Towards a tamely ramified local geometric Langlands correspondence for p-adic groups

For a reductive p-adic group G, Kazhdan-Lusztig prove an isomorphism of the the extended affine Hecke algebra and the G^v-equivariant K-group of the steinberg variety of the Langlands dual group G^v. 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 deformation of Bezrukavnikov's equivalence in Ben-Zvi-Sakellaridis-Venkatesh's framework of the relative Langlands program based on a joint work in preparation with Toan Pham and Milton Lin.