Branching laws for spherical harmonics on superspaces in exceptional cases

It turns out that harmonic analysis on the superspace Rm|2n is quite parallel to the classical theory on the Euclidean space Rm unless the superdimension M:=m−2n is even and non-positive. The underlying symmetry is given by the orthosymplectic superalgebra osp(m|2n) . In this paper, when the symmetry is reduced to osp(m−1|2n) we describe explicitly the corresponding branching laws for spherical harmonics on Rm|2n also in exceptional cases, i.e, when M−1∈−2N0 . In unexceptional cases, these branching laws are well-known and quite analogous as in the Euclidean framework.


Introduction
Lie superalgebras and their representations play an important role in mathematics and physics.For an account of their mathematical theory, see [10,1].Major applications in physics are supersymmetry and supergravity, see [15] and references there.
As a specific application in physics, let us mention the quantum Kepler problem on the superspace R m|2n studied in [17].The underlying symmetry is given by the orthosymplectic superalgebra osp(m|2n).By describing the structure of polynomials on R m|2n under a natural action of osp(m|2n), the energy eigenvalues and the bound states spectrum for the corresponding Schrödinger equation are determined but only when M := m − 2n > 1.
Here M is the so-called superdimension of R m|2n .In particular, to solve the problem two key ingredients are used.The first fact is that the osp(m|2n)module of polynomials on R m|2n is completely reducible when M > 1, and the second one is the branching law for spherical harmonics under the same condition M > 1, see [17,Appendix A].In general, as we recall later on in detail, the first fact is not valid (see Theorems A and B below) and, as we show in this paper, the general branching law is much more complicated, see Theorems 2 and 3 below.
In the framework of Clifford analysis, F. Sommen together with his collaborators has developed function theory on the superspace R m|2n which deals with polynomials or functions depending on m commuting (bosonic) and 2n anticommuting (fermionic) variables [2,3,4,5,6,7], cf.[14].The full symmetry is governed by the Howe dual pair (osp(m|2n), sl(2)), see [2].For an account of Howe dualities, we refer to [8,9,1,13].Early, it turned out that harmonic analysis on the superspace R m|2n is quite parallel to the classical theory on the Euclidean space R m but only in the case when M ∈ −2N 0 .In particular, when M ∈ −2N 0 the osp(m|2n)-module of polynomials on R m|2n is completely reducible, see [7,2], generalizing the result obtained in [17] just for M > 1 we mentioned above.Indeed, when M ∈ −2N 0 , it is shown that the Fischer decomposition is an irreducible decomposition of homogeneous polynomials on R m|2n under the action of osp(m|2n), see [2], cf.[17].
It is well-known that, in the classical case (i.e., when n = 0), a contruction of the so-called Gelfand-Tsetlin (GT) bases of the spaces H k (R m ) of k-homogeneous spherical harmonics on R m is based on knowledge of the Fischer decompositions.Indeed, this enables to describe explicitly branching laws of the spaces H k (R m ) when the symmetry given by Lie algebra so(m) is reduced to so(m − 1), and then, for a given chain of Lie algebras so(m) ⊃ so(m − 1) ⊃ • • • ⊃ so(2), the construction of the corresponding GT bases, see e.g.[11] for more details.On the other hand, in the super case, when the symmetry given by osp(m|2n) is reduced to osp(m − 1|2n) branching laws for spherical harmonics on R m|2n are given in [2], cf.[17], just in unexceptional cases, i.e, when M −1 ∈ −2N 0 .
In [12], for the first time, the Fischer decomposition for scalar valued polynomials on R m|2n was obtained also in the exceptional cases, i.e, when M ∈ −2N 0 and m = 0.In the exceptional cases, the Fischer decomposition is rather an indecomposable (but not necessarily irreducible) decomposition of homogeneous polynomials under the action of osp(m|2n), see Theorems A and B below.In the general case, the osp(m|2n)-module of polynomials on R m|2n was also described recently in [16].
In this paper, when the symmetry is reduced to osp(m−1|2n) we describe branching laws for osp(m|2n)-modules of homogeneous (generalized) spherical harmonics on R m|2n also in the exceptional cases, i.e, when M −1 ∈ −2N 0 , see Theorems 2 and 3 below.To do this we apply the Fischer decompositions in the exceptional cases and the Cauchy-Kovalevskaya extension for polynomials on superspaces, see Theorem 1.These general branching laws enable to construct GT bases of the spaces H k (R m|2n ) of k-homogeneous spherical harmonics on R m|2n for a given chain of Lie superalgebras similarly, as in the classical case, see Section 5. We plan to investigate properties of the GT bases of H k (R m|2n ) in the next paper.
In [4], super Dirac operator is introduced and the corresponding Fischer decomposition is established for spinor valued polynomials on R m|2n but again only when M ∈ −2N 0 .We believe that it is possible to generalize the results obtained in this paper to the spinor valued polynomials on R m|2n .

Harmonic analysis on superspaces
In this section, we briefly recall some known facts from harmonic analysis on superspaces, see [2] and [12] for more details.
Let V = V 0 ⊕V 1 be a finite dimensional superspace endowed with a scalar superproduct g, that is, g is a non-degenerate bilinear form on V such that We view V as the defining representation for the Lie superalgebra g = osp(V, g) and we identify the supersymmetric tensor algebra of V with the space P(V ) of scalar-valued polynomials on V .There is a natural action of g on the superpolynomials P(V ) and it turns out that the hidden symmetry is given by the Lie algebra sl(2) of invariant operators on P(V ) which is generated by the superlaplacian ∆ and the square norm R 2 of the supervariable X ∈ V (that is, R 2 = g(X, X)).
Then the space P = P(R m|2n ) of polynomials on R m|2n is given as where R[x 1 , . . ., x m ] are the polynomials in m commuting (bosonic) variables x 1 , . . ., x m and Λ(θ 1 , . . ., θ 2n ) is the Grassmann algebra generated by 2n anticommuting (fermionic) variables θ 1 , . . ., θ 2n .We assume that the matrix of the scalar superproduct g is the block diagonal matrix where E m is the identity matrix of size m and J 2n is the square matrix of size 2n given by As usual we write osp(m|2n) for osp(V, g).
The basic osp(m|2n)-invariant operators on P are Here we have the supercommutation relations [∂ X i , X j ] = δ ij for 1 ≤ i, j ≤ m+2n, ∆ is the super Laplace operator and E is the super Euler operator on R m|2n .These operators generate the Lie algebra sl(2), that is, the relations hold true where M = m − 2n is the so-called superdimension of R m|2n .See [6,7,2] for details.
Using the operator E we define the space of k-homogeneous polynomials by For an operator A on P, we denote Then the space The Fischer decompositions.(I) In [7, Theorem 3], the Fischer decomposition of polynomials on R m|2n into spherical harmonics is obtained unless M ∈ −2N 0 .In this case, for each k ∈ N 0 , we have that and thus Hence the Fischer decomposition looks like the classical (purely bosonic) one.In Figure 1, all the summands of P k are contained in the k-th column.Each row yields an infinite dimensional representation of sl(2).
Let us remark that, in the case m = 1, we have H k = 0 if and only if k = 0, . . ., 2n + 1 (see [2]).So in this case, for all k > 2n + 1, the k-th rows in Figure 1 are missing.
In [12], the Fischer decomposition for polynomials on R m|2n is described even in the exceptional case when M ∈ −2N 0 and m = 0.In this case, denote the set of exceptional indices by Then, for k ∈ I M , it is known that H k ∩ R 2 P k−2 = ∅ and so the decomposition (1) cannot be valid.In the general case, the following decomposition holds true, see [12], P k = Hk ⊕ R 2 ∆R 2 P k−2 where Hk = Ker k (∆R 2 ∆).Thus we have that Now we recall structure of osp(m|2n)-modules Hk and H k .Denote by L m|2n λ an osp(m|2n)-irreducible module with the highest weight λ.We use the simple root system of osp(m|2n) as in [2,17].This is not the standard choice [10] but is more convenient for our purposes.
and the indecomposable osp(m|2n)-module Hk has a composition series H 0 k ⊂ H k ⊂ Hk with the irreducible quotients Here e.g.H k /H 0 k is the quotient of vector spaces endowed with a natural action of osp(m|2n).
Let us note that some direct summands of the decomposition (4) might be trivial.Indeed, we have Theorem B. ( [12]) Let k ∈ N 0 , N k = {k − 2j| j = 0, . . ., ⌊k/2⌋} and Jk = N k ∩ I M .Then, under the action of osp(m|2n), P k has an indecomposable decomposition where The particular case M = −4 is depicted in Figure 3.The exceptional indices are I −4 = {4, 5, 6}.Notice that the first three rows look like the diagram for the purely fermionic Fischer decomposition with n = 2 (see Figure 2).Other rows are infinite as in the classical case (see Figure 1).The k-th row starts with H k except for the exceptional indices k ∈ I −4 when it starts with Hk .

The Cauchy-Kovalevskaya Extension
For an explicit description of branching laws for (generalized) spherical harmonics on R m|2n when the symmetry is restricted from osp(m|2n) to osp(m − 1|2n) we need to generalize an algebraic version of the so-called Cauchy-Kovalevskaya extension to the super setting.In particular, we show that each polynomial Q k on R m|2n is uniquely determined by its laplacian ∆Q k and the values of Q k and ∂ xm Q k on the hyperplane x m = 0. We identify the hyperplane x m = 0 with R m−1|2n and write P = P(R m−1|2n ) for the polynomials in the supervariable Indeed, we have the following result.
(ii) Then the Cauchy-Kovalevskaya extension operator CK is an invariant isomorphism of P k ⊕ P k−1 ⊕ P k−2 onto P k under the action of osp(m − 1|2n).In addition, we have where Here ∆ is the Laplace operator on R m−1|2n .
Proof.(a) It is clear that we can write polynomials for some polynomials p j , q j ∈ P j .Then Q k is a unique solution of the initial value problem (6) if and only if Indeed, we have that Then the equation ∆Q k = P k−2 is equivalent to the recursion relation (10) by comparing terms with the same powers x j m in the expansions of P k−2 and ∆Q k , see (9) and (11).
(b) For a given ℓ = 0, . . ., k, assume that all the initial data p k−j vanish except possibly for p k−ℓ .Then, by (10), it is easy to see that the solution Q k of the problem ( 6) is given by Q k = X ℓ p k−ℓ .Here the operator X ℓ is defined as in (8).To finish the proof of the general case of the statement (ii), we use linearity of the problem (6).
4 Branching laws for spherical harmonics K. Coulembier [2] (cf.[17]) showed that, under the action of osp(m − 1|2n), the space H k = H k (R m|2n ) of spherical harmonics decomposes as provided the superdimension ).We extend these branching laws also to the exceptional cases.
Therefore, as osp(m − 1|2n)-modules, H k is isomorphic to P k ⊕ P k−1 .Now it is sufficient to decompose P k and P k−1 using the Fischer decomposition (5).Since Under the action of osp(m − 1|2n), we have an irreducible decomposition Moreover, by Theorem A, we know that Therefore, as osp(m − 1|2n)-modules, Hk ≃ P k ⊕ P k−1 ⊕ H 2−M −k .Now it is sufficient to use the 'non-exceptional' Fischer decomposition (1) for P k , P k−1 and the branching law (12) for

GT bases for spherical harmonics
Using the branching laws described in Section 4 it is now easy to construct GT bases of (generalized) spherical harmonics on R m|2n for the chain of Lie superalgebras We shall proceed by induction on m.
(b) Assume that (M − 1) ∈ −2N 0 .Then, for a given k ∈ N 0 , we have Obviously, by Theorem 1, the polynomials (17), (18), ( 19) and (20) form a basis of the space H k .We plan to express elements of the GT bases in terms of classical special polynomials and investigate properties of the GT bases in the next paper.