Symmetric functions and Springer representations

Abstract

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 $\mathrm{GL}(n)$ are $\mathrm{Ext}$-orthogonal to each other, and show that it is compatible with the natural categorification of the ring of symmetric functions.

26pp, v2: corrected an error in the proof of (old) Proposition 2.28, v3: streamlined the exposition, final version

References in corpus (2)

• Affine highest weight categories and affine quasihereditary algebras
• Peter-Weyl, Howe and Schur-Weyl theorems for current groups