Explore the intricate relationship between representation theory of simple affine vertex algebras at elliptic levels and the geometry of homogeneous elliptic affine Springer fibres in this hour-long conference talk. Delve into results and conjectures that connect these complex mathematical concepts, focusing on the association with the Langlands dual affine Lie algebra. Gain insights from the collaborative research conducted by Peng Shan, Dan Xie, Wenbin Yan, and Qixian Zhao, presented at the International Congress of Basic Science (ICBS) 2024.