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

If π is a representation of a p-adic group $G(F)$, and φ is its Langlands parameter, can we use the moduli space of Langlands parameters to find a geometric property of φ that will detect when π 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 φ is open. We also propose an adaptation of Shahidi's enhanced genericity conjecture to ABV-packets: for every Langlands parameter φ for a p-adic group $G(F)$, the ABV-packet ΠABV_φ(G(F)) contains a generic representation if and only if the local adjoint L-function L(s,φ,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 φ coincides with its L-packet. Finally, we prove Vogan's conjecture on A-packets for tempered parameters.