Abstract
We provide a homological construction of unitary simple modules of Cherednik and Hecke algebras of type A via BGG resolutions, solving a conjecture of Berkesch–Griffeth–Sam. We vastly generalize the conjecture and its solution to cyclotomic Cherednik and Hecke algebras over arbitrary ground fields, and calculate the Betti numbers and Castelnuovo–Mumford regularity of certain symmetric linear subspace arrangements.
Highlights
In [1], Bernstein–Gelfand–Gelfand utilise resolutions of simple modules by Verma modules to prove certain beautiful properties of finite-dimensional Lie algebras
For Lie groups, this ongoing project draws on techniques from Dirac cohomology [35], Kazhdan–Lusztig theory [64], and the Langlands Program [58], and has provided profound insights into relativistic quantum mechanics [61]
We show that our resolutions for unitary simples remain stable under reduction modulo p — in other words the beautiful properties of unitary modules extend beyond the confines of characteristic zero to for arbitrary fields
Summary
In [1], Bernstein–Gelfand–Gelfand utilise resolutions of simple modules by Verma modules to prove certain beautiful properties of finite-dimensional Lie algebras Such resolutions ( known as BGG resolutions) have had spectacular applications in the study of the Laplacian on Euclidean space [19], complex representation theory and homology of Kac–Moody algebras [28], statistical mechanics and conformal field theories [27,46,47], and they provide graded free resolutions (in the sense of commutative algebra) for determinantal varieties [20,40]. In our BGG resolution, (ν) appears in homological degree d if and only if ν is obtained from λ by reflecting across d walls
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