We compare the level zero part of the type of a representation of GL(n) over a non-archimedean local field with the tame part of its Langlands parameter restricted to inertia. By normalizing this comparison, we construct canonical $\beta$-extensions of maximal simple characters.
On Extensions of supersingular representations of ${\rm SL}_2(\mathbb{Q}_p)$
Donovan's conjecture and extensions by the centralizer of a defect group
On Bernstein's presentation of Iwahori-Hecke algebras and representations of split reductive groups over non-Archimedean local fields
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