# Why are rational Cherednik algebras so… rational?

The real question is both more serious and somewhat longer than the title.

For the definition of the rational Cherednik algebra attached to a complex reflection group $W$, see for instance 5.1.1 of Rouquier's paper here. It is a flat family of algebras depending on some parameters $h_{H,j}$ indexed by pairs consisting of a $W$-orbit of reflecting hyperplanes $H$ and an integer $0 \leq j \leq e_H-1$, where $e_H$ is the order of the pointwise stabilizer of $H$.

Various features of the structure of the Cherednik algebra turn out to be governed by systems of linear equations with rational coefficients in the parameters. For instance, in this paper Dunkl and Opdam show that the polynomial representation is irreducible exactly if the parameters avoid a certain locally finite system of rational hyperplanes, and in this paper a couple of guys show that it is Morita equivalent to its spherical subalgebra off a certain set of rational hyperplanes. In this paper, Etingof shows (for real reflection groups) that the set of parameters for which the irreducible head of the polynomial representation is finite dimensional is a set described by linear conditions with rational coefficients on the parameters.

Every time someone discovers the set of parameters for which the rational Cherednik algebra satisfies some reasonable properties, it winds up being described linearly in the parameters with rational coefficients. Why?

(I'm asking for a conceptual reason---in each case I mentioned I know the proof. For instance, I'd love to know an a priori proof that the set of parameters where the RCA is not Morita equiv. to its spherical subalgebra is a finite union of rational hyperplanes, without necessarily giving the union explicitly)

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Partial answers, hints, and references to similar phenomena in other places are encouraged! –  GS May 12 '10 at 20:40
By the way,Etingof taught a graduate course at MIT last semester on this very topic and was kind enough to post the lecture notes for the course. We need more cutting edge researchers doing this;kudos to Etingof and many thanks. The notes can be found here: www-math.mit.edu/~etingof/18.735notes.pdf –  Andrew L May 13 '10 at 16:11

The phenomenon seems the same as for affine Kac-Moody algebras, where rational level is where everything special happens, or quantum groups, where roots of unity are the exceptional locus (and of course these examples are related). The parameter for Cherednik algebras is an additive/Lie algebra type parameter, like the KM level, as opposed to the exponentiated version as in the quantum groups case. If you mod out by the action of translation functors, you're asking why finite order points of the parameter are special. I don't know that I can give a completely uniform answer, but it certainly seems reasonable.

For example in type A, modules for Cherednik algebras are realized using twisted D-modules on some stack. The parameter for twisting is an additive one, but the abstract category of twisted D-modules depends only on this parameter mod integral translations (hence translation functors). At integral points (or rather "the" integral points) many special things happen -- there are geometric obstructions to the existence of objects with various supports, and these obstructions vanish integrally -- which is why say category O for Lie algebras becomes much bigger integrally. More generally twisted D-modules can be described as sheaves on a gerbe, which depends only on the twist mod integers. If (and certainly only if) the parameter is rational - ie the twist is a torsion element - then you might expect to represent your gerbe by an Azumaya algebra, or equivalently to have finite rank twisted sheaves. I would imagine this general type of phenomenon is behind the results you mention, though I haven't thought about the specifics. But in any case this is a geometric phenomenon about categories of twisted D-modules in general, and as we know "basically all interesting categories of representations are some categories of twisted D-modules" so this is quite general.

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I guess one obvious issue is that it is really the spherical subalgebra which seems to be most tightly connected to D-modules, so maybe it's not fair to expect those methods to have much to say about the relationship between the full RCA and its spherical subalgebra? On the other hand, maybe you can say something about Varagnolo-Vasserot determination of the spherical, finite dimensional modules for the one-parameter DAHA of a Weyl group? –  GS May 13 '10 at 11:20
Dear Stephen -- yes, sorry, I only think of the spherical subalgebra. I would hope/expect all types to have some kind of D-modulish description eventually.. eg the arguments in the recent Etingof paper (with your appendix) sound awfully (wonderfully) D-modular. But yes at this point this is just an analogy/heuristic outside type A spherical and some variants.. –  David Ben-Zvi May 13 '10 at 13:43
Which is of course exactly what I was asking for. Thanks! Do you have any suggestions for reading up on this sort of thing for KM algebras? BTW, I really appreciated your answer to David Jordan's DAHA question---it's a great resource for stuff I don't know as much about as I should! –  GS May 13 '10 at 17:59