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Let  G  be a real solvable exponential Lie group with Lie algebra  and letf  ∈*${ f\in \mathfrak {g}^* }$   . We take two polarizations  j    ${ \mathfrak {p}_j}$   ,  j  = 1,2, at  f  which meet the Pukanszky condition. LetP  j   :  =  exp   (j   )${ P_j:=\exp ( \mathfrak {p}_j)}$   ,  j  = 1,2, be the associated subgroups in  G . The linear functional  f  defines unitary charactersχ  f    (  exp  X  )   :  =   e   i  〈  f  ,  X  〉${ \chi _f(\exp X):=e^{i \langle { f},{X}\rangle }}$   ,X  ∈j${X\in \mathfrak {p}_j }$   , of  P j  . Letπ  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 thatτ  1    ${ \tau _1 }$    andτ  2    ${ \tau _2 }$    are unitarily equivalent. The description of the intertwining operator of such an equivalence is given via an abstract integralI2   ,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  (    )    ${\mathcal {U}(\mathfrak {g})}$   -invariant dense subspace of the spaceℋ    τ  1    ∞    ${\mathcal {H}_{\tau _1}^\infty }$    of  C  ∞  vectors of π 1  (which is the case when the simple productP  1    P  2${P_1P_2}$    is closed). Finally, given three Pukanszky polarizationsP  i   =  exp   (i   )${ P_i=\exp ( \mathfrak {p}_i)}$   ,  i  = 1,2,3, at  f  of that kind, we accurately determine the composition formula ofI3   ,2∘   I2   ,1∘   I1   ,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.

Intertwining operators of irreducible representations for exponential solvable Lie groups