Gross–Prasad genericity criterion

Establish that for a connected reductive p-adic group G and any Langlands parameter φ, the L-packet Π_φ(G) contains a generic representation if and only if the adjoint L-function L(s, φ, Ad) is regular at s = 1.

Background

The paper centers on genericity in L-packets and connects it to geometric properties of Vogan varieties and perverse sheaves under the p-adic Kazhdan–Lusztig hypothesis. The Gross–Prasad conjecture (GP, Conjecture 2.6) predicts an equivalence between the existence of a generic representation in Π_φ(G) and the regularity of L(s, φ, Ad) at s=1. The authors prove this conjecture under assumptions on the local Langlands correspondence and the p-KLH.

Their approach also leverages a geometric criterion (openness of the associated orbit in the Vogan variety) and reconciles it with analytic conditions on L-functions, thereby linking representation-theoretic genericity to singularity and closure properties in varieties of Langlands parameters.