Canonical Bases of Higher-Level q-Deformed Fock Spaces and Kazhdan-Lusztig Polynomials

Abstract

We define canonical bases of the higher-level q-deformed Fock space modules of the affine Lie algebra sl n generalizing the result of Leclerc and Thibon for the case of level 1. We express the transition matrices between the canonical bases and the natural bases of the Fock spaces in terms of certain affine Kazhdan-Lusztig polynomials.

Quiver Varieties, Category for Rational Cherednik Algebras, and Hecke Algebras

“…His conjectures depend on the work of Uglov on higher level Fock spaces, [48]. They predict an ordering on O h which depends on the choice of a multi-charge depending on h; this multi-charge corresponds to our choice of core corresponding to θ.…”

Canonical bases of higher-level q-deformed Fock spaces

“…Since w + α i is not a weight of F q [s l ], λ l has no removable i-node. This implies, by [U,Thm. 2.4], that λ l is the head of the chain C, and by symmetry µ l is the tail of C. Note that since w + α i is not a weight of…”

“…Note that by [U,Lemma 3.18], the sum above involves only finitely many nonzero terms, hence B m is well-defined. This definition comes from a passage to the limit r → ∞ in the action of the center of the Hecke algebra of S r on q-wedge products of r factors.…”

“…This definition comes from a passage to the limit r → ∞ in the action of the center of the Hecke algebra of S r on q-wedge products of r factors. However, the operators B m do not commute, but by [U,Prop. 4.4 & 4.5], they span a Heisenberg algebra…”

Parabolic Kazhdan–Lusztig polynomials for quasi-minuscule quotients

“…Given any parabolic subgroup W J in a Coxeter system (W, S), Deodhar introduced two Hecke algebra modules (one for each of the two roots q and −1 of the polynomial x 2 − (q − 1)x − q) and two families of polynomials {P J,q u,v (q)} u,v∈W J and {P J,−1 u,v (q)} u,v∈W J indexed by pairs of elements of the set of minimal coset representatives W J . These polynomials are the parabolic analogues of the Kazhdan-Lusztig polynomials: while they are related to their ordinary counterparts in several ways (see, e.g., § 2 and [7], Proposition 3.5), they also play a direct role in several areas such as the geometry of partial flag manifolds [16], the theory of Macdonald polynomials [13], [14], tilting modules [25], [26], generalized Verma modules [5], canonical bases [11], [29], the representation theory of the Lie algebra gl n [22], quantized Schur algebras [30], quantum groups [9], and physics (see, e.g., [12], and the references cited there). The computation of these polynomials is a very difficult task.…”