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We construct two infinite-dimensional irreducible representations for$D(2,1;\alpha)$: a Schr\"odinger model and a Fock model. Further, we alsointroduce an intertwining isomorphism. These representations are similar to theminimal representations constructed for the orthosymplectic Lie supergroup andfor Hermitian Lie groups of tube type. The intertwining isomorphism is theanalogue of the Segal-Bargmann transform for the orthosymplectic Lie supergroupand for Hermitian Lie groups of tube type.

A Schr\"odinger model, Fock model and intertwining Segal-Bargmann transform for the exceptional Lie superalgebra $D(2,1;\alpha)$