Baby Verma modules for rational Cherednik algebras

增强 nilHecke 代数与 baby Verma 模

Enhanced nilHecke algebras and baby Verma modules

本文通过在增强 nilHecke 代数上构造导子得到 p-dg 代数，并利用 p-dg 模范畴上的函子在 Grothendieck 群上实现小量子 sl2 作用，进而将其提升为李代数 sl(2) 的作用，揭示其与 baby Verma 模的同构关系。

CHAMP: A Cherednik Algebra Magma Package

We present a computer algebra package based on Magma for performing computations in rational Cherednik algebras at arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We could determi…

Jantzen sum formula for restricted Verma modules over affine Kac-Moody algebras at the critical level

For a restricted Verma module of an affine Kac-Moody algebra at the critical level we describe the Jantzen filtration and give an alternating sum formula which corresponds to the Jantzen sum formula of a baby Verma module over a modular Lie algebra. This also implies a new proof of the linkage principle which was already deduced by Arakawa and Fiebig.

正特征下有理 Cherednik 代数 $H_{t,c}(S_3,\h)$ 的表示

Representations of the rational Cherednik algebra $H_{t,c}(S_3,\h)$ in positive characteristic

本文系统刻画正特征下 $A_2$ 型有理 Cherednik 代数的不可约范畴 $\mathcal{O}$ 表示，给出任意参数下的 Hilbert 多项式、特征及 Verma 模极大真分次子模生成元。

Representations of Infinitesimal Cherednik Algebras

Infinitesimal Cherednik algebras, first introduced in [EGG], are continuous analogues of rational Cherednik algebras, and in the case of gl_n, are deformations of universal enveloping algebras of the Lie algebras sl_{n+1}. Despite these connections, infinitesimal Cherednik algebras are not widely-studied, and basic questions of intrinsic algebraic and representation theoretical nature remain open. In the first half of this paper, we construct the complete center of H_ζ(gl_n) for the case of n=2 and give one particular generator of the center, the Casimir operator, for general n. We find the ac…

The Dunkl Weight Function for Rational Cherednik Algebras

In this paper we prove the existence of the Dunkl weight function $K_{c, λ}$ for any irreducible representation $λ$ of any finite Coxeter group $W$, generalizing previous results of Dunkl. In particular, $K_{c, λ}$ is a family of tempered distributions on the real reflection representation of $W$ taking values in $\text{End}_\mathbb{C}(λ)$, with holomorphic dependence on the complex multi-parameter $c$. When the parameter $c$ is real, the distribution $K_{c, λ}$ provides an integral formula for Cherednik's Gaussian inner product $γ_{c, λ}$ on the Verma module $Δ_c(λ)$ for the rational Cheredni…

Suzuki functor at the critical level

In this paper we define and study a critical-level generalization of the Suzuki functor, relating the affine general linear Lie algebra to the rational Cherednik algebra of type A. Our main result states that this functor induces a surjective algebra homomorphism from the centre of the completed universal enveloping algebra at the critical level to the centre of the rational Cherednik algebra at t=0. We use this homomorphism to obtain several results about the functor. We compute it on Verma modules, Weyl modules, and their restricted versions. We describe the maps between endomorphism rings i…

Representations for the restricted Lie color algebras

The generalized p-characters for the restricted Lie color are defined. The simplicity of the induced modules for the FP triples are determined. As applications, we have obtained an analogue of the Kac-Weisfiler theorem for the algebraic Lie color algebra cgl(V), also determined the simplicity of the baby Verma modules for cgl(V).

Representations of rational Cherednik algebras of G(m,r,n) in positive characteristic

We study lowest-weight irreducible representations of rational Cherednik algebras attached to the complex reflection groups G(m,r,n) in characteristic p. Our approach is mostly from the perspective of commutative algebra. By studying the kernel of the contravariant bilinear form on Verma modules, we obtain formulas for Hilbert series of irreducible representations in a number of cases, and present conjectures in other cases. We observe that the form of the Hilbert series of the irreducible representations and the generators of the kernel tend to be determined by the value of n modulo p, and ar…

Jantzen filtration and strong linkage principle for modular Lie superalgebras

In this paper, we introduce super Weyl groups, their distinguished elements and properties for basic classical Lie superalgebras. Then we formulate Jantzen filtration for baby Verma modules in graded restricted module categories of basic classical Lie superalgebras over an algebraically closed field of odd characteristic, and prove a sum formula in the corresponding Grothendieck groups. We finally obtain a strong linkage principle in such categories.

Support Varieties, AR-Components, and Good Filtrations

In continuation of work begun in \cite{FR}, we study in this article those Auslander--Reiten components of the algebras $\Dist(G_r)$ that contain simple modules or baby Verma modules, where $\Dist(G_r)$ is the algebra of distributions of the $r$-th Frobenius kernel of the smooth reductive group $G$, defined over an algebraically closed field of positive characteristic.

On the Cohomology of Modular Lie Algebras

In this paper we establish a connection between the cohomology of a modular Lie algebra and its p-envelopes. We also compute the cohomology of Zassenhaus algebras and their minimal p-envelopes with coefficients in generalized baby Verma modules and in simple modules over fields of characteristic p>2.

On structure of graded restricted simple Lie algebras of Cartan type as modules over the Witt algebra

Any graded restricted simple Lie algebra of Cartan type contains a subalgebra isomorphic to the Witt algebra over a field of prime characteristic. As some analogue of study on branching rules for restricted non-classical Lie algebras, it is shown that each graded restricted simple Lie algebra of Cartan type can be decomposed into a direct sum of restricted baby Verma modules and simple modules as an adjoint module over the Witt algebra. In particular, the composition factors are precisely determined.

约化包络代数表示论中的两个问题

Two problems in the representation theory of reduced enveloping algebras

本文研究约化代数群李代数表示论中的两个问题：证明 baby Verma 模张量积具有特定滤过，并在 A 型假设下构造约化包络代数的最小维数模。

Representations of Lie superalgebras in prime characteristic II: The queer series

The modular representation theory of the queer Lie superalgebra q(n) over characteristic p>2 is developed. We obtain a criterion for the irreducibility of baby Verma modules with semisimple p-characters and a criterion for the semisimplicity of the corresponding reduced enveloping algebras. A (2p)-power divisibility of dimensions of q(n)-modules with nilpotent p-characters is established. The representation theory of q(2) is treated in detail. We formulate a Morita super-equivalence conjecture for q(n) with general p-characters which is verified for n=2.

截断电流李代数的模表示

Modular Representations of Truncated current Lie algebras

本文研究正特征域上截断电流代数的结构与表示，分类半单与幂零元，证明 Morita 等价，给出单模维数界，发展 baby Verma 模理论，并在标准 Levi 型下完全分类单模，最后计算 Cartan 不变量。