On two geometric realizations of an affine Hecke algebra

The article is a contribution to the local theory of geometric Langlands duality. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra associated to a reductive group G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G$\end{document} and Grothendieck group of equivariant coherent sheaves on Steinberg variety of Langlands dual group Gˇ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${G\check {\ }}$\end{document}; this isomorphism due to Kazhdan–Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures.