Diagonal Coinvariants and Double Affine Hecke Algebras
نویسنده
چکیده
It was conjectured by Haiman [H] that the space of diagonal coinvariants for a root system R of rank n has a ”natural” quotient of dimension (1 + h) for the Coxeter number h. This space is the quotient C[x, y]/(C[x, y]C[x, y]o ) for the algebra of polynomials C[x, y] with the diagonal action of the Weyl group on x ∈ C ∋ y and the ideal C[x, y]o ⊂ C[x, y] of the W -invariant polynomials without the constant term. In [G], such a quotient was constructed. It appeared to be isomorphic to the perfect representation (in the terminology of [C8]) of the rational double affine Hecke algebra for the simplest nontrivial k = −1− 1/h. We extend Gordon’s theorem to the q, t-case, establishing its direct connection with a fundamental fact that the Weyl algebra of rank n, also called a noncommutative n-torus, has a unique irreducible representation when the center element q is a primitive N -th root of unity and the generators are cyclic of order N. Its dimension is N, which matches the Haiman number as N = 1 + h. It is not by chance. Our theorem results from this fact. Applying the Lusztig-type isomorphism acting from the general DAHA to its rational degeneration, we come to another proof of Gordon’s theorem, simple and entirely algebraic.
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