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

Last update: 15 October 2013

## Central element transfer via Schur-Weyl duality

In Theorem 2.2 and Theorem 2.5, the parameters $z0(ℓ)=ε (id⊗trV) ((12y+γ)ℓ) andZ0(ℓ)= ε(id⊗qtrV) ((zℛ21ℛ)ℓ)$ 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 ${U}_{h}𝔤,$ respectively: $z0(ℓ)∈Z (U𝔤)and Z0(ℓ)∈Z (Uh𝔤).$ The Harish-Chandra isomorphism provides isomorphisms between the centers $Z\left(U𝔤\right)$ or $Z\left({U}_{h}𝔤\right)$ 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 $U={U}_{h}𝔤$ in the form $U={U}^{-}{U}_{0}{U}^{+}\text{.}$ If $U={U}_{h}𝔤$ then ${U}_{0}=\text{span}\left\{{K}^{{\lambda }^{\vee }} | {\lambda }^{\vee }\in {𝔥}_{ℤ}\right\}$ with ${K}^{{\lambda }^{\vee }}{K}^{{\nu }^{\vee }}={K}^{{\lambda }^{\vee }+{\nu }^{\vee }},$ where ${𝔥}_{ℤ}$ is a lattice in $𝔥\text{.}$ Alternatively, $U0=U𝔥=ℂ [h1,…,hr] if U=U𝔤and U-=ℂ [L1±1,…,Lr±1] if U=Uh𝔤,$ where ${h}_{1},\dots ,{h}_{r}$ is a basis of ${𝔥}_{ℤ},$ 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 $evμ(h)= ⟨μ,h⟩ andevμ (Kλ∨)= z⟨μ,λ∨⟩ (3.1)$ for $h\in 𝔥$ and ${K}^{{\lambda }^{\vee }}$ with ${\lambda }^{\vee }\in {𝔥}_{ℤ}\text{.}$ For $\rho =\frac{1}{2}{\sum }_{\alpha \in {R}^{+}}\alpha$ as in (1.16), let ${\sigma }_{\rho }$ be the algebra automorphism given by $σρ(hi)=h +⟨ρ,hi⟩ and σρ (Li)= q⟨ρ,hi⟩ Li. (3.2)$

Define a vector space homomorphism by $π0:U⟶U0 byπ0=ε- ⊗id⊗ε+: U-⊗U0⊗ U+⟶U0, (3.3)$ where ${\epsilon }^{-}:{U}_{-}\to ℂ$ and ${\epsilon }^{+}:{U}_{+}\to ℂ$ are the algebra homomorphisms determined by $ε-(y)=0and ε+(x)=0, for x∈𝔫+ and y∈𝔫-, or ε-(Fi)=0 andε+(Ei) =0,for i=1,…, n.$

The following important theorem says that both the center of $U𝔤$ and the center of ${U}_{h}𝔤$ 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 ${U}_{h}𝔤,$ so that $U0=ℂ[h1,…,hr] if U=U𝔤 andU0=ℂ [L1±1,…,Lr±1] if U=Uh𝔤.$ 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 $U,$ $π0: Z(U) ⟶ σρ(U0W0), z ⟼ σρ(s) is an algebra isomorphism,$ where $s\in {U}_{0}^{{W}_{0}}$ is the symmetric function determined by $zacts on L(μ) byevμ (σρ(s))= evμ+ρ(s), for μ∈𝔥*.$

### 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 $zi(u)+εu- 12= (z0(u)+εu-12) ∏j=1i (u+yj-1) (u+yj+1) (u-yj)2 (u+yj)2 (u-yj+1) (u-yj-1) , (3.4)$ where $zi(u)=∑ℓ∈ℤ≥0 zi(ℓ)u-ℓ, for i=0,1,…,k-1.$ 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 ${𝒲}_{k},$ $ei+1yi+1ℓ ei+1=zi(ℓ) ei+1,for i=0,…,k-1 and ℓ∈ℤ≥0.$

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 ${z}_{V}^{\left(\ell \right)}=\epsilon \left(\text{id}\otimes {\text{tr}}_{V}\right)\left({\left(\frac{1}{2}y+\gamma \right)}^{\ell }\right)$ 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 $y= { 2r, if 𝔤= 𝔰𝔬2r+1, 2r+1, if 𝔤=𝔰𝔭2r , 2r-1, if g=𝔰𝔬2r, ε= { 1, if 𝔤= 𝔰𝔬2r+1, -1, if 𝔤=𝔰𝔭2r , 1, if g=𝔰𝔬2r, V=L(ε1),$ and let ${z}_{V}^{\left(\ell \right)}$ be the central elements in $U𝔤$ defined by $zV(ℓ)=ε (id⊗trV) ((12y+γ)ℓ) ,and writezV (u)=∑i∈ℤ≥0 zV(ℓ)u-ℓ.$ Then $π0 (zV+εu-12)= (εu+12) (u+12y-r) (u-12y+r) σρ ( ∏i=1r (u+hi+12) (u-hi+12) (u+hi-12) (u-hi-12) ) ,$ 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 $e1y1ℓe1 acts on M⊗V⊗2 as zV(ℓ) e1.$ Also $e1 and y1 in 𝒲2 act on M⊗V⊗2 with M=L(0)⊗ V⊗(k-1)$ in the same way that $ek and yk 𝒲k+1 act on M ⊗V⊗(k+1) with M=L(0).$ By Proposition 3.2, ${z}_{k-1}^{\left(\ell \right)}{e}_{k}={e}_{k}{y}_{k}^{\ell }{e}_{k}\text{.}$ Hence, as operators on $L\left(0\right)\otimes {V}^{\otimes \left(k-1\right)},$ $zV(u)+εu- 12=zk-1 (u)+εu-12 (3.5)$ 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 $L(0)⊗ V⊗(k-1)≅ ⨁μL(μ)⊗ 𝒲k-1μ. (3.6)$ As an operator on $L\left(0\right)\otimes V,$ $γ=12 ( ⟨ε1,ε1+2ρ⟩- ⟨ε1,ε1+2ρ⟩+ ⟨0,0+2ρ⟩ ) =0by (1.17),$ and so $z0(ℓ)=ε (i⊗trV) ((12y+γ)ℓ) =ε(id⊗trV) ((12y)ℓ) =εdim(V) (12y)ℓ.$ Therefore, since $\text{dim}\left(V\right)=\epsilon +y,$ $z0(u)= ∑ℓ∈ℤ≥0 zV(ℓ) u-ℓ= ∑ℓ∈ℤ≥0 εdim(V) (12y)ℓ u-ℓ=εdim (V) 11-12yu-1 =1+εy1-12yu-1.$ Thus, as an operator on $L\left(0\right)\otimes V,$ $z0(u)+εu-12= 1+εy1-12yu-1 +εu-12= (εu+12) (u+12y) u-12y . (3.7)$ By the first identity in (1.11) and the definition of $\Phi$ in Theorem 1.2, $yk∈𝒲kacts on L(0)⊗V⊗k= (L(0)⊗V⊗(k-1)) ⊗Vas12y+γ.$ If $L\left(\mu \right)$ is an irreducible $U𝔤\text{-module}$ in $L\left(0\right)\otimes {V}^{\otimes \left(k-1\right)},$ 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 $c\left(\lambda ,\mu \right)=0$ when $\lambda =\mu ,$ and by the constant $c(λ,μ) = 12y+12 ( ⟨μ±εi,μ±εi+2ρ⟩- ⟨μ,μ+2ρ⟩- ⟨ε1,ε1+2ρ⟩ ) = { 12y+c(λ/μ), if μ⊆λ, -12y-c(μ/λ), if μ⊇λ, where λ=μ±εi. (3.8)$ 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 $T=\left({T}^{\left(0\right)},{T}^{\left(1\right)},\dots ,{T}^{\left(k\right)}\right),$ where ${T}^{\left(0\right)}=\varnothing ,$ ${T}^{\left(k\right)}=\mu ,$ and ${T}^{\left(i\right)}$ is a partition obtained from ${T}^{\left(i-1\right)}$ by adding or removing a box (or, in some cases when $𝔤={𝔰𝔬}_{2r+1}$ leaving the partition the same; see (2.21)) and $yivT= { (12y+c(b)) vT, if b=T(i) /T(i-1), (-12y-c(b)) vT, if b=T(i-1) /T(i), 0, if T(i-1) =T(i).$ Thus the product on the right hand side of (3.4) $∏i=1k-1 (u+yi-1) (u+yi+1) (u-yi)2 (u+yi)2 (u-yi+1) (u-yi-1) acts on L(μ)⊗ 𝒲k-1μ in (3.6)$ by $∏i=1k-1 (u+c(T(i),T(i-1))-1) (u+c(T(i),T(i-1))+1) (u-c(T(i),T(i-1)))2 (u+c(T(i),T(i-1)))2 (u-c(T(i),T(i-1))+1) (u-c(T(i),T(i-1))-1) (3.9)$ 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 $∏b∈μ (u+12y+c(b)-1) (u+12y+c(b)+1) (u-12y-c(b))2 (u+12y+c(b))2 (u-12y-c(b)+1) (u-12y-c(b)-1) (3.10)$ (see [Naz1996, Lemma 3.8]). If $\mu =\left({\mu }_{1},\dots ,{\mu }_{r}\right),$ simplifying one row at a time, $∏b∈μ (u+12y+c(b)-1) (u+12y+c(b)+1) (u+12y+c(b))2 = ∏i=1r (u+12y-i) (u+12y+μi-i+1) (u+12y+1-i) (u+12y+μi-i) = u+12y-r u+12y ∏i=1r (u+12y+μi-i+1) (u+12y+μi-i) , (3.11)$ (see the example following this proof). It follows that (3.10) is equal to $(u+12y-r) (u+12y) (u-12y) (u-12y+r) ∏i=1r (u+12y+μi-i+1) (u+12y+μi-i) (u-12y-(μi-i)) (u-12y-(μi-i+1)) = (u+12y-r) (u+12y) (u-12y) (u-12y+r) evμ+ρ ( ∏i=1r (u+hi+12) (u+hi-12) (u-hi+12) (u-hi-12) ) , (3.12)$ since ${\text{ev}}_{\mu +\rho }\left({h}_{i}\right)={\mu }_{i}+{\rho }_{i}={\mu }_{i}+\frac{1}{2}\left(y-2i+1\right)=\frac{1}{2}y+\frac{1}{2}+{\mu }_{i}-i\text{.}$ 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), $zV(u)+εu- 12=(εu+12) (u+12y-r) (u-12y+r) evμ+ρ ( ∏i=1r (u+hi+12) (u+hi-12) (u-hi+12) (u-hi-12) ) .$ By Theorem 3.1, the desired result follows. $\square$

Example. To help illuminate the cancellation done in (3.11), let $\mu =\left(5,5,3,3,1,1\right),$ where the contents of boxes are $0 1 2 3 4 -1 0 1 2 3 -2 -1 0 -3 -2 -1 -4 -5 (3.13)$ In this example, the product over the boxes in the first row of the diagram is $∏b in first row of μ (x+c(b)-1) (x+c(b)+1) (x+c(b)) (x+c(b)) = (x-1) (x+1) (x+0) (x+0) (x+0) (x+2) (x+1) (x+1) (x+1) (x+3) (x+2) (x+2) (x+2) (x+4) (x+3) (x+3) (x+3) (x+5) (x+4) (x+5) = (x-1) (x+5) (x+0) (x+4) ,where x=u+12y.$ Thus, simplifying the product one row at a time, $∏b∈μ (x+c(b)-1) (x+c(b)+1) (x+c(b)) (x+c(b)) = (x-1) (x+5) (x+0) (x+4) (x-2) (x+4) (x-1) (x+3) (x-3) (x+1) (x-2) (x+0) (x-4) (x+0) (x-3) (x-1) (x-5) (x-3) (x-4) (x-4) (x-6) (x-4) (x-5) (x-5) = (x-6)(x+0) (x+5)(x+4) · (x+4)(x+3) · (x+1)(x+0) · (x+0)(x-1) · (x-3)(x-4) · (x-4)(x-5)$ leads to the identity $∏b∈μ (x+c(b)-1) (x+c(b)+1) (x+c(b)) (x+c(b)) =x-rx+0 ∏i=1r x+μi-i+1 x+μi-i ,where μ= (μ1,…,μr).$

### 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 ${Z}_{i}^{\left(\ell \right)},{Z}_{i}^{\left(-\ell \right)}\in {W}_{k}$ for $i=1,\dots ,k-1$ by $Zi+(u)+ z-1q-q-1 -u2u2-1= ( Z0++ z-1q-q-1 -u2u2-1 ) ∏j=1i (u-Yj)2 (u-q-2Yj-1) (u-q2Yj-1) (u-Yj-1)2 (u-q2Yj) (u-q-2Yj) , (3.14) Zi-(u)+ zq-q-1 -u2u2-1= ( Z0-- zq-q-1 -u2u2-1 ) ∏j=1i (u-Yj-1)2 (u-q2Yj) (u-q-2Yj) (u-Yj)2 (u-q-2Yj-1) (u-q2Yj-1) , (3.15)$ where $Zi+(u)= ∑ℓ∈ℤ≥0 Zi(ℓ) u-ℓand Zi-(u)= ∑ℓ∈ℤ≥0 Zi(-ℓ) u-ℓfor i= 0,…,k-1.$ 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 ${W}_{k},$ $Ei+1Yiℓ Ei+1= Zi(ℓ) Ei+1, for i=0,1,… ,k-2 and ℓ∈ℤ.$

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 ${Z}_{V}^{\left(\ell \right)}=\epsilon \left(\text{id}\otimes {\text{qtr}}_{V}\right)\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)$ in the Drinfeld-Jimbo quantum group ${U}_{h}𝔤$ for orthogonal and symplectic Lie algebras $𝔤\text{.}$ By Theorem 2.5 these central elements are natural parameters for the affine BMW algebras.

Let $U={U}_{h}𝔤$ 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 $ℂ\left[{L}_{1}^{±1},\dots ,{L}_{r}^{±1}\right]$ 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 $y= { 2r, if 𝔤= 𝔰𝔬2r+1, 2r+1, if 𝔤=𝔰𝔭2r , 2r-1, if g=𝔰𝔬2r, ε= { 1, if 𝔤= 𝔰𝔬2r+1, -1, if 𝔤=𝔰𝔭2r , 1, if g=𝔰𝔬2r, V=L(ε1),$ and $z=\epsilon {q}^{y}\text{.}$ Let ${Z}_{V}^{\left(\ell \right)}$ be the central elements in ${U}_{h}𝔤$ defined by $ZV(ℓ)=ε (id⊗qtrV) ((zℛ21ℛ)ℓ)$ and write $ZV+(u)= ∑ℓ∈ℤ≥0 ZV(ℓ) u-ℓand ZV-(u)= ∑ℓ∈ℤ≥0 ZV(-ℓ) u-ℓ.$ Then $π0 ( ZV+(u)+ z-1q-q-1 -u2u2-1 ) =(zq-q-1) (u+q) (u-q-1) (u+1) (u-1) (u-εq2r-y) (u-εqy-2r) σρ ( ∏i=1r (u-εLi-2q-1) (u-εLi2q-1) (u-εLi-2q) (u-εLi2q) )$ and $π0 ( ZV-(u)+ zq-q-1 -u2u2-1 ) =-z-1q-q-1 (u-q) (u+q-1) (u+1) (u-1) (u-εqy-2r) (u-εq2r-y) σρ ( ∏i=1r (u-εLi-2q) (u-εLi2q) (u-εLi-2q-1) (u-εLi2q-1) ) ,$ 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 $E1Y1ℓE1 acts on M⊗V⊗2 as ZV(ℓ) E1.$ Also $E1 and Y1 in W2 act on M⊗V⊗2 with M=L(0)⊗ V⊗(k-1)$ in the same way that $Ek and Yk in Wk+1 act on M⊗V⊗(k+1) with M=L(0).$ 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 $L\left(0\right)\otimes {V}^{\otimes \left(k-1\right)},$ $ZV+(u)+ z-1q-q-1 -u2u2-1 = Zk-1+(u)+ z-1q-q-1 -u2u2-1 (3.16)$ and $ZV-(u)- zq-q-1 -u2u2-1 = Zk-1-(u)+ zq-q-1 -u2u2-1 (3.17)$ 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 ${U}_{h}𝔤\otimes {W}_{k-1}\text{-module}$ decomposition $L(0)⊗ V⊗(k-1)≅ ⨁μL(μ) ⊗Wk-1μ. (3.18)$ 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 $ZV(ℓ)=ε (id⊗qtrV) ((zℛ21ℛ)ℓ) =zℓεdimq(V).$ Therefore, since $\epsilon {\text{dim}}_{q}\left(V\right)=\frac{z-{z}^{-1}}{q-{q}^{-1}}=1,$ $ZV+(u)= ∑ℓ∈ℤ≥0 εdimq(V) zℓu-ℓ=ε dimq(V) 11-zu-1= z-z-1+ (q-q-1) (q-q-1) (1-zu-1) .$ A similar computation of ${Z}_{V}^{-}$ yields $ZV-(u)= z-z-1+q-q-1 (q-q-1) (1-z-1u-1) .$ Thus, as operators on $L\left(0\right)\otimes V,$ $ZV++ z-1q-q-1 -u2u2-1= z(q-q-1) (1-z-1u-1) (1-zu-1) (u+q) (u-q-1) (u+1) (u-1) (3.19)$ and $ZV-- zq-q-1 -u2u2-1= -z-1(q-q-1) (1-zu-1) (1-z-1u-1) (u-q) (u+q-1) (u+1) (u-1) . (3.19)$ By (1.29) and the definition of $\Phi$ in Theorem 1.3, $Yk∈Wkacts on L(0)⊗V⊗k= (L(0)⊗V⊗(k-1)) ⊗Vaszℛ21ℛ.$ If $L\left(\mu \right)$ is an irreducible ${U}_{h}𝔤\text{-module}$ in $L\left(0\right)\otimes {V}^{\otimes \left(k-1\right)},$ 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 $\epsilon {q}^{2c\left(\lambda ,\mu \right)}=1·{q}^{0}=1,$ when $𝔤={𝔰𝔬}_{2r+1}$ and $\lambda =\mu ,$ and by the constant $εq2c(λ,μ)= { εqy+2c(λ/μ), if μ⊆λ, εq-y-2c(μ/λ) if μ⊇λ, = { zq2c(λ/μ), if μ⊆λ, z-1 q-2c(μ/λ), if μ⊇λ, ,where λ=μ±εi.$ and $c\left(\lambda ,\mu \right)$ 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 $T=\left\{{T}^{\left(0\right)},{T}^{\left(1\right)},\cdots ,{T}^{\left(k\right)}\right\},$ where ${T}^{\left(0\right)}=\varnothing ,$ ${T}^{\left(k\right)}=\mu ,$ and ${T}^{\left(i\right)}$ is a partition obtained from ${T}^{\left(i-1\right)}$ by adding or removing a box, and $YivT= { zq2c(b)vT, if b=T(i)/ T(i-1), z-1q-2c(b) vT, if b=T(i-1) /T(i), vT, if T(i-1)= T(i).$ Thus $∏i=1k-1 (u-Yi)2 (u-q-2Yi-1) (u-q2Yi-1) (u-Yi-1)2 (u-q2Yi) (u-q-2Yi) acts on L(μ)⊗ Wk-1μ in (3.18)$ by $∏i=1k-1 (u-εq2c(T(i),T(i-1)))2 (u-εq-2q2c(T(i),T(i-1))) (u-εq2q2c(T(i),T(i-1))) (u-εq-2c(T(i),T(i-1)))2 (u-εq2q2c(T(i),T(i-1))) (u-εq-2q2c(T(i),T(i-1))) (3.21)$ 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 $∏b∈μ (u-zq2c(b))2 (u-z-1q-2(c(b)+1)) (u-z-1q-2(c(b)-1)) (u-z-1q-2c(b))2 (u-zq2(c(b)+1)) (u-zq2(c(b)-1)) . (3.22)$ Simplifying one row at a time, $∏b∈μ (u-z-1q-2(c(b)-1)) (u-z-1q-2(c(b)+1)) (u-z-1q-2c(b)) (u-z-1q-2c(b)) = ∏i=1r (u-z-1q-2(-i)) (u-z-1q-2(μi-i+1)) (u-z-1q-2(-(i-1))) (u-z-1q-2(μi-i)) = u-z-1q2r u-z-1q2·0 ∏i=1r u-z-1q-2(μi-i+1) u-z-1q-2(μi-i)$ if $\mu =\left({\mu }_{1},\dots ,{\mu }_{r}\right)\text{.}$ It follows that (3.22) is equal to $(u-z-1q2r) (u-z-1) (u-z) (u-zq-2r) ∏i=1r (u-z-1q-2(μi-i+1)) (u-z-1q-2(μi-i)) · (u-zq2(μi-i)) (u-zq2(μi-i+1)) = (u-εq-(y-2r)) (u-z-1) (u-z) (u-εqy-2r) evμ+ρ ( ∏i=1r (u-εLi-2q-1) (u-εLi2q-1) (u-εLi-2q) (u-εLi2q) ) (3.23)$ since ${z}^{-1}{q}^{2r}=\epsilon {q}^{-y}{q}^{2r}=\epsilon {q}^{2r-y}$ and $evμ+ρ (Li2)= q⟨μ+ρ,2π⟩ =q2μi(y-2i+1) =qy+1+2(μi-i) =εzq2(μi-i)+1.$ 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), $Zk++ z-1q-q-1 -u2u2-1 =z(q-q-1) (u+q) (u-q-1) (u+1) (u-1) (u-εq2r-y) (u-εqy-2r) evμ+ρ ( ∏i=1r (u-εLi-2q-1) (u-εLi2q-1) (u-εLi-2q) (u-εLi2q) ) . (3.24)$ 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 ${U}_{h}𝔤\otimes {W}_{k-1}\text{-module}$ decomposition in (3.18) by $- z-1(q-q-1) (u-q) (u+q-1) (u+1) (u-1) (u-εq2r-y) (u-εqy-2r) evμ+ρ ( ∏i=1r (u-εLi-2q) (u-εLi2q) (u-εLi-2q-1) (u-εLi2q-1) ) .$ 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 $∏i,j 11-xiyj =Ω(XY)= ∑λsaλ(X) saλ(Y), ∏i≤j 11-yiyj ∏i,j 11-xiyj =Ω(XY-sa(2)(Y)) =∑λsaλ(Y) soλ(X), ∏i 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 $saλ ((q-q-1)u-1) = { (q-q-1) u-ℓ (-q-1)ℓ-m qm-1, if λ=(m,1ℓ-m), 0, otherwise. (3.25)$

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 ${Z}_{V}^{\left(\ell \right)}=\epsilon \left(\text{id}\otimes {\text{qtr}}_{L\left({\epsilon }_{1}\right)}\right)\left({\left(z{ℛ}_{21}ℛ\right)}^{\ell }\right)\text{.}$ Let $X$ be the formal alphabet given by $X={\sum }_{i\in \stackrel{ˆ}{V}}{L}_{i}^{2}$ and fix $c=1$ if $\ell$ is even and $c=0$ if $\ell$ is odd. Then for $\ell \ge 1,$ $π0(ZV(ℓ))= σρ ( c+zεℓ∑m=1ℓ (q-q-1) (-1)ℓ-m q-(ℓ-2m+1) s(m,1ℓ-m) (X) )$ where ${s}_{\left(m,{1}^{\ell -m}\right)}\left(X\right)={so}_{\left(m,{1}^{\ell -m}\right)}\left(X\right)$ in the orthogonal cases and ${s}_{\left(m,{1}^{\ell -m}\right)}\left(X\right)={sp}_{\left(m,{1}^{\ell -m}\right)}\left(X\right)$ in the symplectic case.

 Proof. Let $\stackrel{ˆ}{V}$ 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 $π0 ( ZV+(U)+ z-1q-q-1- u2u2-1 ) =σρ ( zq-q-1 u2-q2ε u2-1 ∏j∈Vˆ (1-Lwt(vj)2q-1(εu)-1) (1-Lwt(vj)2q(εu)-1) ) . (3.26)$ By (3.25), $sa(2) ( (q-q-1) (εu)-1 ) =(q2-1) (εu)-2 andsa(12) ( (q-q-1) (εu)-1 ) =(q-2-1) (εu)-2.$ So the Cauchy-Littlewood identities give $(1-q2(εu)-2) 1-(εu)-2 ∏i∈Vˆ (1-Li2q-1(εu)-1) (1-Li2q(εu)-1) =Ω ( X(q-q-1) (εu)-1- sa(2) ( (q-q-1) (εu)-1 ) ) =∑λsaλ ( (q-q-1) (εu)-1 ) soλ(X) =∑ℓ∈ℤ≥0 ( ∑m=1ℓ (q-q-1) (-q-1)ℓ-m qm-1 so(m,1ℓ-m) (X) ) (εu)-ℓ =∑ℓ∈ℤ≥0 eℓ(q-q-1) ( ∑m=1ℓ (q-q-1) (-1)ℓ-m q-(ℓ-2m+1) so(m,1ℓ-m) (X) ) u-ℓ$ in the orthogonal case, and $(1-q-2(εu)-2) 1-(εu)-2 ∏i∈Vˆ (1-Li2q-1(εu)-1) (1-Li2q(εu)-1) =Ω ( X(q-q-1) (εu)-1- sa(12) ( (q-q-1) (εu)-1 ) ) =∑λsaλ ( (q-q-1) (εu)-1 ) spλ(X) =∑ℓ∈ℤ≥0 ( ∑m=1ℓ (q-q-1) (-q-1)ℓ-m qm-1 sp(m,1ℓ-m) (X) ) (εu)-ℓ =∑ℓ∈ℤ≥0 eℓ(q-q-1) ( ∑m=1ℓ (-1)ℓ-m q-(ℓ-2m+1) sp(m,1ℓ-m) (X) ) u-ℓ$ in the symplectic case. The statement now follows by noting that ${u}^{2}/\left({u}^{2}-1\right)=1/\left(1-{u}^{-2}\right)={\sum }_{k\in {ℤ}_{\ge 0}}{u}^{-2k}$ 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.