A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra 𝔬𝔰𝔭(𝑚, 2|2𝑛)

Abstract

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.