Pre-canonical bases on affine Hecke algebras

TL;DR

This paper introduces pre-canonical bases in affine Hecke algebras, which interpolate between standard and canonical bases, revealing simple expansion patterns and conjectured positivity in type A.

Contribution

It defines a new family of bases called pre-canonical bases in affine Hecke algebras, bridging standard and canonical bases with simple expansion properties.

Findings

Expansion of bases is often simple

Conjecture of positivity in type A

Provides new tools for affine Hecke algebra analysis

Abstract

For any affine Weyl group, we introduce the pre-canonical bases. They are a set of bases $\{\mathbf{N}^i\}_{1\leq i \leq m+1} $ (where $m$ is the height of the highest root) of the spherical Hecke algebra that interpolates between the standard basis $\mathbf{N}^1$ and the canonical basis $\mathbf{N}^{m+1}$. The expansion of $\mathbf{N}^{i+1}$ in terms of the $\mathbf{N}^i$ is in many cases very simple and we conjecture that in type $A$ it is positive.