## A minimal representation of the orthosymplectic Lie supergroup

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### DOI

• Sigiswald Barbier, Gent University, Belgium
• Jan Frahm
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.
Original language English International Mathematics Research Notices 1073-7928 https://doi.org/10.1093/imrn/rnz228 E-pub ahead of print - 2019

45 pages

### Research areas

• math.RT, 17B10, 17B60, 22E46, 58C50

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ID: 161612609