We describe joint work (with David Ben-Zvi and David Nadler) that constructs an equivalence between the derived category of smooth representations of $\mathrm{GL}_n(\mathbb{Q}_p)$ and a certain category of coherent sheaves on the moduli stack of Langlands parameters for $\mathrm{GL}_n$. The proof of this equivalence is essentially a reinterpretation of $K$-theoretic results of Kazhdan and Lusztig via derived algebraic geometry. We will also discuss (conjectural) extensions of this work to other quasi-split groups, and to the modular representation theory of $\mathrm{GL}_n$.