Affine and degenerate affine BMW algebras: Actions on tensor space

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

Last update: 15 October 2013

Central element transfer via Schur-Weyl duality

In Theorem 2.2 and Theorem 2.5, the parameters $$z_0^{\left(\ell \right)}=\epsilon (\text{id}\otimes {\text{tr}}_V)\left({(\frac{1}{2}y+\gamma )}^{\ell }\right)\text{and}{Z}_0^{\left(\ell \right)}=\epsilon (\text{id}\otimes {\text{qtr}}_V)\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)$$ of the degenerate affine BMW algebra and affine BMW algebra, respectively, arise naturally from the action on tensor space. It is a consequence of [Dri1990, Prop. 1.2] that these are central elements of the enveloping algebra $U𝔤$ and the quantum group $U_h𝔤,$ respectively: $$z_0^{\left(\ell \right)}\in Z\left(U𝔤\right)\text{and}{Z}_0^{\left(\ell \right)}\in Z\left(U_h𝔤\right)\text{.}$$ The Harish-Chandra isomorphism provides isomorphisms between the centers $Z\left(U𝔤\right)$ or $Z\left(U_h𝔤\right)$ and rings of symmetric functions. In this section we show how to use the recursive formulas of [Naz1996] and [BBl1866492] for the central elements $z_k^{\left(\ell \right)}$ and $Z_k^{\left(\ell \right)}$ in the degenerate affine and affine BMW algebras (formulas (3.4) and (3.14)) to determine the Harish-Chandra images of $z_0^{\left(\ell \right)}$ and $Z_0^{\left(\ell \right)}\text{.}$

Preliminaries on the Harish-Chandra isomorphisms. Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form. The triangular decomposition $𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+}$ (see [Bou1990, VII §8 no. 3 Prop. 9]) yields triangular decompositions of both the enveloping algebra $U=U𝔤$ and the quantum group $U=U_h𝔤$ in the form $U=U^{-}{U}_0{U}^{+}\text{.}$ If $U=U_h𝔤$ then $U_0=\text{span}\left\{K^{{\lambda }^{\vee }}\hspace{0.17em}\right|\hspace{0.17em}{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\}$ with $K^{{\lambda }^{\vee }}{K}^{{\nu }^{\vee }}=K^{{\lambda }^{\vee }+{\nu }^{\vee }},$ where ${𝔥}_{\mathbb{Z}}$ is a lattice in $𝔥\text{.}$ Alternatively, $$U_0=U𝔥=ℂ[h_1,\dots ,h_r]\text{if}\hspace{0.17em}U=U𝔤\text{and}{U}_{-}=ℂ[L_1^{\pm 1},\dots ,L_r^{\pm 1}]\text{if}\hspace{0.17em}U=U_h𝔤,$$ where $h_1,\dots ,h_r$ is a basis of ${𝔥}_{\mathbb{Z}},$ and $L_i=K^{h_i}=q^{h_i}\text{.}$

For $\mu \in {𝔥}^{*},$ define the ring homomorphisms ${\text{ev}}_{\mu }:U_0\to ℂ$ by $$\begin{array}{cc}{\text{ev}}_{\mu }\left(h\right)=⟨\mu ,h⟩\text{and}{\text{ev}}_{\mu }\left(K^{{\lambda }^{\vee }}\right)=z^{⟨\mu ,{\lambda }^{\vee }⟩}& \text{(3.1)}\end{array}$$ for $h\in 𝔥$ and $K^{{\lambda }^{\vee }}$ with ${\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\text{.}$ For $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$ as in (1.16), let ${\sigma }_{\rho }$ be the algebra automorphism given by $$\begin{array}{cc}{\sigma }_{\rho }\left(h_i\right)=h+⟨\rho ,h_i⟩\text{and}\hspace{0.17em}{\sigma }_{\rho }\left(L_i\right)=q^{⟨\rho ,h_i⟩}{L}_i\text{.}& \text{(3.2)}\end{array}$$

Define a vector space homomorphism by $$\begin{array}{cc}{\pi }_0:U⟶U_0\text{by}{\pi }_0={\epsilon }^{-}\otimes \text{id}\otimes {\epsilon }^{+}:U_{-}\otimes U_0\otimes U_{+}⟶U_0,& \text{(3.3)}\end{array}$$ where ${\epsilon }^{-}:U_{-}\to ℂ$ and ${\epsilon }^{+}:U_{+}\to ℂ$ are the algebra homomorphisms determined by $$\begin{array}{c}{\epsilon }^{-}\left(y\right)=0\text{and}{\epsilon }^{+}\left(x\right)=0,\text{for}\hspace{0.17em}x\in {𝔫}^{+}\hspace{0.17em}\text{and}\hspace{0.17em}y\in {𝔫}^{-}\text{, or}\\ {\epsilon }^{-}\left(F_i\right)=0\text{and}{\epsilon }^{+}\left(E_i\right)=0,\text{for}\hspace{0.17em}i=1,\dots ,n\text{.}\end{array}$$

The following important theorem says that both the center of $U𝔤$ and the center of $U_h𝔤$ are isomorphic to rings of symmetric functions.

(Harish-Chandra/Chevalley isomorphism, [Bou1990, VII §8 no. 5 Thm. 2] and [CPr1994, Thm. 9.1.6]). Let $U=U𝔤$ or $U_h𝔤,$ so that $$U_0=ℂ[h_1,\dots ,h_r]\text{if}\hspace{0.17em}U=U𝔤\text{and}{U}_0=ℂ[L_1^{\pm 1},\dots ,L_r^{\pm 1}]\text{if}\hspace{0.17em}U=U_h𝔤\text{.}$$ Let $L\left(\mu \right)$ denote the irreducible $U\text{-module}$ of highest weight $\mu \text{.}$ Then the restriction of ${\pi }_0$ to the center of $U,$ $$\begin{array}{cccc}{\pi }_0:& Z\left(U\right)& ⟶& {\sigma }_{\rho }\left(U_0^{W_0}\right),\\ & z& ⟼& {\sigma }_{\rho }\left(s\right)\end{array}\text{is an algebra isomorphism,}$$ where $s\in U_0^{W_0}$ is the symmetric function determined by $$z\text{acts on}\hspace{0.17em}L\left(\mu \right)\hspace{0.17em}\text{by}{\text{ev}}_{\mu }\left({\sigma }_{\rho }\left(s\right)\right)={\text{ev}}_{\mu +\rho }\left(s\right),\text{for}\hspace{0.17em}\mu \in {𝔥}^{*}\text{.}$$

Central elements $z_V^{\left(\ell \right)}$

Let $z_0^{\left(\ell \right)}$ and $\epsilon$ be the parameters of the degenerate affine BMW algebra ${𝒲}_k\text{.}$ Let $u$ be a variable and define $z_i^{\left(\ell \right)}\in {𝒲}_k$ for $i=1,\dots ,k-1$ by $$\begin{array}{cc}{z}_i\left(u\right)+\epsilon u-\frac{1}{2}=(z_0\left(u\right)+\epsilon u-\frac{1}{2})\prod _{j=1}^i\frac{(u+y_j-1)(u+y_j+1){(u-y_j)}^2}{{(u+y_j)}^2(u-y_j+1)(u-y_j-1)},& \text{(3.4)}\end{array}$$ where $$z_i\left(u\right)=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{z}_i^{\left(\ell \right)}{u}^{-\ell },\text{for}\hspace{0.17em}i=0,1,\dots ,k-1\text{.}$$ The following proposition from [Naz1996, Lemma 3.8] is proved also in [DRV1105.4207, Theorem 3.2 and Remark 3.4]

In the degenerate affine BMW algebra ${𝒲}_k,$ $$e_{i+1}{y}_{i+1}^{\ell }{e}_{i+1}=z_i^{\left(\ell \right)}{e}_{i+1},\text{for}\hspace{0.17em}i=0,\dots ,k-1\hspace{0.17em}\text{and}\hspace{0.17em}\ell \in {\mathbb{Z}}_{\ge 0}\text{.}$$

The following theorem uses the identity (3.4) and the action of the degenerate affine BMW algebra on tensor space to provide a formula for the Harish-Chandra images of the central elements $z_V^{\left(\ell \right)}=\epsilon (\text{id}\otimes {\text{tr}}_V)\left({(\frac{1}{2}y+\gamma )}^{\ell }\right)$ in the enveloping algebra $U𝔤$ for orthogonal and symplectic Lie algebras $𝔤\text{.}$ By Theorem 2.2 these particular central elements are natural parameters for the degenerate affine BMW algebras. The concept of the proof of Theorem 3.3 is, at its core, the same as the pattern taken by Nazarov for the proof of [Naz1996, Theorem 3.9].

Let $𝔤={𝔰𝔬}_{2r+1},$ ${𝔰𝔭}_{2r}$ or ${𝔰𝔬}_{2r},$ use notations for ${𝔥}^{*}$ as in (2.5)-(2.16) and let $h_1,\dots ,h_r$ be the basis of $𝔥$ dual to the orthonormal basis ${\epsilon }_1,\dots ,{\epsilon }_r$ of ${𝔥}^{*}$ (so that $h_i=F_{ii},$ where $F_{ii}$ is as in (2.8)). With respect to the form $⟨,⟩$ in (2.10), let $\gamma ={\sum }_{b}b\otimes b^{*}$ as in (1.15). Let $$y=\{\begin{array}{cc}2r,& \text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1},\\ 2r+1,& \text{if}\hspace{0.17em}𝔤={𝔰𝔭}_{2r},\\ 2r-1,& \text{if}\hspace{0.17em}g={𝔰𝔬}_{2r},\end{array}\epsilon =\{\begin{array}{cc}1,& \text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1},\\ -1,& \text{if}\hspace{0.17em}𝔤={𝔰𝔭}_{2r},\\ 1,& \text{if}\hspace{0.17em}g={𝔰𝔬}_{2r},\end{array}V=L\left({\epsilon }_1\right),$$ and let $z_V^{\left(\ell \right)}$ be the central elements in $U𝔤$ defined by $$z_V^{\left(\ell \right)}=\epsilon (\text{id}\otimes {\text{tr}}_V)\left({(\frac{1}{2}y+\gamma )}^{\ell }\right),\text{and write}{z}_V\left(u\right)=\sum _{i\in {\mathbb{Z}}_{\ge 0}}{z}_V^{\left(\ell \right)}{u}^{-\ell }\text{.}$$ Then $${\pi }_0(z_V+\epsilon u-\frac{1}{2})=(\epsilon u+\frac{1}{2})\frac{(u+\frac{1}{2}y-r)}{(u-\frac{1}{2}y+r)}{\sigma }_{\rho }\left(\prod _{i=1}^r\frac{(u+h_i+\frac{1}{2})(u-h_i+\frac{1}{2})}{(u+h_i-\frac{1}{2})(u-h_i-\frac{1}{2})}\right),$$ where ${\sigma }_{\rho }$ is the algebra automorphism given by ${\sigma }_{\rho }\left(h_i\right)=h_i+⟨\rho ,{\epsilon }_i⟩$ and ${\pi }_0$ is the isomorphism in Theorem 3.1.

		Proof.
	In the definition of the action of the degenerate affine BMW algebra in Theorem 2.2, $y_1$ acts on $M\otimes V$ as $\frac{1}{2}y+\gamma ,$ and $$e_1{y}_1^{\ell }{e}_1\hspace{0.17em}\text{acts on}\hspace{0.17em}M\otimes V^{\otimes 2}\hspace{0.17em}\text{as}\hspace{0.17em}{z}_V^{\left(\ell \right)}{e}_1\text{.}$$ Also $$e_1\hspace{0.17em}\text{and}\hspace{0.17em}{y}_1\hspace{0.17em}\text{in}\hspace{0.17em}{𝒲}_2\hspace{0.17em}\text{act on}\hspace{0.17em}M\otimes V^{\otimes 2}\hspace{0.17em}\text{with}\hspace{0.17em}M=L\left(0\right)\otimes V^{\otimes (k-1)}$$ in the same way that $$e_k\hspace{0.17em}\text{and}\hspace{0.17em}{y}_k\hspace{0.17em}{𝒲}_{k+1}\hspace{0.17em}\text{act on}\hspace{0.17em}M\otimes V^{\otimes (k+1)}\hspace{0.17em}\text{with}\hspace{0.17em}M=L\left(0\right)\text{.}$$ By Proposition 3.2, $z_{k-1}^{\left(\ell \right)}{e}_k=e_k{y}_k^{\ell }{e}_k\text{.}$ Hence, as operators on $L\left(0\right)\otimes V^{\otimes (k-1)},$ $$\begin{array}{cc}{z}_V\left(u\right)+\epsilon u-\frac{1}{2}=z_{k-1}\left(u\right)+\epsilon u-\frac{1}{2}& \text{(3.5)}\end{array}$$ We will use (3.4) to compute the action of this operator on the $L\left(\mu \right)\otimes {𝒲}_{k-1}^{\mu }$ isotypic component in the $U𝔤\otimes {𝒲}_{k-1}\text{-module}$ decomposition $$\begin{array}{cc}L\left(0\right)\otimes V^{\otimes (k-1)}\cong \underset{\mu }{⨁}L\left(\mu \right)\otimes {𝒲}_{k-1}^{\mu }\text{.}& \text{(3.6)}\end{array}$$ As an operator on $L\left(0\right)\otimes V,$ $$\gamma =\frac{1}{2}(⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩-⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩+⟨0,0+2\rho ⟩)=0\text{by (1.17),}$$ and so $$z_0^{\left(\ell \right)}=\epsilon (i\otimes {\text{tr}}_V)\left({(\frac{1}{2}y+\gamma )}^{\ell }\right)=\epsilon (\text{id}\otimes {\text{tr}}_V)\left({\left(\frac{1}{2}y\right)}^{\ell }\right)=\epsilon \text{dim}\left(V\right){\left(\frac{1}{2}y\right)}^{\ell }\text{.}$$ Therefore, since $\text{dim}\left(V\right)=\epsilon +y,$ $$z_0\left(u\right)=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{z}_V^{\left(\ell \right)}{u}^{-\ell }=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\epsilon \text{dim}\left(V\right){\left(\frac{1}{2}y\right)}^{\ell }{u}^{-\ell }=\epsilon \text{dim}\left(V\right)\frac{1}{1-\frac{1}{2}y{u}^{-1}}=\frac{1+\epsilon y}{1-\frac{1}{2}y{u}^{-1}}\text{.}$$ Thus, as an operator on $L\left(0\right)\otimes V,$ $$\begin{array}{cc}{z}_0\left(u\right)+\epsilon u-\frac{1}{2}=\frac{1+\epsilon y}{1-\frac{1}{2}y{u}^{-1}}+\epsilon u-\frac{1}{2}=\frac{(\epsilon u+\frac{1}{2})(u+\frac{1}{2}y)}{u-\frac{1}{2}y}\text{.}& \text{(3.7)}\end{array}$$ By the first identity in (1.11) and the definition of $\Phi$ in Theorem 1.2, $$y_k\in {𝒲}_k\text{acts on}L\left(0\right)\otimes V^{\otimes k}=(L\left(0\right)\otimes V^{\otimes (k-1)})\otimes V\text{as}\frac{1}{2}y+\gamma \text{.}$$ If $L\left(\mu \right)$ is an irreducible $U𝔤\text{-module}$ in $L\left(0\right)\otimes V^{\otimes (k-1)},$ then (1.17), (2.13), and (2.16) give that $y_k$ acts on the $L\left(\lambda \right)$ component of $L\left(\mu \right)\otimes V$ by the constant $c(\lambda ,\mu )=0$ when $\lambda =\mu ,$ and by the constant $$\begin{array}{cc}\begin{array}{ccc}c(\lambda ,\mu )& =& \frac{1}{2}y+\frac{1}{2}(⟨\mu \pm {\epsilon }_i,\mu \pm {\epsilon }_i+2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩)\\ & =& \{\begin{array}{cc}\frac{1}{2}y+c(\lambda /\mu ),& \text{if}\hspace{0.17em}\mu \subseteq \lambda ,\\ -\frac{1}{2}y-c(\mu /\lambda ),& \text{if}\hspace{0.17em}\mu \supseteq \lambda ,\end{array}\text{where}\hspace{0.17em}\lambda =\mu \pm {\epsilon }_i\text{.}\end{array}& \text{(3.8)}\end{array}$$ As in [Naz1996, Theorem 2.6], the irreducible ${𝒲}_k\text{-module}$ ${𝒲}_k^{\mu /0}={𝒲}_k^{\mu }$ has a basis $\left\{v_T\right\}$ indexed by up-down tableaux $T=(T^{\left(0\right)},T^{\left(1\right)},\dots ,T^{\left(k\right)}),$ where $T^{\left(0\right)}=\varnothing ,$ $T^{\left(k\right)}=\mu ,$ and $T^{\left(i\right)}$ is a partition obtained from $T^{(i-1)}$ by adding or removing a box (or, in some cases when $𝔤={𝔰𝔬}_{2r+1}$ leaving the partition the same; see (2.21)) and $$y_i{v}_T=\{\begin{array}{cc}(\frac{1}{2}y+c\left(b\right))v_T,& \text{if}\hspace{0.17em}b=T^{\left(i\right)}/T^{(i-1)},\\ (-\frac{1}{2}y-c\left(b\right))v_T,& \text{if}\hspace{0.17em}b=T^{(i-1)}/T^{\left(i\right)},\\ 0,& \text{if}\hspace{0.17em}{T}^{(i-1)}=T^{\left(i\right)}\text{.}\end{array}$$ Thus the product on the right hand side of (3.4) $$\prod _{i=1}^{k-1}\frac{(u+y_i-1)(u+y_i+1){(u-y_i)}^2}{{(u+y_i)}^2(u-y_i+1)(u-y_i-1)}\text{acts on}\hspace{0.17em}L\left(\mu \right)\otimes {𝒲}_{k-1}^{\mu }\hspace{0.17em}\text{in (3.6)}$$ by $$\begin{array}{cc}\prod _{i=1}^{k-1}\frac{(u+c(T^{\left(i\right)},T^{(i-1)})-1)(u+c(T^{\left(i\right)},T^{(i-1)})+1){(u-c(T^{\left(i\right)},T^{(i-1)}))}^2}{{(u+c(T^{\left(i\right)},T^{(i-1)}))}^2(u-c(T^{\left(i\right)},T^{(i-1)})+1)(u-c(T^{\left(i\right)},T^{(i-1)})-1)}& \text{(3.9)}\end{array}$$ for any up-down tableau $T$ of length $k$ and shape $\mu \text{.}$ If a box is added (or removed) at step $i$ and then removed (or added) at step $j,$ then the $i$ and $j$ factors of this product cancel. Therefore (3.9) is equal to $$\begin{array}{cc}\prod _{b\in \mu }\frac{(u+\frac{1}{2}y+c\left(b\right)-1)(u+\frac{1}{2}y+c\left(b\right)+1){(u-\frac{1}{2}y-c\left(b\right))}^2}{{(u+\frac{1}{2}y+c\left(b\right))}^2(u-\frac{1}{2}y-c\left(b\right)+1)(u-\frac{1}{2}y-c\left(b\right)-1)}& \text{(3.10)}\end{array}$$ (see [Naz1996, Lemma 3.8]). If $\mu =({\mu }_1,\dots ,{\mu }_r),$ simplifying one row at a time, $$\begin{array}{cc}\begin{array}{ccc}\prod _{b\in \mu }\frac{(u+\frac{1}{2}y+c\left(b\right)-1)(u+\frac{1}{2}y+c\left(b\right)+1)}{{(u+\frac{1}{2}y+c\left(b\right))}^2}& =& \prod _{i=1}^r\frac{(u+\frac{1}{2}y-i)(u+\frac{1}{2}y+{\mu }_i-i+1)}{(u+\frac{1}{2}y+1-i)(u+\frac{1}{2}y+{\mu }_i-i)}\\ & =& \frac{u+\frac{1}{2}y-r}{u+\frac{1}{2}y}\prod _{i=1}^r\frac{(u+\frac{1}{2}y+{\mu }_i-i+1)}{(u+\frac{1}{2}y+{\mu }_i-i)},\end{array}& \text{(3.11)}\end{array}$$ (see the example following this proof). It follows that (3.10) is equal to $$\begin{array}{cc}\begin{array}{c}\frac{(u+\frac{1}{2}y-r)}{(u+\frac{1}{2}y)}\frac{(u-\frac{1}{2}y)}{(u-\frac{1}{2}y+r)}\prod _{i=1}^r\frac{(u+\frac{1}{2}y+{\mu }_i-i+1)}{(u+\frac{1}{2}y+{\mu }_i-i)}\frac{(u-\frac{1}{2}y-({\mu }_i-i))}{(u-\frac{1}{2}y-({\mu }_i-i+1))}\\ =\frac{(u+\frac{1}{2}y-r)}{(u+\frac{1}{2}y)}\frac{(u-\frac{1}{2}y)}{(u-\frac{1}{2}y+r)}{\text{ev}}_{\mu +\rho }\left(\prod _{i=1}^r\frac{(u+h_i+\frac{1}{2})}{(u+h_i-\frac{1}{2})}\frac{(u-h_i+\frac{1}{2})}{(u-h_i-\frac{1}{2})}\right),\end{array}& \text{(3.12)}\end{array}$$ since ${\text{ev}}_{\mu +\rho }\left(h_i\right)={\mu }_i+{\rho }_i={\mu }_i+\frac{1}{2}(y-2i+1)=\frac{1}{2}y+\frac{1}{2}+{\mu }_i-i\text{.}$ Combining (3.7) and (3.12), the identity (3.5) gives that, as operators on $L\left(\mu \right)\otimes {𝒲}_{k-1}^{\mu }$ in (3.6), $$z_V\left(u\right)+\epsilon u-\frac{1}{2}=(\epsilon u+\frac{1}{2})\frac{(u+\frac{1}{2}y-r)}{(u-\frac{1}{2}y+r)}{\text{ev}}_{\mu +\rho }\left(\prod _{i=1}^r\frac{(u+h_i+\frac{1}{2})}{(u+h_i-\frac{1}{2})}\frac{(u-h_i+\frac{1}{2})}{(u-h_i-\frac{1}{2})}\right)\text{.}$$ By Theorem 3.1, the desired result follows. $\square$

Example. To help illuminate the cancellation done in (3.11), let $\mu =(5,5,3,3,1,1),$ where the contents of boxes are $$\begin{array}{cc} 0 1 2 3 4 -1 0 1 2 3 -2 -1 0 -3 -2 -1 -4 -5 & \text{(3.13)}\end{array}$$ In this example, the product over the boxes in the first row of the diagram is $$\begin{array}{cc}\prod _{b\hspace{0.17em}\text{in first row of}\hspace{0.17em}\mu }& \frac{(x+c\left(b\right)-1)(x+c\left(b\right)+1)}{(x+c\left(b\right))(x+c\left(b\right))}\\ & =\frac{(x-1)(x+1)}{(x+0)(x+0)}\frac{(x+0)(x+2)}{(x+1)(x+1)}\frac{(x+1)(x+3)}{(x+2)(x+2)}\frac{(x+2)(x+4)}{(x+3)(x+3)}\frac{(x+3)(x+5)}{(x+4)(x+5)}\\ & =\frac{(x-1)(x+5)}{(x+0)(x+4)},\text{where}\hspace{0.17em}x=u+\frac{1}{2}y\text{.}\end{array}$$ Thus, simplifying the product one row at a time, $$\begin{array}{cc}\prod _{b\in \mu }& \frac{(x+c\left(b\right)-1)(x+c\left(b\right)+1)}{(x+c\left(b\right))(x+c\left(b\right))}\\ & =\frac{(x-1)(x+5)}{(x+0)(x+4)}\frac{(x-2)(x+4)}{(x-1)(x+3)}\frac{(x-3)(x+1)}{(x-2)(x+0)}\frac{(x-4)(x+0)}{(x-3)(x-1)}\frac{(x-5)(x-3)}{(x-4)(x-4)}\frac{(x-6)(x-4)}{(x-5)(x-5)}\\ & =\frac{(x-6)}{(x+0)}\frac{(x+5)}{(x+4)}·\frac{(x+4)}{(x+3)}·\frac{(x+1)}{(x+0)}·\frac{(x+0)}{(x-1)}·\frac{(x-3)}{(x-4)}·\frac{(x-4)}{(x-5)}\end{array}$$ leads to the identity $$\prod _{b\in \mu }\frac{(x+c\left(b\right)-1)(x+c\left(b\right)+1)}{(x+c\left(b\right))(x+c\left(b\right))}=\frac{x-r}{x+0}\prod _{i=1}^r\frac{x+{\mu }_i-i+1}{x+{\mu }_i-i},\text{where}\hspace{0.17em}\mu =({\mu }_1,\dots ,{\mu }_r)\text{.}$$

Central elements $Z_V^{\left(\ell \right)}$

Let $Z_0^{\left(\ell \right)},$ $z$ and $q$ be the parameters of the affine BMW algebra $W_k\text{.}$ Let $u$ be a variable and define $Z_i^{\left(\ell \right)},Z_i^{(-\ell )}\in W_k$ for $i=1,\dots ,k-1$ by $$\begin{array}{cc}{Z}_i^{+}\left(u\right)+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1}=(Z_0^{+}+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1})\prod _{j=1}^i\frac{{(u-Y_j)}^2(u-q^{-2}{Y}_j^{-1})(u-q^2{Y}_j^{-1})}{{(u-Y_j^{-1})}^2(u-q^2{Y}_j)(u-q^{-2}{Y}_j)},& \text{(3.14)}\\ Z_i^{-}\left(u\right)+\frac{z}{q-q^{-1}}-\frac{u^2}{u^2-1}=(Z_0^{-}-\frac{z}{q-q^{-1}}-\frac{u^2}{u^2-1})\prod _{j=1}^i\frac{{(u-Y_j^{-1})}^2(u-q^2{Y}_j)(u-q^{-2}{Y}_j)}{{(u-Y_j)}^2(u-q^{-2}{Y}_j^{-1})(u-q^2{Y}_j^{-1})},& \text{(3.15)}\end{array}$$ where $$Z_i^{+}\left(u\right)=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{Z}_i^{\left(\ell \right)}{u}^{-\ell }\text{and}{Z}_i^{-}\left(u\right)=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{Z}_i^{(-\ell )}{u}^{-\ell }\text{for}\hspace{0.17em}i=0,\dots ,k-1\text{.}$$ The following proposition is equivalent to [BBl1866492, Lemma 7.4] and is also proved in [DRV1105.4207, Theorem 3.6 and Remark 3.8].

In the affine BMW algebra $W_k,$ $$E_{i+1}{Y}_i^{\ell }{E}_{i+1}=Z_i^{\left(\ell \right)}{E}_{i+1},\text{for}\hspace{0.17em}i=0,1,\dots ,k-2\hspace{0.17em}\text{and}\hspace{0.17em}\ell \in \mathbb{Z}\text{.}$$

The following theorem uses the identity (3.14) and the action of the affine BMW algebra on tensor space to provide a formula for the Harish-Chandra images of the central elements $Z_V^{\left(\ell \right)}=\epsilon (\text{id}\otimes {\text{qtr}}_V)\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)$ in the Drinfeld-Jimbo quantum group $U_h𝔤$ for orthogonal and symplectic Lie algebras $𝔤\text{.}$ By Theorem 2.5 these central elements are natural parameters for the affine BMW algebras.

Let $U=U_h𝔤$ be the Drinfeld-Jimbo quantum group corresponding to $𝔤={𝔰𝔬}_{2r+1},$ ${𝔰𝔭}_{2r}$ or ${𝔰𝔬}_{2r}$ and use notations for ${𝔥}^{*}$ as in (2.5)-(2.16). Identify $U_0$ as a subalgebra of $ℂ[L_1^{\pm 1},\dots ,L_r^{\pm 1}]$ where ${\text{ev}}_{{\epsilon }_i}\left(L_j\right)=q^{⟨{\epsilon }_i,{\epsilon }_j⟩}=q^{{\delta }_{ij}}$ (so that $L_i=e^{\frac{1}{2}h{F}_{ii}},$ where $F_{ii}$ is as in (2.8)). Let $$y=\{\begin{array}{cc}2r,& \text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1},\\ 2r+1,& \text{if}\hspace{0.17em}𝔤={𝔰𝔭}_{2r},\\ 2r-1,& \text{if}\hspace{0.17em}g={𝔰𝔬}_{2r},\end{array}\epsilon =\{\begin{array}{cc}1,& \text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1},\\ -1,& \text{if}\hspace{0.17em}𝔤={𝔰𝔭}_{2r},\\ 1,& \text{if}\hspace{0.17em}g={𝔰𝔬}_{2r},\end{array}V=L\left({\epsilon }_1\right),$$ and $z=\epsilon q^y\text{.}$ Let $Z_V^{\left(\ell \right)}$ be the central elements in $U_h𝔤$ defined by $$Z_V^{\left(\ell \right)}=\epsilon (\text{id}\otimes {\text{qtr}}_V)\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)$$ and write $$Z_V^{+}\left(u\right)=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{Z}_V^{\left(\ell \right)}{u}^{-\ell }\text{and}{Z}_V^{-}\left(u\right)=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{Z}_V^{(-\ell )}{u}^{-\ell }\text{.}$$ Then $${\pi }_0(Z_V^{+}\left(u\right)+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1})=\left(\frac{z}{q-q^{-1}}\right)\frac{(u+q)(u-q^{-1})}{(u+1)(u-1)}\frac{(u-\epsilon q^{2r-y})}{(u-\epsilon q^{y-2r})}{\sigma }_{\rho }\left(\prod _{i=1}^r\frac{(u-\epsilon L_i^{-2}{q}^{-1})(u-\epsilon L_i^2{q}^{-1})}{(u-\epsilon L_i^{-2}q)(u-\epsilon L_i^{2}q)}\right)$$ and $${\pi }_0(Z_V^{-}\left(u\right)+\frac{z}{q-q^{-1}}-\frac{u^2}{u^2-1})=-\frac{z^{-1}}{q-q^{-1}}\frac{(u-q)(u+q^{-1})}{(u+1)(u-1)}\frac{(u-\epsilon q^{y-2r})}{(u-\epsilon q^{2r-y})}{\sigma }_{\rho }\left(\prod _{i=1}^r\frac{(u-\epsilon L_i^{-2}q)(u-\epsilon L_i^{2}q)}{(u-\epsilon L_i^{-2}{q}^{-1})(u-\epsilon L_i^2{q}^{-1})}\right),$$ where ${\sigma }_{\rho }$ is the algebra automorphism given by ${\sigma }_{\rho }\left(L_i\right)=q^{⟨\rho ,{\epsilon }_i⟩}{L}_i$ and ${\pi }_0$ is the isomorphism in Theorem 3.1.

		Proof.
	In the definition of the action of the affine BMW algebra in Theorem 1.3, $Y_1$ acts on $M\otimes V$ as $z{ℛ}_{21}ℛ,$ and $$E_1{Y}_1^{\ell }{E}_1\hspace{0.17em}\text{acts on}\hspace{0.17em}M\otimes V^{\otimes 2}\hspace{0.17em}\text{as}\hspace{0.17em}{Z}_V^{\left(\ell \right)}{E}_1\text{.}$$ Also $$E_1\hspace{0.17em}\text{and}\hspace{0.17em}{Y}_1\hspace{0.17em}\text{in}\hspace{0.17em}{W}_2\hspace{0.17em}\text{act on}\hspace{0.17em}M\otimes V^{\otimes 2}\hspace{0.17em}\text{with}\hspace{0.17em}M=L\left(0\right)\otimes V^{\otimes (k-1)}$$ in the same way that $$E_k\hspace{0.17em}\text{and}\hspace{0.17em}{Y}_k\hspace{0.17em}\text{in}\hspace{0.17em}{W}_{k+1}\hspace{0.17em}\text{act on}\hspace{0.17em}M\otimes V^{\otimes (k+1)}\hspace{0.17em}\text{with}\hspace{0.17em}M=L\left(0\right)\text{.}$$ By Proposition 3.4, $Z_{k-1}^{\left(\ell \right)}{E}_k=E_k{Y}_k^{\ell }{E}_k$ and so it follows that, as operators on $L\left(0\right)\otimes V^{\otimes (k-1)},$ $$\begin{array}{cc}{Z}_V^{+}\left(u\right)+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1}=Z_{k-1}^{+}\left(u\right)+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1}& \text{(3.16)}\end{array}$$ and $$\begin{array}{cc}{Z}_V^{-}\left(u\right)-\frac{z}{q-q^{-1}}-\frac{u^2}{u^2-1}=Z_{k-1}^{-}\left(u\right)+\frac{z}{q-q^{-1}}-\frac{u^2}{u^2-1}& \text{(3.17)}\end{array}$$ We will use (3.14) and (3.15) to compute the action of these operators on the $L\left(\mu \right)\otimes W_{k-1}^{\mu }$ isotypic component in the $U_h𝔤\otimes W_{k-1}\text{-module}$ decomposition $$\begin{array}{cc}L\left(0\right)\otimes V^{\otimes (k-1)}\cong \underset{\mu }{⨁}L\left(\mu \right)\otimes W_{k-1}^{\mu }\text{.}& \text{(3.18)}\end{array}$$ As an operator on $L\left(0\right)\otimes V,$ $z\left({ℛ}_{21}ℛ\right)=z{q}^{⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩-⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩+⟨0,0+2\rho ⟩}=z\text{.}$ Hence $$Z_V^{\left(\ell \right)}=\epsilon (\text{id}\otimes {\text{qtr}}_V)\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)=z^{\ell }\epsilon {\text{dim}}_q\left(V\right)\text{.}$$ Therefore, since $\epsilon {\text{dim}}_q\left(V\right)=\frac{z-z^{-1}}{q-q^{-1}}=1,$ $$Z_V^{+}\left(u\right)=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\epsilon {\text{dim}}_q\left(V\right)z^{\ell }{u}^{-\ell }=\epsilon {\text{dim}}_q\left(V\right)\frac{1}{1-z{u}^{-1}}=\frac{z-z^{-1}+(q-q^{-1})}{(q-q^{-1})(1-z{u}^{-1})}\text{.}$$ A similar computation of $Z_V^{-}$ yields $$Z_V^{-}\left(u\right)=\frac{z-z^{-1}+q-q^{-1}}{(q-q^{-1})(1-z^{-1}{u}^{-1})}\text{.}$$ Thus, as operators on $L\left(0\right)\otimes V,$ $$\begin{array}{cc}{Z}_V^{+}+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1}=\frac{z}{(q-q^{-1})}\frac{(1-z^{-1}{u}^{-1})}{(1-z{u}^{-1})}\frac{(u+q)(u-q^{-1})}{(u+1)(u-1)}& \text{(3.19)}\end{array}$$ and $$\begin{array}{cc}{Z}_V^{-}-\frac{z}{q-q^{-1}}-\frac{u^2}{u^2-1}=\frac{-z^{-1}}{(q-q^{-1})}\frac{(1-z{u}^{-1})}{(1-z^{-1}{u}^{-1})}\frac{(u-q)(u+q^{-1})}{(u+1)(u-1)}\text{.}& \text{(3.19)}\end{array}$$ By (1.29) and the definition of $\Phi$ in Theorem 1.3, $$Y_k\in W_k\text{acts on}L\left(0\right)\otimes V^{\otimes k}=(L\left(0\right)\otimes V^{\otimes (k-1)})\otimes V\text{as}z{ℛ}_{21}ℛ\text{.}$$ If $L\left(\mu \right)$ is an irreducible $U_h𝔤\text{-module}$ in $L\left(0\right)\otimes V^{\otimes (k-1)},$ then (1.28) and (2.16) give that $Y_k$ acts on the $L\left(\lambda \right)$ component of $L\left(\mu \right)\otimes V$ by the constant $\epsilon q^{2c(\lambda ,\mu )}=1·q^0=1,$ when $𝔤={𝔰𝔬}_{2r+1}$ and $\lambda =\mu ,$ and by the constant $$\epsilon q^{2c(\lambda ,\mu )}=\{\begin{array}{cc}\epsilon q^{y+2c(\lambda /\mu )},& \text{if}\hspace{0.17em}\mu \subseteq \lambda ,\\ \epsilon q^{-y-2c(\mu /\lambda )}& \text{if}\hspace{0.17em}\mu \supseteq \lambda ,\end{array}=\{\begin{array}{cc}z{q}^{2c(\lambda /\mu )},& \text{if}\hspace{0.17em}\mu \subseteq \lambda ,\\ z^{-1}{q}^{-2c(\mu /\lambda )},& \text{if}\hspace{0.17em}\mu \supseteq \lambda ,\end{array},\text{where}\hspace{0.17em}\lambda =\mu \pm {\epsilon }_i\text{.}$$ and $c(\lambda ,\mu )$ is as computed in (3.8). As in [ORa0401317, Theorem 6.3(b)], the irreducible $W_k\text{-module}$ $W^{\mu /0}=W_k^{\mu }$ has a basis $\left\{v_T\right\}$ indexed by up-down tableaux $T=\{T^{\left(0\right)},T^{\left(1\right)},\cdots ,T^{\left(k\right)}\},$ where $T^{\left(0\right)}=\varnothing ,$ $T^{\left(k\right)}=\mu ,$ and $T^{\left(i\right)}$ is a partition obtained from $T^{(i-1)}$ by adding or removing a box, and $$Y_i{v}_T=\{\begin{array}{cc}z{q}^{2c\left(b\right)}{v}_T,& \text{if}\hspace{0.17em}b=T^{\left(i\right)}/T^{(i-1)},\\ z^{-1}{q}^{-2c\left(b\right)}{v}_T,& \text{if}\hspace{0.17em}b=T^{(i-1)}/T^{\left(i\right)},\\ v_T,& \text{if}\hspace{0.17em}{T}^{(i-1)}=T^{\left(i\right)}\text{.}\end{array}$$ Thus $$\prod _{i=1}^{k-1}\frac{{(u-Y_i)}^2(u-q^{-2}{Y}_i^{-1})(u-q^2{Y}_i^{-1})}{{(u-Y_i^{-1})}^2(u-q^2{Y}_i)(u-q^{-2}{Y}_i)}\text{acts on}\hspace{0.17em}L\left(\mu \right)\otimes W_{k-1}^{\mu }\hspace{0.17em}\text{in (3.18)}$$ by $$\begin{array}{cc}\prod _{i=1}^{k-1}\frac{{(u-\epsilon q^{2c(T^{\left(i\right)},T^{(i-1)})})}^2(u-\epsilon q^{-2}{q}^{2c(T^{\left(i\right)},T^{(i-1)})})(u-\epsilon q^2{q}^{2c(T^{\left(i\right)},T^{(i-1)})})}{{(u-\epsilon q^{-2c(T^{\left(i\right)},T^{(i-1)})})}^2(u-\epsilon q^2{q}^{2c(T^{\left(i\right)},T^{(i-1)})})(u-\epsilon q^{-2}{q}^{2c(T^{\left(i\right)},T^{(i-1)})})}& \text{(3.21)}\end{array}$$ for any up-down tableau $T$ of length $k$ and shape $\mu \text{.}$ If a box is added (or removed) at step $i$ and then removed (or added) at step $j,$ then the $i$ and $j$ factors of this product cancel. Therefore (3.21) is equal to $$\begin{array}{cc}\prod _{b\in \mu }\frac{{(u-z{q}^{2c\left(b\right)})}^2(u-z^{-1}{q}^{-2(c\left(b\right)+1)})(u-z^{-1}{q}^{-2\left(c\left(b\right)-1\right)})}{{(u-z^{-1}{q}^{-2c\left(b\right)})}^2(u-z{q}^{2(c\left(b\right)+1)})(u-z{q}^{2(c\left(b\right)-1)})}\text{.}& \text{(3.22)}\end{array}$$ Simplifying one row at a time, $$\begin{array}{ccc}\prod _{b\in \mu }\frac{(u-z^{-1}{q}^{-2(c\left(b\right)-1)})(u-z^{-1}{q}^{-2(c\left(b\right)+1)})}{(u-z^{-1}{q}^{-2c\left(b\right)})(u-z^{-1}{q}^{-2c\left(b\right)})}& =& \prod _{i=1}^r\frac{(u-z^{-1}{q}^{-2(-i)})(u-z^{-1}{q}^{-2({\mu }_i-i+1)})}{(u-z^{-1}{q}^{-2(-(i-1))})(u-z^{-1}{q}^{-2({\mu }_i-i)})}\\ & =& \frac{u-z^{-1}{q}^{2r}}{u-z^{-1}{q}^{2·0}}\prod _{i=1}^r\frac{u-z^{-1}{q}^{-2({\mu }_i-i+1)}}{u-z^{-1}{q}^{-2({\mu }_i-i)}}\end{array}$$ if $\mu =({\mu }_1,\dots ,{\mu }_r)\text{.}$ It follows that (3.22) is equal to $$\begin{array}{cc}\begin{array}{c}\frac{(u-z^{-1}{q}^{2r})}{(u-z^{-1})}\frac{(u-z)}{(u-z{q}^{-2r})}\prod _{i=1}^r\frac{(u-z^{-1}{q}^{-2({\mu }_i-i+1)})}{(u-z^{-1}{q}^{-2({\mu }_i-i)})}·\frac{(u-z{q}^{2({\mu }_i-i)})}{(u-z{q}^{2({\mu }_i-i+1)})}\\ =\frac{(u-\epsilon q^{-(y-2r)})}{(u-z^{-1})}\frac{(u-z)}{(u-\epsilon q^{y-2r})}{\text{ev}}_{\mu +\rho }\left(\prod _{i=1}^r\frac{(u-\epsilon L_i^{-2}{q}^{-1})(u-\epsilon L_i^2{q}^{-1})}{(u-\epsilon L_i^{-2}q)(u-\epsilon L_i^{2}q)}\right)\end{array}& \text{(3.23)}\end{array}$$ since $z^{-1}{q}^{2r}=\epsilon q^{-y}{q}^{2r}=\epsilon q^{2r-y}$ and $${\text{ev}}_{\mu +\rho }\left(L_i^2\right)=q^{⟨\mu +\rho ,2\pi ⟩}=q^{2{\mu }_i(y-2i+1)}=q^{y+1+2({\mu }_i-i)}=\epsilon z{q}^{2({\mu }_i-i)+1}\text{.}$$ Combining (3.19) and (3.23), the identity (3.14) gives that, as operators on $L\left(\mu \right)\otimes W_{k-1}^{\mu }$ in (3.18), $$\begin{array}{cc}\begin{array}{c}{Z}_k^{+}+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1}\\ =\frac{z}{(q-q^{-1})}\frac{(u+q)(u-q^{-1})}{(u+1)(u-1)}\frac{(u-\epsilon q^{2r-y})}{(u-\epsilon q^{y-2r})}{\text{ev}}_{\mu +\rho }\left(\prod _{i=1}^r\frac{(u-\epsilon L_i^{-2}{q}^{-1})(u-\epsilon L_i^2{q}^{-1})}{(u-\epsilon L_i^{-2}q)(u-\epsilon L_i^{2}q)}\right)\text{.}\end{array}& \text{(3.24)}\end{array}$$ Similarly, $Z_k^{-}-\frac{z}{q-q^{-1}}+\frac{1}{u^2-1}$ acts on the $L\left(\mu \right)\otimes W_{k-1}^{\mu }$ isotypic component in the $U_h𝔤\otimes W_{k-1}\text{-module}$ decomposition in (3.18) by $$-\frac{z^{-1}}{(q-q^{-1})}\frac{(u-q)(u+q^{-1})}{(u+1)(u-1)}\frac{(u-\epsilon q^{2r-y})}{(u-\epsilon q^{y-2r})}{\text{ev}}_{\mu +\rho }\left(\prod _{i=1}^r\frac{(u-\epsilon L_i^{-2}q)(u-\epsilon L_i^{2}q)}{(u-\epsilon L_i^{-2}{q}^{-1})(u-\epsilon L_i^2{q}^{-1})}\right)\text{.}$$ By Theorem 3.1, the desired results follow. $\square$

In the following corollary, we shall repackage Theorem 3.5 to give a formula for the Harish-Chandra image of $Z_V^{\left(\ell \right)}$ in terms of “Weyl characters”. To do this we will use the universal characters of [KTe1987] following the notation in [HRa1995, §6]. For a formal alphabet $Y$ let $s{a}_{\lambda }\left(Y\right)$ be the universal Weyl character for ${𝔤𝔩}_r,$ $s{p}_{\lambda }\left(Y\right)$ the universal Weyl character for ${𝔰𝔭}_{2r},$ and $s{o}_{\lambda }\left(Y\right)$ the universal Weyl character for the orthogonal cases.

The Cauchy-Littlewood identities (see [KTe1987, Lemma 1.5.1], [Wey1946, Theorems 7.8FG and 7.9C], and [HRa1995, (6.4) and (6.5)]) are $$\begin{array}{c}\prod _{i,j}\frac{1}{1-x_i{y}_j}=\Omega \left(XY\right)=\sum _{\lambda }s{a}_{\lambda }\left(X\right){sa}_{\lambda }\left(Y\right),\\ \prod _{i\le j}\frac{1}{1-y_i{y}_j}\prod _{i,j}\frac{1}{1-x_i{y}_j}=\Omega (XY-{sa}_{\left(2\right)}\left(Y\right))=\sum _{\lambda }{sa}_{\lambda }\left(Y\right){so}_{\lambda }\left(X\right),\\ \prod _{i<j}\frac{1}{1-y_i{y}_j}\prod _{i,j}\frac{1}{1-x_i{y}_j}=\Omega (XY-{sa}_{\left(1^2\right)}\left(Y\right))=\sum _{\lambda }{sa}_{\lambda }\left(Y\right){sp}_{\lambda }\left(X\right),\end{array}$$ where $\Omega$ is the Cauchy kernel (see [HRa1995, (6.3)]) and the first equality in each line is for the formal alphabets $X={\sum }_i{x}_i$ and $Y={\sum }_j{y}_j\text{.}$ The identity [HRa1995, Lemma 6.7(a)] states $$\begin{array}{cc}{sa}_{\lambda }\left((q-q^{-1})u^{-1}\right)=\{\begin{array}{cc}(q-q^{-1})u^{-\ell }{(-q^{-1})}^{\ell -m}{q}^{m-1},& \text{if}\hspace{0.17em}\lambda =(m,1^{\ell -m}),\\ 0,& \text{otherwise.}\end{array}& \text{(3.25)}\end{array}$$

In the same setting as in Theorem 3.5, let $$y=\{\begin{array}{cc}2r,& \text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1},\\ 2r+1,& \text{if}\hspace{0.17em}𝔤={𝔰𝔭}_{2r},\\ 2r-1,& \text{if}\hspace{0.17em}g={𝔰𝔬}_{2r},\end{array}\epsilon =\{\begin{array}{cc}1,& \text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1},\\ -1,& \text{if}\hspace{0.17em}𝔤={𝔰𝔭}_{2r},\\ 1,& \text{if}\hspace{0.17em}g={𝔰𝔬}_{2r},\end{array}V=L\left({\epsilon }_1\right),$$ $z=\epsilon q^y,$ and let $Z_V^{\left(\ell \right)}$ be the central elements in the Drinfeld-Jimbo quantum group $U_h𝔤$ which are given by $Z_V^{\left(\ell \right)}=\epsilon (\text{id}\otimes {\text{qtr}}_{L\left({\epsilon }_1\right)})\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)\text{.}$ Let $X$ be the formal alphabet given by $X={\sum }_{i\in \hat{V}}{L}_i^2$ and fix $c=1$ if $\ell$ is even and $c=0$ if $\ell$ is odd. Then for $\ell \ge 1,$ $${\pi }_0\left(Z_V^{\left(\ell \right)}\right)={\sigma }_{\rho }(c+z{\epsilon }^{\ell }\sum _{m=1}^{\ell }(q-q^{-1}){(-1)}^{\ell -m}{q}^{-(\ell -2m+1)}{s}_{(m,1^{\ell -m})}\left(X\right))$$ where $s_{(m,1^{\ell -m})}\left(X\right)={so}_{(m,1^{\ell -m})}\left(X\right)$ in the orthogonal cases and $s_{(m,1^{\ell -m})}\left(X\right)={sp}_{(m,1^{\ell -m})}\left(X\right)$ in the symplectic case.

		Proof.
	Let $\hat{V}$ as in (2.6), $L_{-i}=L_i^{-1}$ where $L_i$ is as in the statement of Theorem 3.5, and let $L_{{\epsilon }_0}=1\text{.}$ The identity in Theorem 3.5 can be rewritten as $$\begin{array}{cc}{\pi }_0(Z_V^{+}\left(U\right)+\frac{z^{-1}}{q-q^{-1}}-\frac{u^2}{u^2-1})={\sigma }_{\rho }\left(\frac{z}{q-q^{-1}}\frac{u^2-q^{2\epsilon }}{u^2-1}\prod _{j\in \hat{V}}\frac{(1-L_{\text{wt}\left(v_j\right)}^2{q}^{-1}{\left(\epsilon u\right)}^{-1})}{(1-L_{\text{wt}\left(v_j\right)}^{2}q{\left(\epsilon u\right)}^{-1})}\right)\text{.}& \text{(3.26)}\end{array}$$ By (3.25), $${sa}_{\left(2\right)}\left((q-q^{-1}){\left(\epsilon u\right)}^{-1}\right)=(q^2-1){\left(\epsilon u\right)}^{-2}\text{and}{sa}_{\left(1^2\right)}\left((q-q^{-1}){\left(\epsilon u\right)}^{-1}\right)=(q^{-2}-1){\left(\epsilon u\right)}^{-2}\text{.}$$ So the Cauchy-Littlewood identities give $$\begin{array}{c}\frac{(1-q^2{\left(\epsilon u\right)}^{-2})}{1-{\left(\epsilon u\right)}^{-2}}\prod _{i\in \hat{V}}\frac{(1-L_i^2{q}^{-1}{\left(\epsilon u\right)}^{-1})}{(1-L_i^{2}q{\left(\epsilon u\right)}^{-1})}\\ =\Omega (X(q-q^{-1}){\left(\epsilon u\right)}^{-1}-{sa}_{\left(2\right)}\left((q-q^{-1}){\left(\epsilon u\right)}^{-1}\right))=\sum _{\lambda }{sa}_{\lambda }\left((q-q^{-1}){\left(\epsilon u\right)}^{-1}\right){so}_{\lambda }\left(X\right)\\ =\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\left(\sum _{m=1}^{\ell }(q-q^{-1}){(-q^{-1})}^{\ell -m}{q}^{m-1}{so}_{(m,1^{\ell -m})}\left(X\right)\right){\left(\epsilon u\right)}^{-\ell }\\ =\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{e}^{\ell }(q-q^{-1})\left(\sum _{m=1}^{\ell }(q-q^{-1}){(-1)}^{\ell -m}{q}^{-(\ell -2m+1)}{so}_{(m,1^{\ell -m})}\left(X\right)\right)u^{-\ell }\end{array}$$ in the orthogonal case, and $$\begin{array}{c}\frac{(1-q^{-2}{\left(\epsilon u\right)}^{-2})}{1-{\left(\epsilon u\right)}^{-2}}\prod _{i\in \hat{V}}\frac{(1-L_i^2{q}^{-1}{\left(\epsilon u\right)}^{-1})}{(1-L_i^{2}q{\left(\epsilon u\right)}^{-1})}\\ =\Omega (X(q-q^{-1}){\left(\epsilon u\right)}^{-1}-{sa}_{\left(1^2\right)}\left((q-q^{-1}){\left(\epsilon u\right)}^{-1}\right))=\sum _{\lambda }{sa}_{\lambda }\left((q-q^{-1}){\left(\epsilon u\right)}^{-1}\right){sp}_{\lambda }\left(X\right)\\ =\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\left(\sum _{m=1}^{\ell }(q-q^{-1}){(-q^{-1})}^{\ell -m}{q}^{m-1}{sp}_{(m,1^{\ell -m})}\left(X\right)\right){\left(\epsilon u\right)}^{-\ell }\\ =\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{e}^{\ell }(q-q^{-1})\left(\sum _{m=1}^{\ell }{(-1)}^{\ell -m}{q}^{-(\ell -2m+1)}{sp}_{(m,1^{\ell -m})}\left(X\right)\right)u^{-\ell }\end{array}$$ in the symplectic case. The statement now follows by noting that $u^2/(u^2-1)=1/(1-u^{-2})={\sum }_{k\in {\mathbb{Z}}_{\ge 0}}{u}^{-2k}$ and taking the coefficient of $u^{-\ell }$ on each side of (3.26). $\square$

Notes and references

This is a typed exert of the paper Affine and degenerate affine BMW algebras: Actions on tensor space by Zajj Daugherty, Arun Ram and Rahbar Virk.

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