Arithmetic invariants of discrete Langlands parameters

Let G be a reductive algebraic group over the local field k. The local Langlands conjecture predicts that the irreducible complex representations π of the locally compact group G(k) can be parametrized by objects of an arithmetic nature: homomorphisms φ from the Weil-Deligne group of k to the complex L-group of G, together with an irreducible representation ρ of the component group of the centralizer of φ. In light of this conjecture which has been established for algebraic tori, as well as for some nonabelian groups like GLn(k) [19],[21], and SLn(k) [23] it is reasonable to predict how representation theoretic invariants of π = π(φ, ρ) relate to the arithmetic invariants of its parameters (φ, ρ). An early example of this was the paper [15], which predicts branching laws for the restriction of irreducible representations of the group SOn(k) to the subgroup SOn−1(k), using the ε-factor of a distinguished symplectic representation of the L-group of SOn × SOn−1. These conjectures have now been verified in several cases; see [17] and [18].