Kazhdan and Lusztig identified the affine Hecke algebra 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 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 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 Hellmann
and Zhu). We explain how this refines the more familiar description of
representations, one central character at a time, in terms of categories of
perverse sheaves (as previously observed in local Langlands over R).
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 GLn​(F) into coherent sheaves on the stack of
Langlands parameters.Comment: 62 pages, improvements to main theorems, improved exposition.
Comments appreciated