Classification of graded Hecke algebras for complex reflection groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 22 January 2014

Examples

4A. The symmetric group $G(1,1,n)=S_n$

Let $V$ be an $n$ dimensional vector space with orthonormal basis $v_1,\dots ,v_n$ and let $S_n$ act on $V$ by permuting the $v_i\text{.}$ Let $A$ be a graded Hecke algebra for $S_n\text{.}$ Any element which is a product of two reflections is conjugate to $(1,2,3)$ or $(1,2)(3,4)\text{.}$ The element $(1,2)(3,4)$ has order $2$ and so, in the algebra $A,$ $$[v_i,v_j]=\sum _{k\ne i,j}(a_{(i,j,k)}(v_i,v_j)t_{(i,j,k)}+a_{(j,i,k)}(v_i,v_j)t_{(j,i,k)}),$$ since $v_i$ or $v_j$ is in $V^g=\text{ker}\hspace{0.17em}{a}_g$ for all other three cycles $g\text{.}$ Since, by (1.6), $a_{(j,i,k)}(v_i,v_j)=a_{(i,j,k)}(v_j,v_i)=-a_{(i,j,k)}(v_i,v_j),$ the graded Hecke algebra $A$ is defined by the relations $$\begin{array}{cc}$ [v_i,v_j]$=\beta \sum _{k\ne i,j}(t_{(i,j,k)}-t_{(j,i,k)})\text{and}{t}_w{v}_i=v_{w\left(i\right)}{t}_w,& \text{(4.1)}\end{array}$$ where $w\in S_n,$ $1\le i,j\le n,$ $i\ne j,$ and $\beta =a_{(1,2,3)}(v_1,v_2)\text{.}$

Let $k\in ℂ\text{.}$ Then, with $h$ as in (3.3), $$\begin{array}{cc}⟨v_i,h⟩=\frac{1}{2}\sum _{\ell <m}k⟨v_i,v_{\ell }-v_m⟩t_{(\ell ,m)}=\frac{k}{2}(\sum _{i<\ell }{t}_{(i,\ell )}-\sum _{i>\ell }{t}_{(\ell ,i)})=\frac{k}{2}\sum _{i\ne \ell }\text{sgn}(\ell -i)t_{(i,\ell )}\text{.}& \text{(4.2)}\end{array}$$ If $f\in ℂS_n,$ let $f{|}_{t_g}$ denote the coefficient of $t_g$ in $f\text{.}$ Let $A$ be the graded Hecke algebra defined by the relations in (4.1) with $$\begin{array}{cc}\begin{array}{ccc}\beta & =& a_{(i,j,\ell )}(v_i,v_j)=[⟨v_i,h⟩,⟨v_j,h⟩]{|}_{t_{(i,j,\ell )}}\\ & =& (k^2/4)(t_{(i,\ell )}{t}_{(j,\ell )}+t_{(i,j)}{t}_{(i,\ell )}-t_{(j,\ell )}{t}_{(i,j)}){|}_{t_{(i,j,\ell )}}=k^2/4\text{.}\end{array}& \text{(4.3)}\end{array}$$ If ${\stackrel{\sim }{v}}_i=v_i-⟨v_i,h⟩$ and $s_i$ is the simple reflection $(i,i+1)$ then, by Theorem 3.5, $$\begin{array}{cc}\begin{array}{ccc}{\stackrel{\sim }{v}}_i{\stackrel{\sim }{v}}_j& =& {\stackrel{\sim }{v}}_j{\stackrel{\sim }{v}}_i,t_{s_i}{\stackrel{\sim }{v}}_i={\stackrel{\sim }{v}}_{i+1}{t}_{s_i}+k,t_{s_i}{\stackrel{\sim }{v}}_{i+1}={\stackrel{\sim }{v}}_i{t}_{s_i}-k,\text{and}\\ t_{s_j}{\stackrel{\sim }{v}}_i& =& {\stackrel{\sim }{v}}_i{t}_{s_j},\text{for}\hspace{0.17em}|i-j|>1,\end{array}& \text{(4.4)}\end{array}$$ and the algebra $A$ is the graded Hecke algebra $H_{\text{gr}}$ for $S_n$ which is defined in Section 3. When $k=1,$ the map $$\begin{array}{cc}\begin{array}{ccc}A& ⟶& ℂS_n\\ t_w& ⟼& t_w\\ v_i& ⟼& \frac{1}{2}\sum _{\ell \ne i}{t}_{(i,\ell )}\end{array}& \text{(4.5)}\end{array}$$ is a surjective algebra homomorphism.

4B. The hyperoctahedral group $G(2,1,n)=W{B}_n$

We use the notation from Section 2B so that the group $G(2,1,n)$ is acting by orthogonal matrices on the $n$ dimensional vector space $V$ with orthonormal basis $\{v_1,\dots ,v_n\}\text{.}$ In this case, ${\xi }_i$ denotes the diagonal matrix with all ones on the diagonal except for $-1$ in the $(i,i)\text{th}$ entry.

Let $A$ be a graded Hecke algebra for $G(2,1,n)\text{.}$ If ${\beta }_1=a_{(i,j,k)}(v_i,v_j)$ and ${\beta }_2=a_{{\xi }_1(1,2)}(v_1,v_2),$ then, in the algebra $A,$ $$\begin{array}{cc}[v_i,v_j]={\beta }_2(t_{{\xi }_1(1,2)}-t_{{\xi }_2(1,2)})+{\beta }_1\sum _{\ell \ne i,j}\left(\genfrac{}{}{0ex}{}{t_{(i,j,\ell )}-t_{{\xi }_i,{\xi }_{\ell }(i,j,\ell )}-t_{{\xi }_i,{\xi }_j(i,j,\ell )}+t_{{\xi }_j{\xi }_{\ell }(i,j,\ell )}}{+t_{{\xi }_i{\xi }_j(j,i,\ell )}+t_{{\xi }_j{\xi }_{\ell }(j,i,\ell )}-t_{{\xi }_i{\xi }_{\ell }(j,i,\ell )}-t_{(j,i,\ell )}}\right)\text{.}& \text{(4.6)}\end{array}$$

Let $k_s,k_{\ell }\in ℂ\text{.}$ Then, with $h$ as in (3.3), $$\begin{array}{cc}\begin{array}{ccc}⟨v_i,h⟩& =& \frac{k_s}{2}\sum _{\ell }⟨v_i,2v_{\ell }⟩t_{{\xi }_{\ell }}-\frac{k_{\ell }}{2}\sum _{\ell <m}⟨v_i,v_{\ell }-v_m⟩t_{(\ell ,m)}-\frac{k_{\ell }}{2}\sum _{\ell <m}⟨v_i,v_{\ell }+v_m⟩t_{{\xi }_{\ell }(\ell ,m)}\\ & =& k_2{t}_{{\xi }_i}-\frac{k_{\ell }}{2}(\sum _{i<\ell }(t_{(i,\ell )}+t_{{\xi }_i{\xi }_{\ell }(i,\ell )})+\sum _{i>\ell }(-t_{(i,\ell )}+t_{{\xi }_i{\xi }_{\ell }(i,\ell )}))\text{.}\end{array}& \text{(4.7)}\end{array}$$ If $f\in ℂG(2,1,n),$ let $f{|}_{t_g}$ denote the coefficient of $t_g$ in $f\text{.}$ With notation as in (4.6), let $A$ be the graded Hecke algebra for $G(2,1,n)$ with $$\begin{array}{ccc}{\beta }_1& =& a_{(i,j,\ell )}(v_i,v_j)=[⟨v_i,h⟩,⟨v_j,h⟩]{|}_{t_{(i,j,\ell )}}\\ & =& (k_{\ell }^2/4)(t_{(i,\ell )}{t}_{(j,\ell )}+t_{(i,j)}{t}_{(i,\ell )}-t_{(j,\ell )}{t}_{(i,j)}){|}_{t_{(i,j,\ell )}}=k_{\ell }^2/4,\text{and}\\ {\beta }_2& =& [⟨v_i,h⟩,⟨v_j,h⟩]{|}_{t_{{\xi }_i(i,j)}}\\ & =& (1/2)k_s{k}_{\ell }(-t_{{\xi }_i}{t}_{(i,j)}+t_{(i,j)}{t}_{{\xi }_j}-t_{{\xi }_j}{t}_{{\xi }_i{\xi }_j(i,j)}-t_{{\xi }_i{\xi }_j(i,j)}{t}_{{\xi }_i}){|}_{t_{{\xi }_i(i,j)}}=-k_s{k}_{\ell }\text{.}\end{array}$$ If ${\stackrel{\sim }{v}}_i=v_i-⟨v_i,h⟩,$ then, by Theorem 3.5, the ${\stackrel{\sim }{v}}_i$ commute and the algebra $A$ is the algebra $H_{\text{gr}}$ for $W{B}_n$ defined in Section 3.

4C. The type $D_n$ Weyl group $G(2,2,n)=W{D}_n$

We shall use the notation from Section 2B so that the group $G(2,2,n)$ is acting by orthogonal matrices on the $n$ dimensional vector space $V$ with orthonormal basis $\{v_1,\dots ,v_n\}\text{.}$ This is an index 2 subgroup of $G(2,1,n),$ and our notation is the same as used above for $W{B}_n\text{.}$

Let $A$ be a graded Hecke algebra for $G(2,2,n)\text{.}$ If $\beta =a_{(i,j,k)}(v_i,v_j)$ then, in the algebra $A,$ $$\begin{array}{cc}[v_i,v_j]=\beta \sum _{\ell \ne i,j}\left(\genfrac{}{}{0ex}{}{t_{(i,j,k)}-t_{{\xi }_i{\xi }_{\ell }(i,j,\ell )}-t_{{\xi }_i{\xi }_j(i,j,\ell )}+t_{{\xi }_j{\xi }_{\ell }(i,j,\ell )}}{+t_{{\xi }_i{\xi }_j(j,i,\ell )}+t_{{\xi }_j{\xi }_{\ell }(j,i,\ell )}-t_{{\xi }_i{\xi }_{\ell }(j,i,\ell )}-t_{(j,i,\ell )}}\right)\text{.}& \text{(4.8)}\end{array}$$

Let $k\in ℂ\text{.}$ Then, with $h$ as in (3.3), $$\begin{array}{cc}⟨v_i,h⟩=\frac{k}{2}(\sum _{i<\ell }(t_{(i,\ell )}+t_{{\xi }_i{\xi }_{\ell }(i,\ell )})+\sum _{i>\ell }(-t_{(i,\ell )}+t_{{\xi }_i{\xi }_{\ell }(i,\ell )}))\text{.}& \text{(4.9)}\end{array}$$ If $f\in ℂG(2,2,n),$ let $f{|}_{t_g}$ denote the coefficient of $t_g$ in $f\text{.}$ With notation as in (4.8), let $A$ be the graded Hecke algebra for $G(2,2,n)$ with $$\begin{array}{ccc}\beta & =& a_{(i,j,\ell )}(v_i,v_j)=[⟨v_i,h⟩,⟨v_j,h⟩]{|}_{t_{(i,j,\ell )}}\\ & =& (k^2/4)(t_{(i,\ell )}{t}_{(j,\ell )}+t_{(i,j)}{t}_{(i,\ell )}-t_{(j,\ell )}{t}_{(i,j)}){|}_{t_{(i,j,\ell )}}=k^2/4\text{.}\end{array}$$ If ${\stackrel{\sim }{v}}_i=v_i-⟨v_i,h⟩,$ then, by Theorem 3.5, the ${\stackrel{\sim }{v}}_i$ commute and the algebra $A$ is the algebra $H_{\text{gr}}$ for $W{D}_n$ defined in Section 3.

4D. The dihedral group $I_2\left(r\right)=G(r,r,2)$ of order $2r$

We shall use the notation for $G(r,r,2)$ from Section 2B so that the group $G(r,r,2)$ is acting by unitary matrices on the 2 dimensional vector space $V$ with orthonormal basis $\{v_1,v_2\}\text{.}$ The group $G(r,r,2)$ is realized as a real reflection group by using the basis $${\epsilon }_1=\frac{1}{\sqrt{2}}(v_1+v_2),{\epsilon }_2=\frac{-1}{i\sqrt{2}}(v_1-v_2)\text{.}$$ This basis is also orthonormal and, with respect to this basis, $G(r,r,2)$ acts by the matrices $$\left(\begin{array}{cc}\text{cos}(2\pi m/r)& \mp \text{sin}(2\pi m/r)\\ \text{sin}(2\pi m/r)& \pm \text{cos}(2\pi m/r)\end{array}\right),0\le m\le r-1\text{.}$$

Let $A$ be a graded Hecke algebra for $G(r,r,2)\text{.}$ The conjugacy classes of elements which are products of two reflections are $\{{\xi }_1^k{\xi }_2^{-k},{\xi }_1^{-k}{\xi }_2^k\},$ $0<k<r/2\text{.}$ Then, in the algebra $A,$ $$\begin{array}{cc}[{\epsilon }_1,{\epsilon }_2]=\sum _{0<k<r/2}{\beta }_k(t_{{\xi }_1^k{\xi }_2^{-k}}-t_{{\xi }_1^{-k}{\xi }_2^k}),\text{where}{\beta }_k=a_{{\xi }_1^k{\xi }_2^{-k}}({\epsilon }_1,{\epsilon }_2)\text{.}& \text{(4.10)}\end{array}$$

When $r$ is even, there are two conjugacy classes of reflections $$\left\{{\xi }_1^{2k}{\xi }_2^{-2k}(1,2)\hspace{0.17em}\right|\hspace{0.17em}0\le k<r/2\}\text{and}\left\{{\xi }_1^{2k+1}{\xi }_2^{-(2k+1)}(1,2)\hspace{0.17em}\right|\hspace{0.17em}0\le k<r/2\}\text{.}$$ The reflection ${\xi }_1^m{\xi }_2^{-m}\left(12\right)$ is the reflection in the line perpendicular to the vector $${\alpha }_m=\text{sin}(-2\pi m/2r){\epsilon }_1+\text{cos}(-2\pi m/2r){\epsilon }_2,$$ and the vectors ${\alpha }_m$ can be taken as a root system for $G(r,r,2)\text{.}$ With $h$ as in (3.3) and $k_s,k_{\ell }\in ℂ,$ $$\begin{array}{cc}\begin{array}{ccc}⟨{\epsilon }_1,h⟩& =& \sum _{0\le k<r/2}(k_s\text{sin}(-2k2\pi /2r)t_{{\xi }_1^{2k}{\xi }_2^{-2k}(1,2)}+k_{\ell }\text{sin}(-(2k+1)2\pi /2r)t_{{\xi }_1^{2k+1}{\xi }_2^{-(2k+1)}(1,2)}),\\ ⟨{\epsilon }_2,h⟩& =& \sum _{0\le k<r/2}(k_s\text{cos}(-2k2\pi /2r)t_{{\xi }_1^{2k}{\xi }_2^{-2k}(1,2)}+k_{\ell }\text{cos}(-(2k+1)2\pi /2r)t_{{\xi }_1^{2k+1}{\xi }_2^{-(2k+1)}(1,2)})\text{.}\end{array}& \text{(4.11)}\end{array}$$ If $f\in ℂG(r,r,2),$ let $f{|}_{t_g}$ denote the coefficient of $t_g$ in $f\text{.}$ With notation as in 4.10, let $A$ be the graded Hecke algebra for $G(r,r,2)$ with $$\begin{array}{cc}{\beta }_k=a_{{\xi }_1^k{\xi }_2^{-k}}({\epsilon }_1,{\epsilon }_2)=[⟨{\epsilon }_1,h⟩,⟨{\epsilon }_2,h⟩]{|}_{t_{{\xi }_1^k{\xi }_2^{-k}}}=\{\begin{array}{cc}\text{sin}(k2\pi /2r)r{k}_s{k}_{\ell }& \text{if}\hspace{0.17em}k\hspace{0.17em}\text{is odd}\\ \text{sin}(k2\pi /2r)\frac{r}{2}(k_s^2+k_{\ell }^2)& \text{if}\hspace{0.17em}k\hspace{0.17em}\text{is even.}\end{array}& \text{(4.12)}\end{array}$$ If ${\stackrel{\sim }{\epsilon }}_i={\epsilon }_i-⟨{\epsilon }_i,h⟩,$ then by Theorem 3.5, the ${\stackrel{\sim }{\epsilon }}_i$ commute and the algebra $A$ is the algebra $H_{\text{gr}}$ for $I_2\left(r\right)$ defined in Section 3.

When $r$ is odd, all aspects of the calculation in (4.11) and (4.12) are the same as for the case $r$ even except that there is only one conjugacy class of reflections, $\left\{{\xi }_1^k{\xi }_2^{-k}(1,2)\hspace{0.17em}\right|\hspace{0.17em}0\le k\le r-1\},$ and so $k_s=k_{\ell }\text{.}$

4E. The group $G(r,r/2,2),$ $r/2$ odd

We use the notation from Section 2B, or from above for the group $G(r,r,2)\text{.}$ In this case, the group is not a real reflection group, hence $G(r,r/2,2)$ acts by unitary matrices but not by orthogonal matrices.

Let $A$ be a graded Hecke algebra for $G(r,r/2,2)\text{.}$ The only conjugacy class for which $a_g$ can be nonzero is $\left\{t_{{\xi }_1^k{\xi }_2^{r/2-k}(1,2)}\hspace{0.17em}\right|\hspace{0.17em}0\le k<r\}\text{.}$ Thus, in the algebra $A,$ $$[v_1,v_2]=\beta \sum _k(t_{{\xi }_1^{2k}{\xi }_2^{r/2-2k}(1,2)}-t_{{\xi }_1^{r/2-2k}{\xi }_2^{2k}(1,2)}),\text{where}\beta =a_{{\xi }_2^{r/2}(1,2)}(v_1,v_2)\text{.}$$

Notes and references

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.

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