Irreducible Representations of a Class of Current Algebras of Etingof and Frenkel

The Hilbert Series of the Irreducible Quotient of the Polynomial Representation of the Rational Cherednik Algebra of Type $A_{n-1}$ in Characteristic $p$ for $p|n-1$

We study the irreducible quotient $\mathcal{L}_{t,c}$ of the polynomial representation of the rational Cherednik algebra $\mathcal{H}_{t,c}(S_n,\mathfrak{h})$ of type $A_{n-1}$ over an algebraically closed field of positive characteristic $p$ where $p|n-1$. In the $t=0$ case, for all $c\ne 0$ we give a complete description of the polynomials in the maximal proper graded submodule $\ker \mathcal{B}$, the kernel of the contravariant form $\mathcal{B}$, and subsequently find the Hilbert series of the irreducible quotient $\mathcal{L}_{0,c}$. In the $t=1$ case, we give a complete description of th…

Classification of the finite-dimensional unitary representations of type B rational Cherednik algebras

We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary representations of type $B$ rational Cherednik algebras to show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica.

Unitary representations of cyclotomic rational Cherednik algebras

We classify the irreducible unitary modules in category O for the rational Cherednik algebras of type G(r,1,n) and give explicit combinatorial formulas for their graded characters. More precisely, we produce a combinatorial algorithm determining, for each r-partition of n, the closed semi-linear set of parameters for which the contravariant form on the irreducible representation with the given r-partition as lowest weight is positive definite. We use this algorithm to give a closed form answer for the Cherednik algebra of the symmetric group (recovering a result of Etingof-Stoica and the autho…

量子 loop 代数与 Yangian 的 Theta 级数

Theta series for quantum loop algebras and Yangians

该研究引入并研究了系数在量子loop代数张量平方中的Theta级数，证明其权分量的多项式性，给出O范畴中R矩阵的分解与多项式性结果，并将相关级数推广到Yangian中且通过结合子解释证明其多项式性。

Representations of Rational Cherednik algebras with minimal support and torus knots

We obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen-Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type A_{n-1} for c=m/n, and an expression of its quasispherical part (i.e., the isotypic part of "hooks") in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Et…

Associated varieties of modules over Kac-Moody algebras and $C_2$-cofiniteness of W-algebras

First, we establish the relation between the associated varieties of modules over Kac-Moody algebras \hat{g} and those over affine W-algebras. Second, we prove the Feigin-Frenkel conjecture on the singular supports of G-integrable admissible representations. In fact we show that the associated variates of G-integrable admissible representations are irreducible G-invariant subvarieties of the nullcone of g, by determining them explicitly. Third, we prove the C_2-cofiniteness of a large number of simple W-algebras, including all minimal series principal W-algebras and the exceptional W-algebras …

On Refined Filtration By Supports for Rational Cherednik Categories O

For a complex reflection group $W$ with reflection representation $\mathfrak{h}$, we define and study a natural filtration by Serre subcategories of the category $\mathcal{O}_c(W, \mathfrak{h})$ of representations of the rational Cherednik algebra $H_c(W, \mathfrak{h})$. This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov-Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of …

A Geometric Realization of Spin Representations and Young diagrams from Quiver Varieties

Applying the techniques of an earlier paper with Frenkel, we develop a geometric realization of spin representations and Clifford algebras. In doing so, we give an explicit parametrization of the irreducible components of Nakajima varieties of type D in terms of Young diagrams. We explicity compute the geometric action of the Lie algebra and are able to extend the geometric action to the entire Clifford algebra used in the classical construction of the spin representations.

The Brylinski filtration for affine Kac-Moody algebras and representations of $\mathcal{W}$-algebras

We study the Brylinski filtration induced by a principal Heisenberg subalgebra of an affine Kac-Moody algebra $\mathfrak{g}$, a notion first introduced by Slofstra. The associated graded space of this filtration on dominant weight spaces of integrable highest weight modules of $\mathfrak{g}$ has Hilbert series coinciding with Lusztig's $t$-analogue of weight multiplicities. For the level 1 vacuum module $L(Λ_0)$ of affine Kac-Moody algebras of type $A$, we show that the Brylinski filtration may be most naturally understood in terms of (vertex algebra) representations of the corresponding $\mat…

An $\mathfrak{sl}_2$-type tensor category for the Virasoro algebra at central charge $25$ and applications

Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Vir…

The real loci of Calogero-Moser spaces, representations of rational Cherednik algebras and the Shapiro conjecture

We prove a criterion for the reality of irreducible representations of the rational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterion for the real loci of the Calogero-Moser spaces C_n in terms of the Etingof-Ginzburg finite maps Υ\colon C_n \to C^n/S_n \times C^n/S_n, recovering a result of Mikhin, Tarasov, and Varchenko [MTV2]. As a consequence we obtain a criterion for the real locus of the Wilson's adelic Grassmannian of rank one bispectral solutions of the KP hierarchy. Using Wilson's first parametrisation of the adelic Grassmannian, we give a new proof of a result of [M…

Subspace arrangements and Cherednik algebras

The purpose of this article is to study the relationship between numerical invariants of certain subspace arrangements coming from reflection groups and numerical invariants arising in the representation theory of Cherednik algebras. For instance, we observe that knowledge of the equivariant graded Betti numbers (in the sense of commutative algebra) of any irreducible representation in category O is equivalent to knowledge of the Kazhdan-Lusztig character of the irreducible object (we use this observation in joint work with Fishel-Manosalva). We then explore the extent to which Cherednik algeb…

来自量子伪Kähler平面的三维量子引力

Three-dimensional quantum gravity from the quantum pseudo-Kähler plane

本文通过引入量子伪Kähler平面这一新典范Hopf代数，研究其表示论与模双紧量子二对数，并构造Kashaev变换群的新表示，从而为正宇宙学常数下的三维引力量子化提供新途径。

On representations of rational Cherednik algebras in complex rank

We study a family of abelian categories O_{c, t} depending on complex parameters c, t which are interpolations of the O-category for the rational Cherednik algebra H_c(t) of type A, where t is a positive integer. We define the notion of a Verma object in such a category (a natural analogue of the notion of Verma module). We give some necessary conditions and some sufficient conditions for the existence of a non-trivial morphism between two such Verma objects. We also compute the character of the irreducible quotient of a Verma object for sufficiently generic values of parameters c, t, and prov…

Decomposition of Level-1 Representations of D_4^(1) With Respect to its Subalgebra G_2^(1) in the Spinor Construction

In [FFR] Feingold, Frenkel and Ries gave a spinor construction of the vertex operator para-algebra (abelian intertwining algebra) V = V^0 \oplus V^1 \oplus V^2 \oplus V^3, whose summands are four level-1 irreducible representations of the affine Kac-Moody algebra D_4^(1). The triality group S_3 = < σ,τ| σ^3 = 1 = τ^2, τστ= σ^{-1} > in Aut(V) was constructed, preserving V^0 and permuting the V^i, for i=1,2,3. V is (1/2)Z-graded where V^i_n denotes the n-graded subspace of V^i. Vertex operators Y(v,z) for v in V^0_1 represent D_4^(1) on V, while those for which σ(v) = v represent G_2^(1). We inv…

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=eH_ce. Then U_c is filtered by order of differential operators with associated graded ring gr U_c=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 H_c- or U_c-module M a coherent sheaf on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of U_c and H_c, and relate it to Hilb(n) and to the resolution of singularities from Hilb(n) to h+h*/W. For example, …