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.

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

“…We now have a one-dimensional g 0 -submodule generated by R 2 = 2f 1 f 2 + ζθ and the logical quotient representation to study would be P(K 2|2 )/ R 2 . Since D(2, 1; 1) ∼ = osp(4|2) and D 1 is isomorphic to the spin factor Jordan superalgebra, this was already studied in a more general setting in [14] and [15]…”

“…The representation π λ of g on P(C 2|2 )/I λ has no k-finite vectors. We will remedy this by twisting by a Cayley transform c. So we define our Fock model as the representation ρ λ := π λ • c. The analogue of this Fock model is studied in [10] for classical Lie algebras, and in [15] for osp(m, 2|2n). Proof.…”

“…For example, also the unitary irreducible representations of nilpotent Lie supergroups can be completely understood using the orbit method, [11,12]. Recently, a minimal representation of the orthosymplectic Lie superalgebra has been studied in detail using the Jordan algebra approach, [13,14,15]. This minimal representation of osp(p, q|2n) can be seen as a super version of the minimal representation of o(p, q) which has been studied in detail by Kobayashi and various collaborators, [6,7,8,5] .…”

“…The goal of this paper is to construct a similar representation for the Lie superalgebra D(2, 1; α). In particular we will construct two different models, which are similar to the Schrödinger and Fock model or the orthosymplectic Lie superalgebra studied in [14,15] and an intertwining isomorphism between these two models, which is similar to the Segal-Bargmann transform studied in [10] for the non-super case and in [15] for osp(m, 2|2n).…”

Segal-Bargmann Transforms Associated to a Family of Coupled Supersymmetries

“…Recently, there has been a wide range of Segal-Bargmann-like analysis beyond R n to Lie groups via heat kernel and geometric quantization techniques [17,18,27,20,19,13,7,21]. There have also been Segal-Bargmann transform applications to the study of superalgebras and supersymmetric quantum systems [29,3,2], as well as analogues developed in the quaternionic and Clifford algebra settings [1,8,11,12,14,22].…”