Interpretation of the stronger ramification inequality in the PGL2 case

Determine a natural interpretation, in the context of the PGL_2 case and the associated local Langlands correspondence, of the stronger inequality \(l_0\geq 5d+1\), given that \(l_0>3d\) is sufficient for the relevant torus-theoretic realization.

Background

For an elliptic maximal torus $S$ in $G=\PGL_2$, let $E/F$ be the associated quadratic extension, let $d$ denote its ramification-discriminant parameter, and let $l_0$ be the minimum level among the character twists used to describe the genuine character $\theta_{\pm}$. The paper derives the condition $l_0\geq 5d+1$ from its direct construction of the genuine function via the condition ${\bf (Ram_\alpha)}$.

The authors compare this with the weaker inequality $l_0>3d$, which has a known interpretation in the local Langlands description: it ensures that the relevant supercuspidal representation is induced from a compact open subgroup containing $S(F)$, rather than from one associated with a different elliptic maximal torus. They explicitly state that no natural interpretation is known for the stronger bound $l_0\geq 5d+1$, leaving its conceptual significance unresolved.