Symmetric functions and Springer representations

The characters of the (total) Springer representations are identified with the Green functions by Kazhdan [Israel J. Math. {\bf 28} (1977)], and the latter are identified with Hall-Littlewood’s $Q$-functions by Green [Trans. Amer. Math. Soc. (1955)]. In this paper, we present a purely algebraic proof that the (total) Springer representations of $(n)$ are $\mathrm{E}\mathrm{x}\mathrm{t}$-orthogonal to each other, and show that it is compatible with the natural categorification of the ring of symmetric functions.