Aubert duality and co-tempered data

Let $F$ be a non-archimedean local field of characteristic 0, and let $G$ be either $\Sp_{2n}(F)$ or $\SO_{2n+1}(F)$. We introduce a new algorithm to compute the Aubert dual at the level of Langlands data. This algorithm acts as the dual to the recent Lanard-Mínguez algorithm. It fundamentally differs in two ways: it follows a bottom-up approach rather than a top-down one, and its internal computations strictly preserve the temperedness of the representations. Consequently, this approach naturally yields a new constructive characterization of co-tempered representations.
By operating exclusively within the realm of tempered data, this algorithm enables inductive proofs of new properties for co-tempered representations. In particular, we provide a precise description of their tempered components and establish an explicit duality formula for a large class of tempered representations.