A Fock model and the Segal-Bargmann transform for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$

The minimal representation of a semisimple Lie group is a `small' infinite-dimensional irreducible unitary representation. It is thought to correspond to the minimal nilpotent coadjoint orbit in Kirillov's orbit philosophy. The Segal-Bargmann transform is an intertwining integral transformation between two different models of the minimal representation for Hermitian Lie groups of tube type. In this paper we construct a Fock model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$. We also construct an integral transform which intertwines the Schr\"odinger model for the minimal representation of the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2|2n)$ with this new Fock model.


Introduction
The classical Segal-Bargmann transform is a unitary isomorphism from the space of square integrable functions on R m to the Fock space of entire functions on C m which are square integrable which respect to the weight function exp −|z| 2 . The Segal-Bargmann transform is defined in such a way that it maps the creation (resp anihilation) operators on the Schrödinger space to coordinate multiplication (resp differentiation) on the Fock space. This implies in particular that the harmonic oscillator becomes the much simpler Euler operator on the Fock space [1]. The Segal-Barmann transform can also be interpreted as an intertwining operator between two models of the metaplectic representation (also known as the Segal-Shale-Weil or oscillator representation) of the metaplectic group, a double cover of the symplectic group. See [2] for more on the classical Segal-Bargmann transform and the metaplectic representation. The (even part of the) metaplectic representation is a prominent example of a minimal representation. Another extensively studied example is the minimal representation of O(p, q) [3,4,5,6]. The minimal representation of a semisimple Lie group is the irreducible unitary representation that according to the orbit method should correspond to the minimal nilpotent coadjoint orbit [7]. The Segal-Bargmann has been generalized to this setting of minimal representations. Namely for a Hermitian Lie group of tube type, there exists an explicit integral transform which intertwines the Schrödinger and Fock model of the minimal representation [8].
Lie superalgebras and Lie supergroups are generalisations of Lie algebras and Lie groups. They were introduced to mathematically describe supersymmetry. Their representation theory is an active field of study with still a lot of open questions, for instance a description of the unitary irreducible representations. Since most ingredients of the orbit method still exist in the super setting, it is believed that the orbit method should also in the super setting be a good tool for the study of irreducible representations [9,Chapter 6.3]. For example, the irreducible unitary representations of nilpotent Lie supergroups have been classified that way [10,11].
An ambitious aim in this light would be to construct minimal representations and the intertwining Segal-Bargmann transform for Lie supergroups. Recently a first step in that direction has already been taken. Namely the construction of a minimal representation of the orthosymplectic Lie supergroup OSp(p, q|2n) was accomplished in [12]. In the bosonic case (i.e. n = 0) this realisation corresponds to the Schrödinger model for O(p, q) of [6].
In this article we achieve two further goals. First we construct a Fock model for the minimal representation of the Lie superalgebra osp(m, 2|2n). We also define an integral transform which intertwines the Schrödinger model of the minimal representation of osp(m, 2|2n) with this Fock model. Note that only for q = 2 the Lie group O(p, q) is Hermitian of tube type and thus only in that case do we have a Segal-Bargmann transform. For that reason we have only constructed a Segal-Bargmann transform in the super setting for osp(m, 2|2n). Our main results hold for m − 2n ≥ 4. This restriction comes from [12], where key properties of the integral we use in the definition of the Segal-Bargmann transform are only proven for the case m − 2n ≥ 4.
We will work in this paper always on the algebraic level. So we will work with representations of the Lie superalgebra osp(m, 2|2n) instead of the Lie supergroup OSp(m, 2|2n), and we will act on super-vector spaces defined using superpolynomials instead of using global sections of a supermanifold. This allows us to circumvent the delicate technicalities associated with supergroups and supermanifolds. Note that in the bosonic case, the spaces we work with are dense in certain Hilbert spaces. Using standard techniques one can then integrate the representation to group level and extend to the whole Hilbert superspace, see for example [13,Theorem 2.30] or [8,Theorem 2.17]. These techniques no longer work/exist in the super case. There does exist an abstract way to integrate a so-called admissable (g, K)-module to group level [14] which was for example used in [12] to integrate the minimal representation of osp(p, q|2n) to OSp(p, q|2n). However, this approach is not very concrete.
Explicit examples such as the one constructed in this paper could help to develop such integration tools and to find the correct definitions in the super setting. For example our representations ought to be 'unitary' in some sense. A definition of unitary representations does exists in the supersetting [15,Definition 2]. However, a large class of Lie superalgebras, including osp(p, q|2n), do not allow for any super unitary representation in this sense [11,Theorem 6.2.1]. This highly unsatisfactory situation has inspired the search for a new or extended definition of a unitary representation [16,17]. At the moment it is still unclear what the right definition should be, but we believe that the construction of explicit examples which ought to be 'unitary' could be useful for this endeavour.
1.1. Structure of the paper. We structure the paper as follows. In Section 2 we fix notations and introduce the spaces and algebras we will use throughout the paper. In Section 3 we recall the Schrödinger model of the minimal representation of osp(m, 2|2n) defined in [12]. We also introduce an integral which leads to an osp(m, 2|2n)-invariant, non-degenerate, superhermitian form.
The next three sections contain the main body of this paper. In Section 4 we construct the Fock space as a quotient of the space of superpolynomials. We then define the Bessel-Fischer product, which gives us a non-degenerate, superhermitian form on our Fock space (Propositions 4.7 and 4.12). In the bosonic case (n = 0), this Bessel-Fischer product is equivalent to the inner product coming from an integral on the Fock space [8,Proposition 2.6]. Since we do not longer have this integral in the super setting, we construct a direct proof for the superhermitian property.
This seems new even in the bosonic case. We also show that our Fock space has a reproducing kernel (Theorem 4.11).
In Section 5 we endow this Fock space with an osp(m, 2|2n)-module structure leading to a Fock model of the minimal representation of osp(m, 2|2n). We prove that this is a irreducible representation and obtain a very explicit description (Theorem 5.3). In particular we have complete branching rules for the subalgebras osp(m|2n) and osp(m − 1|2n).
In Section 6, we define an integral transform which maps the space of functions used in the Schrödinger realisation to the space of functions of the Fock realisation (Definition 6.1). We show that this integral is an intertwining isomorphism which preserves the superhermitian form (Theorems 6.3 and 6.6). We also give an explicit inverse (Definition 6.7). As an application we use the Segal-Bargmann transform to define generalised Hermite functions.
In Appendix A we gather some definitions and results on special functions which are used throughout the paper. We have also put the technical and lengthy proof of Theorem 6.3 in Appendix B.

Preliminaries and notations
In this paper Jordan and Lie algebras will be defined over complex numbers C if they have a C in subscript, otherwise they are defined over the field of real numbers R. Function spaces will always be defined over the complex field C. We use the convention N = {0, 1, 2, . . .}.
A super-vector space is defined as a Z 2 -graded vector space, i.e., V = V 0 ⊕ V 1 , with V 0 and V 1 vector spaces. An element v of a super-vector space V is called homogeneous if it belongs to V i , i ∈ Z 2 . We call i the parity of v and denote it by |v|. An homogeneous element v is even if |v| = 0 and odd if |v| = 1. When we use |v| in a formula, we are considering homogeneous elements, with the implicit convention that the formula has to be extended linearly for arbitrary elements. We denote the super-vector space V with V 0 = R m and V 1 = R n as R m|n . We will always assume m ≥ 2.
2.1. Superpolynomials. Let K be either R or C.
Definition 2.1. The space of superpolynomials over K is defined as where P(K m ) denotes the space of complex-valued polynomials over the field K in m variables and Λ(K 2n ) denotes the Grassmann algebra in 2n variables. The variables of P(K m ) and Λ(K 2n ) are called even and odd variables, respectively. They satisfy the commutation relations Let · , · β be a supersymmetric, non-degenerate, even bilinear form on K m|2n . We denote the matrix components by β ij := x i , x j β and denote the components of the inverse matrix by β ij , i.e., β ij is defined such that j β ij β jk = δ ik . Set x j = i x i β ij . The differential operator ∂ i is defined as the unique derivation in End(P(K m|2n )) such that ∂ i (x j ) = δ ij , with δ ij the Kronecker delta. We also define When we are working with both real and complex polynomials at the same time, we will denote ∂ i and ∂ i for the real variable x i as ∂ x i and ∂ xi , respectively. Similarly, we will denote ∂ i and ∂ i for the complex variable z i as ∂ z i and ∂ zi , respectively.
We will make frequent use of the following operators: Here, the operator R 2 is called the square of the radial coordinate and acts through multiplication. The operators E and ∆ are called the Euler operator and the Laplacian, respectively. We have the following lemma.
an sl K (2)-triple. Furthermore they commute in End(P(K m|2n )) with the operators Proof. A straightforward calculation or see, for example, [18].
If we are working with two sets of variables we will add a variable indicator to avoid confusion. For example, we denote and L x ij = x i ∂ x j − (−1) |i||j| x j ∂ x i for the real variables and The orthosymplectic Lie superalgebra. Let K be either R or C. Definition 2.3. The orthosymplectic Lie superalgebra osp K (m|2n, β) is defined as the subalgebra of gl K (m|2n) preserving a supersymmetric non-degenerate even bilinear form β. Thus osp K (m|2n, β) is spanned by X ∈ gl K (m|2n) such that The orthosymplectic Lie superalgebra has a differential operator realisation on P(K m|2n ). A basis in this realisation is given by for i < j, 2.3. Spherical harmonics. Let K be either R or C. The space of homogeneous superpolynomials of degree k is denoted by The space of spherical harmonics of degree k is defined by i.e, it is the space of homogeneous polynomials of degree k which are in the kernel of the Laplacian. The Fischer decomposition gives a decomposition of the space of superpolynomials using these spherical harmonics [18,Theorem 3].
In [19] the following generalisation of the Fischer decomposition was obtained that still holds for the exceptional case M ∈ −2N. Proposition 2.5 (Generalised Fischer decomposition). The superspace P(K m|2n ) decomposes as is the space of generalised spherical harmonics of degree k.
The dimension of the spherical harmonics of degree k is given in [18,Corollary 1].
We will also use the following formula for the dimension of the spherical harmonics of degree k.
Proposition 2.8. The dimension of H k (K m|2n ), for m > 1 is given by then the proposition follows from Proposition 2.7. First suppose 2n ≤ k − 2, then the above equation becomes which is true since the recursive formula of binomial coefficients gives us that for all i ∈ {0, . . . , 2n}. For 2n > k − 2 we have the following extra terms where we ignore the last term if 2n = k − 1. Using basic binomial properties, these terms are clearly equal to zero.
For n = 0 a more insightful reasoning as to why this formula holds is given by Proposition 5 in [21].
2.4. The spin factor Jordan superalgebra J. To each Hermitian Lie group of tube type corresponds an Euclidean Jordan algebra. These Jordan algebras were a crucial ingredient in the unified approach to construct minimal representations and the Segal-Bargmann transform [13,8]. More concretely one can associate with each Jordan algebra certain Lie algebras (the structure algebra and the TKK-algebra), and the Jordan algebra is also used in the construction of spaces on which these Lie algebras act. In this paper we will not use anything directly from Jordan theory, but introducing osp(m, 2|2n) via the spin factor Jordan superalgebra leads to a natural decomposition of osp(m, 2|2n) as well as to some interesting subalgebras that we will use. Definition 2.9. A Jordan superalgebra is a supercommutative superalgebra J satisfying the Jordan identity Here the operator L x is (left) multiplication with x and [· , ·] is the supercommuta- Let K be either R or C. We will define the the spin factor Jordan superalgebra associated with a supersymmetric, non-degenerate, even, bilinear form. Let V K be a super-vector space over K with dim(V K ) = (m − 1|2n) and a supersymmetric, nondegenerate, even, bilinear form · , · β where, for K = R, the even part has signature (m − 1, 0). Recall that we always assume m ≥ 2. We choose a homogeneous basis Definition 2.10. The spin factor Jordan superalgebra is defined as J K := Ke 0 ⊕ V K with |e 0 | = 0. The Jordan product is given by Thus e 0 is the unit of J K . We extend the homogeneous basis ( of J K and extend the bilinear form · , · β as follows. Set Then the corresponding form · , · β is a supersymmetric, non-degenerate, even bilinear form on the super-vector space J K where, for K = R, the even part has signature (m − 1, 1).
Define (β ij ) ij as the inverse of (β ij ) ij . Let (e i ) i be the right dual basis of (e i ) i with respect to the form · , · β , i.e.
In this paper we will assume that the orthosymplectic metric is standardized such that From now on the real spin factor Jordan superalgebra will always be denoted by J or J R and its complexified version by J C .
2.5. The TKK algebra. With each Jordan (super)algebra one can associated a 3-graded Lie (super)algebra via the TKK-construction. There exists different TKK-constructions in the literature, see [22] for an overview, but for the spin factor Jordan superalgebra J K all constructions lead to the orthosymplectic Lie superalgebras osp K (m, 2|2n). We will quickly review the Koecher construction. First consider Inn(J K ), the subalgebra of gl(J K ) of inner derivations. It is generated by the operators [L u , L v ], u, v ∈ J K . If we add the left multiplication operators L u to the inner derivations we obtain the inner structure algebra: Let J + K and J − K be two copies of J K . As a vector space we define the TKK-algebra of J K as The Lie brackets on TKK(J K ) is defined as follows. We interpret istr(J K ) as a subalgebra of gl(J K ) and for homogeneous x, y From [12, Proposition 3.1] we obtain the following Theorem 2.11. We have where the direct sum decomposition is as algebras.
Using the bilinear form we have the following explicit isomorphism of TKK(J K ) with the differential operator realisation of osp K (m, 2|2n): 3. The Schrödinger model of osp(m, 2|2n) In the bosonic case (i.e. n = 0), the Schrödinger model of the minimal representation of a Hermitian Lie group G of tube type can be seen as a representation realized on the Hilbert space L 2 (C) where C is a minimal orbit for the structure group of G, see [8]. In the super setting the classical definition of minimal orbits no longer works. Indeed, supermanifolds, in contrast to ordinary manifolds, are not completely determined by their points. In [12] an orbit was instead defined as the quotient supermanifold of the structure group by a stabilizer subgroup together with an embedding. Using this definition, a minimal orbit C was constructed which can be characterized by R 2 = 0, with R 2 as introduced in (1). We refer to Section 4 of [12] for a detailed description of this minimal orbit. We will now recall the Schrödinger model constructed in [12]. A critical role in this construction is played by the Bessel operators.
3.1. The Bessel operator. The Bessel operator B λ (x k ) is a linear operator acting on P(R m|2n ). It depends on a complex parameter λ and an explicit expression is given by where E and ∆ are the Euler operator and Laplacian introduced in (1).
From Proposition 4.9 of [12] we obtain the following.
Therefore we will only use the Bessel operator with the parameter 2 − M in this paper and we set B(x i ) := B 2−M (x i ). We obtain the following two properties of the Bessel operator from Proposition 4.2 in [23].
As a direct result from the product rule we have In what follows we will mostly use the following slightly modified version of the Bessel operator.
3.3. The (g, k)-module W . We introduce the notations Theorem 3.5. The following isomorphisms hold as algebras.
Proof. This follows from a straightforward verification by, for example, looking at the matrix realisation of g.
In Section 5.2 of [12] the following g-module was constructed for M ≥ 4: where the g-module structure is given by the Schrödinger representation π.
Explicitly, π of g = TKK(J) = J − ⊕ istr(J) ⊕ J + acting on W is given as follows: . . , m + 2n − 1} and where ı is the imaginary unit. For n = 0, our convention corresponds to the Schrödinger realisation given in [8]. It only differs from the Schrödinger realisation given in [12] by a change of basis.
Theorem 5.3 of [12] gives the following decomposition of W . (1) The decomposition of W as a k-module is given by where W j and thus also W are k-finite.
(3) An explicit decomposition of W j into irreducible k 0 -modules is given by Furthermore, if m is even we also have the following k-isomorphism 3.4. The integral and sesquilinear form. In Section 8 of [12] an integral which restricts to W was constructed. We will use a renormalized version of that integral, restricted to x 0 > 0. To give the integral explicitly we consider spherical coordinates in R m−1 by setting . . , m − 1} and We also introduce the following notations . We remark that in [12,Lemma 8.2] it is shown that φ is actually an algebra isomorphism. The Berezin integral on Λ(R 2n ) is defined as where γ ∈ C is the renormalisation factor such that W exp(−4|X|) = 1.
That the integral is well defined follows from Section 8 of [12] together with the fact that γ is non-zero.
where we used the Pochhammer symbol (a) k = a(a + 1)(a + 2) · · · (a + k − 1). Note Proof. Let us denote the integral W not normalized by γ as W ′ . We wish to calculate In [25, Lemma 6.3.5] a very similar integral is calculated if one observes that Using the same calculations as in the proof of [25, Lemma 6.3.5 ], we then obtain provided we take into account that we need extra factors 2 in certain places and that we restrict ourselves to x 0 > 0. Note that in [25, Lemma 6.3.5 ] the tilde in K − 1 2 sometimes mistakenly disappears. If we use Legendre's duplication formula: Definition 3.9. For f, g ∈ W we define the sesquilinear form · , · W as f, g W := W f g.
Theorem 8.13 and Lemma 8.14 in [12] give us the following two properties.
The Schrödinger representation π on W is skew-supersymmetric with respect to · , · W , i.e., for all X ∈ g and f, g ∈ W .
Proposition 3.11. Suppose M = m−2n ≥ 4. The form · , · W defines a sesquilinear, non-degenerate form on W , which is superhermitian, i.e., Note that for both Theorem 8.13 and Lemma 8.14 in [12] there is an extra condition saying that M must be even. However, since we are working in the exceptional case that corresponds with q = 2 in [12], the proofs still hold without this extra condition.

The Fock space
In [8, Section 2.3] an inner product on the polynomial space P(C m ) was introduced, namely the Bessel-Fischer inner product p, q B := p( B)q(z) z=0 , whereq(z) = q(z) is obtained by conjugating the coefficients of the polynomial q. In [8, Proposition 2.6] it is proven that, for polynomials, the Bessel-Fischer inner product is equal to the L 2 -inner product of the Fock space. Since there is no immediate extension of this L 2 -inner product to the super setting, we will use the Bessel-Fischer inner product on polynomials as the starting point to generalize the Fock space to superspace.   Explicitly for p = α a α z α and q = β b β z β we have Note that it is only an inner product in the bosonic case. However, in [16] a new definition of Hilbert superspaces was introduced where the preserved form is no longer an inner product, but rather a non-degenerate, sesquilinear, superhermitian form. We will prove that the Bessel-Fischer product is such a form when restricted to F with M − 2 ∈ −2N. Proposition 4.3 (Sesquilinearity). For p, q, r, s ∈ P(C m|2n ) and α, β, γ, δ ∈ C we have αp + γr, βq + δs B = αβ p, q B + αδ p, s B + γβ r, q B + γδ r, s B .
Proof. This follows from the linearity of the Bessel operators.
Proposition 4.4 (Orthogonality). For p k ∈ P k (C m|2n ) and p l ∈ P l (C m|2n ) with l = k we have p k , p l B = 0.
Proof. This follows from the fact that Bessel operators lower the degree of polynomials by one.
To prove that the Bessel-Fischer product is superhermitian we will use induction on the degree of the polynomials. We will need the following lemma in the induction step of the proof. Lemma 4.6. Suppose we have proven that p, q B = (−1) |p||q| q, p B for all p, q ∈ P(C m|2n ) of degree lower than or equal to k ∈ N. Then for p, q ∈ P l (C m|2n ), l ≤ k we have for all i, j ∈ {1, . . . , m + 2n − 1}.
Proof. We will prove the L ij case explicitly. The L 0i case is entirely analogous. First note that if we combine the given superhermitian property with Proposition 4.5, we get for all p ∈ P(C m|2n ) of degree k or lower and all q ∈ P(C m|2n ) of degree k − 1 or lower. Assume p, q ∈ P l (C m|2n ), l ≤ k. We obtain This gives from which the desired result follows if we prove For the first term in right hand side of this equation we have such that, using similar calculations for the other three terms, Equation (7) can be rewritten as L ij ∆p, ∆q B = −(−1) (|i|+|j|)|p| ∆p, L ij ∆q B .
Since ∆p and ∆q are polynomials of degree lower than l the lemma follows from a straightforward induction argument on l. Proof. Because of the orthogonality we only need to prove the property for p, q homogeneous polynomials of the same degree. Because of the sesquilinearity we may assume that p and q are monomials. We will use induction on the degree k of the polynomials, the case k = 0 being trivial. Suppose we have proven the theorem for all p, q ∈ P k (C m|2n ). We now look at z i p, z j q B for arbitrary i, j ∈ {0, 1, . . . , m+2n−1}. To simplify the following calculations we will restrict ourselves to i, j = 0, the cases i = 0 or j = 0 being similar. Denote c = (M − 2 + 2k). Using the commutation of Equation (2) and Proposition 4.5 we find Using the induction hypothesis together with Proposition 4.5 this becomes Switching the roles of z i p and z j q we also obtain where we used the induction hypothesis on all three terms of the right hand side. If we use Lemma 4.6 on the last term and multiply both sides of this equation with (−1) (|p|+|i|)(|q|+|j|) we get where we made use of β ij = 0 for |i| = |j| in the last step.  for all i, j ∈ {1, . . . , m + 2n − 1} and p, q ∈ P(C m|2n ).
We have the following proposition.
Proposition 4.9. For all p, q ∈ P(C m|2n ) it holds that Thus the Bessel-Fischer product is well-defined on F .
Proof. Because of the superhermitian property we only need to prove p, R 2 q B = 0 holds for all p, q ∈ P(C m|2n ). The complex version of Proposition 3.1 implies that for arbitrary p, q ∈ P(C m|2n ) there exist a q ′ ∈ P(C m|2n ) such that Since the constant term of R 2 q ′ (z) is always zero this proves the theorem.
Note that the above proposition also shows that the Bessel-Fischer product is degenerate on the space P(C m|2n ). In the following subsection we look at the non-degeneracy of the Bessel-Fischer product when we restrict it to F .

4.2.
The reproducing kernel and non-degeneracy. In the bosonic case a reproducing kernel for the Fock space was constructed in Section 2.4 of [8]. We will prove the non-degeneracy of the Bessel-Fischer product on F by first constructing a generalisation of this reproducing kernel in superspace.
where we used the Pochhammer symbol (a) k = a(a + 1)(a + 2) · · · (a + k − 1) and z|w is defined as For all p ∈ P k (C m|2n ) we then have Proof. We have Now suppose p is a monomial, i.e. p(z) = a m+2n−1 l=0 z α l l , with |α| = k and a ∈ C. Iterating the previous calculation and working modulo R 2 w we obtain which gives us the desired result.
Proof. By Lemma 4.10, the orthogonality property and the Fischer decomposition we find that has the desired property.
The non-degeneracy of the Bessel-Fischer product on F is now an almost immediate result. Proof. Suppose p ∈ F is such that p, q B = 0 for all q ∈ F . Using the reproducing kernel we obtain p(w) = p, K(z, w) B = 0. Hence p = 0.
Note that the previous proposition only works when M − 2 ∈ −2N. For the M − 2 ∈ −2N case the Bessel-Fischer product will always be degenerate.
we conclude that such a q = 0 exists.

The Fock model of osp(m, 2|2n)
In [8] the Fock representation ρ := π C • c. is obtained by twisting the complexification of the Schrödinger representation π C with the Cayley transform c. This Cayley transform is an isomorphism of g C which induces a Lie algebra isomorphism between k C and istr(J C ). We will use a similar approach in our construction of the Fock model. We start by defining a Cayley transform c in our setting. 5.1. Definition and properties. Let g C and k C be the complexified versions of g and k introduced in Section 3.3, i.e. Proof. Expanding exp(− ı 2 ad(e − 0 )) exp(−ı ad(e + 0 )) we obtain We complexify the Schrödinger representation π given in Section 3.3 to obtain a representation π C of g C = J − C ⊕ istr(J C ) ⊕ J + C acting on F . Explicitly, π C is given by , with i, j, k ∈ {1, 2, . . . , m + 2n − 1} and l ∈ {0, 1, 2, . . . , m + 2n − 1}. Since L ij , E and B(z l ) map R 2 into R 2 this representation is well defined on F . As in the bosonic case, we will define the Fock representation ρ as the composition of π C and the Cayley transform c, So ρ of g = J − ⊕ istr(J) ⊕ J + acting on F is given as follows with i, j, k ∈ {1, 2, . . . , m + 2n − 1}.
and similarly for k ∈ {0, . . . , m + 2n − 1}. Because of Proposition 4.4 we may assume p and q are homogeneous polynomials the same degree. We now have which proves the theorem 5.2. The (g, k)-module F . We define where the g module structure is given by the Fock representation ρ. We also introduce the notation and reintroduce r 2 from Section 3.2 as its complexified version: We have R 2 = −z 2 0 + r 2 and since we are working modulo R 2 this implies z 2 0 = r 2 . In the following, we will work modulo R 2 but omit R 2 from our notation. For M − 1 ∈ −2N we now find where we made use of the Fischer decomposition (Theorem 2.4) in the last step. In particular, by Proposition 2.8, If M ∈ −2N, then for p ∈ F k the Fischer decomposition on P k (C m|2n ) also gives (1) F k is an irreducible k-module.
(2) F is an irreducible g-module and its k-type decomposition is given by (3) An explicit decomposition of F k into irreducible k 0 -modules is given by Proof. We have the following elements of the action of g: Again by Proposition 2.6 the elements ρ ij and ρ 0i give rise to an irreducible representation of k on H k (C m|2n ) ∼ = F k , which proves (1). For the first two elements we have , which shows that ρ + 0 allows us to go to polynomials of higher degrees while ρ − 0 allows us to go the other direction for M ≥ 3. Therefore we obtain (2).
The following isomorphism is a direct result of this theorem.

The Segal-Bargmann transform
In this section we construct the Segal-Bargmann transform and show that it is an isomorphism from W as defined in Section 3.3 to F as defined in Section 5.2. It will make use of the integral W we defined in Definition 3.7. This integral is only defined for M ≥ 4. Therefore we will always assume M ≥ 4 throughout this section.
6.1. Definition and properties. Let I α (t) be the I-Bessel function as introduced in Appendix A.1. We define an entire function I α on C by Clearly we have I 0 (0) = 1 and We are now able to state the Segal-Bargmann transform that extends the one from the bosonic case obtained in [8].
where x|z is defined as x i β ij z j .
Here we view I α (x|z) as a superfunction in the variables x and z by symbolically replacing t with x|z in the expression Note that I 0 (x|z) is the reproducing kernel K(x|z) of the Fock space we found in Theorem 4.11. Proof. We wish to prove that the integral is convergent for all f ∈ W and that SB(R 2 ) = 0. As shown in the proof of Theorem 8.13 in [12], the elements of W can be decomposed into elements of the form P k K − 1 2 +α1+α2 (2|X|), with P k a homogeneous polynomial of degree k. Here K α is the K-Bessel function introduced in Appendix A.1 interpreted as a radial superfunction as in Section 3.2.
Furthermore α 1 , α 2 ∈ N are subject to the relations k ≥ α 1 + 2α 2 and M ≥ 2α 1 + 2. Also, observe that which is equal to x 0 within the domain of integration of W . Because of all this and Equation (9) it suffices to prove is convergent for all k ∈ N. We will use the explicit description of W given in Equation (5). The morphism φ ♯ leaves the degree of a polynomial unchanged. Hence, we can expand where a j (θ) is a polynomial in P(R 0|2n ) of degree j and b j (ω) is a function depending on the spherical coordinates ω. For c ∈ Z we obtain φ ♯ ( K α (c|X|)) = K α (c|X|), from the proof of Lemma 8.6 in [12]. Here η, ξ and θ 2 were defined in Section 3.4. We introduce the notations We can use Equation (3) with h = I 0 and f = x|z as a function in the x variables to obtain Similarly, using the properties of φ ♯ described in Lemma 8.3 of [12], we now find Combining all this, we see that with β 1 ≥ 0 converges if σ > 2 max{β 2 , 0} + 2 max{β 3 , 0}. This follows from the asymptotic behaviour of the Bessel functions, see Appendix A.1. Therefore we get the following condition with j 1 + j 6 + 2j 2 + 2j 3 + 2j 4 + 2j 7 j 8 = 2n. Taking into account k ≥ α 1 + 2α 2 and M ≥ 2α 1 + 2 the condition reduces to M > 2. We still need that SB(R 2 ) = 0, but this follows easily from (φ ♯ (R 2 )) |s=x0=ρ = (−x 2 0 + s 2 ) |s=x0=ρ = 0. We can now show that SB intertwines the Schrödinger model with the Fock model. Proof. The proof of this theorem is a technical and long but rather straightforward calculation. We refer to Appendix B for more details. Therefore the Segal-Bargmann transform maps a non-zero element of W to a nonzero element of F . It also intertwines the actions π and ρ. Since Theorem 3.6 and Theorem 5.3 give us that W and F are irreducible g-modules, we conclude that SB is an isomorphism of g-modules.
Since they act on a constant with respect to the variable z we get I 0 (x| B(z)) exp( B(z 0 ))1 = I 0 (0) exp(0)1 = 1, which gives if we use Equation (8). Now suppose f, g ∈ W . Since W is an irreducible g-module (Theorem 3.6), there exists a Y ∈ U (g) such that f = π(Y ) exp(−2|X|). Therefore we can reduce the general case to the previous case using the intertwining property (Theorem 6.3) and the fact that the sesquilinear forms forms are skew symmetric for π and ρ (Propositions 3.10 and 5.2): which proves the theorem.
6.2. The inverse Segal-Bargmann transform. Definition 6.7. For p ∈ P(C m|2n ) the inverse Segal-Bargmann transform is defined as Note that both I 0 x| B(z) and exp − B(z 0 ) are infinite power sums of the Bessel operator. However, they are well defined operators on polynomials in z since the Bessel operator lowers the degree of the polynomials and therefore the power operators become zero after a finite number of terms. Thus SB −1 is a well-defined operator on P(C m|2n ). Because of Proposition 3.1 it maps R 2 to zero and thus SB −1 can be restricted to F . That it is also well defined as the inverse of the Segal-Bargmann transform follows from the following proposition. Proof. From Proposition 6.4, we know that SB has an inverse. Suppose the operator A is this inverse. Using Theorem 6.6 we then have the following calculation.
where we used Equation (8) in the last step. Since the sesquilinear form · , · W is non-degenerate we obtain A = SB −1 .
We can make the inverse Segal-Bargmann transform more explicit on the space of homogeneous polynomials. To the best of our knowledge, this explicit expression is also new for the bosonic case.
for α ∈ N m|2n . The generalized Hermite polynomials H α are defined by the equation Proposition 6.11 (Hermite to monomial property). We have Proof. We will use the fact that W is supersymmetric with respect to the Bessel operators [12,Proposition 8.9]: Combining this with Equation (12) of Lemma B.1 we find Because of Lemma 6.5 the theorem follows.
and the K-Bessel function K α (or modified Bessel function of the third kind) by for α, t ∈ C, see [26], Section 4.12. In this paper we will need the following renormalisations Remark that we have the following special case [26,Equation 4.12.4] We then find The calculations for the case B(x k ) I 0 (x|z) is analogous.
This proves the lemma.
We can now prove Theorem 6.3. For convenience we restate it here.