Finite-dimensional representations of rational Cherednik algebras

By Y. Berest, Pavel Etingof, V. Ginzburg. 2002

180 citations recorded by the source corpus.

A complete classification and character formulas for finite-dimensional irreducible representations of the rational Cherednik algebra of type A is given. Less complete results for other types are obtained. Links to the geometry of affine flag manifolds and Hilbert schemes are discussed.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b10c7417bc9b21a61967a9cdaa1c2e71b815f47d

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In Algebra and Number Theory

1. I. Losev
2. I. Gordon
3. V. Ginzburg
4. E. Gorsky
5. Pavel Etingof
6. Stephen Griffeth
7. R. Bezrukavnikov
8. Alexei Oblomkov

In Geometry and Topology

1. E. Gorsky
2. I. Gordon
3. V. Ginzburg
4. I. Losev
5. Alexei Oblomkov
6. Pavel Etingof
7. Stephen Griffeth
8. Vivek V. Shende

In Mathematical Physics

1. E. Gorsky
2. V. Ginzburg
3. I. Gordon
4. I. Losev
5. Alexei Oblomkov
6. Pavel Etingof
7. Stephen Griffeth
8. Vivek V. Shende

Citations in the corpus

• Parabolic induction and restriction functors for rational Cherednik algebras (2008, background): “Proof. It follows from the results of [ BEG2 ] that”
• Cyclic Sieving of Noncrossing Partitions for Complex Reflection Groups (2007, background): “For real reflection groups W , the work of Berest, Etingof and Ginzburg finds such hsop’s using subtle results (including work of Gordon [21]) on the representation theory of rational Cherednik algebras– the authors are very grateful to S. Griffeth, for pointing them to these results.”
• Completions of symplectic reflection algebras (2010, background): “ra H1,c of type A. “Type A” means that Γ = Sn is a symmetric group in nletters, V = h⊕h∗, where h∼=Cn−1 is the reflection Γ-module. In this case cis a single number. It is well known, see, for example [BEG1], that H1,c has a finite dimensional module if and only if c= r n , where n∈Z>1 and ris an i”
• On the quotient ring by diagonal invariants (2002, background): “This conjecture has now been confirmed in [3].”
• Parking Spaces (2012, methodology): “For example, this quotient occurs in the rational Cherednik theory [27, 9, 20, 21] as the finite-dimensional irreducible module L(h+1)/h(1W ) corresponding to the trivial representation 1W of W , and taken at the rational parameter h+1 h .”
• Cherednik algebras and Hilbert schemes in characteristic p (2003, methodology): “if Conjecture 1.4.2 holds for c = 1 n + k,k = 1,2,...,then Γ(Hilb (1) 0 ,V0 ⊗BKR −1(triv H )) is an Hcmoduleobtained by reducingmodulo pthe uniquefinite dimensional irreducibleHc-module constructed in [BEG2] in case of characteristic zero. Remark 1.4.3. The case of c= 1 n <1 is not covered by Theo”
• Rational Cherednik algebras and Hilbert schemes, II: Representations and sheaves (2004, background): “algebra Uc = eHce while the related algebra U − c = e−Hce− is called the anti-spherical subalgebra. By [ BEG2 ,”
• Catalan numbers for complex reflection groups (2009, result): “We now prove our main result, answering a question of the second author, [7, Section 8], and giving a general and case-free construction of the Koszul resolutions produced in [1, 5, 7, 16].”
• Cherednik algebras and differential operators on quasi-invariants (2001, result): “r/n>0. The algebra Hc has a nonzero finite dimensional representation V if and only if ris coprime to n, moreover, in such a case in Theorem 5.9(ii) we have ℓ= 1. This conjecture has been proved in [BEG1]. 6 The eHce-module structure on quasi-invariants In this section we review the definition and ”
• Refined knot invariants and Hilbert schemes (2013, background): “ the representation theory of the rational Cherednik algebra depends crucially on the parameter c. For example, we have the following classification of finite-dimensional representa tions. Theorem 5.2 ([4]). The algebra Hc only has finite-dimensional representations ifc = m n for some gcd(m,n) = 1, in ”
• Representations of Rational Cherednik algebras with minimal support and torus knots (2013, methodology): “: Oc →Oc+1 be the shift functor, which is a right exact functor defined by Shc(V) = Hc(Sn)e⊗ eHc(Sn)e=e −Hc+1(Sn)e− e−V, where e = e0 is the symmetrizer, and e− = en−1 ∈CSn is the antisymmetrizer (see [BEG]). It is shown in [BE] that Shc is an equivalence of categories for c>0. Corollary 6.12. One”
• Torus knots and the rational DAHA (2012, background): “tion. The above desiderata uniquely specify a filtration. The idea of the following construction is due to Etingof and Ginzburg. When m≡ 1 (mod n), this construction appears explicitly in section 7 of [BEG03b] (c.f. [GS05]). Theorem 4.1. Consider Lm/n for m>nand eLm/n for m>0. There is a unique”
• Purity of equivalued affine Springer fibers (2003, unknown): “], Sage [Sag00] and Sommers [Som97]. Quite recently these results have been reconsidered in the context of Cherednik’s double-affine Hecke algebras by Vasserot [Vas] and by Berest, Etingof and Ginzburg [BEG]. A homogeneous (weakly Coxeter) case for GL(n) was analyzed by Laumon and Waldspurger [LW]. It”