Topology of the Hitchin system has been studied for decades, and interesting connections were found to orbital integrals, non-abelian Hodge theory, mirror symmetry etc. I will explain that a large part of the symmetries in these geometries above are “motivated” by algebraic cycles; in particular, I will discuss a proof of the motivic decomposition conjecture for the Hitchin system, and a proof of the motivic “\chi”-independence phenomenon. In the proofs, the desired algebraic cycles are constructed by a combination of the the Fourier-Mukai duality for the derived category of coherent sheaves and Springer theory. Based on joint work with Davesh Maulik and Qizheng Yin.