Journal of Lie Theory 31 (2021), No. 4, 1153--1188
Copyright Heldermann Verlag 2021
A Schrödinger model, Fock model and intertwining Segal-Bargmann transform for the exceptional Lie superalgebra D(2,1;α)
Sigiswald Barbier
Dept. of Electronics and Information Systems, Faculty of Engineering and Architecture, Ghent University, Belgium
Sigiswald.Barbier@UGent.be
Sam Claerebout
Dept. of Electronics and Information Systems, Faculty of Engineering and Architecture, Ghent University, Belgium
Sam.Claerebout@UGent.be
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.
Keywords: Fock model, Schrödinger model, minimal representations, Lie superalgebras, Bessel-Fischer product, Segal-Bargmann transform.
MSC: 17B10, 17B60, 22E46, 58C50.
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