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 z0()=ε (idtrV) ((12y+γ)) andZ0()= ε(idqtrV) ((z21)) 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 Uh𝔤, respectively: z0()Z (U𝔤)and Z0()Z (Uh𝔤). The Harish-Chandra isomorphism provides isomorphisms between the centers Z(U𝔤) or Z(Uh𝔤) and rings of symmetric functions. In this section we show how to use the recursive formulas of [Naz1996] and [BBl1866492] for the central elements zk() and Zk() in the degenerate affine and affine BMW algebras (formulas (3.4) and (3.14)) to determine the Harish-Chandra images of z0() and Z0().

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-𝔥𝔫+ (see [Bou1990, VII §8 no. 3 Prop. 9]) yields triangular decompositions of both the enveloping algebra U=U𝔤 and the quantum group U=Uh𝔤 in the form U=U-U0U+. If U=Uh𝔤 then U0=span{Kλ|λ𝔥} with KλKν=Kλ+ν, where 𝔥 is a lattice in 𝔥. Alternatively, U0=U𝔥= [h1,,hr] ifU=U𝔤and U-= [L1±1,,Lr±1] ifU=Uh𝔤, where h1,,hr is a basis of 𝔥, and Li=Khi=qhi.

For μ𝔥*, define the ring homomorphisms evμ:U0 by evμ(h)= μ,h andevμ (Kλ)= zμ,λ (3.1) for h𝔥 and Kλ with λ𝔥. For ρ=12αR+α as in (1.16), let σρ be the algebra automorphism given by σρ(hi)=h +ρ,hi andσρ (Li)= qρ,hi Li. (3.2)

Define a vector space homomorphism by π0:UU0 byπ0=ε- idε+: U-U0 U+U0, (3.3) where ε-:U- and ε+:U+ are the algebra homomorphisms determined by ε-(y)=0and ε+(x)=0, forx𝔫+ andy𝔫-, or ε-(Fi)=0 andε+(Ei) =0,fori=1,, n.

The following important theorem says that both the center of U𝔤 and the center of Uh𝔤 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 Uh𝔤, so that U0=[h1,,hr] ifU=U𝔤 andU0= [L1±1,,Lr±1] ifU=Uh𝔤. Let L(μ) denote the irreducible U-module of highest weight μ. Then the restriction of π0 to the center of U, π0: Z(U) σρ(U0W0), z σρ(s) is an algebra isomorphism, where sU0W0 is the symmetric function determined by zacts onL(μ) byevμ (σρ(s))= evμ+ρ(s), forμ𝔥*.

Central elements zV()

Let z0() and ε be the parameters of the degenerate affine BMW algebra 𝒲k. Let u be a variable and define zi()𝒲k for i=1,,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)=0 zi()u-, fori=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 𝒲k, ei+1yi+1 ei+1=zi() ei+1,for i=0,,k-1and 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 zV()=ε(idtrV)((12y+γ)) in the enveloping algebra U𝔤 for orthogonal and symplectic Lie algebras 𝔤. 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=𝔰𝔬2r+1, 𝔰𝔭2r or 𝔰𝔬2r, use notations for 𝔥* as in (2.5)-(2.16) and let h1,,hr be the basis of 𝔥 dual to the orthonormal basis ε1,,εr of 𝔥* (so that hi=Fii, where Fii is as in (2.8)). With respect to the form , in (2.10), let γ=bbb* as in (1.15). Let y= { 2r, if𝔤= 𝔰𝔬2r+1, 2r+1, if𝔤=𝔰𝔭2r , 2r-1, ifg=𝔰𝔬2r, ε= { 1, if𝔤= 𝔰𝔬2r+1, -1, if𝔤=𝔰𝔭2r , 1, ifg=𝔰𝔬2r, V=L(ε1), and let zV() be the central elements in U𝔤 defined by zV()=ε (idtrV) ((12y+γ)) ,and writezV (u)=i0 zV()u-. 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 σρ is the algebra automorphism given by σρ(hi)=hi+ρ,εi and π0 is the isomorphism in Theorem 3.1.

Proof.

In the definition of the action of the degenerate affine BMW algebra in Theorem 2.2, y1 acts on MV as 12y+γ, and e1y1e1 acts onMV2 aszV() e1. Also e1andy1 in𝒲2act on MV2with M=L(0) V(k-1) in the same way that ekandyk 𝒲k+1act onM V(k+1) withM=L(0). By Proposition 3.2, zk-1()ek=ekykek. Hence, as operators on L(0)V(k-1), 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(μ)𝒲k-1μ isotypic component in the U𝔤𝒲k-1-module decomposition L(0) V(k-1) μL(μ) 𝒲k-1μ. (3.6)

As an operator on L(0)V, γ=12 ( ε1,ε1+2ρ- ε1,ε1+2ρ+ 0,0+2ρ ) =0by (1.17), and so z0()=ε (itrV) ((12y+γ)) =ε(idtrV) ((12y)) =εdim(V) (12y). Therefore, since dim(V)=ε+y, z0(u)= 0 zV() u-= 0 εdim(V) (12y) u-=εdim (V) 11-12yu-1 =1+εy1-12yu-1. Thus, as an operator on L(0)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 Φ in Theorem 1.2, yk𝒲kacts on L(0)Vk= (L(0)V(k-1)) Vas12y+γ. If L(μ) is an irreducible U𝔤-module in L(0)V(k-1), then (1.17), (2.13), and (2.16) give that yk acts on the L(λ) component of L(μ)V by the constant c(λ,μ)=0 when λ=μ, and by the constant c(λ,μ) = 12y+12 ( μ±εi,μ±εi+2ρ- μ,μ+2ρ- ε1,ε1+2ρ ) = { 12y+c(λ/μ), ifμλ, -12y-c(μ/λ), ifμλ, whereλ=μ±εi. (3.8) As in [Naz1996, Theorem 2.6], the irreducible 𝒲k-module 𝒲kμ/0=𝒲kμ has a basis {vT} indexed by up-down tableaux T=(T(0),T(1),,T(k)), where T(0)=, T(k)=μ, and T(i) is a partition obtained from T(i-1) by adding or removing a box (or, in some cases when 𝔤=𝔰𝔬2r+1 leaving the partition the same; see (2.21)) and yivT= { (12y+c(b)) vT, ifb=T(i) /T(i-1), (-12y-c(b)) vT, ifb=T(i-1) /T(i), 0, ifT(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 onL(μ) 𝒲k-1μin (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 μ. 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 bμ (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 μ=(μ1,,μr), simplifying one row at a time, bμ (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 evμ+ρ(hi)= μi+ρi=μi+ 12(y-2i+1)= 12y+12+μi -i.

Combining (3.7) and (3.12), the identity (3.5) gives that, as operators on L(μ)𝒲k-1μ 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.

Example. To help illuminate the cancellation done in (3.11), let μ=(5,5,3,3,1,1), 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 bin first row ofμ (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, bμ (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 bμ (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 ,whereμ= (μ1,,μr).

Central elements ZV()

Let Z0(), z and q be the parameters of the affine BMW algebra Wk. Let u be a variable and define Zi(),Zi(-)Wk for i=1,,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)= 0 Zi() u-and Zi-(u)= 0 Zi(-) u-fori= 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 Wk, Ei+1Yi Ei+1= Zi() Ei+1, fori=0,1, ,k-2and .

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 ZV()=ε(idqtrV)((z21)) in the Drinfeld-Jimbo quantum group Uh𝔤 for orthogonal and symplectic Lie algebras 𝔤. By Theorem 2.5 these central elements are natural parameters for the affine BMW algebras.

Let U=Uh𝔤 be the Drinfeld-Jimbo quantum group corresponding to 𝔤=𝔰𝔬2r+1, 𝔰𝔭2r or 𝔰𝔬2r and use notations for 𝔥* as in (2.5)-(2.16). Identify U0 as a subalgebra of [L1±1,,Lr±1] where evεi(Lj)=qεi,εj=qδij (so that Li=e12hFii, where Fii is as in (2.8)). Let y= { 2r, if𝔤= 𝔰𝔬2r+1, 2r+1, if𝔤=𝔰𝔭2r , 2r-1, ifg=𝔰𝔬2r, ε= { 1, if𝔤= 𝔰𝔬2r+1, -1, if𝔤=𝔰𝔭2r , 1, ifg=𝔰𝔬2r, V=L(ε1), and z=εqy. Let ZV() be the central elements in Uh𝔤 defined by ZV()=ε (idqtrV) ((z21)) and write ZV+(u)= 0 ZV() u-and ZV-(u)= 0 ZV(-) u-. 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 σρ is the algebra automorphism given by σρ(Li)=qρ,εiLi and π0 is the isomorphism in Theorem 3.1.

Proof.

In the definition of the action of the affine BMW algebra in Theorem 1.3, Y1 acts on MV as z21, and E1Y1E1 acts onMV2 asZV() E1. Also E1andY1 inW2act on MV2with M=L(0) V(k-1) in the same way that EkandYk inWk+1 act onMV(k+1) withM=L(0).

By Proposition 3.4, Zk-1()Ek=EkYkEk and so it follows that, as operators on L(0)V(k-1), 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(μ)Wk-1μ isotypic component in the Uh𝔤Wk-1-module decomposition L(0) V(k-1) μL(μ) Wk-1μ. (3.18)

As an operator on L(0)V, z(21)= z q ε1,ε1+2ρ- ε1,ε1+2ρ+ 0,0+2ρ =z. Hence ZV()=ε (idqtrV) ((z21)) =zεdimq(V). Therefore, since εdimq(V)=z-z-1q-q-1=1, ZV+(u)= 0 εdimq(V) zu-=ε dimq(V) 11-zu-1= z-z-1+ (q-q-1) (q-q-1) (1-zu-1) . A similar computation of ZV- yields ZV-(u)= z-z-1+q-q-1 (q-q-1) (1-z-1u-1) . Thus, as operators on L(0)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 Φ in Theorem 1.3, YkWkacts on L(0)Vk= (L(0)V(k-1)) Vasz21. If L(μ) is an irreducible Uh𝔤-module in L(0)V(k-1), then (1.28) and (2.16) give that Yk acts on the L(λ) component of L(μ)V by the constant εq2c(λ,μ)=1·q0=1, when 𝔤=𝔰𝔬2r+1 and λ=μ, and by the constant εq2c(λ,μ)= { εqy+2c(λ/μ), ifμλ, εq-y-2c(μ/λ) ifμλ, = { zq2c(λ/μ), ifμλ, z-1 q-2c(μ/λ), ifμλ, ,whereλ=μ±εi. and c(λ,μ) is as computed in (3.8). As in [ORa0401317, Theorem 6.3(b)], the irreducible Wk-module Wμ/0=Wkμ has a basis {vT} indexed by up-down tableaux T={T(0),T(1),,T(k)}, where T(0)=, T(k)=μ, and T(i) is a partition obtained from T(i-1) by adding or removing a box, and YivT= { zq2c(b)vT, ifb=T(i)/ T(i-1), z-1q-2c(b) vT, ifb=T(i-1) /T(i), vT, ifT(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 onL(μ) Wk-1μin (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 μ. 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 bμ (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, bμ (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 μ=(μ1,,μr). 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-1q2r= εq-yq2r= εq2r-y and evμ+ρ (Li2)= qμ+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(μ)Wk-1μ 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, Zk--zq-q-1+1u2-1 acts on the L(μ)Wk-1μ isotypic component in the Uh𝔤Wk-1-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.

In the following corollary, we shall repackage Theorem 3.5 to give a formula for the Harish-Chandra image of ZV() 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 saλ(Y) be the universal Weyl character for 𝔤𝔩r, spλ(Y) the universal Weyl character for 𝔰𝔭2r, and soλ(Y) 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), ij 11-yiyj i,j 11-xiyj =Ω(XY-sa(2)(Y)) =λsaλ(Y) soλ(X), i<j 11-yiyj i,j 11-xiyj =Ω(XY-sa(12)(Y)) =λsaλ (Y)spλ(X), where Ω is the Cauchy kernel (see [HRa1995, (6.3)]) and the first equality in each line is for the formal alphabets X=ixi and Y=jyj. The identity [HRa1995, Lemma 6.7(a)] states saλ ((q-q-1)u-1) = { (q-q-1) u- (-q-1)-m qm-1, ifλ=(m,1-m), 0, otherwise. (3.25)

In the same setting as in Theorem 3.5, let y= { 2r, if𝔤= 𝔰𝔬2r+1, 2r+1, if𝔤=𝔰𝔭2r , 2r-1, ifg=𝔰𝔬2r, ε= { 1, if𝔤= 𝔰𝔬2r+1, -1, if𝔤=𝔰𝔭2r , 1, ifg=𝔰𝔬2r, V=L(ε1), z=εqy, and let ZV() be the central elements in the Drinfeld-Jimbo quantum group Uh𝔤 which are given by ZV()=ε(idqtrL(ε1))((z21)). Let X be the formal alphabet given by X=iVˆLi2 and fix c=1 if is even and c=0 if is odd. Then for 1, π0(ZV())= σρ ( c+zεm=1 (q-q-1) (-1)-m q-(-2m+1) s(m,1-m) (X) ) where s(m,1-m)(X)= so(m,1-m)(X) in the orthogonal cases and s(m,1-m)(X)= sp(m,1-m)(X) in the symplectic case.

Proof.

Let Vˆ as in (2.6), L-i=Li-1 where Li is as in the statement of Theorem 3.5, and let Lε0=1. 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 jVˆ (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 iVˆ (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) =0 ( m=1 (q-q-1) (-q-1)-m qm-1 so(m,1-m) (X) ) (εu)- =0 e(q-q-1) ( m=1 (q-q-1) (-1)-m q-(-2m+1) so(m,1-m) (X) ) u- in the orthogonal case, and (1-q-2(εu)-2) 1-(εu)-2 iVˆ (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) =0 ( m=1 (q-q-1) (-q-1)-m qm-1 sp(m,1-m) (X) ) (εu)- =0 e(q-q-1) ( m=1 (-1)-m q-(-2m+1) sp(m,1-m) (X) ) u- in the symplectic case. The statement now follows by noting that u2/(u2-1)= 1/(1-u-2)= k0u-2k and taking the coefficient of u- on each side of (3.26).

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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