Generic representations, open parameters and ABV-packets for $p$-adic groups

Abstract

If $\ddot{I}€$ is a representation of a $p$-adic group $G(F)$, and $\ddot{I}•$ is its Langlands parameter, can we use the moduli space of Langlands parameters to find a geometric property of $\ddot{I}•$ that will detect when $\ddot{I}€$ is generic? In this paper we show that if $G$ is classical or if we assume the Kazhdan-Lusztig hypothesis for $G$, then the answer is yes, and the property is that the orbit of $\ddot{I}•$ is open. We also propose an adaptation of Shahidi's enhanced genericity conjecture to ABV-packets: for every Langlands parameter $\ddot{I}•$ for a $p$-adic group $G(F)$, the ABV-packet ${\hat{I}}_{\ddot{I}}^{\mathrm{ABV}}•(G(F))$ contains a generic representation if and only if the local adjoint L-function $L(s,\ddot{I}•,\mathrm{Ad})$ is regular at $s=1$, and show that this condition is equivalent to the "open parameter" condition above. We show that this genericity conjecture for ABV-packets follows from other standard conjectures and we verify its validity with the same conditions on $G$. We show that, in this case, the ABV-packet for $\ddot{I}•$ coincides with its $L$-packet. Finally, we prove Vogan's conjecture on $A$-packets for tempered parameters.