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A real affine-Hecke parameter greater than one has infinite quantum characteristic

Statement

Let $q > 1$ be a real affine-Hecke parameter, viewed as an element of $C^{\times}$ , and let $ξ \neq 1$ be a root of unity in $C$ . With quantum characteristic as in The quantum characteristic of an invertible Hecke parameter, $e (C, q) = \infty$ and $e (C, ξ)$ is the finite multiplicative order of $ξ$ . In particular $q \neq ξ$ . The statement compares scalar parameter regimes; it asserts no isomorphism, Morita equivalence, or classification relation between the corresponding algebras.

Facts & Assumptions

Given: $q \in R$ , $q > 1$ , and a root of unity $ξ \in C^{\times}$ with $ξ \neq 1$ .

[F1]

The quantum characteristic is the least integer $e \geq 2$ for which $1 + u + \dots + u^{e - 1} = 0$ , and is $\infty$ if no such integer exists. If $u \neq 1$ , finite $e$ exists exactly when $u$ is a root of unity, and then it is its multiplicative order (The quantum characteristic of an invertible Hecke parameter).

Proof

Given: The scalars $q$ and $ξ$ above.

1.1F1givenalgebra

For every integer $n \geq 2$ , $1 + q + \dots + q^{n - 1}$ is a sum of positive real numbers and is therefore positive. No finite $n$ satisfies the defining vanishing condition in [F1], so $e (C, q) = \infty$ .

2.1F1givenstep 1.1algebra∎

Since $ξ \neq 1$ , the identity $(1 - ξ) (1 + ξ + \dots + ξ^{n - 1}) = 1 - ξ^{n}$ shows that the geometric sum vanishes exactly when $ξ^{n} = 1$ . Its least such positive exponent is the multiplicative order, which is at least $2$ ; hence [F1] gives $e (C, ξ) = ord (ξ) < \infty$ . Also $∣ ξ ∣ = 1$ while $∣ q ∣ = q > 1$ , so $q \neq ξ$ .

Depends on

The quantum characteristic of an invertible Hecke parameter

Used by

Dependency tree · two levels

4 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)