[Paper Review] On representations of rational Cherednik algebras in complex rank

Inna Entova-Aizenbud|arXiv (Cornell University)|Jan 1, 2013

Algebraic structures and combinatorial models19 references3 citations

TL;DR

This paper introduces and studies Verma objects in a family of abelian categories ${\underline{\mathcal{O}}}_{c,\nu}$, which are interpolations of the category $\mathcal{O}$ for rational Cherednik algebras of type $A$ in complex rank. It establishes necessary and sufficient conditions for non-trivial morphisms between Verma objects, computes characters of irreducible quotients for generic $c,\nu$, and proves that infinite-length Verma objects exist only when $c \in \mathbb{Q}_{<0}$, confirming a degeneration phenomenon conjectured by Etingof.

ABSTRACT

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 prove that a Verma object of infinite length exists in O_{c, t} only if c is rational and c &lt; 0. We also show that for every rational c &lt; 0 there exists a rational t &lt; 0 such that there exists a Verma object of infinite length in O_{c, t}. The latter result is an example of a degeneration phenomenon which can occur in rational values of t, as was conjectured by P. Etingof.

Motivation & Objective

• To define and study a family of abelian categories ${\underline{\mathcal{O}}}_{c,\nu}$ that interpolate the classical category $\mathcal{O}$ of rational Cherednik algebras $H_c(n)$ for $n \in \mathbb{Z}_+$.
• To introduce and analyze Verma objects $M_{c,\nu}(\lambda)$ as analogues of classical Verma modules in this interpolated setting.
• To determine conditions under which non-trivial morphisms exist between Verma objects in ${\underline{\mathcal{O}}}_{c,\nu}$.
• To compute the character of the irreducible quotient of a Verma object for generic $c,\nu$.
• To investigate the existence of Verma objects of infinite length and identify the parameter values where this occurs, confirming a degeneration phenomenon in rational $\nu$.

Proposed method

• Constructs the category ${\underline{\mathcal{O}}}_{c,\nu}$ as a full subcategory of ${\mathrm{ind}}\text{-}{\underline{\mathrm{Rep}}}(S_\nu)$, equipped with morphisms $x,y$ modeling the action of $x_i, y_i$ in $H_c(n)$.
• Defines Verma objects $M_{c,\nu}(\lambda)$ as graded ind-objects in ${\mathrm{ind}}\text{-}{\underline{\mathrm{Rep}}}(S_\nu)$, with lowest-degree component given by the indecomposable object $X_\lambda$ corresponding to a Young diagram $\lambda$.
• Uses Pieri’s rule and representation-theoretic tools in Deligne’s category ${\underline{\mathrm{Rep}}}(S_\nu)$ to analyze composition factors and morphisms between Verma objects.
• Applies linearity and algebraic conditions on parameters $c, \nu$ to derive equations characterizing when morphisms exist, leading to the definition of algebraic loci $\mathcal{L}_{\tau,\mu^j,m^j}$.
• Employs bounds on $n(\mu)$ and $n(\mu^\vee)$ to show that only finitely many Young diagrams $\mu$ can satisfy the derived conditions, proving finite composition factors for Verma objects under genericity.
• Analyzes the structure of the equations governing morphisms and uses the fact that $c \notin \mathbb{Q}$ to deduce that multiple loci must coincide, leading to constraints on $C, C'$ and ultimately finiteness.

Experimental results

Research questions

• RQ1 For which complex parameters $c, \nu$ does a non-trivial morphism exist between two Verma objects $M_{c,\nu}(\lambda)$ and $M_{c,\nu}(\mu)$?
• RQ2 What is the character of the irreducible quotient of a Verma object $M_{c,\nu}(\lambda)$ for generic $c, \nu$?
• RQ3 Under what conditions does a Verma object in ${\underline{\mathcal{O}}}_{c,\nu}$ have infinite length?
• RQ4 Can the degeneration phenomenon observed at rational $\nu$ values be systematically explained, particularly in relation to negative rational $c$?
• RQ5 Is the number of composition factors of a Verma object finite, and under what parameter conditions?

Key findings

• A non-trivial morphism between Verma objects $M_{c,\nu}(\tau)$ and $M_{c,\nu}(\mu)$ exists only if the parameters $c, \nu$ satisfy a system of algebraic equations derived from the structure of $S^{m^j}\mathfrak{h}_0 \otimes \tau$ and Pieri’s rule.
• For sufficiently generic $c, \nu$, the character of the irreducible quotient of a Verma object $M_{c,\nu}(\lambda)$ is computed as a direct sum of irreducible objects in ${\underline{\mathrm{Rep}}}(S_\nu)$, with multiplicities determined by the structure of the associated Young diagrams.
• A Verma object of infinite length exists in ${\underline{\mathcal{O}}}_{c,\nu}$ only if $c \in \mathbb{Q}_{<0}$, establishing a sharp condition on the parameter $c$.
• For every $c \in \mathbb{Q}_{<0}$, there exists $\nu \in \mathbb{Q}_{<0}$ such that a Verma object of infinite length exists in ${\underline{\mathcal{O}}}_{c,\nu}$, confirming a degeneration phenomenon at rational $\nu$.
• The number of composition factors of any Verma object $M_{c,\nu}(\tau)$ is finite when $c \notin \mathbb{Q}$, due to the boundedness of $|\mu|$ and $\ell(\mu)$ under the derived constraints.
• The analysis shows that the set of Young diagrams $\mu$ for which morphisms exist is finite under genericity, due to the interplay between the degree $m^j$, the size $|\mu|$, and the $n(\mu^\vee)$ term, which grows quadratically in $\ell(\mu)$.