A note on Standard Modules and Vogan L-packets

Abstract: Let $F$ be a non-Archimedean local field of characteristic $0$, let $G$ be the group of $F$-rational points of a connected reductive group defined over $F$ and let $G'$ be the group of $F$-rational points of its quasi-split inner form. Given standard modules $I(\tau ,\nu )$ and $I(\tau ',\nu ')$ for $G$ and $G'$ respectively with $\tau '$ a generic tempered representation, such that the Harish-Chandra's $\mu $-functions of a representation in the supercuspidal support of $\tau $ and of a generic essentially square-integral representation in some Jacquet module of $\tau '$ agree (after a suitable identification of the underlying spaces under which $\nu =\nu '$), we show that $I(\tau ,\nu )$ is irreducible whenever $I(\tau ',\nu ')$ is. The conditions are satisfied if the Langlands quotients $J(\tau ,\nu )$ and $J(\tau ',\nu ')$ of respectively $I(\tau ,\nu )$ and $I(\tau ',\nu ')$ lie in the same Vogan $L$-packet (whenever this Vogan $L$-packet is defined), proving that, for any Vogan $L$-packet, all the standard modules whose Langlands quotient is equal to a member of the Vogan $L$-packet are irreducible, if and only if this Vogan $L$-packet contains a generic representation. The result for generic Vogan $L$-packets of quasi-split orthogonal and symplectic groups was proven by Moeglin-Waldspurger and used in their proof of the general case of the local Gan-Gross-Prasad conjectures for these Groups.