A minimal representation of the orthosymplectic Lie supergroup

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A minimal representation of the orthosymplectic Lie supergroup. / Barbier, Sigiswald; Frahm, Jan.

In: International Mathematics Research Notices, 2019.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

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Barbier, Sigiswald ; Frahm, Jan. / A minimal representation of the orthosymplectic Lie supergroup. In: International Mathematics Research Notices. 2019.

Bibtex

@article{a535a80134214478a0dd9e87f4cdd625,
title = "A minimal representation of the orthosymplectic Lie supergroup",
abstract = "We construct a minimal representation of the orthosymplectic Lie supergroup $OSp(p,q|2n)$, generalising the Schr\{"}odinger model of the minimal representation of $O(p,q)$ to the super case. The underlying Lie algebra representation is realized on functions on the minimal orbit inside the Jordan superalgebra associated with $\mathfrak{osp}(p,q|2n)$, so that our construction is in line with the orbit philosophy. Its annihilator is given by a Joseph-like ideal for $\mathfrak{osp}(p,q|2n)$, and therefore the representation is a natural generalization of a minimal representations to the context of Lie superalgebras. We also calculate its Gelfand--Kirillov dimension and construct a non-degenerate sesquilinear form for which the representation is skew-symmetric and which is the analogue of an $L^2$-inner product in the supercase. ",
keywords = "math.RT, 17B10, 17B60, 22E46, 58C50",
author = "Sigiswald Barbier and Jan Frahm",
note = "45 pages",
year = "2019",
doi = "10.1093/imrn/rnz228",
language = "English",
journal = "International Mathematics Research Notices",
issn = "1073-7928",
publisher = "Oxford University Press",

}

RIS

TY - JOUR

T1 - A minimal representation of the orthosymplectic Lie supergroup

AU - Barbier, Sigiswald

AU - Frahm, Jan

N1 - 45 pages

PY - 2019

Y1 - 2019

N2 - We construct a minimal representation of the orthosymplectic Lie supergroup $OSp(p,q|2n)$, generalising the Schr\"odinger model of the minimal representation of $O(p,q)$ to the super case. The underlying Lie algebra representation is realized on functions on the minimal orbit inside the Jordan superalgebra associated with $\mathfrak{osp}(p,q|2n)$, so that our construction is in line with the orbit philosophy. Its annihilator is given by a Joseph-like ideal for $\mathfrak{osp}(p,q|2n)$, and therefore the representation is a natural generalization of a minimal representations to the context of Lie superalgebras. We also calculate its Gelfand--Kirillov dimension and construct a non-degenerate sesquilinear form for which the representation is skew-symmetric and which is the analogue of an $L^2$-inner product in the supercase.

AB - We construct a minimal representation of the orthosymplectic Lie supergroup $OSp(p,q|2n)$, generalising the Schr\"odinger model of the minimal representation of $O(p,q)$ to the super case. The underlying Lie algebra representation is realized on functions on the minimal orbit inside the Jordan superalgebra associated with $\mathfrak{osp}(p,q|2n)$, so that our construction is in line with the orbit philosophy. Its annihilator is given by a Joseph-like ideal for $\mathfrak{osp}(p,q|2n)$, and therefore the representation is a natural generalization of a minimal representations to the context of Lie superalgebras. We also calculate its Gelfand--Kirillov dimension and construct a non-degenerate sesquilinear form for which the representation is skew-symmetric and which is the analogue of an $L^2$-inner product in the supercase.

KW - math.RT

KW - 17B10, 17B60, 22E46, 58C50

U2 - 10.1093/imrn/rnz228

DO - 10.1093/imrn/rnz228

M3 - Journal article

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

ER -