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The rank-one DAHA difference-reflection operators satisfy the presentation

Statement

Let $P = K [z, z^{- 1}]$ . Define operators on $P$ by

(Π f) (z) = f (q^{- 1} z^{- 1}), (σ_{1} f) (z) = f (z^{- 1}), (σ_{0} f) (z) = f (q^{- 2} z^{- 1}),

(T_{i} f) (z) = t (σ_{i} f) (z) + (t - t^{- 1}) \frac{(σ_{i} f) (z) - f (z)}{u_{i} (z) - 1}, u_{1} (z) = z^{- 2}, u_{0} (z) = q^{2} z^{2} .

Then $Z \mapsto M_{z}$ , $Z^{- 1} \mapsto M_{z^{- 1}}$ , $π \mapsto Π$ , and $T_{i} \mapsto T_{i}$ satisfy every defining relation of The equal-parameter double-affine Hecke algebra of type A1. Consequently they define a unital representation of $H_{q, t} (A_{1})$ on $P$ .

Facts & Assumptions

Given: $K = Q (q, t)$ , the rank-one substitutions and Bernstein polynomials, and the DAHA quotient presentation.

[F1]

The rank-one root datum and parameter conventions for DAHA: $s_{1} (z^{m}) = z^{- m}$ , $s_{0} (z^{m}) = q^{- 2 m} z^{- m}$ , $π (z^{m}) = q^{- m} z^{- m}$ , and $u_{1} = z^{- 2}$ , $u_{0} = q^{2} z^{2}$ .

[F2]

The equal-parameter double-affine Hecke algebra of type A1: the required relations are the Laurent inverse, the $π$ -relations, the quadratic relations, and the Bernstein cross-relations for all $m \in Z$ .

[F3]

The rank-one DAHA Bernstein quotients are Laurent polynomials: $(z^{m} - s_{i} (z^{m})) / (1 - u_{i} (z)) = B_{i, m} (z) \in K [z, z^{- 1}]$ for both affine simple roots and every integer $m$ .

[F4]

The free associative R-algebra on a set and descent of relations: an assignment of the free generators to an associative $K$ -algebra that kills all defining relators descends uniquely to a homomorphism from the presented quotient.

[F5]

The rank-one DAHA polynomial representation: the candidate generator assignment sends $Z$ to $M_{z}$ , $Z^{- 1}$ to $M_{z^{- 1}}$ , $π$ to the stated substitution, and $T_{i}$ to the displayed difference-reflection operators; the relation checks below establish that it is an algebra action.

Proof

1.1F1F2givenalgebra

Multiplication by $z$ and $z^{- 1}$ are inverse operators, so the Laurent-inverse relator holds. Direct substitution gives $Π^{2} = 1$ and $Π M_{z} Π = M_{q^{- 1} z^{- 1}}$ , which is the $π Z π$ relation. It also gives $Π σ_{0} Π = σ_{1}$ and $Π M_{u_{0}} Π = M_{u_{1}}$ ; since both Hecke formulas use the same $t$ , conjugating the formula for $T_{0}$ yields $Π T_{0} Π = T_{1}$ .

1.2F1F2algebra

Put $c = t - t^{- 1}$ , $a_{i} = (t u_{i} - t^{- 1}) / (u_{i} - 1)$ , and $b_{i} = - c / (u_{i} - 1)$ . Since $σ_{i} (u_{i}) = u_{i}^{- 1}$ , one has $T_{i} = a_{i} σ_{i} + b_{i}$ , $σ_{i} (a_{i}) = (t^{- 1} u_{i} - t) / (u_{i} - 1)$ , and $σ_{i} (b_{i}) = c u_{i} / (u_{i} - 1)$ . Multiplication in the crossed-product operator algebra gives $a_{i} σ_{i} (b_{i}) + b_{i} a_{i} = c a_{i}$ and $a_{i} σ_{i} (a_{i}) + b_{i}^{2} = 1 + c b_{i}$ , so $T_{i}^{2} = c T_{i} + 1$ , equivalent to $(T_{i} - t) (T_{i} + t^{- 1}) = 0$ .

1.3F2F3algebra

For multiplication $M_{f}$ by a Laurent polynomial, direct expansion of $T_{i} M_{f} - M_{s_{i} (f)} T_{i}$ gives multiplication by $c (f - s_{i} (f)) / (1 - u_{i})$ . With $f = z^{m}$ , [F3] identifies this Laurent polynomial with $c B_{i, m}$ , which is exactly each Bernstein relation in [F2]. The finite-sum formulas in [F3] also show that $T_{i}$ preserves $P$ .

2.1step 1.1step 1.2step 1.3F2F4F5algebra∎

The assignments in [F5], checked in steps 1.1, 1.2, and 1.3, satisfy all relators of the free-algebra quotient. By [F4] they descend to a unital $K$ -algebra homomorphism $H_{q, t} (A_{1}) \to {End}_{K} (P)$ .

Depends on

Used by

Nothing in the library uses this result yet.

Cited to discharge well-definedness by The rank-one DAHA polynomial representation.

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Sources

Alexander A. Kirillov Jr., Lectures on the affine Hecke algebras and Macdonald conjectures (standard reference, not scraped)
Pavel Etingof and Xiaoguang Ma, Lecture Notes on Cherednik Algebras (standard reference, not scraped)