Affine Hecke Algebras Beyond Type A: Examples and Counterexamples

Since $3>1$, [F1] gives $e(\mathbb{C},3)=\mathrm{\infty }$.

The first candidate geometric sums for $i$ are $1+i\mathrm{\ne }0$, $1+i+i^2=i\mathrm{\ne }0$, and $1+i+i^2+i^3=0$. By the least-exponent definition [F2], $e(\mathbb{C},i)=4$.

Thus the p-adic positive-real parameter has infinite quantum characteristic and the finite type-A root-of-unity parameter has finite quantum characteristic $4$. Since $\mathrm{\mid }3\mathrm{\mid }=3$ and $\mathrm{\mid }i\mathrm{\mid }=1$, they are different scalars. These calculations establish only the stated parameter distinction.

The vectors $e_1$ and $\frac{1}{2}(e_1+e_2)$ form a basis of $X$, while $e_1-e_2$ and $2e_2$ form a basis of $Y$: an integer pair $(a,b)$ has even coordinate sum exactly when $(a,b)=a(e_1-e_2)+\frac{a+b}{2}(2e_2)$. The dot-product matrix between these two bases is the identity, so the pairing is perfect and integral. The roots lie in $X$, the coroots lie in $Y$, and direct pairings on ${\alpha }_1,{\alpha }_2$ give the Cartan matrix $\left(\begin{array}{cc}2& -2\\ -1& 2\end{array}\right)$. The positive roots are ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2=e_1,{\alpha }_1+2{\alpha }_2=e_1+e_2$, so $\mathrm{\Delta }$ is a base and the root set is reduced. The displayed root set is the union of the signed-permutation orbits of ${\alpha }_1$ and ${\alpha }_2$. The prescribed coroot map is equivariant under signed permutations: it doubles each short root and fixes each long-root vector, and signed permutations preserve root length. Thus the simple-root checks give $⟨\alpha ,{\alpha }^{\vee }⟩=2$ for every listed root, and the generated reflections preserve both root sets. The signed-permutation group generated by $s_1,s_2$ is finite. Therefore [F1] applies and these data are a reduced based $B_2$ root datum.

The simple walls of ${\mathcal{A}}_0$ are $x_1=x_2$, $x_2=0$, and $x_1+x_2=1$. In its interior, $0<x_2<1$, $0<x_1-x_2<1$, and $0<x_1+x_2<1$; the remaining positive-root function satisfies $0<x_1<1$ because $x_2>0$ and $x_1+x_2<1$. Thus no other integer root wall cuts it. Their reflections are respectively the displayed $s_1,s_2,s_0$. The product $s_1{s}_2$ is $(x_1,x_2)\mapsto (-x_2,x_1)$, a quarter-turn of order $4$. The product $s_2{s}_0$ is $(x_1,x_2)\mapsto (1-x_2,x_1-1)$; it fixes $(1,0)$ and its linear part about that point is the same quarter-turn, so it has order $4$. The product $s_0{s}_1$ is $(x_1,x_2)\mapsto (1-x_1,1-x_2)$, a half-turn of order $2$. Hence the affine Coxeter labels are $m(s_1,s_2)=m(s_2,s_0)=4$ and $m(s_1,s_0)=2$.

The affine roots defining $s_0$ and $s_1$ are long and the one defining $s_2$ is short. The displayed basis of $Y$ is ${\alpha }_1^{\vee },{\alpha }_2^{\vee }$, so $Y=Q^{\vee }$ and $\mathrm{\Omega }=1$ by [F2]. Thus conjugacy in the extended group is conjugacy in the affine Coxeter group. The Coxeter labels just computed are all even, so the odd-edge graph has three singleton components; by [F4] the three assigned positive parameters are constant on every required conjugacy class. In the symmetric convention the parameters are $v_{s_0}=v_{s_1}=\sqrt{2}$ and $v_{s_2}=\sqrt{3}$, with the quadratic relations of [F2].

Here $Y=Q^{\vee }$, so the extended group has trivial length-zero quotient and $\mathrm{\Omega }=\{1\}$. Applying [L1] to the affine Coxeter system identified in step 1.2, with the valid affine Coxeter parameters from step 2.1, shows that $\{T_w:w\in W_{\mathrm{a}\mathrm{f}\mathrm{f}}\}$ is a $\mathbb{C}$-basis. This verifies the claimed unequal-parameter affine $B_2$ example.