The Iwahori–Matsumoto affine Hecke algebra of a based root datum

Let $R=(X,\mathrm{\Phi },Y,{\mathrm{\Phi }}^{\vee },\mathrm{\Delta })$ be a reduced based root datum: $X,Y$ are finite-rank free lattices in perfect integral duality, $\mathrm{\Phi }\subset X$ is the finite root set, ${\mathrm{\Phi }}^{\vee }\subset Y$ its coroot set, $\mathrm{\Delta }$ a base, and $W_0$ the finite Weyl group (Abstract root data and their Weyl groups). Its finite simple-reflection set is $S_0:=\{s_{\alpha }:\alpha \in \mathrm{\Delta }\}\subset W_0$; the simple reflections in a chosen base generate $W_0$ (Combinatorics of a reduced root datum). Write $Q^{\vee }=\mathbb{Z}{\mathrm{\Phi }}^{\vee }\subseteq Y$ for the coroot lattice.

Put $V=Y{\otimes }_{\mathbb{Z}}\mathbb{R}$. The canonical affine walls are $H_{\alpha ,j}=\{v:⟨\alpha ,v⟩=j\}$, with reflection $r_{\alpha ,j}(v)=v-(⟨\alpha ,v⟩-j){\alpha }^{\vee }$. The base alcove $A$ is the unique alcove in the positive chamber with $0$ in its closure; equivalently $A$ is defined by $0<⟨\alpha ,v⟩<1$ for every positive root $\alpha$. If there are no roots, put $A=V$. For a nonsemisimple root datum the alcoves include the whole common root kernel as a factor. Define $W_{\mathrm{a}\mathrm{f}\mathrm{f}}=Q^{\vee }⋊W_0$, let $S_{\mathrm{a}\mathrm{f}\mathrm{f}}$ be its reflections in the walls of $A$, put $W_{\mathrm{e}}=Y⋊W_0$, and let $\mathrm{\Omega }={\mathrm{Stab}}_{W_{\mathrm{e}}}(A)$. The canonical affine Coxeter system and the decomposition $W_{\mathrm{e}}=W_{\mathrm{a}\mathrm{f}\mathrm{f}}⋊\mathrm{\Omega }$, including central lattice directions, are proved in Canonical affine geometry and wall parameters of a root datum ↗. In particular $S_0\subset S_{\mathrm{a}\mathrm{f}\mathrm{f}}$ with the finite Weyl product orders. Define length on $W_{\mathrm{e}}$ as the number of walls separating $A$ from its translate; on $W_{\mathrm{a}\mathrm{f}\mathrm{f}}$ it is Coxeter length, and its zero set is $\mathrm{\Omega }$. Thus no arbitrary nongeometric affine Coxeter extension is silently compared with the lattice below.

Choose a parameter function $q:S_{\mathrm{a}\mathrm{f}\mathrm{f}}\to {\mathbb{R}}_{>0}$ such that $q_s=q_t$ whenever $s,t\in S_{\mathrm{a}\mathrm{f}\mathrm{f}}$ are conjugate as elements of $W_{\mathrm{e}}$. This condition compares only elements in the domain of $q$; conjugation by all of $W_{\mathrm{e}}$ need not preserve $S_{\mathrm{a}\mathrm{f}\mathrm{f}}$. In particular, every $\omega \in \mathrm{\Omega }$ preserves $S_{\mathrm{a}\mathrm{f}\mathrm{f}}$ and satisfies $q_{\omega s{\omega }^{-1}}=q_s$, so its action preserves the quadratic relations. For $s,t\in S_{\mathrm{a}\mathrm{f}\mathrm{f}}$ with finite Coxeter label $m(s,t)$, impose the alternating braid relation with $m(s,t)$ factors. Define ${\mathcal{H}}_{\mathrm{a}\mathrm{f}\mathrm{f}}(R,q)$ to be the quotient of the free associative $\mathbb{C}$-algebra on $(T_s{)}_{s\in S_{\mathrm{a}\mathrm{f}\mathrm{f}}}$ by those braid relations and

$$(T_s-q_s)(T_s+1)=0(s\in S_{\mathrm{a}\mathrm{f}\mathrm{f}})\mathrm{.}$$

The extended Iwahori–Matsumoto affine Hecke algebra is the crossed product ${\mathcal{H}}_{\mathrm{I}\mathrm{M}}(R,q):={\mathcal{H}}_{\mathrm{a}\mathrm{f}\mathrm{f}}(R,q)⋊\mathrm{\Omega }$. Its underlying vector space is ${\mathcal{H}}_{\mathrm{a}\mathrm{f}\mathrm{f}}(R,q){\otimes }_{\mathbb{C}}\mathbb{C}[\mathrm{\Omega }]$, with multiplication $(h{U}_{\omega })(h^{\mathrm{\prime }}{U}_{\eta })=h\omega (h^{\mathrm{\prime }})U_{\omega \eta }$. Thus it adjoins symbols $U_{\omega }$ with $U_{\omega }{U}_{\eta }=U_{\omega \eta }$ and $U_{\omega }{T}_s{U}_{\omega }^{-1}=T_{\omega s{\omega }^{-1}}$. The stated parameter condition makes each $\mathrm{\Omega }$-conjugation an algebra automorphism. If $v_s=\sqrt{q_s}>0$ and $N_s=q_s^{-1\mathrm{/}2}{T}_s$, then $(N_s-v_s)(N_s+v_s^{-1})=0$, the symmetric normalization in Universal parameters, the generic Coxeter Hecke algebra and generator conjugacy.

This definition asserts neither an Iwahori–Matsumoto basis nor an equivalence with a Bernstein presentation. It does not identify an arbitrary reductive $p$-adic group's Iwahori-fixed category with modules over this algebra; that requires a separate group-specific equivalence.