A Generalized Global Cartan Decomposition: A Basic Example

Abstract

Suppose G 1 ⊆ G are complex linear simple Lie groups. Let 𝔤 1 ⊆ 𝔤 be the corresponding pair of Lie algebras. For the Killing-orthogonal 𝔭 of 𝔤 1 in 𝔤 we have a vector space direct sum 𝔤 = 𝔤 1⊕ 𝔭, which generalizes the classical Cartan decomposition on the Lie algebras level. In this article we study the corresponding problem of a ‘generalized global Cartan decomposition’ on the Lie groups level for the pair of groups ( G , G 1) = (SL (4,ℂ),Sp (2,ℂ)); here 𝔤 = 𝔰 𝔩(4,ℂ), 𝔤 1 = 𝔰 𝔭(2,ℂ), and 𝔭 = {X ∈ 𝔤 | X ♯ = X}, where X↦ X ♯ is the symplectic involution. We prove that G = G 1exp 𝔭 ∪ i G 1exp 𝔭. The key point of the proof is to study in detail the set exp 𝔭; and for that purpose we introduce the J-twisted Pfaffian of size 2n defined on the set of all 2n × 2n matrices X satisfying X ♯ = X, which is here a natural counterpart of the standard Pfaffian.