Iain Gordon

On the quotient ring by diagonal invariants

For a finite Coxeter group, W, and its reflection representation 𝔥, we find the character and Hilbert series for a quotient ring of ℂ[𝔥⊕𝔥*] by an ideal containing the W–invariant polynomials without constant term. This confirms conjectures of Haiman.

Rational Cherednik algebras and Hilbert schemes. II: representations and sheaves

Let H_c be the rational Cherednik algebra of type A_{n-1} with spherical subalgebra U_c = e H_c e. Then U_c is filtered by order of differential operators with associated graded ring gr U_c = \mathbb{C} [ \mathfrac{h} ⊕ \mathfrac{h}^* ]^W, where is the n-th symmetric group. Using the Z-algebra construction from [GS], it is also possible to associate to a filtered H_c - or U_c - module \hat{Φ}(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of U_c and H_c, and we relate it to Hilb(n) and to the resolution of singularities τ : Hilb(n) → \mathfrac{h} ⊕ \mathfrac{h}^* / W. For example, we prove the following. • If c=1/n so that L_c(triv) is the unique one-dimensional simple H_c-module, then \hat{Φ}(e L_c(triv)) ≅ \mathcal{O}_{Z_n}, where Z_n = τ^{-1}(0) is the punctual Hilbert scheme. • If c = 1/n+k for k \in \mathbb{N}, then under a canonical filtration on the finite-dimensional module L_c(triv), gr e L_c(triv) has a natural bigraded structure that coincides with that on H^0( Z_n, \mathscr{L}^k), where \mathscr{L} ≅ \mathcal{O}_{Hilb(n)}(1); this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3]. • Under mild restrictions on c, the characteristic cycle of \hat{Φ}(e Δ_c(μ)) equals \sum_λ K_{μλ}[Z_λ], where K_{μλ} are Kostka numbers and the Z_λ are (known) irreducible components of τ^{-1}(\mathfrak{h}/W)

On category

We study equivalences for category Op of the rational Cherednik algebras Hp of type G`(n) = (μ`) n o Sn: a highest weight equivalence between Op and Oσ(p) for σ ∈ S` and an action of S` on a non-empty Zariski open set of parameters p; a derived equivalence between Op and Op′ whenever p and p′ have integral difference; a highest weight equivalence between Op and a parabolic category O for the general linear group, under a non-rationality assumption on the parameter p. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.

\mathcal{O}

We construct

for cyclotomic rational Cherednik algebras

(q,t)

Catalan numbers for complex reflection groups

-Catalan polynomials and

Block representation type of reduced enveloping algebras

q

A REMARK ON RATIONAL CHEREDNIK ALGEBRAS AND DIFFERENTIAL OPERATORS ON THE CYCLIC QUIVER

-Fuss-Catalan polynomials for any irreducible complex reflection group

Rational Cherednik Algebras

W

Monodromy of Partial KZ Functors for Rational Cherednik Algebras

. The two main ingredients in this construction are Rouquiers formulation of shift functors for the rational Cherednik algebras of