A minimal representation of the orthosymplectic Lie supergroup

TY - JOUR

T1 - A minimal representation of the orthosymplectic Lie supergroup

AU - Barbier, Sigiswald

AU - Frahm, Jan

N1 - 45 pages

PY - 2021/11/1

Y1 - 2021/11/1

N2 - We construct a minimal representation of the orthosymplectic Lie supergroup OSp(p, q|2n) for p + q even, generalizing the Schr dinger 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 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 osp(p, q|2n), and therefore the representation is a natural generalization of a minimal representation to the context of Lie superalgebras. We also calculate its Gelfand Kirillov dimension and construct a nondegenerate sesquilinear form for which the representation is skew-symmetric and which is the analogue of an L2-inner product in the supercase.

AB - We construct a minimal representation of the orthosymplectic Lie supergroup OSp(p, q|2n) for p + q even, generalizing the Schr dinger 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 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 osp(p, q|2n), and therefore the representation is a natural generalization of a minimal representation to the context of Lie superalgebras. We also calculate its Gelfand Kirillov dimension and construct a nondegenerate sesquilinear form for which the representation is skew-symmetric and which is the analogue of an L2-inner product in the supercase.

KW - math.RT

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

UR - http://www.scopus.com/inward/record.url?scp=85122182642&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnz228

DO - 10.1093/imrn/rnz228

M3 - Journal article

SN - 1073-7928

VL - 2021

SP - 16357

EP - 16420

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 21

ER -