Mathematics – Representation Theory
Scientific paper
2011-11-20
Mathematics
Representation Theory
42pp, v2: added Cor. 3.9, v3: generalized definitions/results in \S 2. v4: added Cor. 5.8/Appendix B, and changed some termino
Scientific paper
We interpret the orthogonality relation of Kostka polynomials arising from complex reflection groups (c.f. [Shoji, Invent. Math. 74 (1983), J. Algebra 245 (2001)] and [Lusztig, Adv. Math. 61 (1986)]) in terms of homological algebra. This leads us to the notion of Kostka system, which can be seen as a categorical counter-part of Kostka polynomials. Then, we show that every generalized Springer correspondence [Lusztig, Invent. Math. 75 (1984)] (in good characteristic) gives rise to a Kostka system. This enables us to see the top-term generation property of the (twisted) homology of generalized Springer fibers, and the transition formula of Kostka polynomials between two generalized Springer correspondences of type $\mathsf{BC}$. The latter provides an inductive algorithm to compute Kostka polynomials by upgrading [Ciubotaru-Kato-K, Invent. Math. 178 (2012)] \S 3 to its graded version. In the appendices, we present a purely algebraic proofs that Kostka systems exist for type $\mathsf{A}$ and asymptotic type $\mathsf{BC}$ cases, and therefore one can skip geometric sections \S 3--5 to see the key ideas and basic examples/techniques.
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