lemmas 22 and 23 in [FL1], one can see that these are rows with entries being N (Z) −1 · t l+n/2 µ−n/2 ν−n/2 (Z −1 ), . . . , N (Z) −1 · t l+n/2 µ−n/2 ν+n/2 (Z −1 ).

n-Regular Functions on H ×
In this section we show that, if a (left or right) n-regular function is defined on all of H × , then the operators (deg +m), m = 2, . . . , n, can be inverted. Thus the operator D n = (deg +n) · · · (deg +2) can be inverted as well. This will be needed, for example, when we define the invariant bilinear pairing for such functions. We start with a left n-regular function f : H × → S ⊙ · · · ⊙ S and derive some properties of such functions. Of course, right n-regular functions g : H × → S ′ ⊙· · ·⊙S ′ have similar properties. Let 0 < r < R, then, by the Cauchy-Fueter formula for n-regular functions (Theorem 9), Proposition 13. Let f : H × → S ⊙ · · · ⊙ S be a left n-regular function. Then f (X) = f + (X) + f − (X), for all X ∈ H × .
Proof. The proof is the same as that of Proposition 13 in [FL4]. Let f * = f − f + − f − , we want to show that f * ≡ 0. Note that f * : H × → S ⊙ · · · ⊙ S is a left n-regular function such that D n f * ≡ 0, hence f * is a sum of homogeneous functions of degrees −2, −3, . . . , −n: Since the operators ∇ + k , k = 1, . . . , n lower the degree of homogeneity by one, each f −j : H × → S ⊙ · · · ⊙ S is left n-regular. We will show that each f −j , j = 2, 3, . . . , n, is identically zero. Let then f −n−1 is a left n-regular function that is homogeneous of degree −(n + 1). By the Cauchy-Fueter formulas for n-regular functions (Theorem 9), where R, r > 0 are such that r 2 < N (X) < R 2 . By Liouville's theorem (Corollary 10), the first integral defines a left n-regular function on H that is either constant or unbounded. On the other hand, the second integral defines a left n-regular function on H × that decays at infinity at a rate ∼ N (W ) −1−n/2 . We conclude that f −n−1 ≡ 0, hence f −n ≡ 0 as well.
Definition 14. Let f : H × → S ⊙ · · · ⊙ S be a left n-regular function. We define From the previous discussion we immediately obtain: Proposition 15. Let f : H × → S ⊙ · · ·⊙ S be a left n-regular function and g : H × → S ′ ⊙ · · · ⊙ S ′ a right n-regular function. Then, for m = 1, 2, . . . , n, From the expansions of the Cauchy-Fueter kernel (13) we immediately obtain an analogue of Laurent series expansion for n-regular functions.
Corollary 16. Let f : H × → S ⊙ · · · ⊙ S be a left n-regular function, write f = f + + f − as in Proposition 13. Then the functions f + and f − can be expanded as series l,µ,ν (X) .
If g : H × → S ′ ⊙ · · · ⊙ S ′ is a right n-regular function, then it can be expressed as g = g + + g − in a similar way, and the functions g + and g − can be expanded as series Formulas expressing the coefficients a l,µ,ν , b l,µ,ν , c l,µ,ν and d l,µ,ν will be given in Corollary 21.

Invariant Bilinear Pairing for n-Regular Functions
We define a pairing between left and right n-regular functions as follows. If f (Z) and g(Z) are left and right n-regular functions on H × respectively, then, by the results of the previous section, D −1 n f and D −1 n g are well defined, and we set where S 3 R ⊂ H is the sphere of radius R centered at the origin Note that we use the subscript R n in f, g Rn to indicate that the pairing is between left and right n-regular functions; however, this is not a pairing between the spaces R n and R ′ n , since the functions f and g are not allowed to have singularities away from the origin. Recall that by Lemma 6 in [FL1] the 3-form Dz restricted to S 3 R becomes Z dS/R, where dS is the usual Euclidean volume element on S 3 R . Thus we can rewrite (14) as Since ∇ + k D −1 n f = 0 and, by Lemma 6, the integrand of (14) is a closed 3-form. Thus, by Lemma 7, the contour of integration can be continuously deformed. In particular, this pairing does not depend on the choice of R > 0.
Proposition 17. If f (Z) and g(Z) are left and right n-regular functions respectively on H × , then Proof. Since the expression is independent of the choice of R > 0, we have: for all m ∈ Z. From this (15) follows.
Corollary 18. If f (Z) and g(Z) are left and right n-regular functions on H C respectively and W ∈ D + R (open domains D ± R were defined by equation (22) in [FL3]), the Cauchy-Fueter formulas for n-regular functions (Theorem 9) can be rewritten as We can rewrite the bilinear pairing (14) in a more symmetrical way. Let 0 < r < R and 0 < r 1 < R < r 2 . Using the Cauchy-Fueter formulas for n-regular functions (Theorem 9), substituting into (14) and shifting contours of integration, we obtain: As usual, we realize U (2) × U (2) as a diagonal subgroup of GL(2, H C ): where a * and d * denote the matrix adjoints of a and d in the standard realization of H C as 2 × 2 complex matrices. Then SU (2) × SU (2) is realized as Proposition 19. The bilinear pairing (14) is SU (2) × SU (2) and gl(2, H C )-invariant.
Proof. Observe that the functions F l,µ,ν are homogeneous of degree −(2l + n + 2). On the one hand, the bilinear pairing (14) is independent of the choice of radius R > 0. Recall that, by Lemma 6 in [FL1], the 3-form Dz restricted to S 3 R equals Z dS/R. It follows that if f is homogeneous of degree d f and g is homogeneous of degree d g , then either f, g Rn = 0 or d f + d g = −(n + 2). This proves (20) and Then (19) follows from the expansions (13), the Cauchy-Fueter formulas (Theorem 9) and equation (15).
7 Spaces of n-Regular Functions as Representations of the Conformal Lie Algebra In this section we identify the irreducible components and the K-types of the spaces of left and right n-regular functions on H × regarded as representations of the conformal Lie algebra gl(2, H C ). Let where l = 0, 1 2 , 1, 3 2 , . . . , µ ∈ Z + l + n/2, ν ∈ Z + l, −l − n/2 ≤ µ ≤ l + n/2, −l ≤ ν ≤ l.
Proof. The descriptions of F + n , G + n , F + n ⊕ F − n and G + n ⊕ G − n follow from Corollaries 16 and 21. Note that the inversion 0 1 1 0 ∈ GL(2, H C ) preserves C[z 0 , z 1 , z 2 , z 3 , N (Z) −1 ] ⊗ (S ⊙ · · · ⊙ S) as well as F + n ⊕ F − n . Comparing the degrees of homogeneity, we see that 0 1 1 0 switches F + n and F − n , and the description of F − n follows. Finally, the case of G − n is similar to that of F − n .
Proof. To show that F + n is invariant under the π nl -action of gl(2, H C ), we need to show that if F (n) l,µ,ν (Z) ∈ F + n and A B C D ∈ gl(2, H C ), then π nl A B C D F (n) l,µ,ν (Z) ∈ F + n . Indeed, consider first a special case of A = B = D = 0, then F (n) l,µ,ν (Z) is homogeneous of degree 2l and π nl 0 0 C 0 F (n) l,µ,ν (Z) is homogeneous of degree 2l+1. By the Cauchy-Fueter formula for n-regular functions (Theorem 9), where S 3 R ⊂ H is the sphere centered at the origin of radius R large enough so that W ∈ D + R . On the other hand, recall the expansion (13) of the Cauchy-Fueter kernel k n/2 (Z − W ). The integral l,µ,ν (Z) = 0 unless l ′ = l + 1/2 (see the proof of Proposition 20 based on the independence of the integral of R > 0). Then expansion (13) together with (23) show that π nl 0 0 C 0 F (n) l,µ,ν (W ) is a finite linear combination of F (n) l+1/2,µ ′ ,ν ′ (W )'s and hence an element of F + n . The other cases can be proved similarly.
As suggested by the referee, another proof of this proposition can be obtained from Theorem 5 and Proposition 22.
Proof. We will show that F ± n are irreducible. Then the irreducibility of G ± n is immediate from Propositions 19 and 20. It is easy to see from the description of the K-types of F ± n given by Proposition 24 that it is sufficient to show that any non-zero vector in F + n (d) also generates non-zero vectors in F + n (d − 1) as long as F + n (d − 1) = {0} and F + n (d + 1) as long as F + n (d + 1) = {0} (and similarly for F − n (d)). This follows from the observation that if v d ∈ F ± n (d), then π nl 0 B 0 0 v d ∈ F ± n (d − 1), π nl 0 0 C 0 v d ∈ F ± n (d + 1), for each 0 B 0 0 , 0 0 C 0 ∈ sl(2, H C ), Lemma 2 describing π nl 0 B 0 0 and the fact that conjugation by ( 0 1 1 0 ) ∈ GL(2, H C ) switches 0 B 0 0 and 0 0 C 0 .
Let us consider pairings on F ± n and G ± n : (f 1 , f 2 ) Rn = 1 2π 2 Z∈S 3 f + 2 (Z) · (D −1 n f 1 )(Z) dS, f 1 , f 2 ∈ F ± n , (g 1 , g 2 ) R ′ n = 1 2π 2 Z∈S 3 (D −1 n g 1 )(Z) · g + 2 (Z) dS, g 1 , g 2 ∈ G ± n , where S 3 ⊂ H ⊂ H C is the unit sphere centered at the origin. Clearly, these pairings are complex anti-linear, positive definite on F + n and G + n , and either positive or negative definite on F − n and G − n , depending on the effect of the operator D −1 n .