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

A different graded Hecke algebra for $G(r,1,n)$

The classification of graded Hecke algebras for complex reflection groups in Section 2 shows that there do not exist graded Hecke algebras $A\cong S\left(V\right)\otimes ℂG$ for the groups $G=G(r,1,n),$ $r>2,$ $n>3\text{.}$ In this section, we define a different “semidirect product” of the symmetric algebra $S\left(V\right)$ and the group algebra $ℂG$ for the groups $G(r,1,n)\text{.}$ These algebras are not graded Hecke algebras in the sense of Section 1, but they do have a structure similar to what we would expect from experience with graded Hecke algebras for real reflection groups. Is it possible that there is a general definition of graded Hecke algebras, different from that given in Section 1, which includes the algebras defined below as examples for the groups $G(r,1,n)\text{?}$

We shall use the notation for the groups $G(r,1,n)$ as in Section 2B so that the group $G(r,1,n)$ is acting by monomial matrices on a vector space $V$ of dimension $n$ with orthonormal basis $\{v_1,\dots ,v_n\}\text{.}$ Let $s_i$ denote the permutation $(i,i+1)\in G(r,1,n)\text{.}$

Define $H_{r,1,n}^{*}$ to be the algebra generated by the group algebra $ℂG(r,1,n)$ and $V$ with relations $$\begin{array}{cc}\begin{array}{cccc}{v}_i{v}_j& =& v_j{v}_i,& \text{for all}\hspace{0.17em}1\le i,j\le n,\\ t_{{\xi }_i}{v}_j& =& v_j{t}_{{\xi }_i},& \text{for all}\hspace{0.17em}1\le i,j\le n,\\ t_{s_i}{v}_k& =& v_k{t}_{s_i},& \text{if}\hspace{0.17em}k\notin \{i,i+1\},\\ t_{s_i}{v}_{i+1}& =& v_i{t}_{s_i}+\sum _{\ell =0}^{r-1}{t}_{{\xi }_i^{\ell }{\xi }_{i+1}^{-\ell }},& \text{for}\hspace{0.17em}1\le i\le n-1\text{.}\end{array}& \text{(5.1)}\end{array}$$ The following proposition establishes a "evaluation homomorphism" for the algebras $H_{r,1,n}^{*}$ which is a generalization of the homomorphism in (4.5).

Define elements ${\bar{v}}_k$ in the group algebra $ℂG(r,1,n)$ by setting ${\bar{v}}_1=0$ and $${\bar{v}}_k=\frac{1}{r}\sum _{i<k}\sum _{0\le \ell \le r-1}{t}_{{\xi }_i^{\ell }{\xi }_k^{-\ell }(i,k)},\text{for}\hspace{0.17em}2\le k\le n\text{.}$$ Then there is a surjective algebra homomorphism $$\begin{array}{ccc}{H}_{r,1,n}^{*}& ⟶& ℂG(r,1,n)\\ t_g& ⟼& t_g\\ v_k& ⟼& {\bar{v}}_k\end{array}$$

		Proof.
	We must check that the defining relations (5.1) of $H_{r,1,n}^{*}$ hold with the $v_k$ replaced by the ${\bar{v}}_k\text{.}$ For each $1\le k\le n,$ let $${\bar{z}}_k={\bar{v}}_1+\cdots +{\bar{v}}_k=\frac{1}{r}\sum _{\underset{0\le \ell \le r-1}{1\le i<j\le k}}{t}_{{\xi }_i^{\ell }{\xi }_j^{-\ell }(i,j)}\text{.}$$ Then, for each $k,$ ${\bar{z}}_k\in Z\left(ℂG(r,1,k)\right)$ since it is the sum of the elements of the conjugacy class of reflections $t_{{\xi }_i^{\ell }{\xi }_j^{-\ell }(i,j)}$ in $G(r,1,k)\text{.}$ So ${\bar{z}}_k$ commutes with ${\bar{z}}_1,\dots ,{\bar{z}}_k$ and therefore ${\bar{z}}_1,\dots ,{\bar{z}}_n$ commute. Since ${\bar{v}}_k={\bar{z}}_k-{\bar{z}}_{k-1},$ it follows that ${\bar{v}}_1,\dots ,{\bar{v}}_n$ also commute. If $m>k$ then $t_{{\xi }_m}$ clearly commutes with ${\bar{z}}_k\text{.}$ If $m\le k$ then $t_{{\xi }_m}$ commutes with ${\bar{z}}_k$ since ${\bar{z}}_k\in Z\left(G(r,1,k)\right)\text{.}$ So $t_{{\xi }_m}$ commutes with ${\bar{z}}_1,\dots ,{\bar{z}}_n$ and hence with ${\bar{v}}_1,\dots ,{\bar{v}}_n\text{.}$ Since $$\begin{array}{ccc}{t}_{s_k}{\bar{v}}_k{t}_{s_k}& =& t_{s_k}\left(\sum _{\underset{0\le \ell \le r-1}{i<k}}{t}_{{\xi }_i^{\ell }{\xi }_k^{-\ell }(i,k)}\right)t_{s_k}=\sum _{\underset{0\le \ell \le r-1}{i<k}}{t}_{{\xi }_i^{\ell }{\xi }_{k+1}^{-\ell }(i,k+1)}\\ & =& \sum _{\underset{0\le \ell \le r-1}{i<k+1}}{t}_{{\xi }_i^{\ell }{\xi }_{k+1}^{-\ell }(i,k+1)}-\sum _{0\le \ell \le r-1}{t}_{{\xi }_k^{\ell }{\xi }_{k+1}^{-\ell }(k,k+1)}\\ & =& {\bar{v}}_{k+1}-\sum _{0\le \ell \le r-1}{t}_{{\xi }_k^{\ell }{\xi }_{k+1}^{-\ell }}{t}_{s_k},\end{array}$$ it follows that $${\bar{v}}_k{t}_{s_k}=t_{s_k}{\bar{v}}_{k+1}-\sum _{0\le \ell \le r-1}{t}_{s_k}{t}_{{\xi }_k^{\ell }{\xi }_{k+1}^{-\ell }}{t}_{s_k}=t_{s_k}{\bar{v}}_{k+1}-\sum _{0\le \ell \le r-1}{t}_{{\xi }_k^{-\ell }{\xi }_{k+1}^{\ell }}=t_{s_k}{\bar{v}}_{k+1}-\sum _{0\le \ell \le r-1}{t}_{{\xi }_k^{\ell }{\xi }_{k+1}^{-\ell }}\text{.}$$ $\square$

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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