On certain supercuspidal representations of $SL_n(F)$ associated with tamely ramified extensions: the formal degree conjecture and the root number conjecture

TL;DR

This paper explicitly constructs certain supercuspidal representations of SL_n(F) associated with tamely ramified extensions, verifying the formal degree and root number conjectures within the local Langlands framework.

Contribution

It provides explicit constructions of supercuspidal representations of SL_n(F) linked to tamely ramified extensions and verifies key conjectures using the local Langlands correspondence.

Findings

Verification of the formal degree conjecture for constructed representations

Verification of the root number conjecture for these representations

Explicit parametrization of supercuspidal representations via hyper special compact groups

Abstract

Based upon the general theory, developed by the author, on the parametrization of the irreducible representations of the hyper special compact groups corresponding to the regular adjoint orbit, supercuspidal representations of $SL_n(F)$ are explicitly constructed for which the formal degree conjecture and the root number conjecture are verified with respect to certain $L$-parameter defined, by means of Kaletha, that is, the local Langlands correspondence of tori and the Langlands-Schelstad procedure, by the data parametrizing the irreducible representations of the hyper special compact subgroup $SL_n(O_F)$.