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The Bernstein cross-relation quotients are Laurent polynomials

Statement

Let $R = (X, Φ, Y, Φ^{\lor}, Δ)$ be the reduced based root datum and let $C [Y]$ and the reflection action be as in The formal Bernstein presentation for a reduced based root datum. For every $α \in Δ$ and $y \in Y$ ,

\frac{θ_{y} - θ_{s_{α} (y)}}{1 - θ_{- α^{\lor}}} \in C [Y] if α \notin 2 X,

and

\frac{θ_{y} - θ_{s_{α} (y)}}{1 - θ_{- 2 α^{\lor}}} \in C [Y] if α \in 2 X .

Facts & Assumptions

Given: A reduced based root datum $R = (X, Φ, Y, Φ^{\lor}, Δ)$ , a simple root $α \in Δ$ , and an element $y \in Y$ .

[F1]

The root datum has an integral perfect pairing, and its simple reflection acts by $s_{α} (y) = y - ⟨ α, y ⟩ α^{\lor}$ with $⟨ α, y ⟩ \in Z$ (Abstract root data and their Weyl groups).

[F2]

The group algebra $C [Y]$ has basis ${θ_{z} : z \in Y}$ and multiplication $θ_{z} θ_{z^{'}} = θ_{z + z^{'}}$ (The formal Bernstein presentation for a reduced based root datum).

Proof

Given: The data above. Put $n = ⟨ α, y ⟩ \in Z$ and $u = θ_{- α^{\lor}}$ .

1.1F1F2givenalgebra

By [F1], $s_{α} (y) = y - n α^{\lor}$ , so by [F2] the numerator is $θ_{y} (1 - u^{n})$ . If $n > 0$ , then $(1 - u^{n}) / (1 - u) = 1 + u + \dots + u^{n - 1}$ ; if $n = 0$ , the quotient is zero; and if $n = - k < 0$ , then $(1 - u^{n}) / (1 - u) = - u^{- k} (1 + u + \dots + u^{k - 1})$ . Each expression is a finite Laurent polynomial, proving the first assertion.

2.1F1F2step 1.1algebra∎

If $α \in 2 X$ , write $α = 2 β$ with $β \in X$ . Then $n = ⟨ 2 β, y ⟩ = 2 ⟨ β, y ⟩ = 2 m$ for an integer $m$ . With $u = θ_{- α^{\lor}}$ , the numerator is $θ_{y} (1 - u^{2 m})$ . For $m > 0$ , $(1 - u^{2 m}) / (1 - u^{2}) = 1 + u^{2} + \dots + u^{2 m - 2}$ ; for $m = 0$ it is zero; and for $m < 0$ the same finite negative-power identity from step 1.1, applied to $u^{2}$ , is a Laurent polynomial. Since $1 - u^{2} = 1 - θ_{- 2 α^{\lor}}$ , the second assertion follows.

Depends on

Used by

General Bernstein PBW and the Iwahori–Matsumoto comparison Theorem

Cited to discharge well-definedness by The formal Bernstein presentation for a reduced based root datum.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Maarten Solleveld, Affine Hecke Algebras and Their Representations (standard reference, not scraped)
Dan Ciubotaru, Tutorial on Affine Hecke Algebras (standard reference, not scraped)