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Usually (for $p$-adic groups, real groups or automorphic forms) you can write an explicit "intertwining operator" between the two inductions. This is a very basic construction in representation theory - as basis as parabolic induction itself in some sense. Usually it is given by some sort of integral which converges only in some range and one needs to work in order to prove its meromorphic continuation (it does have both zeros and poles; because of this parabolic induction is only generically independent of the choice of representation of $L$).