Last update: 22 January 2014
A reflection is an element of $GL\left(V\right)$ that has exactly one eigenvalue not equal to $1\text{.}$ The reflecting hyperplane of a reflection is the $(n-1)\text{-dimensional}$ subspace which is fixed pointwise. A complex reflection group $G$ is a finite subgroup of $GL\left(V\right)$ generated by reflections. The group $G$ is irreducible if $V$ cannot be written in the form $V={V}_{1}\oplus {V}_{2}$ where ${V}_{1}$ and ${V}_{2}$ are $G\text{-invariant}$ subspaces. The group $G$ is a real reflection group if $V=\u2102{\otimes}_{\mathbb{R}}{V}_{\mathbb{R}}$ for a real vector space ${V}_{\mathbb{R}}$ and $G\subseteq GL\left({V}_{\mathbb{R}}\right)\text{.}$
The following facts about reflection groups are well known.
Let $G$ be an irreducible reflection group.
(a) | [STo1954, Theorem 5.3] The number of elements $g\in G$ such that $\text{codim}\left({V}^{g}\right)=2$ is ${\sum}_{i<j}{m}_{i}{m}_{j}$ where ${m}_{1},\dots ,{m}_{n}$ are the exponents of $G\text{.}$ |
(b) | [Car1972, Lemma 2] If $G$ is a real reflection group and $g\in G$ with $\text{codim}\left({V}^{g}\right)=2,$ then $g$ is the product of two reflections. |
(c) | [OTe1992, Theorem 6.27] For any $g\in G,$ the space ${V}^{g}$ is the intersection of reflecting hyperplanes. |
Remark. The statement of Lemma 2.1b does not hold for complex reflection groups. Consider the exceptional complex reflection group ${G}_{4}$ of rank $2,$ in the notation of Shephard and Todd [STo1954]. All the reflections have order $3$ and $-1\in {G}_{4}\text{.}$ Suppose $-1=rs$ for two reflections $r$ and $s\text{.}$ If $s$ has eigenvalues $1$ and $\omega ,$ where $\omega $ is a primitive cube root of unity, then ${r}^{-1}=-s$ has eigenvalues $-1$ and $-\omega ,$ a contradiction to the assumption that $r$ is a reflection. Thus $-1\in {G}_{4}$ is not a product of two reflections.
Let $G\subseteq GL\left(V\right)$ be a complex reflection group. Let $A$ be a graded Hecke algebra for $G$ and let $g\in G\text{.}$ Let ${V}^{G}=\{v\in V\hspace{0.17em}|\hspace{0.17em}gv=v\hspace{0.17em}\text{for all}\hspace{0.17em}g\in G\}$ be the invariants in $V\text{.}$
(a) | If $g=1$ and $\text{dim}\hspace{0.17em}{V}^{G}\le 1,$ then ${a}_{g}=0\text{.}$ |
(b) | If the order of $g$ is $2,$ then ${a}_{g}=0\text{.}$ |
Proof. | |
(a) Let $\u27e8,\u27e9:V\times V\to \u2102$ be a nondegenerate $G\text{-invariant}$ Hermitian form on $V$ and write $V={V}^{G}\oplus {\left({V}^{G}\right)}^{\perp}$ where ${\left({V}^{G}\right)}^{\perp}=\{v\in V\hspace{0.17em}|\hspace{0.17em}\u27e8v,w\u27e9=0\hspace{0.17em}\text{for all}\hspace{0.17em}w\in {V}^{G}\}\text{.}$ Since $\text{dim}\left({V}^{G}\right)\le 1$ and ${a}_{1}$ is skew symmetric, ${a}_{1}$ restricted to ${V}^{G}$ is $0\text{.}$ There is a basis ${\alpha}_{1},\dots ,{\alpha}_{k}$ of ${\left({V}^{G}\right)}^{\perp}$ and constants ${\xi}_{1},\dots ,{\xi}_{k}\in \u2102,$ ${\xi}_{i}\ne 1,$ such that the reflections ${s}_{1},\dots ,{s}_{k}$ given by $${s}_{i}v=v+({\xi}_{i}-1)\frac{\u27e8v,{\alpha}_{i}\u27e9}{\u27e8{\alpha}_{i},{\alpha}_{i}\u27e9}{\alpha}_{i},\phantom{\rule{2em}{0ex}}\text{for}\hspace{0.17em}v\in V,$$ are in $G\text{.}$ Equation (1.6) implies that, for any $v\in V,$ $${a}_{1}({\alpha}_{i},v)={a}_{1}({s}_{i}{\alpha}_{i},{s}_{i}v)={a}_{1}({\xi}_{i}{\alpha}_{i},v+({\xi}_{i}-1)\frac{\u27e8v,{\alpha}_{i}\u27e9}{\u27e8{\alpha}_{i},{\alpha}_{i}\u27e9}{\alpha}_{i})={\xi}_{i}{a}_{1}({\alpha}_{i},v),$$ since ${a}_{1}({\alpha}_{i},{\alpha}_{i})=0$ (as ${a}_{1}$ is skew symmetric). Since ${\xi}_{i}\ne 1,$ ${a}_{1}({\alpha}_{i},v)=0$ for $1\le i\le k\text{.}$ Thus $\text{ker}\hspace{0.17em}{a}_{1}=V\text{.}$ (b) Since ${g}^{2}=1,$ all eigenvalues of $g$ are $\pm 1\text{.}$ If $\text{codim}\left({V}^{g}\right)\ne 2,$ then ${a}_{g}=0$ by Theorem 1.9b. If $\text{codim}\left({V}^{g}\right)=2,$ then $$g-{\text{id}}_{{V}^{g}}\oplus (-{\text{id}}_{{\left({V}^{g}\right)}^{\perp}})$$ as a linear transformation on $V\text{.}$ By [Ste1964, Theorem 1.5], [Bou1981 V, §5 Ex. 8], the stabilizer, $\text{Stab}\left({V}^{g}\right),$ of ${V}^{g}$ is a reflection subgroup of $G$ and so there is a reflection $s\in \text{Stab}\left({V}^{g}\right)$ that is the identity on ${V}^{g}\text{.}$ So $s\in {Z}_{G}\left(g\right)$ and $\text{det}\left(s\right)=\text{det}\left({s}^{\perp}\right)\ne 1,$ where ${s}^{\perp}$ is $s$ restricted to ${\left({V}^{g}\right)}^{\perp}\text{.}$ Thus, by Theorem 1.9b, ${a}_{g}=0\text{.}$ $\square $ |
2A. Real reflection groups
If $G\subseteq GL\left(V\right)$ is a real reflection group then $V=\u2102{\otimes}_{\mathbb{R}}{V}_{\mathbb{R}}$ and $G\subseteq GL\left({V}_{\mathbb{R}}\right),$ where ${V}_{\mathbb{R}}$ is a real vector space. We shall assume that $G$ is irreducible.
Let us recall some basic facts about real reflection groups which can be found in [Hum1990] or [Bou1981]. The action of $G$ on ${V}_{\mathbb{R}}$ has fundamental chambers $wC$ indexed by $w\in G\text{.}$ The roots for $G$ are vectors $\alpha \in {V}_{\mathbb{R}}$ such that the reflections in $G$ are the reflections ${s}_{\alpha}$ in the hyperplanes $${H}_{\alpha}=\{v\in {V}_{\mathbb{R}}\hspace{0.17em}|\hspace{0.17em}\u27e8v,\alpha \u27e9=0\}\text{.}$$ For each fundamental chamber $C,$ the reflections ${s}_{1},{s}_{2},\dots ,{s}_{n}$ in the hyperplanes ${H}_{{\alpha}_{1}},{H}_{{\alpha}_{2}},\dots ,{H}_{{\alpha}_{n}}$ that bound $C$ form a set of simple reflections for $G\text{.}$ The simple reflections obtained from a different choice of fundamental chamber $wC$ are $w{s}_{1}{w}^{-1},\dots ,w{s}_{n}{w}^{-1}\text{.}$
Let $G\subseteq GL\left({V}_{\mathbb{R}}\right)$ be a real reflection group. Let ${s}_{1},\dots ,{s}_{n}$ be a set of simple reflections in $G$ and let ${m}_{ij}$ be the order of ${s}_{i}{s}_{j}\text{.}$ Then $g\in G$ satisfies ${g}^{2}\ne 1,$ $\text{codim}\left({V}^{g}\right)=2,$ and $\text{det}\left({h}^{\perp}\right)=1$ for all $h\in {Z}_{G}\left(g\right)$ (the conditions in Theorem 1.9c) if and only if $g$ is conjugate to $${\left({s}_{i}{s}_{j}\right)}^{k},\phantom{\rule{2em}{0ex}}\text{with}\hspace{0.17em}0<k<\frac{{m}_{ij}}{2},$$ for some $1\le i,j\le n\text{.}$
Proof. | |
$\u27f9\text{:}$ Let $\alpha $ and $\beta $ be two roots such that ${V}^{g}={H}_{\alpha}\cap {H}_{\beta}$ (see Lemma 2.1c). Then ${H}_{\alpha}\cap {H}_{\beta}$ has nontrivial intersection with some fundamental chamber $C$ for $W,$ and we may assume that ${H}_{\alpha}$ and ${H}_{\beta}$ are walls of the chamber $C$ (since $C$ is a cone in ${\mathbb{R}}^{n}\text{).}$ Since choosing simple reflections with respect to a different chamber $wC$ corresponds to conjugation by $w,$ we may assume that the reflections in the hyperplanes ${H}_{\alpha}$ and ${H}_{\beta}$ are simple reflections and $\alpha ={\alpha}_{1}$ and $\beta ={\alpha}_{2}\text{.}$ The element $g$ is an element of the stabilizer $\text{Stab}\left({V}^{g}\right),$ which is a reflection group by [Ste1964, Theorem 1.5]. Since $\text{codim}\left({V}^{g}\right)=2,$ $\text{Stab}\left({V}^{g}\right)$ is a rank two real reflection group, and therefore a dihedral group. This dihedral group is generated by the two simple reflections ${s}_{1}$ and ${s}_{2}$ in the hyperplanes ${H}_{{\alpha}_{1}}$ and ${H}_{{\alpha}_{2}}$ (restricted to ${\left({V}^{g}\right)}^{\perp}\text{)}$ and all reflections have determinant $-1\text{.}$ Let ${g}^{\perp}$ be the element $g$ restricted to ${\left({V}^{g}\right)}^{\perp}\text{.}$ Since $g\in {Z}_{G}\left(g\right),$ $\text{det}\left({g}^{\perp}\right)=1,$ and so $g$ must be a product of an even number of reflections. Thus $g={\left({s}_{1}{s}_{2}\right)}^{k}$ or $g={\left({s}_{2}{s}_{1}\right)}^{k},$ for some $0<k\le m/2,$ where $m$ is the order of ${s}_{1}{s}_{2}\text{.}$ Since ${g}^{2}\ne 1,$ $k\ne m/2,$ and so $g$ is conjugate to ${\left({s}_{1}{s}_{2}\right)}^{k}$ for some $0<k<m/2\text{.}$ $\u27f8\text{:}$ Assume that $g={\left({s}_{i}{s}_{j}\right)}^{k}$ for some $0<k<{m}_{ij}/2\text{.}$ Then ${V}^{g}={H}_{{\alpha}_{i}}\cap {H}_{{\alpha}_{j}}$ and so $\text{codim}\left({V}^{g}\right)=2\text{.}$ Since $g$ is a product of an even number of reflections, $\text{det}\left({g}^{\perp}\right)=1\text{.}$ The only elements of $O\left({V}_{\mathbb{R}}\right)\cong {O}_{2}\left(\mathbb{R}\right)$ that are diagonalizable in $GL\left({V}_{\mathbb{R}}\right)\cong {GL}_{2}\left(\mathbb{R}\right)$ are $\pm 1$ and elements with determinant $-1\text{.}$ Thus, the eigenvectors of the element ${g}^{\perp}$ (which has distinct eigenvalues since it is not $\pm 1\text{)}$ do not lie in ${V}_{\mathbb{R}},$ only in $V=\u2102{\otimes}_{\mathbb{R}}\mathbb{R}\text{.}$ Let $h\in {Z}_{G}\left(g\right)$ and let ${h}^{\perp}\in O\left({V}_{\mathbb{R}}\right)\cong {O}_{2}\left(\mathbb{R}\right)$ denote $h$ restricted to ${\left({V}^{g}\right)}^{\perp}\text{.}$ Since ${h}^{\perp}$ commutes with ${g}^{\perp}$ and ${g}^{\perp}$ has distinct eigenvalues, ${g}^{\perp}$ and ${h}^{\perp}$ have the same eigenvectors. Hence, $\text{det}\hspace{0.17em}{h}^{\perp}=1\text{.}$ $\square $ |
Using Theorem 2.3 and Theorem 1.9b, we can read off the graded Hecke algebras for the irreducible real reflection groups from the Dynkin diagrams. For each irreducible real reflection group, label a set of simple reflections ${s}_{1},\dots ,{s}_{n}$ using the Dynkin diagrams below. If nodes $i$ and $j$ and nodes $j$ and $k$ are connected by single edges, then ${s}_{i}{s}_{j}$ is conjugate to ${s}_{j}{s}_{k}$ via the element ${s}_{i}{s}_{j}{s}_{k}\text{.}$
The following table gives representatives of the conjugacy classes of $g\in G$ that may have ${a}_{g}\ne 0$ for some graded Hecke algebra $A\text{.}$ We assume that the reflection group $G$ is acting on its irreducible reflection representation $V\text{.}$ When $G$ is the symmetric group ${S}_{n}$ acting on an $n\text{-dimensional}$ vector space $V$ by permutation matrices, then $\text{dim}\left({V}^{G}\right)=1$ and, by Lemma 2.2a and Theorem 2.3, ${a}_{g}\ne 0$ for some graded Hecke algebra $A$ only if $g$ is conjugate to the three cycle $(1,2,3)={s}_{1}{s}_{2}$ (this example is analyzed in Section 3). $$\begin{array}{c}\begin{array}{|cc|}\hline \text{Type}& \genfrac{}{}{0ex}{}{\text{Representative}\hspace{0.17em}g}{\text{with}\hspace{0.17em}{a}_{g}\ne 0}\\ {A}_{n-1}& {s}_{1}{s}_{2}\\ {B}_{n}& {s}_{1}{s}_{2},\hspace{0.17em}{s}_{2}{s}_{3}\\ {D}_{n}& {s}_{2}{s}_{3}\\ {E}_{6},{E}_{7},{E}_{8}& {s}_{1}{s}_{4}\\ {F}_{4}& {s}_{1}{s}_{2},\hspace{0.17em}{s}_{2}{s}_{3},\hspace{0.17em}{s}_{3}{s}_{4}\\ {H}_{3},{H}_{4}& {s}_{1}{s}_{2},\hspace{0.17em}{\left({s}_{1}{s}_{2}\right)}^{2},\hspace{0.17em}{s}_{2}{s}_{3}\\ {I}_{2}\left(m\right)& {\left({s}_{1}{s}_{2}\right)}^{k},\hspace{0.17em}0<k<m/2\\ \hline\end{array}\\ \text{Table 1.}\hspace{0.17em}\text{Graded Hecke algebras for real reflection groups.}\end{array}$$ $$\begin{array}{c}\begin{array}{cc}{A}_{n-1}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n1\n2\nn-2\nn-1\n\n\end{array}& {B}_{n}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n1\n2\n3\nn-1\nn\n\n\end{array}\\ {D}_{n}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n1\n2\n3\n4\nn-1\nn\n\n\end{array}& {E}_{6}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n2\n3\n1\n4\n5\n6\n\n\end{array}\\ {E}_{7}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n2\n3\n1\n4\n5\n6\n7\n\n\end{array}& {E}_{8}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n2\n3\n1\n4\n5\n6\n7\n8\n\n\end{array}\\ {F}_{4}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n1\n2\n3\n4\n\n\end{array}& {H}_{3}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n1\n2\n3\n5\n\n\end{array}\\ {H}_{4}\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n1\n2\n3\n4\n5\n\n\end{array}& {I}_{2}\left(m\right)\begin{array}{c}\n\n\n\n\n\n\n\n\n\n1\n2\nm\n\n\end{array}\end{array}\\ \text{Figure 1.}\hspace{0.17em}\text{Coxeter-Dynkin diagrams for real reflection groups.}\end{array}$$
2B. Complex reflection groups
The irreducible complex reflection groups were classified by Shephard and Todd [STo1954]. There is one infinite family denoted $G(r,p,n)$ and a list of exceptional complex reflection groups denoted ${G}_{4},\dots ,{G}_{35}\text{.}$ In this subsection, we classify the graded Hecke algebras for the groups $G(r,p,n)\text{.}$
Let $r,$ $p$ and $n$ be positive integers with $p$ dividing $r$ and let $\xi ={e}^{2\pi i/r}\text{.}$ Let ${S}_{n}$ be the symmetric group of $n\times n$ matrices and let $${\xi}_{j}=\text{diag}(1,1,\dots ,1,\xi ,1,\dots ,1),$$ where $\xi $ appears in the $j\text{th}$ entry. Then $$G(r,p,n)=\left\{{\xi}_{1}^{{\lambda}_{1}}\cdots {\xi}_{n}^{{\lambda}_{n}}w\hspace{0.17em}\right|\hspace{0.17em}w\in {S}_{n},\hspace{0.17em}0\le {\lambda}_{i}\le r-1,\hspace{0.17em}{\lambda}_{1}+\cdots +{\lambda}_{n}=0\hspace{0.17em}\text{mod}\hspace{0.17em}p\}\text{.}$$ For $\lambda =({\lambda}_{1},\dots ,{\lambda}_{n})\in {(\mathbb{Z}/r\mathbb{Z})}^{n},$ let ${\xi}^{\lambda}={\xi}_{1}^{{\lambda}_{1}}\cdots {\xi}_{n}^{{\lambda}_{n}}\text{.}$ Then the multiplication in $G(r,p,n)$ is described by the relations $${\xi}^{\lambda}{\xi}^{\mu}={\xi}^{\lambda +\mu}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}w{\xi}^{\lambda}={\xi}^{w\lambda}w,\phantom{\rule{2em}{0ex}}\text{for}\hspace{0.17em}\lambda ,\mu \in {(\mathbb{Z}/r\mathbb{Z})}^{n},\hspace{0.17em}w\in {S}_{n},$$ where ${S}_{n}$ acts on ${(\mathbb{Z}/r\mathbb{Z})}^{n}$ by permuting the factors. Let ${v}_{i}$ be the column vector with $1$ in the ${i}^{\text{th}}$ entry and all other entries $0\text{.}$ The group $G(r,p,n)$ acts on $V\u2254{\u2102}^{n}$ with orthonormal basis $\{{v}_{1},\dots ,{v}_{n}\}$ as a complex reflection group.
Every real reflection group is a complex reflection group and several of these are special cases of the groups $G(r,p,n)\text{.}$ In particular,
(a) | $G(1,1,n)$ is the symmetric group ${S}_{n},$ |
(b) | $G(2,1,n)$ is the Weyl group $W{B}_{n}$ of type ${B}_{n},$ |
(c) | $G(2,2,n)$ is the Weyl group $W{D}_{n}$ of type ${D}_{n},$ and |
(d) | $G(r,r,2)$ is the dihedral group ${I}_{2}\left(r\right)$ of order $2r\text{.}$ |
The reflections in $G(r,p,n)$ are $$\begin{array}{cc}{\xi}_{i}^{kp},& 1\le i\le n,\hspace{0.17em}0\le k\le (r/p)-1,\hspace{0.17em}\text{and}\\ {\xi}_{i}^{k}{\xi}_{j}^{-k}(i,j),& 1\le i<j\le n,\hspace{0.17em}0\le k\le r-1,\end{array}$$ where $(i,j)$ is the transposition in ${S}_{n}$ that switches $i$ and $j\text{.}$
Conjugacy in $G(r,p,n)\text{.}$ Each element of $G(r,p,n)$ is conjugate by elements of ${S}_{n}$ to a disjoint product of cycles of the form $${\xi}_{i}^{{\lambda}_{i}}\cdots {\xi}_{k}^{{\lambda}_{k}}(i,i+1,\dots ,k)\text{.}$$ By conjugating this cycle by ${\xi}_{i}^{-c}{\xi}_{i+1}^{{\lambda}_{i}}{\xi}_{i+2}^{{\lambda}_{i}+{\lambda}_{i+1}}\cdots {\xi}_{k}^{{\lambda}_{i}+\cdots +{\lambda}_{k-1}}\in G(r,r,n),$ we have $${\xi}_{i}^{-c}{\xi}_{k}^{c+{\lambda}_{i}+\cdots +{\lambda}_{k}}(i,\dots ,k),\phantom{\rule{2em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}c=(k-i){\lambda}_{i}+(k-i-1){\lambda}_{i+1}+\cdots +{\lambda}_{k-1}\text{.}$$ If ${i}_{1},{i}_{2},\dots ,{i}_{\ell}$ denote the minimal indices of the cycles and ${c}_{1},\dots ,{c}_{\ell}$ are the numbers $c$ for the various cycles, then after conjugating by ${\xi}_{{i}_{1}}^{{c}_{1}}\cdots {\xi}_{{i}_{\ell -1}}^{{c}_{\ell -1}}{\xi}_{{i}_{\ell}}^{-({c}_{1}+\cdots +{c}_{\ell -1})}\in G(r,r,n),$ each cycle becomes $${\xi}_{k}^{{\lambda}_{i}+\cdots +{\lambda}_{k}}(i,\dots ,k)\phantom{\rule{2em}{0ex}}\text{except the last, which is}\phantom{\rule{2em}{0ex}}{\xi}_{{i}_{\ell}}^{-a}{\xi}_{n}^{b}({i}_{\ell},\dots ,n),$$ where $a={c}_{1}+\cdots +{c}_{\ell}$ and $b=a+{\lambda}_{{i}_{\ell}}+\cdots +{\lambda}_{n}\text{.}$ If $k=n-{i}_{\ell}+1$ is the length of the last cycle, then conjugating the last cycle by ${\xi}_{{i}_{\ell}}^{k-1}{\xi}_{{i}_{\ell}+1}^{-1}\cdots {\xi}_{n}^{-1}\in G(r,r,n)$ gives $${\xi}_{{i}_{\ell}}^{-a+k}{\xi}_{n}^{b-k}({i}_{\ell},\dots ,n)\text{.}$$ If we conjugate the last cycle by ${\xi}_{{i}_{\ell}}^{p}\in G(r,p,n),$ we have $${\xi}_{{i}_{\ell}}^{-a+p}{\xi}_{n}^{b-p}({i}_{\ell},\dots ,n)\text{.}$$ In summary, any element $g$ of $G(r,p,n)$ is conjugate to a product of disjoint cycles where each cycle is of the form $$\begin{array}{cc}{\xi}_{k}^{a}(i,i+1,\dots ,k),\phantom{\rule{2em}{0ex}}0\le a\le r-1,& \text{(2.4a)}\end{array}$$ except possibly the last cycle, which is of the form $$\begin{array}{cc}{\xi}_{{i}_{\ell}}^{a}{\xi}_{n}^{b}({i}_{\ell},{i}_{\ell}+1,\dots ,n),\phantom{\rule{2em}{0ex}}\text{with}\hspace{0.17em}0\le a\le \text{gcd}(p,k)-1,& \text{(2.4b)}\end{array}$$ where $k=n-{i}_{\ell}+1$ is the length of the last cycle.
Centralizers in $G(r,p,n)\text{.}$ Let ${Z}_{G(r,p,n)}\left(g\right)=\{h\in G(r,p,n)\hspace{0.17em}|\hspace{0.17em}hg=gh\}$ denote the centralizer of $g\in G(r,p,n)\text{.}$ Since $G(r,p,n)$ is a subgroup of $G(r,1,n),$ $${Z}_{G(r,p,n)}\left(g\right)={Z}_{G(r,1,n)}\left(g\right)\cap G(r,p,n),$$ for any element $g\in G(r,p,n)\text{.}$ Suppose that $g$ is an element of $G(r,1,n)$ which is a product of disjoint cycles of the form ${\xi}_{k}^{a}(i,\dots ,k)$ and that $h\in G(r,1,n)$ commutes with $g\text{.}$ Conjugation by $h$ effects some combination of the following operations on the cycles of $g\text{:}$
(a) | permuting cycles of the same type, ${\xi}_{k}^{a}(i,\dots ,k)$ and ${\xi}_{m}^{b}(j,\dots ,m)$ with $b=a$ and $k-i=m-j,$ |
(b) | conjugating a single cycle ${\xi}_{k}^{a}(i,\dots ,k)$ by powers of itself, and |
(c) | conjugating a single cycle ${\xi}_{k}^{a}(i,\dots ,k)$ by ${\xi}_{i}^{b}\cdots {\xi}_{k}^{b},$ for any $0\le b\le r-1\text{.}$ |
Determining the graded Hecke algebras for $G(r,p,n)\text{.}$ It follows from Lemma 1.8a that if $g={\xi}_{i}^{a+b}{\xi}_{k}^{-a}(i,\dots ,k),$ then ${\left({V}^{g}\right)}^{\perp}$ has basis $$\{{v}_{k}-{v}_{k-1},{v}_{k-1}-{v}_{k-2},\dots ,{v}_{i+1}-{\xi}^{a}{v}_{i}\}\phantom{\rule{1em}{0ex}}\text{if}\hspace{0.17em}b=0,\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\{{v}_{i},\dots ,{v}_{k}\}\phantom{\rule{1em}{0ex}}\text{if}\hspace{0.17em}b\ne 0\text{.}$$ Thus, if $g\in G(r,p,n)$ and $\text{codim}\left({V}^{g}\right)=2,$ then $g$ is conjugate to one of the following elements: $$\begin{array}{cccc}b& =& {\xi}_{1}^{a}{\xi}_{3}^{-a}(1,2,3),& 0\le a\le \text{gcd}(p,3)-1,\\ c& =& {\xi}_{1}^{a+\ell}{\xi}_{2}^{-a}(1,2),& \ell \ne 0\hspace{0.17em}\text{(so}\hspace{0.17em}r\ne 1\text{),}\\ d& =& {\xi}_{1}^{{\ell}_{1}}{\xi}_{2}^{{\ell}_{2}},& {\ell}_{1}\ne 0,\hspace{0.17em}{\ell}_{2}\ne 0\hspace{0.17em}\text{(so}\hspace{0.17em}r\ne 1\text{),}\\ e& =& (1,2){\xi}_{3}^{\ell},& \ell \ne 0,\\ f& =& (1,2){\xi}_{3}^{a}{\xi}_{4}^{-a}(3,4)\text{.}\end{array}$$ It is interesting to note that these elements are also representatives of the conjugacy classes of elements in $G(r,p,n)$ which can be written as a product of two reflections.
We determine conditions on the above elements and on $r,$ $p,$ and $n$ to give nontrivial graded Hecke algebras:
(z) | The center of $G(r,p,n)$ is $$Z\left(G(r,p,n)\right)=\left\{{\xi}_{1}^{\ell}\cdots {\xi}_{n}^{\ell}\hspace{0.17em}\right|\hspace{0.17em}n\ell =0\hspace{0.17em}\text{mod}\hspace{0.17em}p\}\text{.}$$ Since ${\xi}_{1}^{p}\cdots {\xi}_{n}^{p}\in Z\left(G(r,p,n)\right),$ it follows that $p=r$ or $p=r/2$ whenever $Z\left(G(r,p,n)\right)\subseteq \left\{\pm 1\right\}=\{{\xi}_{1}^{0}\cdots {\xi}_{n}^{0},{\xi}_{1}^{r/2}\cdots {\xi}_{n}^{r/2}\}\text{.}$ |
(b1) | If $n\ge 4,$ the element ${\xi}_{1}{\xi}_{2}{\xi}_{3}{\xi}_{4}^{-3}\in {Z}_{G}\left(b\right)$ and has determinant ${\xi}^{2}$ on ${\left({V}^{b}\right)}^{\perp}=\text{span-}\{{v}_{3}-{v}_{2},{v}_{2}-{\xi}^{a}{v}_{1}\}\text{.}$ |
(b2) | If $n=3$ and $p=0$ mod $3,$ the element ${\xi}_{1}^{p/3}{\xi}_{2}^{p/3}{\xi}_{3}^{p/3}\in {Z}_{G}\left(b\right)$ and has determinant ${\xi}^{2p/3}$ on ${\left({V}^{b}\right)}^{\perp}\text{.}$ |
(c1) | If $n\ge 3,$ the element ${\xi}_{1}{\xi}_{2}{\xi}_{3}^{-2}\in {Z}_{G}\left(c\right)$ and has determinant ${\xi}^{2}$ on ${\left({V}^{c}\right)}^{\perp}=\text{span-}\{{v}_{1},{v}_{2}\}\text{.}$ |
(c2) | If $n=2,$ $p=r/2$ and $p$ is odd, the element ${\xi}_{1}^{p/4}{\xi}_{2}^{p/4}\in {Z}_{G}\left(c\right)$ and has determinant ${\xi}^{r/2}$ on ${\left({V}^{c}\right)}^{\perp}\text{.}$ |
(d1) | If $n\ge 3,$ the element ${\xi}_{1}{\xi}_{3}^{-1}\in {Z}_{G}\left(d\right)$ and has determinant $\xi $ on ${\left({V}^{d}\right)}^{\perp}=\text{span-}\{{v}_{1},{v}_{2}\}\text{.}$ |
(d2) | If $p=r/2,$ the element ${\xi}_{1}^{r/2}\in {Z}_{G}\left(d\right)$ and has determinant ${\xi}^{r/2}$ on ${\left({V}^{d}\right)}^{\perp}\text{.}$ |
(ef) | The elements $e$ and $f$ have order $2\text{.}$ |
We arrive at the following enumeration of the nontrivial graded Hecke algebras for complex reflection groups. (The tensor product algebra $S\left(V\right)\otimes \u2102G$ always exists and corresponds to the case when all of the skew symmetric forms ${a}_{g}$ are zero). The table below gives representatives of the conjugacy classes of $g\in G$ that may have ${a}_{g}\ne 0$ for some graded Hecke algebra $A\text{.}$ $$\begin{array}{c}\begin{array}{|cc|}\hline \text{Group}& \genfrac{}{}{0ex}{}{\text{Representative}\hspace{0.17em}g}{\text{with}\hspace{0.17em}{a}_{g}\ne 0}\\ G(1,1,n)={S}_{n}& (1,2,3)\\ G(2,1,n)=W{B}_{n},\phantom{\rule{1em}{0ex}}n\ge 3& {\xi}_{1}(1,2),\hspace{0.17em}(1,2,3)\\ G(2,2,n)=W{D}_{n},\phantom{\rule{1em}{0ex}}n\ge 3& (1,2,3)\\ G(r,r,2)={I}_{2}\left(r\right)& {\xi}_{1}^{k}{\xi}_{2}^{r-k},\hspace{0.17em}0<k<r/2\\ G(r,r/2,2),\phantom{\rule{1em}{0ex}}r/2\hspace{0.17em}\text{odd}& {\xi}_{2}^{r/2}(1,2)\\ G(r,r,3),\phantom{\rule{1em}{0ex}}r\ne 0\hspace{0.17em}\text{mod}\hspace{0.17em}3& (1,2,3)\\ G(r,r/2,3),\phantom{\rule{1em}{0ex}}r/2\ne 0\hspace{0.17em}\text{mod}\hspace{0.17em}3,\hspace{0.17em}r\ne 2& (1,2,3)\\ \hline\end{array}\\ \text{Table 2.}\hspace{0.17em}\text{Graded Hecke algebras for the groups}\hspace{0.17em}G(r,p,n)\text{.}\end{array}$$
2C. Exceptional complex reflection groups
The irreducible exceptional complex reflection groups $G$ are denoted ${G}_{4},\dots ,{G}_{35}$ in the classification of Shephard and Todd. From Table VII in [STo1954], one sees that the center of $G$ is $\pm 1$ only in the cases ${G}_{4},$ ${G}_{12},$ ${G}_{24}$ and ${G}_{33}\text{.}$ By Schur’s lemma, the center of an irreducible complex reflection group consists of multiples of the identity. Thus, by Corollary 1.11, the only exceptional complex reflection groups that could have a nontrivial graded Hecke algebra (i.e., with some ${a}_{g}\ne 0\text{)}$ are ${G}_{4},$ ${G}_{12},$ ${G}_{24}$ and ${G}_{33}$ (we exclude the real groups). We determine the graded Hecke algebras for these groups using Theorem 1.9b and Lemma 2.2.
The rank $2$ group ${G}_{4}$ has order $24$ and seven conjugacy classes. The following data concerning these conjugacy classes are obtained from the computer software GAP [Sch1995] using the package CHEVIE [GHL1996]. In the following table, $\omega $ is a primitive cube root of unity and $C\left(g\right)$ denotes the conjugacy class of $g\text{.}$ $$\begin{array}{|cccccccc|}\hline \multicolumn{8}{c}{\text{Conjugacy class representatives for}\hspace{0.17em}{G}_{4}}\\ \text{Order}\left(g\right)& 1& 4& 3& 6& 6& 3& 2\\ \text{det}\left(g\right)& 1& 1& \omega & \omega & {\omega}^{-1}& {\omega}^{-1}& 1\\ \left|\mathcal{C}\left(g\right)\right|& 1& 6& 4& 4& 4& 4& 1\\ \left|{Z}_{G}\left(g\right)\right|& 24& 4& 6& 6& 6& 6& 24\\ \hline\end{array}$$ The elements with determinant $1$ and order more than $2$ in ${G}_{4}$ all have order $4\text{.}$ If $g$ is an element of order $4,$ then $\left|{Z}_{G}\left(g\right)\right|=4$ and every element of ${Z}_{G}\left(g\right)$ has determinant $1$ since ${Z}_{G}\left(g\right)$ is generated by $g\text{.}$ Hence, by Theorem 1.9b and Lemma 2.2, ${a}_{g}$ can be nonzero for a graded Hecke algebra for ${G}_{4}$ exactly when $g$ has order $4\text{.}$ Thus, the dimension of the space of parameters for graded Hecke algebras of ${G}_{4}$ is $1\text{.}$
The rank $2$ group ${G}_{12}$ has order $48\text{.}$ The computer software GAP provides the following information about the conjugacy classes of ${G}_{12}\text{.}$ $$\begin{array}{|ccccccccc|}\hline \multicolumn{9}{c}{\text{Conjugacy class representatives for}\hspace{0.17em}{G}_{12}}\\ \text{Order}\left(g\right)& 1& 2& 8& 6& 8& 2& 3& 4\\ \text{det}\left(g\right)& 1& -1& -1& 1& -1& 1& 1& 1\\ \left|\mathcal{C}\left(g\right)\right|& 1& 12& 6& 8& 6& 1& 8& 6\\ \left|{Z}_{G}\left(g\right)\right|& 48& 4& 8& 6& 8& 48& 6& 8\\ \hline\end{array}$$ If $g$ is an element in ${G}_{12}$ with order more than $2$ and determinant $1,$ then $g$ has order $3,$ $4,$ or $6\text{.}$ Let $h$ be any element of order $8\text{.}$ Then $h$ has determinant $-1$ and commutes with ${h}^{2}$ of order $4\text{.}$ Hence, by Theorem 1.9b, if $g$ has order $4,$ then ${a}_{g}=0\text{.}$ Let ${g}_{6}$ be a representative from the class of elements of order $6\text{.}$ Since $\left|{Z}_{G}\left({g}_{6}\right)\right|=6,$ ${Z}_{G}\left({g}_{6}\right)$ is generated by ${g}_{6}$ and hence every element of ${Z}_{G}\left({g}_{6}\right)$ has determinant $1\text{.}$ We can choose ${g}_{6}^{2}$ as a representative for the conjugacy class of elements of order $3\text{.}$ As ${g}_{6}$ and ${g}_{6}^{2}$ commute, $\u27e8{g}_{6}\u27e9\subset {Z}_{G}\left({g}_{6}^{2}\right)\text{.}$ But $\left|\u27e8{g}_{6}\u27e9\right|=6=\left|{Z}_{G}\left({g}_{3}\right)\right|,$ so ${Z}_{G}\left({g}_{6}^{2}\right)$ is generated by ${g}_{6}$ and every element of ${Z}_{G}\left({g}_{3}\right)$ has determinant $1\text{.}$ Thus, ${a}_{g}$ can be nonzero for a graded Hecke algebra $A$ for ${G}_{12}$ exactly when $g$ has order $3$ or $6\text{.}$ Thus, the dimension of the space of parameters of graded Hecke algebras for ${G}_{12}$ is $2\text{.}$
The rank $3$ group ${G}_{24}$ has order $336\text{.}$ Note that $-1\in {G}_{24}$ since $Z\left(G\right)=\{\pm 1\}\text{.}$ Up to $G\text{-orbits,}$ there are two codimension $2$ subspaces, $L$ and $M,$ that are equal to ${V}^{g}$ for some $g\in {G}_{24}$ (see [OTe1992, App. C, Table C.5]). Furthermore, $\text{Stab}\left(L\right)\cong {A}_{2}$ and $\text{Stab}\left(M\right)\cong {B}_{2}\text{.}$ We need only consider elements of order $3$ in $\text{Stab}\left(L\right)\cong {A}_{2}$ and of order $4$ in $\text{Stab}\left(M\right)\cong {B}_{2}$ (as the rest have order $1$ or $2\text{).}$ In ${G}_{24},$ there is only one conjugacy class of elements of order $3$ and only one conjugacy class of elements of order $4$ and determinant $1\text{.}$ The table below (obtained using GAP) records information about these classes. $$\begin{array}{|ccc|}\hline \multicolumn{3}{c}{\text{Conjugacy class representatives for}\hspace{0.17em}{G}_{24}}\\ \text{Order}\left(g\right)& 3& 4\\ \text{det}\left(g\right)& 1& 1\\ \left|\mathcal{C}\left(g\right)\right|& 56& 42\\ \left|{Z}_{G}\left(g\right)\right|& 6& 8\\ \hline\end{array}$$ If $g$ has order $3,$ ${Z}_{G}\left(g\right)$ must contain $1,g,{g}^{2},$ and $-1,$ and hence ${Z}_{G}\left(g\right)$ is generated by these elements since $\left|{Z}_{G}\left(g\right)\right|=6\text{.}$ Thus all elements of ${Z}_{G}\left(g\right)$ have determinant $1$ on ${\left({V}^{g}\right)}^{\perp}\text{.}$ If $g$ has order $4$ and determinant $1,$ then ${Z}_{G}\left(g\right)$ must contain $1,$ $g,$ ${g}^{2},$ ${g}^{3},$ and $-1,$ elements which all have determinant $1$ on ${\left({V}^{g}\right)}^{\perp}\text{.}$ Since $\left|{Z}_{G}\left(g\right)\right|=8,$ these elements generate ${Z}_{G}\left(g\right)$ and so every element of ${Z}_{G}\left(g\right)$ has determinant $1$ on ${\left({V}^{G}\right)}^{\perp}\text{.}$ Hence, ${a}_{g}$ can be nonzero for a graded Hecke algebra of ${G}_{24}$ exactly when $g$ has order $3$ or $g$ has order $4$ and determinant $1\text{.}$ Thus, the dimension of the space of parameters for graded Hecke algebras for ${G}_{24}$ is $2\text{.}$
The group ${G}_{33}$ is the only exceptional complex reflection group of rank $5\text{.}$ It has order $72\xb76!$ and degrees $4,$ $6,$ $10,$ $12,$ $18\text{.}$ There are $45$ reflecting hyperplanes and the corresponding reflections all have order $2\text{.}$ Up to $G\text{-orbits,}$ there are two codimension $2$ subspaces, $L$ and $M,$ that are equal to ${V}^{g}$ for some $g\in {G}_{33}$ (see [OTe1992, App. C, Table C.14]). Furthermore, $\text{Stab}\left(L\right)\cong {A}_{1}\times {A}_{1}$ and $\text{Stab}\left(M\right)\cong {A}_{2}\text{.}$ We need not consider the case where ${V}^{g}=L$ since then $g$ has order $2$ and hence ${a}_{g}=0$ for any graded Hecke algebra by Proposition 2.2b.
We use a presentation for ${G}_{33}$ in six coordinates from [STo1954]: Let $V\u2254\left\{({x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6})\hspace{0.17em}\right|\hspace{0.17em}{x}_{i}\in \u2102\}$ and consider the group generated by order $2$ reflections about the hyperplanes ${H}_{1}=\{{x}_{2}-{x}_{3}=0\},$ ${H}_{2}=\{{x}_{3}-{x}_{4}=0\},$ ${H}_{3}=\{{x}_{1}-{x}_{2}=0\},$ ${H}_{4}=\{{x}_{1}-\omega {x}_{2}=0\},$ ${H}_{5}=\{{x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5}+{x}_{6}=0\},$ where $\omega $ is a primitive cube root of unity. The fixed point space of this (reducible) group is $Y={H}_{1}\cap \cdots \cap {H}_{5}=\left\{(0,0,0,0,x,-x)\hspace{0.17em}\right|\hspace{0.17em}x\in \u2102\},$ and ${G}_{33}$ is just the restriction to ${Y}^{\perp}\text{.}$ Let ${s}_{i}$ be the order $2$ reflection about ${H}_{i}\text{.}$ Let $g={s}_{1}{s}_{3}\text{.}$ Then ${V}^{g}={H}_{1}\cap {H}_{3}$ and $\text{Stab}\left({V}^{g}\right)\cong {A}_{2}\text{.}$ Let $h={\left({s}_{1}{s}_{3}{s}_{4}\right)}^{2},$ the diagonal matrix with diagonal $\{\omega ,\omega ,\omega ,1,1,1\}\text{.}$ Then $h$ acts as $\omega $ times the identity on ${\left({V}^{g}\right)}^{\perp}$ as ${\left({V}^{g}\right)}^{\perp}\subseteq \u2102\text{-span}\{{x}_{1},{x}_{2},{x}_{3}\}\text{.}$ Hence, $h$ commutes with $g\text{.}$ But ${\left({V}^{g}\right)}^{\perp}$ has dimension $2$ and $h$ has determinant ${\omega}^{2}\ne 1$ on ${\left({V}^{g}\right)}^{\perp}\text{.}$ Thus, by Theorem 1.9b and Lemma 2.2, ${a}_{g}=0$ for any graded Hecke algebra. The same argument applied to ${Y}^{\perp}$ shows that ${G}_{33}$ has no nontrivial graded Hecke algebras. In summary, the dimension of the space of parameters for graded Hecke algebras for ${G}_{33}$ is zero. $$\begin{array}{c}\begin{array}{|cc|}\hline \text{Group}& g\hspace{0.17em}\text{with}\hspace{0.17em}{a}_{g}\ne 0\\ {G}_{4}& \text{Order}\left(g\right)=4\\ {G}_{12}& \text{Order}\left({g}_{1}\right)=3\hspace{0.17em}\text{and}\hspace{0.17em}\text{Order}\left({g}_{2}\right)=6\\ {G}_{24}& \text{Order}\left({g}_{1}\right)=3\hspace{0.17em}\text{and}\hspace{0.17em}\text{Order}\left({g}_{2}\right)=4,\hspace{0.17em}\text{det}\left({g}_{2}\right)=1\\ \hline\end{array}\\ \text{Table 3.}\hspace{0.17em}\text{Graded Hecke algebras for exceptional complex reflection groups.}\end{array}$$
This is a typed version of Classification of graded Hecke algebras for complex reflection groups by Arun Ram and Anne V. Shepler.
Research of the first author supported in part by the National Security Agency and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. Research of the second author supported in part by National Science Foundation grant DMS-9971099.