arXiv subjects

Sam Claerebout

Publications and source records attributed to Sam Claerebout.

5 recordsLinked to original sources

arXiv2023

Minimal representations of the metaplectic Lie supergroup and the super Segal-Bargmann transform

We construct a Schrödinger model and a Fock model of a minimal representation of the metaplectic Lie supergroup $\mathrm{Mp}(2m|2n,2n)$. Then, we show that the Schrödinger model of the minimal representation leads to an already known Schrödinger model of the metaplectic representation of $\mathrm{Mp}(2m|2n,2n)$. Therefore, the Fock model of the minimal representation allows us to construct a Fock model of this metaplectic representation. We then construct an intertwining super Segal-Bargmann transform which extends the classical Segal-Bargmann transform.

math.RT↗

arXiv2023

A general approach to constructing minimal representations of Lie supergroups

In this paper we describe an approach to generalise minimal representations to the super setting for Lie superalgebras obtained from Jordan superalgebras using the TKK construction. This approach was used successfully to construct a Fock model, a Schrödinger model and intertwining Segal-Bargmann transform for the orthosymplectic Lie supergroup $\mathop{OSp}(p,q|2n)$ and the exceptional Lie supergroup $\mathbb{D}(2,1;α)$. We also describe some obstacles to use this approach for the periplectic and queer Lie superalgebras.

math.RT↗

arXiv2023

A superunitary Fock model of the exceptional Lie supergroup $\mathbb{D}(2,1;α)$

We construct a Fock model of the minimal representation of the exceptional Lie supergroup $\mathbb{D}(2,1, α)$. 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.

math.RT↗

arXiv2021

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

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.

math.RT↗

arXiv2020

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

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

math.RT↗