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The rank-one DAHA Bernstein quotients are Laurent polynomials

Statement

With the root data and $B_{i, m}$ from The rank-one root datum and parameter conventions for DAHA, for every integer $m$ ,

B_{1, m} = \frac{z^{m} - z^{- m}}{1 - z^{- 2}}, B_{0, m} = \frac{z^{m} - q^{- 2 m} z^{- m}}{1 - q^{2} z^{2}}

as elements of $K [z, z^{- 1}]$ . In particular, the rank-one Bernstein cross-relations use actual Laurent polynomials and require no localization of the coefficient algebra.

Facts & Assumptions

Given: The field $K = Q (q, t)$ , the Laurent variable $z$ , the actions $s_{0} (z^{m}) = q^{- 2 m} z^{- m}$ and $s_{1} (z^{m}) = z^{- m}$ , and the finite sums defining $B_{i, m}$ .

[F1]

The rank-one root datum and parameter conventions for DAHA: for $m > 0$ , $B_{1, m} = \sum_{j = 0}^{m - 1} z^{m - 2 j}$ and $B_{0, m} = - \sum_{j = 0}^{m - 1} q^{- 2 j - 2} z^{m - 2 - 2 j}$ ; negative indices are defined by the corresponding finite sums there.

Proof

1.1F1givenalgebra

For $m > 0$ , multiplying $\sum_{j = 0}^{m - 1} z^{m - 2 j}$ by $1 - z^{- 2}$ cancels all intermediate terms and leaves $z^{m} - z^{- m}$ . The negative-index formula is its negative, so it gives $(z^{- m} - z^{m}) / (1 - z^{- 2})$ ; the case $m = 0$ has zero numerator. Hence the first quotient equals $B_{1, m}$ for every integer $m$ .

2.1F1givenalgebra∎

For $m > 0$ , the geometric identity $z^{2 m} - q^{- 2 m} = (z^{2} - q^{- 2}) \sum_{j = 0}^{m - 1} z^{2 (m - 1 - j)} q^{- 2 j}$ and $1 - q^{2} z^{2} = - q^{2} (z^{2} - q^{- 2})$ give $\frac{z^{m} - q^{- 2 m} z^{- m}}{1 - q^{2} z^{2}} = - \sum_{j = 0}^{m - 1} q^{- 2 j - 2} z^{m - 2 - 2 j} = B_{0, m}$ . For $m = - r < 0$ , $z^{- r} - q^{2 r} z^{r} = z^{- r} (1 - (q^{2} z^{2})^{r})$ , so division by $1 - q^{2} z^{2}$ gives $\sum_{j = 0}^{r - 1} q^{2 j} z^{- r + 2 j} = B_{0, - r}$ . The zero case is again zero. Thus the second quotient is Laurent for every integer $m$ .

Depends on

The rank-one root datum and parameter conventions for DAHA

Used by

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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)