Abstract: Representations of many remarkable objects, such as Lie algebras, Hecke algebras, algebraic groups, etc., has deep relationships with geometry of some algebraic varieties. A manifestation of such relationship is that sometimes we can realise the center of representation categories as cohomology ring of certain algebraic varieties. In this talk, we will survey some important examples in this direction and explain a new example relating center of G1T-modules and cohomology of affine Springer fibre (joint work with Eric Vasserot).