A Schrödinger model, Fock model and intertwining Segal-Bargmann transform for the exceptional Lie superalgebra $D(2,1;α)$

Abstract

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