Why does the intertwining integral such as the one defined in A. W. Knapp's paper "Intertwining operators for semisimple groups" depend only on an element w of a Weyl group?


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Intertwining Integral defined on a Weyl group?
It is not true that the intertwiner depends only upon the Weyl element but also upon the parabolic subgroup $P$. Moreover, it is better to consider the intertwiner to depend upon a set of complex parameters. Its analytic properties give alot of information about the irreducibility, growth of matrix coeffecients, and their like...


But I guess, you are more looking for an explanation, why the Weyl group enters:


1.) The Weyl group of a parabolic $P$ is the normalizer of a Levi subgroup $M$ inside $G$ modulo the inner automorphisms of the Levi subgroup, I guess it will actually be the group of outer automorphisms of $M$.


2.) Most (unitary) representations are subrepresentation/subquotient of parabolic induced representations $\pi$ of $M$. Here $\pi$ is not necessary unitary, even though the induced or subquotients/subrepresentations might be. This goes under the name Langlands classification theorem and Harish-Chandras philosophy of cusp forms.


3.) An intertwiner of the induced representation will come from an Weyl element by the Frobenius reciprocity together with Mackeys restriction prinicple and Bruhat decomposition $G//P =W$. For simplicity, assume that you are working with the group $G=GL(n,F)$. where $F$ is a finite field. For two ireducible reps $\pi, \pi\_0$ of $M$, this yields that
$$Hom\_G( Ind\_P^G \pi , Ind\_P^G \pi\_0) \cong \bigoplus\_{w \in W} Hom\_M( \pi^w , \pi\_0),$$
where the later is zero, if and only if $\pi \cong \pi\_0^w$ for some $w$. In principle that suggests at least, that every nontrivial intertwiner should come from Weyl elements.


