The Real Jacobi Group Revisited

Abstract

The real Jacobi group GJ₁(ℝ), defined as the semi-direct product of the group SL(2, ℝ) with the Heisenberg group H₁, is embedded in a 4×4 matrix realisation of the group Sp(2, ℝ). The left-invariant one-forms on GJ₁(ℝ) and their dual orthogonal left-invariant vector fields are calculated in the S-coordinates (x,y,θ,p,q,κ), and a left-invariant metric depending on 4 parameters (α,β,γ,δ) is obtained. An invariant metric depending on (α,β) in the variables (x,y,θ) on the Sasaki manifold SL(2, ℝ) is presented. The well-known Kähler balanced metric in the variables (x,y,p,q) of the four-dimensional Siegel-Jacobi upper half-plane XJ₁=GJ₁(ℝ)SO(2)×ℝ≈X₁×ℝ² depending on (α,γ) is written down as a sum of the squares of four invariant one-forms, where X₁ denotes the Siegel upper half-plane. The left-invariant metric in the variables (x,y,p,q,κ) depending on (α,γ,δ) of a five-dimensional manifold X~J₁=GJ₁(ℝ)SO(2)≈X₁×ℝ³ is determined.