# Graded category O for for rational Cherednik algebras, but at t=0

The paper [1] introduced the category $$\mathcal{O}$$ for rational Cherednik algebras $$H_{t,c}(W)$$. This construction is tailored for the $$t=1$$ case (equivalently, the $$t\neq 0$$ case). The general setup for the structural theory introduced is that of an algebra with a triangular decomposition $$A=\overline{B}\otimes H\otimes B$$ and an inner grading, given by an element $$\partial$$, i.e. $$A_i=\lbrace a\in A\mid [\partial,a]=ia\rbrace$$.

If $$\partial$$ exists, it can be shown that the category $$\mathcal{O}$$ (of locally nilpotent modules, which can be spanned by some generalized weight spaces), is a highest weight category (in the sense of [2]).

• This holds for $$H_{1,c}(W)$$ (Theorem 2.19 in [1])
• This does not hold for $$H_{0,c}(W)$$ as one can already see in the $$A_1$$ case, where the is no unique simple head for the standard modules (consider the ideals $$(x^2-a)$$. There cannot exist an inner grading element as the element $$x^2$$ is central of degree 0.

However, we can consider a category of graded modules that are locally nilpotent. This definition does not involve the inner grading element.

Question:

1. Is the graded category of locally nilpotent $$H_{0,c}(W)$$-modules a highest weight category? Is this written down somewhere?

2. What about more general $$\infty$$-dim. algebras with triangular decomposition?

(I believe that the answer to 1. is "yes". All central elements of non-zero degree, such as $$x^2$$, $$y^2$$ in $$A_1$$-case have to act by zero on graded simples. 2. could involve something like requiring no central elements of non-zero degree.)

[1] Ginzburg, Victor; Guay, Nicolas; Opdam, Eric; Rouquier, Raphaël. On the category $$\scr O$$ for rational Cherednik algebras. Invent. Math. 154 (2003), no. 3, 617--651.

[2] Cline, E.; Parshall, B.; Scott, L. Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391 (1988), 85--99.

This is extremely false: for generic $\mathbf{c}$, the algebra $H_{0,\mathbf{c}}(W)$ is Morita equivalent to the functions on Calogero-Moser space, a finite dimensional smooth affine variety, and category $\mathcal O$ is the subcategory of coherent sheaves which are supported set-theoretically on a subvariety isomorphic to a disjoint union of affine spaces. In particular, the graded simples have $\mathrm{Ext}(L,L)$ isomorphic to an exterior algebra, like any skyscraper sheaf in a smooth variety.
• Thank you for this nice geometric answer. Can I just check that I understand correctly what the contradiction is? For dimension of $L$ larger than 1, $Ext^1(L,L)\neq 0$ but $L\nless L$ so we contradict Lemma 3.2(b) in CPS [1]. Hence the category cannot be highest weight as $\dim L=|W|$ on the Azumaya locus. Sep 30 '15 at 10:19