We characterize the Local Langlands Correspondence $\left( \text{LLC} \right)$ for inner forms of $\text{G}{{\text{L}}_{n}}$ via the Jacquet–Langlands Correspondence $\left( \text{JLC} \right)$ and compatibility with the Langlands Classification. We show that $\text{LLC}$ satisfies a natural compatibility with parabolic induction and characterize $\text{LLC}$ for inner forms as a unique family of bijections $\prod \left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)\,\to \,\Phi \left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)$ for each $r$, (for a fixed $D$), satisfying certain properties. We construct a surjective map of Bernstein centers $\mathfrak{Z}\left( \text{G}{{\text{L}}_{n}}\left( F \right) \right)\,\to \,\mathfrak{Z}\left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)$ and show this produces pairs of matching distributions in the sense of Haines. Finally, we construct explicit Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke algebras of $\text{G}{{\text{L}}_{r}}\left( D \right)$, and thereby produce many explicit pairs of matching functions.

Let $G$ be a connected semisimple split group over a $p$-adic field. We establish the explicit link between principal nilpotent orbits and the irreducible constituents of principal series in terms of $L$-group objects.

Let F a locally compact non-Archimedean field, of residue characteristic p, and ψ a nontrivial additive character of F. Let σ, σ′ be irreducible representations of the absolute Weil group of F, each of degree a power of p and not induced from a nontrivial unramified extension of F. We give a formula for the value at $s=½ of the ϵ-factor ϵ (σ ⊗ σ ',ψ,s)$, modulo roots of unity in ${\Bbb C}$ of order a power of p. Via the Langlands correspondence, we get an analogous formula for supercuspidal representations of ${\rm GL}_n(F)$.