## The center of the affine BMW algebra ${W}_{k}$

The affine BMW algebra ${W}_{k}$ is an algebra over the commutative ring $C$ and the polynomial ring $C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]$ is a subalgebra of ${W}_{k}$. The symmetric group ${S}_{k}$ acts on $C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]$ by permuting the variables and the ring of symmetric functions is $C[ Y1±1 ,…, Yk±1 ] Sk = { f∈ C[y1,…, yk] | wf=f, for w∈Sk }.$ A classical fact (see, for example [GV, Proposition 2.1]) is that the center of the affine Hecke algebra ${H}_{k}$ is $Z(ℋk) = C[ Y1±1 ,…, Yk±1 ] Sk .$ The following theorem gives an analogous characterization of the center of the degenerate affine BMW algebra.

The center of the degenerate affine BMW algebra ${W}_{k}$ is $Rk = {f∈ C[ Y1±1 ,…, Yk±1 ] Sk | f(Y1, Y1-1, Y3, …,Yk) = f(1,1,Y3, …,Yk)}.$

Proof.
Step 1: $f\in {W}_{k}$ commutes with all ${y}_{i}$ $⇔$ $f\in C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]$:
Assume $f\in {W}_{k}$ and write $f= ∑ cd n1,…, nk Td n1,…, nk$ in terms of the basis in Theorem 3.1?????????. Let $d\in {D}_{k}$ with the maximal number of crossings such that ${c}_{d}^{{n}_{1},\dots ,{n}_{k}}\ne 0$ and suppose there is an edge $\left(i,j\right)$ of $d$ such that $j\ne i\prime$. Then, by (4.22) and (4.23), $the coefficient of Yi Td n1,…, nk in Yif is cd n1,…, nk$ and $the coefficient of Yi Td n1,…, nk in fYi is 0.$ If ${Y}_{i}f=f{Y}_{i}$, it follows that there is no such edge, and so $d=1$ (and therefore ${T}_{d}=1$). Thus $f\in C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]$. Conversely, if $f\in C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]$ then ${Y}_{i}f=f{Y}_{i}$.

Step 2. $f\in C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]$ commutes with all ${T}_{i}$ $⇔$ $f\in {R}_{k}$ :
Assume $f\in C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]$ and write

 $f=\sum _{a,b\in ℤ}{Y}_{1}^{a}{Y}_{2}^{b}{f}_{a,b}$,    where ${f}_{a,b}\in C\left[{Y}_{3}^{±1},\dots ,{Y}_{k}^{±1}\right]$.
Then $f\left(1,1,{Y}_{3},\dots {Y}_{k}\right)=\sum _{a,b\in ℤ}{f}_{a,b}$ and
 $f\left({Y}_{1},{Y}_{1}^{-1},{Y}_{3},\dots ,{Y}_{k}\right)=\sum _{a,b\in ℤ}{Y}_{1}^{a-b}{f}_{a,b}=\sum _{\ell \in ℤ}{Y}_{1}^{\ell }\left(\sum _{b\in ℤ}{f}_{\ell +b,b}\right)$. (4.25)
By direct computation using (3.31) and (3.33),
 ${T}_{1}{Y}_{1}^{a}{Y}_{2}^{b}={Y}_{1}^{a}{Y}_{1}^{a}{T}_{1}{Y}_{2}^{b-a}={s}_{1}\left({Y}_{1}^{a}{Y}_{2}^{b}\right){T}_{1}+\left(q-{q}^{-1}\right)\frac{{Y}_{1}^{a}{Y}_{2}^{b}-{s}_{1}\left({Y}_{1}^{a}{Y}_{2}^{b}\right)}{1-{Y}_{1}{Y}_{2}^{-1}}+{ℰ}_{b-a}$,
where
 ${ℰ}_{\ell }=\left\{\begin{array}{ll}-\left(q-{q}^{-1}\right)\sum _{r=1}^{\ell }{Y}_{1}^{\ell -r}{E}_{1}{Y}_{1}^{-r},& \text{if}\phantom{\rule{0.5em}{0ex}}\ell >0,\\ \left(q-{q}^{-1}\right)\sum _{r=1}^{-\ell }{Y}_{1}^{\ell +r-1}{E}_{1}{Y}_{1}^{r-1},& \text{if}\phantom{\rule{0.5em}{0ex}}\ell <0,\\ 0,& \text{if}\phantom{\rule{0.5em}{0ex}}\ell =0.\end{array}$
It follows that
 ${T}_{1}f=\left({s}_{1}f\right){T}_{1}+\left(q-{q}^{-1}\right)\frac{f-{s}_{1}f}{1-{Y}_{1}{Y}_{2}^{-1}}+\sum _{\ell \in {ℤ}_{\ne 0}}{ℰ}_{\ell }\left(\sum _{b\in ℤ}{f}_{\ell +b,b}\right)$. (4.26)
Thus if $f\left({Y}_{1},{Y}_{1}^{-1},{Y}_{3},\dots ,{Y}_{k}\right)=f\left(1,1,{Y}_{3},\dots ,{Y}_{k}\right)$ then, by (4.25),
 $\sum _{b\in ℤ}{f}_{\ell +b,b}=0,\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}\ell \ne 0$. (4.27)
Hence, if $f\in {C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]}^{{S}_{k}}$ and $f\left({Y}_{1},{Y}_{1}^{-1},{Y}_{3},\dots ,{Y}_{k}\right)=f\left(1,1,{Y}_{3},\dots ,{Y}_{k}\right)$ then ${s}_{1}f=f$ and (4.27) holds so that, by (4.26), ${T}_{1}f=f{T}_{1}$. Similarly, $f$ commutes with all ${T}_{i}$.
Conversely, if $f\in C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]$ and ${T}_{i}f=f{T}_{i}$ then
 ${s}_{i}f=f$     and     $\sum _{b\in ℤ}{f}_{\ell +b,b}=0,\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}\ell \ne 0$,
so that $f\in {C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]}^{{S}_{k}}$ and $f\left({Y}_{1},{Y}_{1}^{-1},{Y}_{3},\dots ,{Y}_{k}\right)=f\left(1,1,{Y}_{3},\dots ,{Y}_{k}\right)$. □

The symmetric group ${S}_{k}$ acts on ${ℤ}^{k}$ by permuting the factors. The ring

 ${C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]}^{{S}_{k}}$     has basis     $\left\{{m}_{\lambda }\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}\lambda \in {ℤ}^{k}\phantom{\rule{0.5em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}{\lambda }_{1}\ge {\lambda }_{2}\ge \cdots {\lambda }_{k}\right\},$
where
 ${m}_{\lambda }=\sum _{\gamma \in {S}_{k}\lambda }{Y}_{1}^{{\gamma }_{1}}\cdots {Y}_{k}^{{\gamma }_{k}}$.
The elementary symmetric functions are
 ${e}_{r}={m}_{\left({1}^{r},{0}^{k-r}\right)}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{e}_{-r}={m}_{\left({0}^{k-r},{\left(-1\right)}^{r}\right)},\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}r=0,1,\dots ,k,$
and the power sum symmetric functions are
 ${p}_{r}={m}_{\left(r,{0}^{k-1}\right)}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{p}_{-r}={m}_{\left({0}^{k-1},-r\right)},\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}r\in {ℤ}_{\ge 0}$.
The Newton identities (see [Mac, Ch. I (2.11′)]) say
 $\ell {e}_{\ell }=\sum _{r=1}^{\ell }{\left(-1\right)}^{r-1}{p}_{r}{e}_{\ell -r}$     and     $\ell {e}_{-\ell }=\sum _{r=1}^{\ell }{\left(-1\right)}^{r-1}{p}_{-r}{e}_{-\left(\ell -r\right)}$,
where the second equation is obtained from the first by replacing ${Y}_{i}$ with ${Y}_{i}^{-1}$. For $\ell \in ℤ$ and $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{k}\right)\in {ℤ}^{k}$,
 ${e}_{k}^{\ell }{m}_{\lambda }={m}_{\lambda +\left({\ell }^{k}\right)}$,     where  $\lambda +\left({\ell }^{k}\right)=\left({\lambda }_{1}+\ell ,\dots ,{\lambda }_{k}+\ell \right)$.
In particular,
 ${e}_{-r}={e}_{k}^{-1}{e}_{k-r},\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}r=0,1,\dots k$. (4.29)
Define
 ${p}_{i}^{+}={p}_{i}+{p}_{-i},\phantom{\rule{2em}{0ex}}{p}_{i}^{-}={p}_{i}-{p}_{-i},\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}i\in {ℤ}_{>0}$.
The consequence of (4.29) and (4.28) is that
 $\begin{array}{rl}{ℚ\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]}^{{S}_{k}}& =ℚ\left[{e}_{k}^{±1},{e}_{1},\dots ,{e}_{k-1}\right]\\ & =ℚ\left[{e}_{k}^{±1}\right]\left[{e}_{1},{e}_{2},\dots {e}_{⌊\frac{k}{2}⌋},{e}_{k}{e}_{-⌊\frac{k-1}{2}⌋},{e}_{k}{e}_{-2},{e}_{k}{e}_{-1}\right]\\ & =ℚ\left[{e}_{k}^{±1}\right]\left[{e}_{1},{e}_{2},\dots {e}_{⌊\frac{k}{2}⌋},{e}_{-⌊\frac{k-1}{2}⌋},{e}_{-2},{e}_{-1}\right]\\ & =ℚ\left[{e}_{k}^{±1}\right]\left[{p}_{1},{p}_{2},\dots {p}_{⌊\frac{k}{2}⌋},{p}_{-⌊\frac{k-1}{2}⌋},{p}_{-2},{p}_{-1}\right]\\ & =ℚ\left[{e}_{k}^{±1}\right]\left[{p}_{1}^{+},{p}_{2}^{+},\dots {p}_{⌊\frac{k}{2}⌋}^{+},{p}_{⌊\frac{k-1}{2}⌋}^{-},{p}_{2}^{-},{p}_{1}^{-}\right].\end{array}$
For $\nu \in {ℤ}^{k}$ with ${\nu }_{1}\ge \cdots \ge {\nu }_{\ell }>0$ define
 ${p}_{\nu }^{+}={p}_{{\nu }_{1}}^{+}\cdots {p}_{{\nu }_{\ell }}^{+}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{p}_{\nu }^{-}={p}_{{\nu }_{1}}^{-}\cdots {p}_{{\nu }_{\ell }}^{-}$.
Then
 ${C\left[{Y}_{1}^{±1},\dots ,{Y}_{k}^{±1}\right]}^{{S}_{k}}\phantom{\rule{2em}{0ex}}\text{has basis}\phantom{\rule{2em}{0ex}}\left\{{e}_{k}^{\ell }{p}_{\lambda }^{+}{p}_{\mu }^{-}\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}\ell \in ℤ,\phantom{\rule{0.5em}{0ex}}\ell \left(\lambda \right)\le ⌊\frac{k}{2}⌋,\ell \left(\mu \right)\le ⌊\frac{k-1}{2}⌋\right\}$. (4.29)
In analogy with (4.12) we expect that if ${R}_{k}$ is as in Theorem 4.6 then
 ${R}_{k}=C\left[{e}_{k}^{±1}\right]\left[{p}_{1}^{-},{p}_{2}^{-},\dots \right]$. (4.30)

The left ideal of ${W}_{2}$ generated by ${E}_{1}$ is $C\left[{Y}_{1}^{±1}\right]{E}_{1}$. This is an infinite dimensional (generically irreducible) ${W}_{2}$-module on which $Z\left({W}_{2}\right)$ acts by constants. Thus, as noted by IS THIS IN RUI SI?????, it follows that ${W}_{2}$ is not finitely generated as a $Z\left({W}_{2}\right)$-module.

## Notes and references

This page is the result of joint work with Zajj Daugherty and Rahbar Virk [DRV]. The characterisation of the center of the affine BMW algebra given here is analogous to that for the degenerate affine BMW algebra as found in (???). PUT A REMARK ABOUT the solution of [St. Exercise 7.7] as found on [St., p. 497] in THIS CONTEXT???

## References

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