# 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)

• 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 – The Mathemagician May 13 '10 at 16:11