Orbit correspondences for real reductive dual pairs

Shu-Yen Pan

Vol. 248 (2010), No. 2, 403–427

DOI: 10.2140/pjm.2010.248.403

Abstract

Suppose that one of the real vector spaces V and W is symplectic and the other is quadratic. Let g₁ and g₂ denote the Lie algebras of the groups of isometries of the two spaces, and let τ_i : V ⊗_ℝW → g_i be their respective moment maps for i = 1,2. Suppose that 𝒪 and 𝒬 are nilpotent orbits in g₁ and g₂, respectively. We prove that τ₂(τ₁⁻¹(𝒪)) and τ₁(τ₂⁻¹(𝒬))) are each the union of at most two closures of nilpotent orbits in g₁ and g₂, respectively (where 𝒫 denotes the closure of a nilpotent orbit 𝒫).

Keywords

orbit correspondence, nilpotent orbit, Young diagram

Mathematical Subject Classification 2000

Primary: 22E60

Secondary: 22E47, 22E30

Milestones

Received: 29 June 2008

Revised: 15 September 2010

Accepted: 23 September 2010

Published: 1 December 2010