Classification and double commutant property for dual pairs in an orthosymplectic Lie supergroup

2022/08/20 by Merino, Allan, Salmasian, Hadi · 1 citation
#22E30 #FOS: Mathematics #Primary: 22E45 #Representation Theory (math.RT) #Secondary: 22E46

paper · doi:10.48550/arxiv.2208.09746

Abstract

In this paper, we obtain a full classification of reductive dual pairs in a, real or complex, Lie superalgebra \mathfrakspo(E) and Lie supergroup SpO(E). Moreover, by looking at the natural action of the orthosymplectic Lie supergroup SpO(E) on the Weyl-Clifford algebra WC(E), we prove that for a reductive dual pair (\mathscrG , \mathscrG') = ((G , \mathfrakg) , (G' , \mathfrakg')) in SpO(E), the superalgebra WC(E)^\mathscrG consisting of \mathscrG-invariant elements in WC(E) is generated by the Lie superalgebra \mathfrakg'. We obtain a full classification of reductive dual pairs in the (real or complex) Lie superalgebra \mathfrakspo(\mathrm E) and the Lie supergroup SpO(\mathrm E). Using this classification we prove that for a reductive dual pair (\mathscrG , \mathscrG') = ((\mathrm G , \mathfrakg) , (\mathrm G' , \mathfrakg')) in SpO(\mathrm E), the superalgebra WC(\mathrm E)^\mathscrG consisting of \mathscrG-invariant elements in the Weyl-Clifford algebra WC(\mathrm E), equipped with the natural action of the orthosymplectic Lie supergroup SpO(\mathrm E), is generated by the Lie superalgebra \mathfrakg'. As an application, we prove that Howe duality holds for the dual pairs (SpO(2n|1) , OSp(2k|2l)) ⊆ SpO(ℂ^2k|2l ⊗ ℂ^2n|1).

Cited by

• Howe duality for the dual pair (SpO(2n|1) , \mathfrakosp(2|2))

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