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

Symplectic and orthogonal higher Casimir elements

Our final goal in this paper will be to connect the central elements appearing naturally as parameters of the affine and degenerate affine BMW algebras (see Theorems 2.2 and 2.5) to higher Casimir elements for orthogonal and symplectic Lie algebras and quantum groups. In the degenerate case, we explain how the generating function for $z_V^{\left(\ell \right)}$ derived in Theorem 3.3 can be matched up with the generating functions for central elements given by Perelomov-Popov in [PPo0214703, PPo0205621]. Expositions of the Perelomov-Popov results are also in [Mol2355506, §7.1] and [Zeh1973, §127]. In the affine case we show how the formula for $Z_V^{\left(\ell \right)}$ in Corollary 3.6 can be derived as a special case of a remarkable identity for central elements in quantum groups discovered by Baumann [Bau1620662, Thm. 1].

The central elements $z_V^{\left(\ell \right)}$ as higher Casimir elements

Returning to the notation developed in the preliminaries of Section 2, let $𝔤={𝔤𝔩}_r$ with nondegenerate ad-invariant form $⟨,⟩$ as in (2.1) and operator $\gamma ={\gamma }^{𝔤𝔩}$ as in (2.2). Then $$(\text{id}\otimes {\text{tr}}_V)\left({\gamma }^{\ell }\right)=\sum _{i_1,i_2,\dots ,i_{\ell }}{E}_{i_1{i}_2}{E}_{i_2{i}_3}\cdots E_{i_{\ell }{i}_1}$$ are the central elements of $U{𝔤𝔩}_n$ found, for example, in Gelfand [Gel1950, (3)]. Perelomov-Popov [PPo0214703, PPo0205621] generalized this construction to $𝔤={𝔰𝔬}_{2r+1},$ ${𝔰𝔭}_{2r},$ and ${𝔰𝔬}_{2r},$ by letting $F_{ij}$ be the natural spanning set for $𝔤$ given in (2.7), viewing $F={\left(F_{ij}\right)}_{i,j\in \hat{V}}$ as a matrix with entries in $𝔤,$ and writing $$\begin{array}{cc}\text{tr}{F}^k=\sum _{i_1,i_2,\dots ,i_k\in \hat{V}}{F}_{i_1{i}_2}{F}_{i_2{i}_3}\cdots F_{i_k{i}_1}\text{for}\hspace{0.17em}k\in {\mathbb{Z}}_{>0},& \text{(4.1)}\end{array}$$ as an element of the enveloping algebra $U𝔤$ (see [Mol2355506, Thm. 7.1.7]). These elements are central in $U𝔤$ and Perelomov-Popov gave the following generating function formula for their Harish-Chandra images (see [Zeh1973, §127]). The proof we give below shows that the result of Perelemov-Popov is equivalent to Theorem 3.3 (which we obtained from the degenerate affine BMW algebra and Schur-Weyl duality). A proof of Theorem 4.1 using the theory of twisted Yangians is given in [Mol2355506, §7.1].

(Perelomov-Popov) [Mol2355506, Cor. 7.1.8] Let $𝔤={𝔰𝔬}_{2r+1}$ or ${𝔰𝔭}_{2r}$ or ${𝔰𝔬}_{2r},$ use notations for ${𝔥}^{*}$ as in Section 2 and let $h_1,\dots ,h_r$ be the basis of $𝔥$ dual to the orthonormal basis ${\epsilon }_1,\dots ,{\epsilon }_r$ of ${𝔥}^{*}\text{.}$ Let ${\rho }_r^{\prime }=\frac{1}{2}-\frac{1}{2}y,$ $l_0=0$ in the case that $𝔤={𝔰𝔬}_{2r+1}$ and let $$l_i=-l_{-i}=h_i+{\rho }_i,\text{for}\hspace{0.17em}i=1,2,\dots ,r,\text{where}\hspace{0.17em}{\rho }_i=\frac{1}{2}(y-2i+1)\text{.}$$ Then $$\begin{array}{cc}{\pi }_0(1+\frac{x+\frac{1}{2}}{x+\frac{1}{2}-\frac{1}{2}\epsilon }\left(\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\frac{{(-1)}^{\ell }\text{tr}\left(F^k\right)}{{(x+{\rho }_r^{\prime })}^{\ell +1}}\right))=\prod _{i\in \hat{V}}\frac{x+l_i+1}{x+l_i}\text{.}& \text{(4.2)}\end{array}$$

		Proof.
	By (2.11), $$\gamma =\frac{1}{2}\sum _{i,j\in \hat{V}}{F}_{ij}\otimes F_{ji}\text{.}$$ Let $\eta :𝔤\to \text{End}\left(V\right)$ be the defining representation. Since $F_{-j,-i}=-{\theta }_{ij}{F}_{ij},$ $$\begin{array}{ccc}(\text{id}\otimes \eta )\left(\gamma \right)& =& \frac{1}{2}\sum _{i,j\in \hat{V}}{F}_{ij}\otimes (E_{ji}-{\theta }_{ji}{E}_{-i,-j})=\frac{1}{2}\sum _{i,j\in \hat{V}}(F_{ij}\otimes E_{ji}-{\theta }_{ji}{F}_{-j,-i}\otimes E_{ji})\\ & =& \frac{1}{2}\sum _{i,j\in \hat{V}}(F_{ij}-{\theta }_{ji}{F}_{-j,-i})\otimes E_{ji}=\frac{1}{2}\sum _{i,j\in \hat{V}}(F_{ij}+F_{ij})\otimes E_{ji}\\ & =& \sum _{i,j\in \hat{V}}{F}_{ij}\otimes E_{ji}=F^t=-\theta F\theta ,\text{where}\theta =\left(\begin{array}{cc}\epsilon \text{id}& 0\\ 0& \text{id}\end{array}\right)\text{.}\end{array}$$ Thus $$\begin{array}{cc}(\text{id}\otimes {\text{tr}}_V)\left({\gamma }^k\right)=\text{tr}\left({\left(F^t\right)}^k\right)=\text{tr}\left({(-\theta F\theta )}^k\right)={(-1)}^k\text{tr}\left({\theta }^2{F}^k\right)={(-1)}^k\text{tr}\left(F^k\right),& \text{(4.3)}\end{array}$$ which provides the connection of the elements $z_0^{\left(\ell \right)}$ appearing in Theorems 2.2 and 3.3 to the elements in (4.1). In order to transform the generating function for the elements $(\text{id}\otimes {\text{tr}}_V)\left({\gamma }^{\ell }\right)$ into the generating function for the elements $(\text{id}\otimes {\text{tr}}_V)\left({(\frac{1}{2}y+\gamma )}^{\ell }\right),$ notice $$\begin{array}{c}\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}(\text{id}\otimes {\text{tr}}_V)\left({(\frac{1}{2}y+\gamma )}^{\ell }\right)u^{-\ell }=(\text{id}\otimes {\text{tr}}_V)\left(\frac{1}{1-(\frac{1}{2}y+\gamma )u^{-1}}\right)\\ =(\text{id}\otimes {\text{tr}}_V)\left(\frac{1}{1-\frac{1}{2}y{u}^{-1}-\gamma u^{-1}}\right)=(\text{id}\otimes {\text{tr}}_V)\left(\left(\frac{1}{1-\frac{1}{2}y{u}^{-1}}\right)\left(\frac{1}{1-\gamma \frac{u^{-1}}{1-\frac{1}{2}y{u}^{-1}}}\right)\right)\\ =(\text{id}\otimes {\text{tr}}_V)\left(\frac{1}{1-\frac{1}{2}y{u}^{-1}}\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\frac{{\gamma }^{\ell }{u}^{-\ell }}{{(1-\frac{1}{2}y{u}^{-1})}^{\ell }}\right)=u\left(\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\frac{(\text{id}\otimes {\text{tr}}_V)\left({\gamma }^{\ell }\right)}{{(u-\frac{1}{2}y)}^{\ell +1}}\right)\\ =\left(\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\frac{(\text{id}\otimes {\text{tr}}_V)\left({\gamma }^{\ell }\right)}{{(u-\frac{1}{2}+{\rho }_r^{\prime })}^{\ell +1}}\right),\text{where}\hspace{0.17em}{\rho }_r^{\prime }=\frac{1}{2}-\frac{1}{2}y\text{.}\end{array}$$ Then Theorem 3.3 is equivalent to $$\begin{array}{c}{\pi }_0(1+\frac{\epsilon u}{\epsilon u-\frac{1}{2}}\left(\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\frac{(\text{id}\otimes {\text{tr}}_V)\left({\gamma }^{\ell }\right)}{{(u-\frac{1}{2}+{\rho }_r^{\prime })}^{\ell +1}}\right))={\pi }_0(1+\frac{\epsilon }{\epsilon u-\frac{1}{2}}\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}(\text{id}\otimes {\text{tr}}_V)\left({(\frac{1}{2}y+\gamma )}^{\ell }\right)u^{-\ell })\\ =\frac{(\epsilon u+\frac{1}{2})}{(\epsilon u-\frac{1}{2})}\frac{(u+\frac{1}{2}y-r)}{(u-\frac{1}{2}y+r)}{\sigma }_{\rho }\left(\prod _{i=1}^r\frac{(u+h_i+\frac{1}{2})}{(u+h_i-\frac{1}{2})}\frac{(u-h_i+\frac{1}{2})}{(u-h_i-\frac{1}{2})}\right)\\ =\frac{(\epsilon u+\frac{1}{2})}{(\epsilon u-\frac{1}{2})}\frac{(u+\frac{1}{2}y-r)}{(u-\frac{1}{2}y+r)}\left(\prod _{i=1}^r\frac{(u+h_i+{\rho }_i+\frac{1}{2})}{(u+h_i-{\rho }_i+\frac{1}{2})}\frac{(u-h_i-{\rho }_i+\frac{1}{2})}{(u-h_i-{\rho }_i-\frac{1}{2})}\right)\\ =\frac{(u-\frac{1}{2}+\frac{1}{2}\epsilon +\frac{1}{2})}{(u-\frac{1}{2}-\frac{1}{2}\epsilon +\frac{1}{2})}\frac{(u-\frac{1}{2}+\frac{1}{2}y-r+\frac{1}{2})}{(u-\frac{1}{2}-\frac{1}{2}y+r+\frac{1}{2})}\left(\prod _{i=1}^r\frac{(u-\frac{1}{2}+h_i+{\rho }_i+1)}{(u-\frac{1}{2}+h_i+{\rho }_i)}\frac{(u-\frac{1}{2}-h_i-{\rho }_i+1)}{(u-\frac{1}{2}-h_i-{\rho }_i)}\right)\text{.}\end{array}$$ Replacing $x=u-\frac{1}{2},$ $$\begin{array}{c}{\pi }_0(1+\frac{x+\frac{1}{2}}{x+\frac{1}{2}-\frac{1}{2}\epsilon }\left(\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\frac{(\text{id}\otimes {\text{tr}}_V)\left({\gamma }^{\ell }\right)}{{(x+{\rho }_r^{\prime })}^{\ell +1}}\right))\\ =\frac{(x+\frac{1}{2}\epsilon +\frac{1}{2})}{(x-\frac{1}{2}\epsilon +\frac{1}{2})}\frac{(x+\frac{1}{2}y-r+\frac{1}{2})}{(x-\frac{1}{2}y+r+\frac{1}{2})}\left(\prod _{i=1}^r\frac{(x+h_i+{\rho }_i+1)}{(x+h_i+{\rho }_i)}\frac{(x-h_i-{\rho }_i+1)}{(x-h_i-{\rho }_i)}\right)\end{array}$$ Since $$\frac{(x+\frac{1}{2}\epsilon +\frac{1}{2})}{(x-\frac{1}{2}\epsilon +\frac{1}{2})}\frac{(x+\frac{1}{2}y-r+\frac{1}{2})}{(x-\frac{1}{2}y+r+\frac{1}{2})}=\{\begin{array}{cc}\frac{x+l_0+1}{x+l_0},& \text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1},\\ 1,& \text{if}\hspace{0.17em}𝔤={𝔰𝔭}_{2r}\hspace{0.17em}\text{or}\hspace{0.17em}{𝔰𝔬}_{2r},\end{array}$$ it follows that $${\pi }_0(1+\frac{x+\frac{1}{2}}{x+\frac{1}{2}-\frac{1}{2}\epsilon }\left(\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}\frac{(\text{id}\otimes {\text{tr}}_V)\left({\gamma }^{\ell }\right)}{{(x+{\rho }_r^{\prime })}^{\ell +1}}\right))=\prod _{i\in \hat{V}}\frac{x+l_i+1}{x+l_i}\text{.}$$ In combination with (4.3), this demonstrates the equivalence of the Perelomov-Popov theorem and Theorem 3.3. $\square$

Since ${\pi }_0$ and $\rho ={\rho }_1{\epsilon }_1+\cdots +{\rho }_n{\epsilon }_n$ are defined via a specific choice of positive roots, each side of (4.2) depends on that choice, though the identity does not. The preferred choice of positive roots in [Mol2355506, p. 139] differs from our preferred choice in (2.12) by the action of the Weyl group element $w=(1,-r)(2,-(r-1))\cdots (r-1,-2)(r,-1),$ in cycle notation.

The central elements $Z_V^{\left(\ell \right)}$ as quantum higher Casimir elements

In this section, we show how the formula for the central elements $Z_V^{\left(\ell \right)}$ in Corollary 3.6 is related to an identity for central elements in quantum groups discovered by Baumann in [Bau1620662, Thm. 1]. To do this we rewrite the Baumann identity for $𝔤={𝔰𝔭}_{2r},$ ${𝔰𝔬}_{2r+1}$ and ${𝔰𝔬}_{2r}$ and $\lambda ={\epsilon }_1$ in terms of Weyl characters indexed by partitions. Then a theorem of Turaev and Wenzl computing $(\text{id}\otimes {\text{qtr}}_{L\left(\nu \right)})\left({ℛ}_{21}ℛ\right)$ provides a conversion between the expansion in Corollary 3.6 and the expansion obtained from Baumann’s identity.

For $\lambda \in {𝔥}^{*}$ define the Weyl character $$s_{\lambda }=\frac{a_{\lambda +\rho }}{a_{\rho }},\text{where}{a}_{\mu }=\sum _{w\in W_0}\text{det}\left(w\right)e^{w\mu }\text{.}$$ The expressions $s_{\lambda }$ and $a_{\mu }$ are elements of the group algebra of ${𝔥}^{*},$ $ℂ\left[{𝔥}^{*}\right]=ℂ\text{-span}\left\{e^{\nu }\hspace{0.17em}\right|\hspace{0.17em}\nu \in {𝔥}^{*}\}$ with $e^{\mu }{e}^{\nu }=e^{\mu +\nu }\text{.}$ If $w\in W_0$ then $$\begin{array}{cc}{a}_{w\mu }=\text{det}\left(w\right)a_{\mu }\text{and}{s}_{w\circ \mu }=\text{det}\left(w\right)s_{\mu },& \text{(4.4)}\end{array}$$ where the dot action of $W_0$ on ${𝔥}^{*}$ is given by $$\begin{array}{cc}w\circ \mu =w(\mu +\rho )-\rho ,\text{for}\hspace{0.17em}w\in W_0,\mu \in {𝔥}^{*}\text{.}& \text{(4.5)}\end{array}$$ The $W_0\text{-invariants}$ in $ℂ\left[{𝔥}^{*}\right]$ are $ℂ{\left[{𝔥}^{*}\right]}^{W_0}=ℂ\text{-span}\left\{s_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \hspace{0.17em}\text{dominant integral}\}\text{.}$ For $\ell \in {\mathbb{Z}}_{\ge 0}$ let $$\begin{array}{cc}\begin{array}{cccc}{\Psi }_{\ell }:& ℂ{\left[{𝔥}^{*}\right]}^{W_0}& ⟶& Z\left(U_h𝔤\right)\\ & s_{\nu }& ⟼& (\text{id}\otimes {\text{qtr}}_{L\left(\nu \right)})\left({\left({ℛ}_{21}ℛ\right)}^{\ell }\right)\end{array}\text{.}& \text{(4.6)}\end{array}$$ By [Dri1990, Prop. 1.2], the map ${\Psi }_{\ell }$ is a vector space isomorphism.

[Bau1620662, Thm. 1] For $\ell \in {\mathbb{Z}}_{\ge 0}$ define $$m_{\lambda }^{\left(\ell \right)}=\sum _{w\in W_0}{q}^{2\ell ⟨w\lambda ,\rho ⟩}{s}_{w\lambda }\text{.}\text{Then}{\Psi }_{\ell }\left(m_{\lambda }^{\left(\ell \right)}\right)={\Psi }_1\left(m_{\ell \lambda }^{(1/\ell )}\right)\text{.}$$

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 $Z_V^{\left(\ell \right)}=\epsilon (\text{id}\otimes {\text{qtr}}_{L\left({\epsilon }_1\right)})\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)\text{.}$ Then, for $\ell \ge 1,$ $$Z_V^{\left(\ell \right)}={\epsilon }^{\ell }{\Psi }_1\left(\begin{array}{c}c+z\sum _{m=\text{max}(\ell -r+1,1)}^{\ell }{(-1)}^{\ell -m}{q}^{-(\ell -2m+1)}{s}_{(m,1^{\ell =m+1-1})}\\ +z\sum _{m=\text{max}(y-\ell +1,1)}^{\ell -y+r}{q}^{-(\ell -2m+1)}{(-1)}^{m-\ell +y}{s}_{(m,1^{m-\ell +y-1})}\end{array}\right),$$ where $c$ is given by $$c=\{\begin{array}{cc}1& \text{if}\hspace{0.17em}\ell \hspace{0.17em}\text{is even and}\hspace{0.17em}\ell <y,\\ & \text{or}\hspace{0.17em}\ell \ge y\hspace{0.17em}\text{and}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1},\\ 0& \text{otherwise.}\end{array}$$

		Proof.
	The Weyl group $W_0$ for $𝔤={𝔰𝔬}_{2r+1}$ or $𝔤={𝔰𝔭}_{2r}$ is the group of signed permutations. With positive roots as in (2.12), the simple reflections are $s_i=(i,i+1)$ (the transposition switching εi and ${\epsilon }_{i+1},$ for $i=1,2,\dots ,r-1\text{)}$ and $s_r=(r,-r)$ (the transposition switching the sign of ${\epsilon }_r\text{).}$ For $𝔤={𝔰𝔬}_{2r},$ the Weyl group $W_0$ consists of signed permutations with an even number of signs, with simple reflections $s_i=(i,i+1)$ for $i=1,2,\dots ,r-1,$ and $s_r=(r-1,-r)(r,-(r-1))\text{.}$ To prove the desired identity, we will use the second identity in (4.4) to relabel the Weyl characters $s_{w\lambda }$ appearing in $$m_{{\epsilon }_1}^{\left(\ell \right)}=\sum _{w\in W_0}{q}^{2\ell ⟨w{\epsilon }_1,\rho ⟩}{s}_{w{\epsilon }_1}\text{and}{m}_{\ell {\epsilon }_1}^{(1/\ell )}=\sum _{w\in W_0}{q}^{2⟨w{\epsilon }_1,\rho ⟩}{s}_{w\ell {\epsilon }_1}$$ by dominant integral weights. By (2.13), ${\rho }_{i+1}={\rho }_i-1$ and ${\rho }_r=\frac{1}{2}(y-2r+1)\text{.}$ So if $\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_r{\epsilon }_r$ then $$s_i\circ \mu ={\mu }_1+\cdots +{\mu }_{i-1}{\epsilon }_{i-1}+({\mu }_{i+1}-1){\epsilon }_i+({\mu }_i+1){\epsilon }_{i+1}+{\mu }_{i+2}{\epsilon }_{i+2}+\cdots +{\mu }_r{\epsilon }_r$$ for $i=1,2,\dots ,r-1,$ and $$\begin{array}{c}{s}_r\circ \mu ={\mu }_1{\epsilon }_1+\cdots {\mu }_{r-1}{\epsilon }_{r-1}+(-{\mu }_r-(y-2r+1)){\epsilon }_r,\text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1}\hspace{0.17em}\text{or}\hspace{0.17em}{𝔰𝔭}_{2r}\text{, and}\\ s_r\circ \mu ={\mu }_1{\epsilon }_1+\cdots {\mu }_{r-1}{\epsilon }_{r-2}+(-{\mu }_r-1){\epsilon }_{r-1}+(-{\mu }_{r-1}-1){\epsilon }_r,\text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r}\text{.}\end{array}$$ In particular, $s_{\ell {\epsilon }_i}=0$ if $0<\ell <i,$ and $$\begin{array}{cc}{s}_{\ell {\epsilon }_i}=s_{s_1{s}_2\cdots s_{i-1}\circ \ell {\epsilon }_i}={(-1)}^{i-1}{s}_{(\ell -i+1){\epsilon }_1+{\epsilon }_2+\cdots +{\epsilon }_i}={(-1)}^{i-1}{s}_{(\ell -i+1,1^{i-1})}\text{if}\hspace{0.17em}0<i\le \ell \text{.}& \text{(4.7)}\end{array}$$ Furthermore, if $𝔤={𝔰𝔬}_{2r+1}$ or ${𝔰𝔭}_{2r},$ then $$\begin{array}{cc}\begin{array}{c}{s}_i\cdots s_{r-1}{s}_r{s}_{r-1}\cdots s_i\circ (-\ell {\epsilon }_i)=s_i\cdots s_{r-1}{s}_r\circ (-({\epsilon }_i+\cdots +{\epsilon }_{r-1})-(\ell -(r-i)){\epsilon }_r)\\ =s_i\cdots s_{r-1}\circ (-({\epsilon }_i+\cdots +{\epsilon }_{r-1})+(\ell -(r-i)-(y-2r+1)){\epsilon }_r)\\ =(\ell -(r-i)-(y-2r+1)-(r-i)){\epsilon }_i=(\ell +2i-y-1){\epsilon }_i\text{.}\end{array}& \text{(4.8)}\end{array}$$ Similarly, if $𝔤={𝔰𝔬}_{2r},$ then $$\begin{array}{cc}\begin{array}{c}{s}_i\cdots s_{r-2}{s}_r{s}_{r-1}\cdots s_i\circ (-\ell {\epsilon }_i)=s_i\cdots s_{r-2}{s}_r\circ (-({\epsilon }_i+\cdots +{\epsilon }_{r-1})-(\ell -(r-i)){\epsilon }_r)\\ =s_i\cdots s_{r-2}\circ (-({\epsilon }_i+\cdots +{\epsilon }_{r-2})+(\ell -(r-i)-1){\epsilon }_{r-1})\\ =(\ell -(r-i)-1-(r-i-1)){\epsilon }_i=(\ell +2i-y-1){\epsilon }_i\text{.}\end{array}& \text{(4.9)}\end{array}$$ So, letting $W_{{\epsilon }_1}$ be the stabilizer of ${\epsilon }_1,$ $\left|W_{{\epsilon }_1}\right|=2^{r-1}(r-1)!,$ and combining (4.7) and (4.8) gives $$\begin{array}{cc}\begin{array}{ccc}\frac{1}{\left|W_{{\epsilon }_1}\right|}{m}_{{\epsilon }_1}^{\left(\ell \right)}& =& \sum _{i=1}^r{q}^{2\ell ⟨{\epsilon }_i,\rho ⟩}{s}_{{\epsilon }_i}+q^{2\ell ⟨-{\epsilon }_i,\rho ⟩}{s}_{-{\epsilon }_i}=\{\begin{array}{cc}{q}^{2\ell ⟨{\epsilon }_1,\rho ⟩}{s}_{{\epsilon }_1}+q^{-2e^{\lambda }⟨{\epsilon }_r,\rho ⟩}{s}_{{\epsilon }_{-r}}& \text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1},\\ q^{2\ell ⟨{\epsilon }_1,\rho ⟩}{s}_{{\epsilon }_1}& \text{if}\hspace{0.17em}𝔤={𝔰𝔭}_{2r}\hspace{0.17em}\text{or}\hspace{0.17em}𝔤={𝔰𝔬}_{2r}\text{.}\end{array}\\ & =& q^{\ell (y-1)}{s}_{{\epsilon }_1}-a\text{where}\hspace{0.17em}a=\{\begin{array}{cc}{q}^{-\ell }{s}_0& \text{if}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1}\\ 0& \text{otherwise,}\end{array}\end{array}& \text{(4.10)}\end{array}$$ since $1+2i-y-1=0$ has a solution exactly if $i=r$ and $𝔤={𝔰𝔬}_{2r+1}\text{.}$ If $\ell >0$ then using $$\text{det}\left(s_i\cdots s_{r-1}{s}_r{s}_{r-1}\cdots s_i\right)=-1=-\text{det}\left(s_i\cdots s_{r-2}{s}_r{s}_{r-1}\cdots s_i\right),$$ $\ell +2i-y-1-(i-1)=\ell -y+i,$ and (4.7), equations (4.8) and (4.9) give $$q^{2⟨-{\epsilon }_i,\rho ⟩}{s}_{-\ell {\epsilon }_i}=\{\begin{array}{cc}-q^{-\ell }{s}_0,& \text{if}\hspace{0.17em}\ell =y+1-2i\hspace{0.17em}\text{and}\hspace{0.17em}𝔤={𝔰𝔬}_{2r+1}\hspace{0.17em}\text{or}\hspace{0.17em}{𝔰𝔭}_{2r},\\ q^{-\ell }{s}_0,& \text{if}\hspace{0.17em}\ell =y+1-2i\hspace{0.17em}\text{and}\hspace{0.17em}𝔤={𝔰𝔬}_{2r},\\ {(-1)}^i{q}^{-(y-2i+1)}{s}_{(\ell -y+i,1^{i-1})},& \text{if}\hspace{0.17em}\ell -y+i\ge 1,\\ 0,& \text{otherwise.}\end{array}$$ Since $\ell -y+i\ge 1$ when $i\ge y-\ell +1,$ $$\begin{array}{c}\begin{array}{ccc}\frac{1}{\left|W_{{\epsilon }_1}\right|}{m}_{\ell {\epsilon }_1}^{(1/\ell )}& =& \sum _{i=1}^r{q}^{2⟨{\epsilon }_i,\rho ⟩}{s}_{\ell {\epsilon }_i}+q^{2⟨-{\epsilon }_i,\rho ⟩}{s}_{-\ell {\epsilon }_i}\\ & =& \left(\sum _{i=1}^{\text{min}(\ell ,r)}{q}^{y-2i+1}{(-1)}^{i-1}{s}_{(\ell -i+1,1^{i-1})}\right)-b+\left(\sum _{i=\text{max}(y-\ell +1,1)}^r{q}^{-(y-2i+1)}{(-1)}^i{s}_{(\ell -y+u,1^{i-1})}\right)\end{array}\\ \text{where}\hspace{0.17em}b=0\hspace{0.17em}\text{if}\hspace{0.17em}\ell \ge y\text{, and if}\hspace{0.17em}\ell <y\hspace{0.17em}\text{then}\hspace{0.17em}b\hspace{0.17em}\text{is given by}\begin{array}{ccc}& \ell \hspace{0.17em}\text{is odd}& \ell \hspace{0.17em}\text{is even}\\ 𝔤={𝔰𝔬}_{2r+1}& q^{-\ell }{s}_0& 0\\ 𝔤={𝔰𝔭}_{2r}& 0& q^{-\ell }{s}_0\\ 𝔤={𝔰𝔬}_{2r}& 0& -q^{-\ell }{s}_0\end{array}\end{array}$$ (so that $b$ is nonzero exactly when $\ell =y+1-2i$ has a solution with Then reindexing (with $m=\ell -i+1$ in the first sum and $m=\ell -y+i$ in the second sum) gives $$\begin{array}{ccc}\frac{1}{\left|W_{{\epsilon }_1}\right|}{m}_{\ell {\epsilon }_1}^{(1/\ell )}& =& -b+\sum _{m=\text{max}(\ell -r+1,1)}^{\ell }{(-1)}^{\ell -m}{q}^{y-2(\ell -m)-1}{s}_{(m,1^{\ell -m+1-1})}\\ & & +\sum _{m=\text{max}(y-\ell +1,1)}^{\ell -y+r}{q}^{y-2(\ell -m)-1}{(-1)}^{m-\ell +y}{s}_{(m,1^{m-\ell +y-1})}\text{.}\end{array}$$ Notice that the last sum appears only if $\ell >y-r\text{.}$ Theorem 4.3 applied in the case that $\lambda ={\epsilon }_1$ gives $$\begin{array}{ccc}{Z}_V^{\left(\ell \right)}& =& \epsilon (\text{id}\otimes {\text{qtr}}_V)\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)=\epsilon z^{\ell }{\Psi }_{\ell }\left(s_{{\epsilon }_1}\right)=\epsilon {\left(\epsilon q^y\right)}^{\ell }{\Psi }_{\ell }\left(q^{-\ell (y-1)}(\frac{1}{\left|W_{{\epsilon }_1}\right|}{m}_{{\epsilon }_1}^{\left(\ell \right)}+a)\right)\\ & =& {\epsilon }^{\ell +1}{a}^{\ell }(\frac{1}{\left|W_{{\epsilon }_1}\right|}{\Psi }_{\ell }\left(m_{{\epsilon }_1}^{\left(\ell \right)}\right)+a)={\epsilon }^{\ell +1}{q}^{\ell }(\frac{1}{\left|W_{{\epsilon }_1}\right|}{\Psi }_1\left(m_{\ell {\epsilon }_1}^{(1/\ell )}\right)+a)\\ & =& {\epsilon }^{\ell }{\Psi }_1\left(\epsilon q^{\ell }(\frac{m_{\ell {\epsilon }_1}^{(1/\ell )}}{\left|W_{{\epsilon }_1}\right|}+a)\right)\\ & =& {\epsilon }^{\ell }{\Psi }_1\left(\begin{array}{c}\epsilon q^{\ell }(a-b)+z\sum _{m=\text{max}(\ell -r+1,1)}^{\ell }{(-1)}^{\ell -m}{q}^{-(\ell -2m+1)}{s}_{(m,1^{\ell -m+1-1})}\\ +z\sum _{m=\text{max}(\ell -r+1,1)}^{\ell -y+r}{q}^{-(\ell -2m+1)}{(-1)}^{m-\ell +y}{s}_{(m,1^{m-\ell +y-1})}\end{array}\right),\end{array}$$ since $z=\epsilon q^y\text{.}$ The result follows since $c=\epsilon q^{\ell }(a-b)\text{.}$ $\square$

We would like to connect Corollary 4.4 to the Harish-Chandra images of the parameters $Z_V^{\left(\ell \right)}$ computed in Corollary 3.6. In order to do so, we will use the following result from [TWe1217386, Lemma 3.5.1] (also see [Dri1990, Prop. 5.3]).

Let $𝔤$ be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form and let $U=U_h𝔤$ be the corresponding Drinfeld-Jimbo quantum group with $R\text{-matrix}$ $ℛ\text{.}$ Let $\nu$ be a dominant integral weight so that the irreducible module $L\left(\nu \right)$ of highest weight $\nu$ is finite-dimensional and let $s_{\nu }$ be the Weyl character of $L\left(\nu \right)\text{.}$ Then $$(\text{id}\otimes {\text{qtr}}_{L\left(\nu \right)})\left({ℛ}_{21}ℛ\right)\text{acts on}\hspace{0.17em}L\left(\mu \right)\hspace{0.17em}\text{by}{\text{ev}}_{2(\mu +\rho )}\left(s_{\nu }\right){\text{id}}_{L\left(\mu \right)},$$ where ${\text{ev}}_{\gamma }:ℂ\left[{𝔥}^{*}\right]\to ℂ$ are the algebra homomorphisms given by ${\text{ev}}_{\gamma }\left(e^{\tau }\right)=q^{⟨\gamma ,\tau ⟩}$ for $\gamma ,\tau \in {𝔥}^{*}\text{.}$

For $𝔤={𝔰𝔬}_{2r+1},$ ${𝔰𝔭}_{2r}$ or ${𝔰𝔬}_{2r}$ the Turaev-Wenzl identity almost provides an inverse to the Harish-Chandra homomorphism. With ${\epsilon }_1,\dots ,{\epsilon }_r$ as in (2.9), converting variable alphabets from $$Y=\sum _{i\in \hat{V}}{e}^{{\epsilon }_i}\text{to}X=\sum _{i\in \hat{V}}{L}_i^2,\text{then}{\text{ev}}_{2\lambda }\left(s_{\mu }\left(Y\right)\right)={\text{ev}}_{\lambda }\left(s_{\mu }\left(X\right)\right)\text{.}$$ Thus, Theorem 4.5 in combination with the Harish-Chandra isomorphism in Theorem 3.1 says that ${\text{ev}}_{\mu }\left({\pi }_0\left({\Psi }_1\left(s_{\nu }\right)\right)\right)={\text{ev}}_{2(\mu +\rho )}{s}_{\nu }\left(Y\right)={\text{ev}}_{\mu +\rho }{s}_{\nu }\left(X\right)={\text{ev}}_{\mu }\left({\sigma }_{\rho }\left(s_{\nu }\left(X\right)\right)\right)\text{.}$ Hence $$\begin{array}{cc}{\pi }_0\left({\Psi }_1\left(s_{\nu }\right)\right)={\sigma }_{\rho }\left(s_{\nu }\left(X\right)\right)\text{.}& \text{(4.11)}\end{array}$$ The modification rules of [KTe1987, §2.4] are used to convert the universal Weyl characters appearing in Corollary 3.6 to actual Weyl characters $s_{\lambda }\text{.}$ In general, either ${sp}_{\lambda }\left(X\right)=0$ or there is a unique dominant weight $\mu$ and a uniquely determined sign such that ${sp}_{\lambda }\left(X\right)=\pm s_{\mu },$ and similarly for the orthogonal cases. In particular, if $\ell \left(\lambda \right)<r$ then ${sp}_{\lambda }\left(X\right)=s_{\lambda }$ in the symplectic case and ${so}_{\lambda }\left(X\right)=s_{\lambda }$ in the orthogonal case [KTe1987, Prop. 2.2.1]. In view of (4.11), the conversion from universal Weyl characters to actual Weyl characters provides the equivalence between Corollary 4.4 and Corollary 3.6.

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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