2004/10/12 by Iain J. Gordon, I. Gordon, Gordon, I. +2 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0410293
arxiv created 2004/10/12 · openalex publication_date 2004/10/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Hc be the rational Cherednik algebra of type An-1 with spherical subalgebra Uc=eHce. Then Uc is filtered by order of differential operators with associated graded ring gr Uc=C[h + h*]W, where W is the n-th symmetric group. Using the Z-algebra construction from our earlier paper (math.RA/0407516) it is also possible to associate to a filtered Hc- or Uc-module M a coherent sheaf on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of Uc and Hc, and relate it to Hilb(n) and to the resolution of singularities from Hilb(n) to h+h*/W. For example, we prove: (1) If c=1/n, so that Lc(triv) is the unique one-dimensional simple Hc-module, then Lc(triv) corresponds to the structure sheaf of the punctual Hilbert scheme. (2) If c=1/n+k (for k some natural number) then, under a canonical filtration on the finite dimensional module Lc(triv), gr eLc(triv) has a natural bigraded structure which coincides with that on the global sections of certain ample line bundles on the punctual Hilbert scheme; this confirms conjectures of Berest, Etingof and Ginzburg, and relates representations of Hc and Uc with Haiman's combinatorial work on the Hilbert scheme. (3) Under mild restrictions on c, the characteristic cycle of the standard Hc-modules are described in terms of certain irreducible subvarieties of the Hilbert scheme (appearing originally in work of Grojnowski) with multiplicities given by Kostka numbers.