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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.

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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.

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    $\begingroup$ 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. $\endgroup$ – Zahlendreher Sep 30 '15 at 10:19

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