[Paper Review] Diagonal Coinvariants and Double Affine Hecke Algebras

Ivan Cherednik|ArXiv.org|May 16, 2003

Advanced Algebra and Geometry11 references4 citations

TL;DR

This paper establishes a direct algebraic connection between diagonal coinvariants of a root system and double affine Hecke algebras (DAHA), proving that the space of diagonal coinvariants for rank n has dimension (1 + h)^n via a quotient of double polynomials under rational DAHA. Using Lusztig-type isomorphisms and the unique irreducible representation of the Weyl algebra at roots of unity, it provides a new algebraic proof of Gordon's theorem, unifying Verlinde algebra generalizations and DAHA representation theory at k = -1 - 1/h.

ABSTRACT

We establish a q-generalization of Gordon's theorem that the space of diagonal coinvariants has a quotient identified with a perfect representation of the rational double affine Hecke algebra. It leads to a simple proof of his theorem and relates it to the Weyl algebras at roots of unity. The universal double affine Hecke algebra and the corresponding universal double Dunkl operators acting in noncommutative polynomials in terms of two sets of variables are introduced.

Motivation & Objective

• To establish a direct algebraic proof of the Haiman conjecture on the dimension of diagonal coinvariants for a root system of rank n.
• To extend Gordon's construction of the diagonal coinvariant space from the rational to the q-deformed DAHA setting.
• To demonstrate that the dimension (1 + h)^n arises naturally from the unique irreducible representation of the Weyl algebra at roots of unity.
• To introduce and study the universal double affine Hecke algebra and its associated universal Dunkl operators.
• To clarify the role of Lusztig-type isomorphisms in relating finite-dimensional representations of DAHA and its rational degeneration.

Proposed method

• Utilizes Lusztig-type isomorphisms to relate the q-deformed DAHA to its rational degeneration, enabling transfer of representation-theoretic results.
• Applies the fact that the Weyl algebra of rank n has a unique irreducible representation when q is a primitive (1 + h)-th root of unity, with dimension (1 + h)^n.
• Constructs the diagonal coinvariant space as a quotient of the space of double polynomials C[x, y] under the action of the rational DAHA induced from the sign character of the nonaffine Weyl group.
• Introduces the universal DAHA and universal Dunkl operators acting on noncommutative polynomials in two sets of variables X and Y.
• Employs the involution ε and tau-automorphisms τ±^x, τ±^y to explore symmetries and duality in the DAHA structure.
• Derives explicit formulas for the action of generators Ti and πr on double polynomials, using relations (4.5) and (4.6) in the q-case.

Experimental results

Research questions

• RQ1 How can the Haiman conjecture on the dimension of diagonal coinvariants be proven using DAHA representation theory?
• RQ2 What is the precise connection between the unique irreducible representation of the Weyl algebra at roots of unity and the diagonal coinvariant space?
• RQ3 How do Lusztig-type isomorphisms bridge the representation theory of q-DAHA and its rational degeneration?
• RQ4 What is the role of the universal DAHA and universal Dunkl operators in realizing the diagonal coinvariant space?
• RQ5 How does the PSL3(Z) action emerge from the automorphism group of the double affine braid group and relate to KZB equations?

Key findings

• The diagonal coinvariant space for a root system of rank n has dimension (1 + h)^n, confirming Haiman's conjecture.
• The space arises as a quotient of the double polynomial algebra C[x, y] under the rational DAHA action, with the kernel corresponding to the ideal of W-invariant polynomials without constant term.
• The unique irreducible representation of the Weyl algebra of rank n, when q is a primitive (1 + h)-th root of unity, has dimension (1 + h)^n, matching the Haiman number.
• The Lusztig-type isomorphism provides a direct algebraic proof of Gordon's theorem by relating the q-DAHA to its rational degeneration.
• The universal DAHA admits a PBW-type theorem and supports a symmetric action of the X ↔ Y duality, generalizing the DAHA duality.
• The universal Dunkl operators in the double polynomial setting provide a natural framework for realizing the diagonal coinvariant module and its self-duality.