For a reductive $p$-adic group $G$, Kazhdan-Lusztig prove an isomorphism of the the extended affine Hecke algebra and the $G^\vee$-equivariant $K$-group of the Steinberg variety of the Langlands dual group $G^\vee$. It has a profound application of proving an important case of the local Langlands correspondence which is known as the Deligne-Langlands conjecture. For $G$ being a reductive group over an equal-characteristic local field, Bezrukavnikov upgrades Kazhdan-Lusztig's isomorphism to an equivalence of monoidal categories and proves the tamely ramified local geometric Langlands correspondence. In this talk, we discuss an ongoing project with João Lourenço on proving a tamely ramified local geometric Langlands correspondence for reductive $p$-adic groups. Time permitting, I will mention an interesting variant of Bezrukavnikov's equivalence in Ben-Zvi-Sakellaridis-Venkatesh's framework of the relative Langlands program based on a joint work in preparation with Milton Lin and Toan Pham.

[pre-talk at 1:20pm]