Micro-packets containing generic representations

TL;DR

This paper generalizes a known criterion for the existence of generic representations in L-packets to micro-packets, and proves the Enhanced Shahidi Conjecture for real groups, linking genericity to triviality of certain parameters.

Contribution

It extends the criterion for generic representations from L-packets to micro-packets and proves the Enhanced Shahidi Conjecture for real groups.

Findings

The equivalence between genericity and open orbits holds for micro-packets.

The Enhanced Shahidi Conjecture is proved for real groups.

Generic representations exist in Arthur packets if and only if the A-parameter is trivial on SL_2.

Abstract

For a real group $G$, it is known from the work of Kostant and Vogan that the L-packet associated with an L-parameter $\varphi$ of $G$ contains a \emph{generic} representation if and only if the ${}^{\vee}G$-orbit in the variety of geometric parameters corresponding to $\varphi$ is open. In these notes, we generalize this result slightly by proving that the same equivalence holds when the L-packet of $\varphi$ is replaced by the micro-packet attached to $\varphi$ by Adams-Barbasch-Vogan. As a corollary, we deduce the Enhanced Shahidi Conjecture for real groups: the Arthur packet attached to an A-parameter $\psi$ of $G$ contains a generic representation if and only if $\psi|_{\mathrm{SL}_2}$ is trivial.