A homological study of Green polynomials*

Abstract

We interpret the orthogonality relation of Kostka polynomials arising from complex reflection groups ([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 counterpart of Kostka polynomials. Then, we show that every generalized Springer correspondence ([Lusztig, Invent. Math. 75 (1984)]) in a 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 BC. The latter provides an inductive algorithm to compute Kostka polynomials by upgrading [Ciubotaru-Kato-K, Invent. Math. 178 (2012)] §3 to its graded version. In the appendices, we present purely algebraic proofs that Kostka systems exist for type A and asymptotic type BC cases, and therefore one can skip geometric sections §3-5 to see the key ideas and basic examples/techniques. * The word "green" means 'midori' in Japanese. † French translation: Uneétude homologique des polynômes de Green

Spectral transfer morphisms for unipotent affine Hecke algebras

“…The same is known to be true for the additional contributions to packet Y W 0 r coming from the extraspecial STMs (see [10], [27,Section 4]). These remarkable facts should be considered as an aspect of Langlands duality.…”

“…In turn, these generic central characters are parameterized by pairs σ − , σ + ) of the Slooten symbols (with defects D ± = m ± ) covering (u − , u + ). By the results of [10,42], [27,Section 4], the top graded part with respect to Slooten's functions a m ± [63] of the corresponding graded Hecke algebra module is the irreducible W (C n − ) × W (C n + )-module corresponding to (σ − , σ + ), via the generalized Springer correspondence of [46]. Via Proposition 4.7, the spectral correspondences of the standard STMs to H I M (G) together exhaust the set of pairs of Slooten symbols (σ − , σ + ).…”

Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups

“…Acknowledgments. First of all, we would like to acknowledge that we learned key ideas about the relation of W -equivariant coherent sheaves on t with constructible sheaves on the nilpotent cone from the works of Syu Kato [41] and Laura Rider [63]. In particular, the key equivalence (0.0.4) was first proved in [63] (we reprove it in a different way to get a more explicit functor realizing this equivalence).…”

Affine highest weight categories and affine quasihereditary algebras

“…Then the algebra A W := CW ⋉ C[h * ] is graded so that deg(w) = 0 for all w ∈ W and deg(x) = 2 for x ∈ h * . It is essentially shown in [23] that the category A W -mod of finitey generated graded A W -modules is P-highest weight. In particular, by Theorem 6.7, the algebra A W is P-quasihereditary.…”