Micro-packets containing generic representations
Nicolas Arancibia Robert
TL;DR
The paper extends the Kostant–Vogan open-orbit criterion for generic representations from $L$-packets to ABV micro-packets attached to $L$-parameters, establishing a precise equivalence between openness of the corresponding $G^ $-orbit and the presence of a generic member in the micro-packet. It develops a robust geometric framework based on characteristic cycles and the Weyl group action to relate microlocal geometry to representation theory, utilizing the translation principle to handle singular infinitesimal characters. A main result proves that a micro-packet $\Pi_S^{\mathrm{mic}}(G/\mathbb{R})$ contains a generic representation if and only if the orbit $S$ is open, and as a corollary, proves the Enhanced Shahidi Conjecture for real groups: the Arthur packet attached to an $A$-parameter $\psi$ has a generic member precisely when $\psi|_{\mathrm{SL}_2}$ is trivial. This work provides a pathway to compute micro-packets for particular groups (e.g., type $G_2$) and tightens the link between microlocal geometry and Langlands-type correspondences, with implications for temperedness and the relationship between ABV micro-packets and classical $L$-packets.
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 $ψ$ of $G$ contains a generic representation if and only if $ψ|_{\mathrm{SL}_2}$ is trivial.