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

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