A note on Standard Modules and Vogan L-packets

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 $I(τ,ν)$ and $I(τ',ν')$ for $G$ and $G'$ respectively with $τ'$ a generic tempered representation, such that the Harish-Chandra's $μ$-functions of a representation in the supercuspidal support of $τ$ and of a generic essentially square-integral representation in some Jacquet module of $τ'$ agree (after a suitable identification of the underlying spaces under which $ν=ν'$), we show that $I(τ,ν)$ is irreducible whenever $I(τ',ν')$ is. The conditions are satisfied if the Langlands quotients $J(τ,ν)$ and $J(τ',ν')$ of respectively $I(τ,ν)$ and $I(τ',ν')$ 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.

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