Affine Hecke algebra

In mathematics, an affine Hecke algebra is the Hecke algebra of an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials.

Let ${\displaystyle V}$ be a Euclidean space of a finite dimension and ${\displaystyle \Sigma }$ an affine root system on ${\displaystyle V}$. An affine Hecke algebra is a certain associative algebra that deforms the group algebra ${\displaystyle \mathbb {C} [W]}$ of the Weyl group ${\displaystyle W}$ of ${\displaystyle \Sigma }$ (the affine Weyl group). It is usually denoted by ${\displaystyle H(\Sigma ,q)}$, where ${\displaystyle q:\Sigma \rightarrow \mathbb {C} }$ is multiplicity function that plays the role of deformation parameter. For ${\displaystyle q\equiv 1}$ the affine Hecke algebra ${\displaystyle H(\Sigma ,q)}$ indeed reduces to ${\displaystyle \mathbb {C} [W]}$.

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