The closedness of the product set of two Pukanszky polarizations of exponential Lie groups

Abstract

Let 𝐺 = exp 𝔤 be an exponential solvable Lie group with Lie algebra 𝔤 and 𝔤^* the dual vector space of 𝔤. We take two (real) polarizations 𝔥₁, 𝔥₂ of 𝔤 at 𝑓 ∈ 𝔤^* which satisfy the Pukanszky condition and define a unitary character 𝜒_𝑓 of 𝐻_𝑗 = exp(𝔥_𝑗) (𝑗 = 1, 2) by the formula 𝜒_𝑓(exp 𝑋) = 𝑒^{𝑖𝑓(𝑋)} (𝑋 ∈ 𝔥_𝑗). Then, it is well known that the two monomial representations 𝜋_𝑗 = ind^𝐺_𝐻𝑗 𝜒_𝑓 (𝑗 = 1, 2) are mutually equivalent and we even have a natural candidate of the intertwining operator between them. In order to verify that it is a true intertwining operator, the principal obstacle is the convergence of the integral in question. A property which assures this convergence, is the closedness of the simple product set 𝐻₂ 𝐻₁ in 𝐺. In this paper, we establish this property, generalizing then a previous proof in the particular case when one of the polarizations is of Vergne type.