## Affine and degenerate affine BMW algebras: Actions on tensor space

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 $(id⊗trV) (γℓ)= ∑i1,i2,…,iℓ Ei1i2 Ei2i3⋯ Eiℓi1$ 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 $F={\left({F}_{ij}\right)}_{i,j\in \stackrel{ˆ}{V}}$ as a matrix with entries in $𝔤,$ and writing $trFk= ∑i1,i2,…,ik∈Vˆ Fi1i2 Fi2i3⋯ Fiki1 for k∈ℤ>0, (4.1)$ 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 $li=-l-i=hi +ρi,for i =1,2,…,r,where ρi=12 (y-2i+1).$ Then $π0 ( 1+ x+12 x+12-12ε ( ∑ℓ∈ℤ≥0 (-1)ℓtr(Fk) (x+ρr′)ℓ+1 ) ) =∏i∈Vˆ x+li+1 x+li . (4.2)$

 Proof. By (2.11), $γ=12 ∑i,j∈Vˆ Fij⊗Fji.$ Let $\eta :𝔤\to \text{End}\left(V\right)$ be the defining representation. Since ${F}_{-j,-i}=-{\theta }_{ij}{F}_{ij},$ $(id⊗η)(γ) = 12∑i,j∈Vˆ Fij⊗ ( Eji-θji E-i,-j ) =12∑i,j∈Vˆ ( Fij⊗Eji- θjiF-j,-i ⊗Eji ) = 12∑i,j∈Vˆ ( Fij-θji F-j,-i ) ⊗Eji =12∑i,j∈Vˆ (Fij+Fij) ⊗Eji = ∑i,j∈Vˆ Fij⊗Eji= Ft=-θFθ, whereθ= ( εid0 0id ) .$ Thus $(id⊗trV) (γk)= tr((Ft)k) =tr((-θFθ)k) =(-1)ktr (θ2Fk)= (-1)ktr (Fk), (4.3)$ 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 $\left(\text{id}\otimes {\text{tr}}_{V}\right)\left({\gamma }^{\ell }\right)$ into the generating function for the elements $\left(\text{id}\otimes {\text{tr}}_{V}\right)\left({\left(\frac{1}{2}y+\gamma \right)}^{\ell }\right),$ notice $∑ℓ∈ℤ≥0 (id⊗trV) ((12y+γ)ℓ) u-ℓ=(id⊗trV) (11-(12y+γ)u-1) =(id⊗trV) (11-12yu-1-γu-1) =(id⊗trV) ( (11-12yu-1) (11-γu-11-12yu-1) ) =(id⊗trV) ( 11-12yu-1 ∑ℓ∈ℤ≥0 γℓu-ℓ (1-12yu-1)ℓ ) =u ( ∑ℓ∈ℤ≥0 (id⊗trV)(γℓ) (u-12y)ℓ+1 ) = ( ∑ℓ∈ℤ≥0 (id⊗trV)(γℓ) (u-12+ρr′)ℓ+1 ) ,where ρr′= 12-12y.$ Then Theorem 3.3 is equivalent to $π0 ( 1+εuεu-12 ( ∑ℓ∈ℤ≥0 (id⊗trV)(γℓ) (u-12+ρr′)ℓ+1 ) ) =π0 ( 1+εεu-12 ∑ℓ∈ℤ≥0 (id⊗trV) ((12y+γ)ℓ) u-ℓ ) = (εu+12) (εu-12) (u+12y-r) (u-12y+r) σρ ( ∏i=1r (u+hi+12) (u+hi-12) (u-hi+12) (u-hi-12) ) = (εu+12) (εu-12) (u+12y-r) (u-12y+r) ( ∏i=1r (u+hi+ρi+12) (u+hi-ρi+12) (u-hi-ρi+12) (u-hi-ρi-12) ) = (u-12+12ε+12) (u-12-12ε+12) (u-12+12y-r+12) (u-12-12y+r+12) ( ∏i=1r (u-12+hi+ρi+1) (u-12+hi+ρi) (u-12-hi-ρi+1) (u-12-hi-ρi) ) .$ Replacing $x=u-\frac{1}{2},$ $π0 ( 1+ x+12 x+12-12ε ( ∑ℓ∈ℤ≥0 (id⊗trV)(γℓ) (x+ρr′)ℓ+1 ) ) = (x+12ε+12) (x-12ε+12) (x+12y-r+12) (x-12y+r+12) ( ∏i=1r (x+hi+ρi+1) (x+hi+ρi) (x-hi-ρi+1) (x-hi-ρi) )$ Since $(x+12ε+12) (x-12ε+12) (x+12y-r+12) (x-12y+r+12) = { x+l0+1x+l0, if 𝔤=𝔰𝔬2r+1, 1, if 𝔤=𝔰𝔭2r or 𝔰𝔬2r,$ it follows that $π0 ( 1+ x+12 x+12-12ε ( ∑ℓ∈ℤ≥0 (id⊗trV)(γℓ) (x+ρr′)ℓ+1 ) ) =∏i∈Vˆ x+li+1 x+li .$ 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 $w=\left(1,-r\right)\left(2,-\left(r-1\right)\right)\cdots \left(r-1,-2\right)\left(r,-1\right),$ 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 $\left(\text{id}\otimes {\text{qtr}}_{L\left(\nu \right)}\right)\left({ℛ}_{21}ℛ\right)$ 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 $sλ=aλ+ρaρ ,whereaμ= ∑w∈W0det(w) ewμ.$ The expressions ${s}_{\lambda }$ and ${a}_{\mu }$ are elements of the group algebra of ${𝔥}^{*},$ $ℂ\left[{𝔥}^{*}\right]=ℂ\text{-span}\left\{{e}^{\nu } | \nu \in {𝔥}^{*}\right\}$ with ${e}^{\mu }{e}^{\nu }={e}^{\mu +\nu }\text{.}$ If $w\in {W}_{0}$ then $awμ=det(w) aμand sw∘μ=det (w)sμ, (4.4)$ where the dot action of ${W}_{0}$ on ${𝔥}^{*}$ is given by $w∘μ=w(μ+ρ) -ρ,for w∈ W0,μ∈𝔥*. (4.5)$ The ${W}_{0}\text{-invariants}$ in $ℂ\left[{𝔥}^{*}\right]$ are $ℂ{\left[{𝔥}^{*}\right]}^{{W}_{0}}=ℂ\text{-span}\left\{{s}_{\lambda } | \lambda \text{dominant integral}\right\}\text{.}$ For $\ell \in {ℤ}_{\ge 0}$ let $Ψℓ: ℂ[𝔥*]W0 ⟶ Z(Uh𝔤) sν ⟼ (id⊗qtrL(ν))((ℛ21ℛ)ℓ) . (4.6)$ By [Dri1990, Prop. 1.2], the map ${\Psi }_{\ell }$ is a vector space isomorphism.

[Bau1620662, Thm. 1] For $\ell \in {ℤ}_{\ge 0}$ define $mλ(ℓ)= ∑w∈W0 q2ℓ⟨wλ,ρ⟩ swλ.Then Ψℓ (mλ(ℓ))= Ψ1 (mℓλ(1/ℓ)) .$

In the same setting as in Theorem 3.5, let $y= { 2r, if 𝔤= 𝔰𝔬2r+1, 2r+1, if 𝔤=𝔰𝔭2r , 2r-1, if g=𝔰𝔬2r, ε= { 1, if 𝔤= 𝔰𝔬2r+1, -1, if 𝔤=𝔰𝔭2r , 1, if g=𝔰𝔬2r, V=L(ε1),$ $z=\epsilon {q}^{y},$ and let ${Z}_{V}^{\left(\ell \right)}$ be the central elements in the Drinfeld-Jimbo quantum group ${U}_{h}𝔤$ which are given by ${Z}_{V}^{\left(\ell \right)}=\epsilon \left(\text{id}\otimes {\text{qtr}}_{L\left({\epsilon }_{1}\right)}\right)\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)\text{.}$ Then, for $\ell \ge 1,$ $ZV(ℓ)=εℓΨ1 ( c+z∑m=max(ℓ-r+1,1)ℓ (-1)ℓ-m q-(ℓ-2m+1) s(m,1ℓ=m+1-1) +z∑m=max(y-ℓ+1,1)ℓ-y+r q-(ℓ-2m+1) (-1)m-ℓ+y s(m,1m-ℓ+y-1) ) ,$ where $c$ is given by $c= { 1 if ℓ is even and ℓ

 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 ${s}_{i}=\left(i,i+1\right)$ (the transposition switching εi and ${\epsilon }_{i+1},$ for $i=1,2,\dots ,r-1\text{)}$ and ${s}_{r}=\left(r,-r\right)$ (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 ${s}_{i}=\left(i,i+1\right)$ for $i=1,2,\dots ,r-1,$ and ${s}_{r}=\left(r-1,-r\right)\left(r,-\left(r-1\right)\right)\text{.}$ To prove the desired identity, we will use the second identity in (4.4) to relabel the Weyl characters ${s}_{w\lambda }$ appearing in $mε1(ℓ) =∑w∈W0 q2ℓ⟨wε1,ρ⟩ swε1and mℓε1(1/ℓ) =∑w∈W0 q2⟨wε1,ρ⟩ swℓε1$ by dominant integral weights. By (2.13), ${\rho }_{i+1}={\rho }_{i}-1$ and ${\rho }_{r}=\frac{1}{2}\left(y-2r+1\right)\text{.}$ So if $\mu ={\mu }_{1}{\epsilon }_{1}+\cdots +{\mu }_{r}{\epsilon }_{r}$ then $si∘μ=μ1+⋯+ μi-1εi-1+ (μi+1-1)εi +(μi+1)εi+1 +μi+2εi+2 +⋯+μrεr$ for $i=1,2,\dots ,r-1,$ and $sr∘μ=μ1ε1 +⋯μr-1 εr-1+ (-μr-(y-2r+1)) εr,if 𝔤= 𝔰𝔬2r+1 or 𝔰𝔭2r, and sr∘μ=μ1ε1+ ⋯μr-1εr-2+ (-μr-1)εr-1 +(-μr-1-1) εr,if 𝔤= 𝔰𝔬2r.$ In particular, ${s}_{\ell {\epsilon }_{i}}=0$ if $0<\ell and $sℓεi= ss1s2⋯si-1∘ℓεi =(-1)i-1 s(ℓ-i+1)ε1+ε2+⋯+εi =(-1)i-1 s(ℓ-i+1,1i-1) if 0 Furthermore, if $𝔤={𝔰𝔬}_{2r+1}$ or ${𝔰𝔭}_{2r},$ then $si⋯sr-1 srsr-1⋯ si∘(-ℓεi) =si⋯sr-1 sr∘ ( -(εi+⋯+εr-1) -(ℓ-(r-i))εr ) = si⋯sr-1∘ ( -(εi+⋯+εr-1) +(ℓ-(r-i)-(y-2r+1)) εr ) = ( ℓ-(r-i)- (y-2r+1)- (r-i) ) εi= (ℓ+2i-y-1)εi. (4.8)$ Similarly, if $𝔤={𝔰𝔬}_{2r},$ then $si⋯sr-2 srsr-1⋯ si∘(-ℓεi) =si⋯sr-2 sr∘ ( -(εi+⋯+εr-1) -(ℓ-(r-i))εr ) = si⋯sr-2∘ ( -(εi+⋯+εr-2) +(ℓ-(r-i)-1) εr-1 ) = ( ℓ-(r-i)- 1- (r-i-1) ) εi= (ℓ+2i-y-1)εi. (4.9)$ So, letting ${W}_{{\epsilon }_{1}}$ be the stabilizer of ${\epsilon }_{1},$ $|{W}_{{\epsilon }_{1}}|={2}^{r-1}\left(r-1\right)!,$ and combining (4.7) and (4.8) gives $1|Wε1| mε1(ℓ) = ∑i=1r q2ℓ⟨εi,ρ⟩ sεi+ q2ℓ⟨-εi,ρ⟩ s-εi= { q2ℓ⟨ε1,ρ⟩ sε1+q-2eλ⟨εr,ρ⟩ sε-r if 𝔤=𝔰𝔬2r+1, q2ℓ⟨ε1,ρ⟩ sε1 if 𝔤=𝔰𝔭2r or 𝔤=𝔰𝔬2r. = qℓ(y-1) sε1-awhere a= { q-ℓs0 if 𝔤=𝔰𝔬2r+1 0 otherwise, (4.10)$ since $1+2i-y-1=0$ has a solution exactly if $i=r$ and $𝔤={𝔰𝔬}_{2r+1}\text{.}$ If $\ell >0$ then using $det ( si⋯sr-1sr sr-1⋯si ) =-1=-det ( si⋯sr-2 srsr-1⋯si ) ,$ $\ell +2i-y-1-\left(i-1\right)=\ell -y+i,$ and (4.7), equations (4.8) and (4.9) give $q2⟨-εi,ρ⟩ s-ℓεi= { -q-ℓs0, if ℓ=y+1-2i and 𝔤= 𝔰𝔬2r+1 or 𝔰𝔭2r, q-ℓs0, if ℓ=y+1-2i and 𝔤=𝔰𝔬2r, (-1)i q-(y-2i+1) s(ℓ-y+i,1i-1), if ℓ-y+i≥1, 0, otherwise.$ Since $\ell -y+i\ge 1$ when $i\ge y-\ell +1,$ $1|Wε1| mℓε1(1/ℓ) = ∑i=1r q2⟨εi,ρ⟩ sℓεi+ q2⟨-εi,ρ⟩ s-ℓεi = ( ∑i=1min(ℓ,r) qy-2i+1 (-1)i-1 s(ℓ-i+1,1i-1) ) -b+ ( ∑i=max(y-ℓ+1,1)r q-(y-2i+1) (-1)i s(ℓ-y+u,1i-1) ) where b=0 if ℓ ≥y, and if ℓ (so that $b$ is nonzero exactly when $\ell =y+1-2i$ has a solution with $1\le i\le r\text{).}$ Then reindexing (with $m=\ell -i+1$ in the first sum and $m=\ell -y+i$ in the second sum) gives $1|Wε1| mℓε1(1/ℓ) = -b+ ∑m=max(ℓ-r+1,1)ℓ (-1)ℓ-m qy-2(ℓ-m)-1 s(m,1ℓ-m+1-1) + ∑m=max(y-ℓ+1,1)ℓ-y+r qy-2(ℓ-m)-1 (-1)m-ℓ+y s(m,1m-ℓ+y-1).$ Notice that the last sum appears only if $\ell >y-r\text{.}$ Theorem 4.3 applied in the case that $\lambda ={\epsilon }_{1}$ gives $ZV(ℓ) = ε(id⊗qtrV) ((zℛ21ℛ)ℓ) =εzℓΨℓ (sε1)=ε (εqy)ℓ Ψℓ ( q-ℓ(y-1) ( 1|Wε1| mε1(ℓ)+a ) ) = εℓ+1aℓ ( 1|Wε1| Ψℓ (mε1(ℓ))+a ) =εℓ+1qℓ ( 1|Wε1| Ψ1 (mℓε1(1/ℓ)) +a ) = εℓΨ1 ( εqℓ ( mℓε1(1/ℓ) |Wε1| +a ) ) = εℓΨ1 ( εqℓ(a-b)+z ∑m=max(ℓ-r+1,1)ℓ (-1)ℓ-m q-(ℓ-2m+1) s(m,1ℓ-m+1-1) +z∑m=max(ℓ-r+1,1)ℓ-y+r q-(ℓ-2m+1) (-1)m-ℓ+y s(m,1m-ℓ+y-1) ) ,$ since $z=\epsilon {q}^{y}\text{.}$ The result follows since $c=\epsilon {q}^{\ell }\left(a-b\right)\text{.}$ $\square$

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 $U={U}_{h}𝔤$ 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 $(id⊗qtrL(ν)) (ℛ21ℛ)acts on L(μ) by ev2(μ+ρ) (sν)idL(μ),$ 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 $Y=∑i∈Vˆ eεito X=∑i∈Vˆ Li2,then ev2λ(sμ(Y)) =evλ(sμ(X)).$ Thus, Theorem 4.5 in combination with the Harish-Chandra isomorphism in Theorem 3.1 says that ${\text{ev}}_{\mu }\left({\pi }_{0}\left({\Psi }_{1}\left({s}_{\nu }\right)\right)\right)={\text{ev}}_{2\left(\mu +\rho \right)}{s}_{\nu }\left(Y\right)={\text{ev}}_{\mu +\rho }{s}_{\nu }\left(X\right)={\text{ev}}_{\mu }\left({\sigma }_{\rho }\left({s}_{\nu }\left(X\right)\right)\right)\text{.}$ Hence $π0(Ψ1(sν))= σρ(sν(X)). (4.11)$ 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 ${sp}_{\lambda }\left(X\right)=±{s}_{\mu },$ and similarly for the orthogonal cases. In particular, if $\ell \left(\lambda \right) 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.