LAUSR

Abstract The real Jacobi group G n J ( R ) , defined as the semidirect product of the Heisenberg group H n ( R ) with the symplectic group Sp ( n , R ) , admits a matrix embedding in Sp ( n + 1 , R ) . The modified pre-Iwasawa decomposition of Sp ( n , R ) allows us to introduce a convenient coordinatization S n of G n J ( R ) , which for G 1 J ( R ) coincides with the S -coordinates. Invariant one-forms on G n J ( R ) are determined. The formula of the 4-parameter invariant metric on G 1 J ( R ) obtained as sum of squares of 6 invariant one-forms is extended to G n J ( R ) , n ∈ N . We obtain a three parameter invariant metric on the extended Siegel–Jacobi upper half space X n J ≈ X n J × R by adding the square of an invariant one-form to the two-parameter balanced metric on the Siegel–Jacobi upper half space X n J = G n J ( R ) U ( n ) × R .