2019/10/16 by Marisa Gaetz, Gaetz, Marisa
Mathematics · #22E10 (Primary) #22E60 (Secondary) #Advanced Topics in Algebra #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1910.07592
openalex publication_date 2019/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In Roger Howe's 1989 paper, ``Remarks on classical invariant theory," Howe introduces the notion of a dual pair of Lie subalgebras: a pair (\mathfrakg1, \mathfrakg2) of reductive Lie subalgebras of a Lie algebra \mathfrakg such that \mathfrakg1 and \mathfrakg2 are each other's centralizers in \mathfrakg. This notion has a natural analog for algebraic groups: a dual pair of subgroups is a pair (G1, G2) of reductive subgroups of an algebraic group G such that G1 and G2 are each other's centralizers in G. In this paper, we classify the dual pairs in the complex classical groups (GL(n,ℂ), SL(n,ℂ), Sp(2n,ℂ), O(n,ℂ), and SO(n,ℂ)) and in the corresponding Lie algebras (\mathfrakgl(n,ℂ), \mathfraksl(n,ℂ), \mathfraksp(2n,ℂ), and \mathfrakso(n,ℂ)). We also present substantial progress towards classifying the dual pairs in the projective counterparts of the complex classical groups (PGL(n,ℂ), PSp(2n,ℂ), PO(n,ℂ), and PSO(n,ℂ)).