  Kazhdan and Lusztig identified the affine Hecke algebra $\mathcal{H}$ with an
equivariant $K$-group of the Steinberg variety, and applied this to prove the
Deligne-Langlands conjecture, i.e., the local Langlands parametrization of
irreducible representations of reductive groups over nonarchimedean local
fields $F$ with an Iwahori-fixed vector. We apply techniques from derived
algebraic geometry to pass from $K$-theory to Hochschild homology and thereby
identify $\mathcal{H}$ with the endomorphisms of a coherent sheaf on the stack
of unipotent Langlands parameters, the coherent Springer sheaf. As a result the
derived category of $\mathcal{H}$-modules is realized as a full subcategory of
coherent sheaves on this stack, confirming expectations from strong forms of
the local Langlands correspondence (including recent conjectures of
Fargues-Scholze, Hellmann and Zhu).
  In the case of the general linear group our result allows us to lift the
local Langlands classification of irreducible representations to a categorical
statement: we construct a full embedding of the derived category of smooth
representations of $\mathrm{GL}_n(F)$ into coherent sheaves on the stack of
Langlands parameters.
