A superunitary Fock model of the exceptional Lie supergroup $\mathbb{D}(2,1;\alpha)$

We construct a Fock model of the minimal representation of the exceptional Lie supergroup $\mathbb{D}(2,1, \alpha)$. Explicit expressions for the action are given by integrating to group level a Fock model of the Lie superalgebra $D(2,1, \alpha)$ constructed earlier by the authors. It is also shown that the representation is superunitary in the sense of de Goursac--Michel.


Introduction
The main result of this paper is the construction of a Fock model of the minimal representation of the Lie supergroup D(2, 1, α). We do this by integrating to group level the representation of the Lie superalgebra D(2, 1, α) considered in [1]. In that sense, this paper can be seen as a sequel to [1]. We also show that this Fock model is superunitary in the sense of [2].
Minimal representations of Lie groups have a long tradition and can be constructed in many settings [3,4,5,6,7,8,9]. In the philosophy of the orbit method, minimal representations correspond to the minimal nilpotent orbit of the coadjoint action of the Lie group on the dual Lie algebra. They are 'small' infinite-dimensional representations, or more technically, they attain the smallest Gelfand-Kirillov dimension of all possible infinite-dimensional representations [5]. This implies that there are a lot of symmetries in their realisations which leads to a rich representation theory.
Recently, there has been an effort to generalize the framework of minimal representations to the setting of Lie supergroups and Lie superalgebras [10,11,12,1]. Although this is a logical next step, there are lot of technical and conceptual hurdles. For instance, a lot of tools used for Lie groups are not yet developed or become much more complex in the super setting. Another obstacle is the fact that in [13] it is shown that there are no superunitary representations for a large class of Lie supergroups in the standard definition [14] of a superunitary representation. This has lead to the development of alternative definitions of what should be a superunitary representation [2,15,16]. However, at the moment no satisfactory definition has been found. Therefore it is important to construct concrete models of representations that 'ought' to be superunitary, as we will do in this paper.
For the orthosymplectic Lie supergroup OSp(p, q|2n) a Schrödinger model of the minimal representation was constructed in [11] using the framework of Jordan (super)algebras developed in [6]. This generalizes the minimal representation of O(p, q) considered in [7,17,18,19]. Later, also a Fock model and intertwining Segal-Bargmann transform for OSp(p, q|2n) were obtained in [12].
Recently, a Schrödinger model, Fock model and Segal-Bargmann transform of the Lie superalgebra D(2, 1, α) were constructed [1]. The paper [1] works entirely on algebra level. In particular it does not say anything about unitarity. It does, however, show that there exists a superhermitian product for which the Fock model is invariant. The goal of this paper is to integrate the Fock model considered in [1] to group level. We will show that the superhermitian product can be extended to a Hilbert space and that our representation extend to a superunitary representation in the sense of [2].
1.1. Contents. Let us now describe the contents of this paper. We start in Section 2 by recalling the definition of the Lie superalgebra D(2, 1, α) and giving an explicit expression of the Fock model considered in [1]. In Section 3, we introduce the Lie supergroup D(2, 1, α) and deduce some properties we need to integrate the representation, while in Section 4, we recall the necessary properties of the polynomial Fock space considered in [1] and complete it to a Hilbert superspace.
Section 5 contains the main content of this paper. We start by giving an explicit form of the representation of the Lie supergroup D(2, 1, α) in Theorem 5.1. We also give two alternatives way to present this representation (Corollaries 5.2 and 5.3). Note, however, that for one generating element of D(2, 1, α) we were only able to give an explicit form if α > 0.
We recall the definition of a superunitary representation (SUR) as introduced in [2] in Subsection 5.3 and show that the Fock model is such a SUR if α < 0 (Theorem 5.10). In [2], also the concept of a strong SUR is defined. However, we show that the Fock model is never a strong SUR (Theorem 5.12).

1.2.
Notations. The field K will always mean the real numbers R or the complex numbers C. Function spaces will always be defined over C. We use the convention N = {0, 1, 2, . . .} and denote the complex unit by ı.
A supervector space is a Z/2Z-graded vector space . We call i its parity and denote it by |v|. 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. If dim(V i ) = d i , then we write dim(V ) = (d 0 |d 1 ). We denote the super-vector space V with V 0 = K m and V 1 = K n as K m|n . A superalgebra is a supervector space A = A 0 ⊕ A 1 for which A is an algebra and A i A j ⊆ A i+j .
Consider a two-dimensional vector space V with basis u + and u -. Introduce a non-degenerate skew-symmetric bilinear form ψ by ψ(u + , u -) = 1. We will need three copies (V i , ψ i ), i = 1, 2, 3 of (V, ψ) and the corresponding Lie algebra sl(V i ) = sp(ψ i ) of linear transformations preserving ψ i .
Consider the matrices They give a realisation of sl(V i ) where the vector space V i is given by We also obtain p from For the odd basis elements u 1 ± ⊗ u 2 ± ⊗ u 3 ± of D(2, 1; α) we introduce a more compact notation We have the following realisation of sl(2) in D(2, 1, α) The corresponding three-grading by the eigenspaces of ad(H 2 + H 3 ) is given by 2.2. The Fock representation. In [1] a Fock representation dρ depending on a parameter λ ∈ {1, α} was constructed on the so called polynomial Fock space F λ . We will briefly reconstruct it here and refer to [1, Section 4.3] for further details. Note that from now on we will exclude α = 0 and α = 1 since for that case the picture becomes quite different. Remark that α = 1 correspond to the non-deformed case D(2, 1, 1) = osp(4|2), while for α = 0, the algebra D(2, 1, 0) is not simple. See [1, Section 4.2] for a more detailed explanation. Let z 1 , z 2 and z 3 , z 4 be the even resp. odd representatives of the coordinates on P(C 2|2 ). We define with I λ := P(C 2|2 )V λ .
As shown in [1, Section 5.1], if p ∈ F α , then there exists p i,k ∈ C such that The explicit expression for the Fock representation dρ on F λ is as follows.

The Lie supergroup D(2, 1; α)
In this section we define the supergroup D(2, 1; α) which has D(2, 1; α) as its Lie superalgebra. We also give some basic results of SL(V ), which we will need later on. Note that in this section we will work over the field R of real numbers.
3.1. Definition of D(2, 1; α). We will use the characterisation of Lie supergroups based on pairs, see for example [22,Chapter 7] for more details.
Since σ extends the adjoint representation of G 0 on g 0 we call it the adjoint representation of G 0 on g and denote it by Ad.
Note that these Lie supergroups are called super Harish-Chandra pairs in [2]. The term Lie supergroup is then used for a supermanifold endowed with a group structure for which the multiplication is a smooth map. However, as is mentioned in [2] these two structures are categorically equivalent.

3.2.
Properties of SL(V ). Define the following one-dimensional subgroups of SL(V i ) for i ∈ {1, 2, 3} On the one hand we have the Cartan decomposition of SL(V i ).
Theorem 3.2 (Cartan decomposition). We have a decomposition SL(V ) = KAK, i.e., every g ∈ sl(V ) can be written as g = kak ′ with k, k ′ ∈ K and a ∈ A.
This decomposition implies that a representation of SL(V i ) is fully determined by its restriction to K i and A i . On the other hand we have an explicit integration of sl(V i ) to SL(V i ).
Proof. This follows immediately from the multinomial theorem and the fact that (A + B + C) 2 = A 2 + B 2 + C 2 is a sum of three commuting variables.
There exists an X ∈ Lie(SL(V i )) such that g = exp(X) if and only if g is the identity or for some a, k, l ∈ R such that ρ := √ a 2 + l 2 − k 2 = 0. In this case we have for ρ = 0. For ρ = 0 this calculation gives us exp(X) = I.
Note that in particular, we have This implies that from an explicit representation of sl(V i ) we can obtain an explicit action of elements in K i and A i when integrated to the group level. Because of the Cartan decomposition this then defines an action of SL(V i ).
Since we can write every element of SL(V i ) as a finite product of exponentials of elements of sl(V i ), we obtain the following corollary for D(2, 1; α).
can be written as a finite product of exponentials of elements of g 0 , i.e., for all g ∈ G 0 we have for some X i ∈ g 0 and n ∈ N.

The Fock space F
In this section we introduce the notion of a Hilbert superspace as defined in [2]. We also extend the polynomial Fock space F λ to the Fock space F and show it is such a Hilbert superspace when combined with the Bessel-Fischer product.
From now on we will restrict ourselves to the case α ∈ R \ N since only then the Bessel-Fischer product will be non-degenerate. Furthermore, we also choose λ = α and denote the polynomial Fock space F λ by F. Recall from Subsection 2.3 that the case λ = 1 is always equivalent to a representation with λ = α.  From [1, Proposition 5.6.] we obtain the following explicit form of the Bessel-Fischer product.
Then the only non-zero evaluations of p, q B are where we used the Pochhammer symbol (a) k = a(a + 1)(a + 2) · · · (a + k − 1).
From this explicit form we can easily see that the Bessel-Fischer product is degenerate if and only if α ∈ N, which is why we assume α ∈ R \ N.
According to the propositions in [1, Section 5], the polynomial Fock space F endowed with the Bessel-Fischer product · , · B is such a Hermitian superspace. for all x, y ∈ H is an inner product on H.
For F we find the following condition on its fundamental symmetries with respect to the Bessel-Fischer product.
Proof. Suppose J is an arbitrary fundamental symmetry of F, then we have for an ǫ > 0. The other three cases are similar.
Based on this condition, we define the endomorphism S of F by the linear extension of Then, one can easily verify that S is a fundamental symmetry of F with respect to the Bessel-Fischer product. Proposition 4.6. Suppose p, q ∈ {z k 1 , z k 1 z 2 , z k 1 z 3 , z k 1 z 4 }, with k ∈ N. Then the only non-zero evaluations of (p, q) S are where we used the Pochhammer symbol (a) k = a(a + 1)(a + 2) · · · (a + k − 1).
Proof. This follows immediately from Proposition 4.2. Note that the choice of a fundamental symmetry does not matter for the topology, thanks to [2,Theorem 3.4].
Denote by F the completion of F with respect to (· , ·) S , then (F , · , · B ) is a Hilbert superspace, which we call the Fock space. Define f S := (f, f ) S , then we have The condition f S < ∞ on f is equivalent to the condition that the sums

The superunitary representation ρ 0
In this section we explicitly integrate the differential action dρ of D(2, 1, α) on F to an action ρ 0 of D(2, 1, α) on F . We also introduce the notion of superunitary representations as defined in [2]. Then, we prove that our action defines a superunitary representation on F for α < 0.
Recall from Section 4 that we assume α ∈ R \ N.

Definition and explicit form.
We define ρ 0 (exp(X)) := exp(dρ(X)) for all X ∈ g 0 . Because of Corollary 3.5 this defines a representation of all of G 0 . We will now describe this representation more explicitly. Note that we omit the action of A 2 (a 2 ) from our explicit representation. This case will be discussed in Section 5.2.

Proof.
(3) We have and therefore This gives us ).
Corollary 5.2. The representation ρ 0 acting on f ∈ F given by The second method is as follows. Denote by P k (C m|n ) the space of homogeneous superpolynomials of degree k in m even variables and n odd variables. Then , defines an isomorphism between F λ and the space of even degree superpolynomials in the even variable ℓ 1 and the two odd variables ℓ 2 , ℓ 3 . Here the "even" in P even (C 1|2 ) refers to the degree and not the parity of the superpolynomial terms.
Note that the symbolic change of odd variables ℓ 2 and ℓ 3 to the constant 1 is only well defined if we use the convention that every instance of ℓ 3 ℓ 2 in f is first rewritten as −ℓ 2 ℓ 3 .

5.2.
The action of A 2 (a 2 ). For the action ρ 0 (A 2 (a 2 )) we were unable to find an explicit form if α < 0. For α > 0 we can write F in terms of a Generalised Laguerre polynomial basis, are the generalised Laguerre polynomials and U(a, b, c) is the confluent hypergeometric function of the second kind. Note that this does not define a basis of F if α < 0, since then exp(−z 1 ) S < ∞. We can now give the actions of A 2 (a 2 ) with respect to this basis.
Proposition 5.4. For α > 0 we have Proof. We have Since and for j ∈ {3, 4}, we obtain the desired result.
Despite not having an explicit form α < 0, we can show that this action is unitary if and only if α < 0.

Superunitary representations.
The following definitions can be found in [2].
Definition 5.6. Let (H 1 , · , · 1 ) and (H 2 , · , · 2 ) be Hilbert superspaces and suppose T : H 1 → H 2 is a linear operator. We call T a bounded operator between H 1 and H 2 if it is continuous with respect to their Hilbert topologies. The set of bounded operators is denoted by B (H 1 , H 2 ) and B(H 1 ) := B (H 1 , H 1 ).
Here H ∞ is the space of smooth vectors of the representation π 0 and Ad is the adjoint representation of G 0 on g.
Using this definition of a superunitary representation we can now prove the following result. Proof. Thanks to Corollary 3.5, we only need to consider the representation ρ 0 on elements of the form g = exp(X 1 ) · · · exp(X n ) ∈ G 0 , with X i ∈ g 0 and n ∈ N. We now prove the different conditions of Definition 5.9.
• For all f ∈ F , the maps ρ f 0 : g → ρ 0 (g)f are continuous on G 0 : We need to prove the following Since is a neighbourhood of g for all r > 0, it suffices to prove For A 2 we know from Proposition 5.5 that the actions is unitary if α < 0. Since unitarity implies continuity, we are done.
For K 3 we have (e ıδ f 1,k z 1 + e −ıδ f 2,k z 2 + f 3,k z 3 + f 4,k z 4 ) and therefore which goes to 0 as δ goes to 0. For A 3 we have and therefore which goes to 0 as δ goes to 0. For K 2 we have and therefore Using Lebesgue's dominated convergence theorem we now find as desired. Lastly, the K 1 and A 1 cases are analogous to the K 3 and A 3 cases.
The assumption α < 0 is only used to prove the continuity of ρ 0 (A 2 (δ)). Note that Proposition 5.5 only implies that the actions are not unitary if α > 0. It tells us nothing about the continuity in this case. It is possible that Theorem 5.10 holds even without the assumption α < 0.
From the discussion in Section 2.3, we can at least conclude that for every α there always exists a superunitary representation of D(2, 1; α). Indeed, if α > 0, we can look at the Fock representation of D(2, 1; −1−α) instead of the Fock representation of D(2, 1; α).

Strong superunitary representation.
In [2,Section 4.4] the notion of a strong superunitary representation is also defined. However, it is easy to prove that our superunitary representation is not a strong superunitary representation.

5.5.
Harish-Chandra supermodules. We will end this paper by giving an alternative, non-explicit, way to integrate the algebra representation of D(2, 1, α) to group level. We do this by using the framework of Harish-Chandra supermodules developed in [24]. It would be interesting to know if this abstract integration gives the same representation as the explicit integration of Theorem 5.1, but we were unable to verify this.
Definition 5.13. [24, Definition 4.1] Let V be a complex super-vector space, G = (G 0 , g) a Lie supergroup and K a maximal compact subgroup of G 0 . Then V is a (g, K)-module if it is a locally finite K-representation that has also a compatible g-module structure, that is, the derived action of K agrees with the Lie(K)-module structure: dπ 0 (X)(v) = d dt π 0 (exp(tX))(v) t=0 = dπ(X)(v) for all X ∈ Lie(K), v ∈ V and π 0 (k)(dπ(X)(v)) = dπ(Ad(k)(X))(π 0 (k)(v)), for all k ∈ K, X ∈ g, v ∈ V, where π 0 is the K-representation and dπ the g-representation. A (g, K)-module is a Harish-Chandra supermodule if it is finitely generated over U (g) and is K-multiplicity finite.
The maximal compact subgroup of SL(V i ) is K i . The maximal compact subgroup of G 0 is therefore the 3-Torus K := K 1 × K 2 × K 3 .
Proposition 5.14. The module F is a Harish-Chandra supermodule.
Corollary 5.15. The (g, K)-module F integrates to a unique smooth Fréchet representation of moderate growth for the Lie supergroup D(2, 1; α).