Intertwining operators of irreducible representations for exponential solvable Lie groups

Abstract

Abstract Let G be a real solvable exponential Lie group with Lie algebra 𝔤 and let f ∈ 𝔤 * ${ f\in \mathfrak {g}^* }$ . We take two polarizations 𝔭 j ${ \mathfrak {p}_j}$ , j = 1,2, at f which meet the Pukanszky condition. Let P j : = exp ( 𝔭 j ) ${ P_j:=\exp ( \mathfrak {p}_j)}$ , j = 1,2, be the associated subgroups in G. The linear functional f defines unitary characters χ f ( exp X ) : = e i 〈 f , X 〉 ${ \chi _f(\exp X):=e^{i \langle { f},{X}\rangle }}$ , X ∈ 𝔭 j ${X\in \mathfrak {p}_j }$ , of Pj . Let π j : = ind P j G χ f ${\pi _j :=\hbox{ind}_{ P_j }^{G} \chi _f}$ , j = 1,2, be the corresponding induced representations, which are unitary and irreducible. It is well known that τ 1 ${ \tau _1 }$ and τ 2 ${ \tau _2 }$ are unitarily equivalent. The description of the intertwining operator of such an equivalence is given via an abstract integral I 𝔭 2 , 𝔭 1 ${ I_{\mathfrak {p}_2,\mathfrak {p}_1} }$ . In this paper, we show that this formal integral gives us a concrete intertwining operator provided that it converges absolutely on a G- and 𝒰 ( 𝔤 ) ${\mathcal {U}(\mathfrak {g})}$ -invariant dense subspace of the space ℋ τ 1 ∞ ${\mathcal {H}_{\tau _1}^\infty }$ of C ∞ vectors of π1 (which is the case when the simple product P 1 P 2 ${P_1P_2}$ is closed). Finally, given three Pukanszky polarizations P i = exp ( 𝔭 i ) ${ P_i=\exp ( \mathfrak {p}_i)}$ , i = 1,2,3, at f of that kind, we accurately determine the composition formula of I 𝔭 3 , 𝔭 2 ∘ I 𝔭 2 , 𝔭 1 ∘ I 𝔭 1 , 𝔭 3 ${ I_{\mathfrak {p}_3,\mathfrak {p}_2}\circ I_{\mathfrak {p}_2,\mathfrak {p}_1}\circ I_{\mathfrak {p}_1,\mathfrak {p}_3}}$ using Maslov's index.