Fargues' categorical conjecture for elliptic parameters for SL(n)

(joint work with Eunsu Hur) Fargues and Scholze give a geometric construction of L-parameters attached to smooth irreducible representations of p-adic groups. They furthermore predict an enhancement to a category equivalence, following the philosophy of the geometric Langlands program. I will explain how to use Fu's recent result on stability to prove Fargues and Scholze's functor is an equivalence on elliptic parameters for SL(n). In particular, we prove Fargues and Scholze's construction gives a bijection between supercuspidal representations of SL(n) and (enhanced) elliptic L-parameters.