The rank-one root datum and parameter conventions for DAHA

Work over the rational function field $K=\mathbb{Q}(q,t)$, where $q$ and $t$ are algebraically independent indeterminates. The rank-one reduced root system is $\mathrm{\Phi }=\{\pm \alpha \}$ with $\alpha =2\rho$, weight lattice $P=\mathbb{Z}\rho$, coroot lattice $Q^{\vee }=\mathbb{Z}{\alpha }^{\vee }$, and pairing $⟨\rho ,{\alpha }^{\vee }⟩=1$. Put $z=X^{\rho }$, so the lattice algebra is $K[z,z^{-1}]$; the finite Weyl reflection is $s_1(z)=z^{-1}$.

Let $\delta$ be the affine degree with $z^{\delta }=q^{-2}$, and set ${\alpha }_0=-2\rho +\delta$. The affine reflections and the length-zero symmetry act on Laurent functions by

Thus $s_0^2=s_1^2={\pi }^2=1$, $\pi s_0\pi =s_1$, and the affine denominators are $u_0=z^{-{\alpha }_0}=q^2{z}^2$ and $u_1=z^{-{\alpha }_1}=z^{-2}$. The affine Coxeter group generated by $s_0,s_1$ is infinite dihedral, with no finite braid relation between these reflections. The extended affine group is $W_{\mathrm{a}\mathrm{f}\mathrm{f}}⋊\mathrm{\Omega }$, where $\mathrm{\Omega }=\{1,\pi \}$ acts by swapping $s_0,s_1$; every element has a unique form $w{\pi }^{ϵ}$ with $w$ an alternating reduced word in $s_0,s_1$ and $ϵ\in \{0,1\}$.

For $m\in \mathbb{Z}$, define the Laurent polynomials $B_{1,m},B_{0,m}$ by

They are the finite Laurent-polynomial versions of the Bernstein quotients with $s_1(z^m)=z^{-m}$ and $s_0(z^m)=q^{-2m}{z}^{-m}$. The symmetric Hecke normalization uses the same nonzero parameter $t$ at both affine simple roots: $(T_i-t)(T_i+t^{-1})=0$. The DAHA parameter $q$ is the lattice-translation parameter; it is distinct from the finite Hecke parameter $t$ and from the multiplicative type-A normalization $Q$.

This item selects type $A_1$ for the original design's “selected root system” scope. No specialization of $q$ or $t$ is part of this definition.