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 osp(m, 2|2n). We also construct an integral transform which intertwines the Schrödinger model for the minimal representation of the orthosymplectic Lie superalgebra osp(m, 2|2n) with this new Fock model.

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

We construct two infinite-dimensional irreducible representations for D(2, 1; α): a Schrödinger model and a Fock model. Further, we also introduce an intertwining isomorphism. These representations are similar to the minimal representations constructed for the orthosymplectic Lie supergroup and for Hermitian Lie groups of tube type. The intertwining isomorphism is the analogue of the Segal-Bargmann transform for the orthosymplectic Lie supergroup and for Hermitian Lie groups of tube type. Contents 1. Introduction 1.1. Contents 1.2. Notations 2. The Lie superalgebra D(2, 1; α) 2.1. The construction of D(2, 1; α) 2.2. Roots of D(2, 1; α) 2.3. Three grading 2.4. Real forms 3. The Jordan superalgebra D α 3.1. Definition 3.2. The structure algebra str(D α ) 3.3. The TKK-construction for D α 4. Two polynomial realisations 4.1. The Bessel operator 4.2. A polynomial realisation 4.3. The Fock representation 4.4. The Schrödinger representation 5. The Fock space and Bessel-Fischer product 5.1. The Bessel-Fischer product 5.2. Reproducing kernel 6. Properties of the Fock Representation 6.1. Skew-symmetric 6.2. The (g, k)-module F λ 6.3. The Gelfand-Kirillov dimension