The formal Bernstein presentation for a reduced based root datum

Let $R=(X,\mathrm{\Phi },Y,{\mathrm{\Phi }}^{\vee },\mathrm{\Delta })$ be as in Abstract root data and their Weyl groups, with finite Weyl group $W_0$. Use the finite simple-reflection set $S_0=\{s_{\alpha }:\alpha \in \mathrm{\Delta }\}$ specified in The Iwahori–Matsumoto affine Hecke algebra of a based root datum, where $S_0$ generates $W_0$ and is contained in $S_{\mathrm{a}\mathrm{f}\mathrm{f}}$. For $s,t\in S_0$, let $m(s,t)$ be the order of $st$ in $W_0$, which is finite because $W_0$ is finite. The group algebra $\mathbb{C}[Y]$ is the complex vector space with basis $\{{\theta }_y:y\in Y\}$ and multiplication ${\theta }_y{\theta }_z={\theta }_{y+z}$; its identity is ${\theta }_0$. The simple reflection corresponding to $\alpha \in \mathrm{\Delta }$ acts on $Y$ by $s_{\alpha }(y)=y-⟨\alpha ,y⟩{\alpha }^{\vee }$.

Write $2X=\{2x:x\in X\}$. For each simple root $\alpha \in \mathrm{\Delta }\cap 2X$, also supply a positive scalar $v_{\alpha }^{\mathrm{\prime }}$; it is the second parameter in the doubled-root rank-one Bernstein relation. This additional scalar is formal input to the presentation. To compare this presentation with a specified ${\mathcal{H}}_{\mathrm{I}\mathrm{M}}(R,q)$, its parameters must be attached: require $v_{\alpha }^{\mathrm{\prime }}=v(H_{\alpha ,1})$, where $v(H)$ is the positive square root of the parameter of its wall reflection, well defined by Canonical affine geometry and wall parameters of a root datum. The finite parameter is already $v_{\alpha }=v(H_{\alpha ,0})$. Without this tie one is defining a formal Bernstein algebra, not asserting comparison with the fixed $q$. When $\alpha \mathrm{\notin }2X$, no second parameter is used; the wall-parameter lemma proves the two wall classes have equal parameters.

Define the formal Bernstein algebra $\mathcal{B}(R,v,v^{\mathrm{\prime }})$ as the quotient of the free associative $\mathbb{C}$-algebra on $\{{\theta }_y:y\in Y\}$ and $\{N_s:s\in S_0\}$ by ${\theta }_0=1$, ${\theta }_y{\theta }_z={\theta }_{y+z}$ for $y,z\in Y$, the finite Hecke relations, and the following cross-relation for each $\alpha \in \mathrm{\Delta }$ and $y\in Y$. Here $v_{\alpha }:=v_{s_{\alpha }}=\sqrt{q_{s_{\alpha }}}>0$ from The Iwahori–Matsumoto affine Hecke algebra of a based root datum, and the finite Hecke relations are $(N_s-v_s)(N_s+v_s^{-1})=0$ for $s\in S_0$ and equality of the alternating products of $N_s,N_t$ of length $m(s,t)$ for distinct $s,t\in S_0$ (Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy):

$${\theta }_y{N}_{s_{\alpha }}-N_{s_{\alpha }}{\theta }_{s_{\alpha }(y)}=\{\begin{array}{ll}(v_{\alpha }-v_{\alpha }^{-1}){\displaystyle \frac{{\theta }_y-{\theta }_{s_{\alpha }(y)}}{1-{\theta }_{-{\alpha }^{\vee }}}},& \alpha \mathrm{\notin }2X,\\ ((v_{\alpha }-v_{\alpha }^{-1})+{\theta }_{-{\alpha }^{\vee }}(v_{\alpha }^{\mathrm{\prime }}-(v_{\alpha }^{\mathrm{\prime }}{)}^{-1})){\displaystyle \frac{{\theta }_y-{\theta }_{s_{\alpha }(y)}}{1-{\theta }_{-2{\alpha }^{\vee }}}},& \alpha \in 2X\mathrm{.}\end{array}$$

The quotients are interpreted in $\mathbb{C}[Y]$; their well-definedness is proved in The Bernstein cross-relation quotients are Laurent polynomials ↗. The quotient construction defines an associative algebra. It does not assert that $\mathcal{B}(R,v,v^{\mathrm{\prime }})$ is nonzero, has a PBW basis, or is isomorphic to the Iwahori–Matsumoto algebra of The Iwahori–Matsumoto affine Hecke algebra of a based root datum.