The W-goodness finite-dimensional quotient conjecture

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Let WW be a real reflection group with reflection representation VV. A positive integer rr is WW-good if, for every parameter cc with ∣c∣=r|c|=r, the Cherednik algebra Hc(W)H_c(W) admits a finite-dimensional module of the form S(V)/S(V)VrS(V)/S(V)V_r, where Vr⊂Sr(V)V_r\subset S^r(V) is an irreducible WW-submodule of dimension dim⁡V\dim V. Let V(r;c)V^{(r;c)} be the distinguished copy of the defining module in the degree-rr quasiharmonics at parameter cc. W-goodness quotient conjecture. If rr is WW-good, then

S(V)/S(V)V(r;c)S(V)/S(V)V^{(r;c)}

is finite-dimensional. The conjecture connects the universal quasiharmonic submodule V(r;c)V^{(r;c)} with the finite-dimensional representations guaranteed by WW-goodness. The source provides examples of WW-good integers for types AA, BB, DD, and I2(m)I_2(m), but does not establish the conjecture in general.

References

Primary source

Arkady Berenstein and Yurii Burman, “Quasiharmonic polynomials for Coxeter groups and representations of Cherednik algebras”, arXiv:math/0505173 (2007).

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