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

Last update: 13 October 2013

## Actions of classical type tantalizers

In this section, we define the affine Birman-Murakami-Wenzl (BMW) algebra ${W}_{k}$ and its degenerate version ${𝒲}_{k},$ exactly following our treatment in [DRV1105.4207]. Just as the affine BMW algebras ${W}_{k}$ and the affine Hecke algebras ${H}_{k}$ are quotients of the group algebra of affine braid group $C{B}_{k},$ the degenerate affine BMW algebras ${𝒲}_{k}$ and the degenerate affine Hecke algebras ${ℋ}_{k}$ are quotients of ${ℬ}_{k}\text{.}$ Moreover, the tensor space actions defined in Theorems 1.2 and Theorem 1.3 factor through these quotients in important cases. The affine and degenerate affine BMW algebras arise when $𝔤$ is ${𝔰𝔬}_{n}$ or ${𝔰𝔭}_{n}$ and $V$ is the first fundamental representation; similarly, the affine and degenerate affine Hecke algebras arise when $𝔤$ is ${𝔤𝔩}_{n}$ or ${𝔰𝔩}_{n}$ and $V$ is the first fundamental representation. In the case when $M$ is the trivial representation and $𝔤$ is ${𝔰𝔬}_{n},$ the Jucys-Murphy elements ${y}_{1},\dots ,{y}_{k}$ in ${ℬ}_{k}$ become the Jucys-Murphy elements for the Brauer algebras used in [Naz1996]; in the case that $𝔤={𝔰𝔩}_{n},$ these become the classical Jucys-Murphy elements in the group algebra of the symmetric group.

In defining the affine and degenerate affine BMW algebras, we must make a choice of infinite families of parameters, ${Z}_{0}^{\left(\ell \right)}$ and ${z}_{0}^{\left(\ell \right)},$ respectively. In order to avoid choices which yield the zero algebra, we choose parameters in the ground ring $C=Z\left(U\right)$ which arise naturally in each of the action theorems below. As we will see in the proofs of Theorem 2.2 and Theorem 2.5 (specifically, the calculations in (2.33) and (2.44)), the natural actions of ${ℬ}_{k}$ and $C{B}_{k}$ on tensor space in Theorems 1.2 and Theorem 1.3 force the parameters to be $z0(ℓ)=ε (id⊗trV) ((12y+γ)ℓ) andZ0(ℓ) =ε(id⊗qtrV) ((zℛ21ℛ)ℓ).$

Preliminaries on classical type combinatorics. Let $V={ℂ}^{r}\text{.}$ The Lie algebras $𝔤={𝔤𝔩}_{r}$ and ${𝔰𝔩}_{r}$ are given by $𝔤𝔩r=End(V) and𝔰𝔩r= {x∈𝔤𝔩r | tr(x)=0},$ with bracket $\left[x,y\right]=xy-yx\text{.}$ Then $𝔤𝔩rhas basis {Eij | 1≤i,j≤r},$ where ${E}_{ij}$ is the matrix with 1 in the $\left(i\text{.}j\right)$ entry and 0 elsewhere. A Cartan subalgebra of ${𝔤𝔩}_{r}$ is $𝔥𝔤𝔩= {x∈𝔤𝔩r | x is diagonal} with basis {E11,E22,…,Err},$ and the dual basis $\left\{{\epsilon }_{1},\dots ,{\epsilon }_{r}\right\}$ of ${𝔥}_{𝔤𝔩}^{*}$ is specified by $εi:𝔥𝔤𝔩→ℂ given byεi (Ejj)=δij.$ The form $⟨,⟩:𝔤⊗𝔤→ℂ given by⟨x,y⟩= trV(xy) (2.1)$ is a nondegenerate ad-invariant symmetric bilinear form on $𝔤$ such that the restriction to ${𝔥}_{𝔤𝔩}$ is a nondegenerate form $⟨,⟩:{𝔥}_{𝔤𝔩}\otimes {𝔥}_{𝔤𝔩}\to ℂ\text{.}$ Since $⟨,⟩$ is nondegenerate, the map $\nu :{𝔥}_{𝔤𝔩}\to {𝔥}_{𝔤𝔩}^{*}$ given by $\nu \left(h\right)=⟨h,·⟩$ is a vector space isomorphism which induces a nondegenerate form $⟨,⟩$ on ${𝔥}_{𝔤𝔩}^{*}\text{.}$ Further, ${E11,…,Err} and {ε1,…,εr} are orthonormal bases of 𝔥𝔤𝔩 and 𝔥𝔤𝔩*,$ respectively. A Cartan subalgebra of ${𝔰𝔩}_{r}$ is $h𝔰𝔩= (E11+⋯+Err)⊥ = { x∈𝔥𝔤𝔩 | ⟨x,E11+⋯+Err⟩ =0 } ,$ the orthogonal subspace to $ℂ\left({E}_{11}+\cdots +{E}_{rr}\right)\text{.}$ The dominant integral weights for ${𝔤𝔩}_{r},$ $P+= { λ1ε1+⋯+ λrεr | λi∈ℤ,λ1≥ ⋯≥λr } ,$ index the irreducible finite-dimensional representations $L\left(\lambda \right)$ of ${𝔤𝔩}_{r}\text{.}$ The irreducible finite-dimensional representations of ${𝔰𝔩}_{r}$ are $L(λ‾)= Res𝔰𝔩r𝔤𝔩r (L(λ)),$ where $\stackrel{‾}{\lambda }$ is the orthogonal projection of $\lambda$ to ${𝔥}_{𝔰𝔩}^{*}={\left({\epsilon }_{1}+\cdots +{\epsilon }_{r}\right)}^{\perp }\text{.}$

The matrix units $\left\{{E}_{ij} | 1\le i,j\le r\right\}$ form a basis of ${𝔤𝔩}_{r}$ for which the dual basis with respect to the form in (2.1) is $\left\{{E}_{ji} | 1\le i,j\le r\right\}\text{.}$ So $γ𝔤𝔩= ∑1≤i,j≤r Eij⊗Eji= ∑1≤i,j≤ri≠j Eij⊗Eji+ ∑i=1Eii ⊗Eii,and (2.2) γ𝔰𝔩= γ𝔤𝔩-E+ ⊗E+,where E+=E11+⋯+Err. (2.3)$ If the Casimir for ${𝔤𝔩}_{r},$ $κ𝔤𝔩= ∑1≤i,j≤r EijEji, acts onL(λ) by the constant κ𝔤𝔩(λ)$ then the Casimir for ${𝔰𝔩}_{r},$ $κ𝔰𝔩=κ𝔤𝔩- E+E+, acts onL(λ‾) by the constantκ𝔤𝔩(λ) -1r|λ|2, (2.4)$ where $|\lambda |={\lambda }_{1}+\cdots +{\lambda }_{r}\text{.}$

Let $V={ℂ}^{N}\text{.}$ The Lie algebras $𝔤={𝔰𝔬}_{N}$ or ${𝔰𝔭}_{N}$ are given by $𝔤= { x∈𝔤𝔩(V) | (xv1,v2)+ (v1,xv2)=0 for all v1,v2∈V } ,$ where $\left(,\right):V\otimes V\to ℂ$ is a nondegenerate bilinear form satisfying $(v1,v2)=ε (v2,v1), whereε= { 1, if 𝔤=𝔰𝔬2r+1, -1, if 𝔤=𝔰𝔭2r, 1, if 𝔤=𝔰𝔬2r. (2.5)$ Choose $a basis {vi | i∈Vˆ} of V, whereVˆ= { {-r,…,-1,0,1,…,r} , if 𝔤=𝔰𝔬2r+1, {-r,…,-1,1,…,r} , if 𝔤=𝔰𝔭2r, {-r,…,-1,1,…,r} , if 𝔤=𝔰𝔬2r. (2.6)$ so that the matrix of the bilinear form $\left(,\right):V\otimes V\to ℂ$ is $J= ( 0 1 ⋰ 1 ε ⋰ ε 0 ) ,and𝔤= { x∈𝔤𝔩N | xtJ+Jx=0 } ,$ where $N=\text{dim}\left(V\right)$ and ${x}^{t}$ is the transpose of $x\text{.}$ Then, as in Molev [Mol2355506, (7.9)] and [Bou1990, Ch. 8 §13 2.I, 3.I, 4.I], $𝔤=span{Fij | i,j∈Vˆ} whereFij=Eij -θijE-j,-i, (2.7)$ where ${E}_{ij}$ is the matrix with 1 in the $\left(i,j\right)\text{-entry}$ and 0 elsewhere and $θij= { 1, if 𝔤=𝔰𝔬2r+1, sgn(i)· sgn(j), if 𝔤=𝔰𝔭2r, 1, if 𝔤=𝔰𝔬2r.$

A Cartan subalgebra of $𝔤$ is $𝔥=span{Fii | i∈Vˆ} with basis {F11,F22,…,Frr} (2.8)$ The dual basis $\left\{{\epsilon }_{1},\dots ,{\epsilon }_{r}\right\}$ of ${𝔥}^{*}$ is specified by $εi:𝔥→ℂ given byεi(Fjj) =δij. (2.9)$ The form $⟨,⟩:𝔤⊗𝔤→ℂ given by⟨x,y⟩ =12trV(xy) (2.10)$ is a nondegenerate ad-invariant symmetric bilinear form on $𝔤$ such that the restriction to $𝔥$ is a nondegenerate form $⟨,⟩:𝔥\otimes 𝔥\to ℂ$ on $𝔥$ Since $⟨,⟩$ is nondegenerate, the map $\nu :𝔥\to {𝔥}^{*}$ given by $\nu \left(h\right)=⟨h,·⟩$ is a vector space isomorphism which induces a nondegenerate form $⟨,⟩$ on ${𝔥}^{*}\text{.}$ Let $⟨,⟩:{𝔥}^{*}\otimes {𝔥}^{*}\to ℂ$ be the form on ${𝔥}^{*}$ induced by the form on $𝔥$ and the vector space isomorphism $\nu :𝔥\to {𝔥}^{*}$ given by $\nu \left(h\right)=⟨h,·⟩\text{.}$ Further, $⟨F11,…,Frr⟩ and {ε1,…,εr} are orthonormal bases of 𝔥 and 𝔥*.$

With ${F}_{ij}$ as in (2.7), $𝔤$ has basis ${Fi,i | 0 With respect to the nondegenerate ad-invariant symmetric bilinear form $⟨,⟩:𝔤\otimes 𝔤\to ℂ$ given in (2.10), $⟨x,y⟩=\frac{1}{2}{\text{tr}}_{V}\left(xy\right),$ the dual basis with respect to $⟨,⟩$ is $Fij*=Fji if i≠-j,and Fi,-i*=12 F-i,i.$ The sets ${F-i,-i | 0 are alternate bases, and ${F}_{i,-i}=0$ when $𝔤={𝔰𝔬}_{2r+1}$ or $𝔤={𝔰𝔬}_{2r}\text{.}$ So $2γ=∑i,j∈V Fij⊗Fji* +∑i∈Vˆ Fi,-i⊗ Fi,-i*= ∑i,j∈Vˆ Fij⊗Fji. (2.11)$

To compute the value $\frac{1}{2}⟨\lambda ,\lambda +2\rho ⟩$ in (1.17) choose positive roots $R+= { {εi±εj | 1≤i Since $∑1≤i it follows that $2ρ=∑i=1r (y-2i+1)εi, wherey= ⟨ε1,ε1+2ρ⟩= { 2r if 𝔤=𝔰𝔬2r+1, 2r+1, if 𝔤=𝔰𝔭2r, 2r-1, if 𝔤=𝔰𝔬2r, (2.13)$ is the value by which the Casimir $\kappa$ acts on $L\left({\epsilon }_{1}\right)\text{.}$ Set $q={e}^{h/2}\text{.}$ The quantum dimension of $V$ is $dimq(V)=trV (ehρ)= ε+[y],where [y]= qy-q-y q-q-1 , (2.14)$ since, with respect to a weight basis of $V,$ the eigenvalues of the diagonal matrix ${e}^{h\rho }$ are ${e}^{\frac{1}{2}h\left(y-2i+1\right)}={q}^{\left(y-2i+1\right)}\text{.}$

Identify a weight $\lambda ={\lambda }_{1}{\epsilon }_{1}+\cdots +{\lambda }_{r}{\epsilon }_{r}$ with the configuration of boxes with ${\lambda }_{i}$ boxes in row $i\text{.}$ If $b$ is a box in position $\left(i,j\right)$ of $\lambda$ then the content of $b$ is $c(b)=j-i= the diagonal number of b,so that 0 1 2 -1 0 1 -2 (2.15)$ are the contents of the boxes of $\lambda =2{\epsilon }_{1}+3{\epsilon }_{2}+{\epsilon }_{3}\text{.}$ If $\lambda ={\lambda }_{1}{\epsilon }_{1}+\cdots +{\lambda }_{n}{\epsilon }_{n},$ then $⟨λ,λ+2ρ⟩- ⟨λ-εi,λ-εi+2ρ⟩= 2λi+2ρi-1=y+2 λi-2i=y+2c (λ/λ-),$ where $\lambda /{\lambda }^{-}$ is the box at the end of row $i$ in $\lambda \text{.}$ By induction, $⟨λ,λ+2ρ⟩ =y|λ|+2 ∑b∈λc(b), (2.16)$ for $\lambda ={\lambda }_{1}{\epsilon }_{1}+\cdots +{\lambda }_{r}{\epsilon }_{r}$ with ${\lambda }_{i}\in ℤ\text{.}$

Let $L\left(\lambda \right)$ be the irreducible highest weight $𝔤\text{-module}$ with highest weight $\lambda ,$ and let $V=L\left({\epsilon }_{1}\right)\text{.}$ Then, for $𝔤={𝔰𝔬}_{2r+1},{𝔰𝔭}_{2r}$ or ${𝔰𝔬}_{2r},$ $V≅V*andV⊗ V≅L(0)⊕L (2ε1)⊕L (ε1+ε2). (2.17)$ For each component in the decomposition of $V\otimes V$ the values by which $\gamma ={\sum }_{b\in B}b\otimes {b}^{*}$ acts (see (1.17)) are $⟨0,0+2ρ⟩- ⟨ε1,ε1+2ρ⟩- ⟨ε1,ε1+2ρ⟩= 0-y-y=-2y, ⟨2ε1,2ε1+2ρ⟩- ⟨ε1,ε1+2ρ⟩- ⟨ε1,ε1+2ρ⟩= 4+2(y-1)-y-y=2, ⟨ε1+ε2,ε1+ε2+2ρ⟩- ⟨ε1,ε1+2ρ⟩- ⟨ε1,ε1+2ρ⟩= 2+(y-1)+(y-3) -y-y=-2. (2.18)$ The second symmetric and exterior powers of $V$ are $S2(V)= { L(ep1)⊕L(0) if 𝔤=𝔰𝔬2r+1 or 𝔰𝔬2r, L(2ε1), if 𝔤=𝔰𝔭2r, (2.19)$ and $Λ2(V)= { L(ε1+ε2), if 𝔤= 𝔰𝔬2r+1 or 𝔰𝔬2r, L(ε1+ε2)⊕ L(0), if 𝔤=𝔰𝔭2r. (2.20)$ For all dominant integral weights $\lambda$ $L(λ)⊗V= { L(λ)⨁ ( ⨁λ± L(λ±) ) , if 𝔤=𝔰𝔬2r+1 and λr>0, ⨁λ±L (λ±), if 𝔤=𝔰𝔭2r, 𝔤=𝔰𝔬2r, or if 𝔤=𝔰𝔬2r+1 and λr=0, (2.21)$ where the sum over ${\lambda }^{±}$ denotes a sum over all dominant weights obtained by adding or removing a box from $\lambda$ (by a routine check using the product formula for Weyl characters in [Bou1990, VIII §9 Prop. 2]). If $𝔤={𝔰𝔬}_{2r}$ then addition and removal of a box should include the possibility of addition and removal of a box marked with a $-$ sign, and removal of a box from row $r$ when ${\lambda }_{r}=\frac{1}{2}$ changes ${\lambda }_{r}$ to $-\frac{1}{2}\text{.}$

Preliminaries on quantum trace. This paragraph provides a brief review of quantum traces and quantum dimensions (see also [CPr1994, 4.2.9]) in the form suitable to our needs for the proofs of the main theorems of this section. If $𝔤$ is a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form, and ${U}_{h}𝔤$ is the Drinfel’d-Jimbo quantum group corresponding to $𝔤,$ then both $U=U𝔤 with ℛ=1 ⊗1 and v=1 andU=Uh𝔤 withv=e-hρu (2.22)$ are ribbon Hopf algebras ([LRa1977, Corollary (2.15)]). For $U=U𝔤$ or ${U}_{h}𝔤,$ let $V$ be a finite-dimensional $U\text{-module,}$ and let ${V}^{*}$ be the dual module. Define $ev: V*⊗V ⟶ 1 ϕ⊗v ⟼ ϕ(v) and coev: 1 ⟶ V⊗V* 1 ⟼ ∑ivi⊗vi$ where $\left\{{v}_{1},\dots ,{v}_{n}\right\}$ is a basis of $V$ and $\left\{{v}^{1},\dots ,{v}^{n}\right\}$ is the dual basis in ${V}^{*}\text{.}$ Let ${E}_{V}$ be the composition $EV:V⊗V* ⟶v-1⊗1V⊗ V*⟶ŘVV* V*⊗V⟶ev1 ⟶coevV⊗V*,$ so that ${E}_{V}$ is a $U\text{-module}$ homomorphism with image a submodule of $V\otimes {V}^{*}$ isomorphic to the trivial representation of $U\text{.}$

Let $M$ be a $U\text{-module}$ and let $\psi \in {\text{End}}_{U}\left(M\otimes V\right)\text{.}$ Then,as operators on $M\otimes V\otimes {V}^{*},$ $(id⊗EV) (ψ⊗id) (id⊗EV)= (id⊗qtrV) (ψ)⊗EV, (2.23)$ where the quantum trace $\left(\text{id}\otimes {\text{qtr}}_{V}\right)\left(\psi \right):M\to M$ is given by $(id⊗qtrV)(ψ)= (id⊗trV) ((1⊗uv-1)ψ). (2.24)$ The special case when $M=1$ and $\psi ={\text{id}}_{V}$ is the quantum dimension of $V,$ $dimq(V)=qtrV (idV),so that EV2=dimq(V) EV. (2.25)$ If ${C}_{V}:V\to V$ is the map defined in (1.26), then $(id⊗qtrV) (ŘVV)= CV-1 (2.26)$ (see, for example, [LRa1977, Prop. 3.11]). In the case $U=U𝔤,$ the ribbon element $v=1,$ so that ${\text{qtr}}_{V}\left(\phi \right)={\text{tr}}_{V}\left(\phi \right)$ and ${C}_{V}={\text{id}}_{V}\text{.}$

The identity (2.23) and the second identity in (2.25) are the source of the connection between quantum traces, the Jones basic construction and conditional expectations (see [GHJ1989, Def. 2.6.6]). These tools are extremely powerful for the study of Temperley-Lieb algebras, Brauer algebras, BMW algebras, and other algebras which arise as tantalizer algebras (tensor power centralizer algebras).

### The degenerate affine BMW algebra action

Define ${e}_{i}$ in the degenerate affine braid algebra ${ℬ}_{k}$ by $tsiyi= yi+1tsi -(1-ei), for i=1,2,…,k-1, (2.27)$ so that, with ${\gamma }_{i,i+1}$ as in (1.6), $γi,i+1tsi =1-ei. (2.28)$ By definition, the algebra ${ℬ}_{k}$ is an algebra over a commutative base ring $C\text{.}$ Fix constants $ε=±1and z0(ℓ)∈C for ℓ∈ℤ≥0.$ The degenerate affine Birman-Wenzl-Murakami (BMW) algebra ${𝒲}_{k}$ (with parameters $\epsilon$ and ${z}_{0}^{\left(\ell \right)}\text{)}$ is the quotient of the degenerate affine braid algebra ${ℬ}_{k}$ by the relations $eitsi= tsiei= εei,ei tsi-1ei =eitsi+1 ei=εei, (2.29) e1y1ℓe1= z0(ℓ)e1, ei(yi+yi+1) =0=(yi+yi+1)ei. (2.30)$ The degenerate affine Hecke algebra ${ℋ}_{k}$ is the quotient of ${𝒲}_{k}$ by the relations $ei=0,for i= 1,…,k-1. (2.31)$

Let $\Phi :{ℬ}_{k}\to {\text{End}}_{𝔤}\left(M\otimes {V}^{\otimes k}\right)$ be the representation defined in Theorem 1.2.

 (a) Let $𝔤$ be ${𝔰𝔬}_{2r+1},$ ${𝔰𝔭}_{2r}$ or ${𝔰𝔬}_{2r}$ and $\gamma ={\sum }_{b}b\otimes {b}^{*}$ as in (2.11). Use notations for irreducible representations as in (2.21). 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 $z0(ℓ)=ε (id⊗trV) ((12y+γ)ℓ), for ℓ∈ℤ≥0.$ Then $\Phi :{ℬ}_{k}\to {\text{End}}_{U}\left(M\otimes {V}^{\otimes k}\right)$ is a representation of the degenerate affine BMW algebra ${𝒲}_{k}\text{.}$ (b) If $𝔤={𝔤𝔩}_{r},$ $\gamma ={\sum }_{b}b\otimes {b}^{*}$ is as in (2.2), and $V=L\left({\epsilon }_{1}\right)$ then $\Phi :{ℬ}_{k}\to {\text{End}}_{U𝔤}\left(M\otimes {V}^{\otimes k}\right)$ is a representation of the degenerate affine Hecke algebra. If $𝔤={𝔰𝔩}_{r},$ $\gamma ={\sum }_{b}b\otimes {b}^{*}$ is as in (2.3), and $V=L\left(\stackrel{‾}{{\epsilon }_{1}}\right)$ then $\Phi \prime :{ℬ}_{k}\to {\text{End}}_{U𝔤}\left(M\otimes {V}^{\otimes k}\right)$ given by $Φ′(tsi)= Φ(tsi), Φ′(γℓ,m) =1r+Φ(γℓ,m) ,andΦ′ (κi)=Φ(κi)$ extends to a representation of the degenerate affine Hecke algebra.

 Proof. (a) The action of $\gamma$ on the tensor product of two simple modules is given in (1.17), so the computations in (2.18) determine the action of $\gamma$ on the components of $V\otimes V\text{.}$ The decompositions of the second symmetric and exterior powers in (2.19) and (2.20) determine the action of ${t}_{{s}_{1}}$ on $V\otimes V\text{.}$ The operator $\Phi \left({e}_{1}\right)$ is determined from $\Phi \left({t}_{{s}_{1}}\right)$ and $\Phi \left(\gamma \right)$ via (2.28), $Φ(γ) Φ(ts1)= 1-Φ(e1).$ In summary, $\Phi \left({t}_{{s}_{1}}\right),$ $\Phi \left({e}_{1}\right),$ and $\Phi \left(\gamma \right)$ act on the components of $V\otimes V$ by $L(0) L(2ε1) L(ε1+ε2) Φ(γ1,2) -y 1 -1 Φ(ts1) ε 1 -1 Φ(e1) 1+εy 0 0$ where $y$ and $\gamma$ are as in (2.13) and (1.15), respectively. The first relation in (2.29) follows. Since $\text{dim}\left(V\right)=\epsilon +y,$ the first identity in (2.25) gives that $Φ(e1)= εEV. (2.32)$ By (2.22), (2.23), and (2.26), $Φ(eitsi-1ei) = ε(1⊗EV) (ŘVV⊗1) (1⊗EV)ε= (id⊗trV) (ŘVV)⊗ EV = CV-1⊗EV =id⊗EV=ε Φ(ei),$ which establishes the second relation in (2.29). Since $y=⟨{\epsilon }_{1},{\epsilon }_{1}+2\rho ⟩=\Phi \left({\kappa }_{i}\right),$ it follows from (1.11) that $\Phi \left({y}_{1}\right)=\Phi \left(\frac{1}{2}{\kappa }_{1}+{\gamma }_{0,1}\right)=\frac{1}{2}y+\gamma ,$ and by (2.23), $Φ(e1y1ℓe1) = ε(id⊗EV) (12y+γ)ℓε (id⊗EV)= (id⊗trV) ((12y+γ)ℓ) ⊗EV = ε(id⊗trV) ((12y+γ)ℓ) Φ(e1)= z0(ℓ)Φ (e1), (2.33)$ which gives the first relation in (2.30). Since the ${y}_{i}$ commute and ${t}_{{s}_{i}}\left({y}_{i}+{y}_{i+1}\right)=\left({y}_{i}+{y}_{i+1}\right){t}_{{s}_{i}},$ it follows that $ei(yi+yi+1)= (tsiyi-yi+1tsi+1) (yi+yi+1)= (yi+yi+1) (tsiyi-yi+1tsi+1) = (yi+yi+1)ei.$ For $b\in U𝔤$ or $U𝔤\otimes U𝔤,$ let ${b}_{i}$ and ${b}_{i,i+1}$ denote the action of an element $b$ on the $i\text{th},$ respectively $i\text{th}$ and $\left(i+1\right)\text{st},$ factors of $V$ in $M\otimes {V}^{\otimes \left(i+1\right)}\text{.}$ Then, as operators on $M\otimes {V}^{\otimes \left(i+1\right)},$ $(yi+yi+1)ei = ( 12κi+ ∑r=0i-1 γr,i+12 κi+1+ ∑r=0i γr,i+1 ) ei= ( 12Δ (κ)i,i+1+ ∑r=0i-1 (γr,i+γr,i+1) ) ei = ( 12Δ(κ)i,i+1+ ∑r=0i-1 ∑bb⊗Δ (b*)i,i+1 ) ei=0,$ because ${e}_{i}$ is a projection onto $L\left(0\right)$ and the action of ${b}^{*}$ and $\kappa$ on $L\left(0\right)$ is 0. (b) In the case where $𝔤={𝔤𝔩}_{r}$ and $V=L\left({\epsilon }_{1}\right),$ $V⊗V=L(2ε1) ⊕L(ε1+ε2) ,with Λ2 (V)=L(ε1+ε2) andS2(V)= L(2ε1).$ So by (1.17), $L(2ε1) L(ε1+ε2) Φ(γ1,2) 1 -1 Φ(ts1) 1 -1 andΦ(e1) =Φ(γ)-Φ (ts1)=0. (2.34)$ In the case where $𝔤={𝔰𝔩}_{r}$ and $V=L\left({\stackrel{‾}{\epsilon }}_{1}\right),$ $V⊗V=L(2ε1‾) ⊕L(ε1+ε2‾), Λ2(V)= L(ε1+ε2‾) andS2(V)= L(2ε1‾).$ As the map $\varphi :{ℬ}_{k}\to {ℬ}_{k}$ given by $tsi↦tsi, γi,j↦γi,j-a, κi↦κi,for fixed a∈C$ is an automorphism, the result follows from (2.34) and (2.4). $\square$

Fix ${b}_{1},\dots ,{b}_{r}\in C\text{.}$ The degenerate cyclotomic BMW algebra ${𝒲}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$ is the degenerate affine BMW algebra with the additional relation $(y1-b1)⋯ (y1-br)=0. (2.35)$ The degenerate cyclotomic Hecke algebra ${ℋ}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$ is the degenerate affine Hecke algebra ${ℋ}_{k}$ with the additional relation (2.35). In Theorem 2.2, if $\Phi \left({y}_{1}\right)$ has eigenvalues ${u}_{1},\dots ,{u}_{r}$ then $\Phi$ is a representation of ${𝒲}_{r,k}\left({u}_{1},\dots ,{u}_{r}\right)$ or ${ℋ}_{r,k}\left({u}_{1},\dots ,{u}_{r}\right)\text{.}$

In general, for any constants ${a}_{0},$ $a,$ and $c,$ the map $\varphi :{ℬ}_{k}\to {ℬ}_{k}$ given by $tsi↦tsi, γi,j↦γi,j-c, κ0↦κ0-a0, andκj↦κj-a, for j=1,…,k,$ is an automorphism. So, following the proof of Theorem 2.2(b), $\Phi \prime :{ℬ}_{k}\to {\text{End}}_{U𝔤}\left(M\otimes {V}^{\otimes k}\right)$ given by $Φ′(tsi)= Φ(tsi), Φ′(γℓ,m) =1r+Φ(γℓ,m), Φ′(κ0)= a0+Φ(κ0), andΦ′ (κj)=a+Φ (κj)for j= 1,…,k,$ also extends to a representation of ${ℋ}_{k}$ when $𝔤={𝔰𝔩}_{r}\text{.}$ When $M=L\left(\mu \right)$ is a finite-dimensional highest weight module taking ${a}_{0}=\frac{|\mu |}{r}$ and $a=\frac{1}{r}$ is combinatorially convenient.

### The affine BMW algebra action

Let $C$ be a commutative ring and let $C{B}_{k}$ be the group algebra of the affine braid group. Fix constants $q,z∈Cand Z0(ℓ)∈C, for ℓ∈ℤ,$ with $q$ and $z$ invertible. Let ${Y}_{i}=z{X}^{{\epsilon }_{i}}$ so that $Y1=zXε1, Yi=Ti-1 Yi-1Ti-1, andYiYj= YjYi,for 1 ≤i,j≤k. (2.36)$ In the affine braid group $TiYiYi+1= YiYi+1Ti. (2.37)$ Assume $q-{q}^{-1}$ is invertible in $C\text{.}$ Define ${E}_{i}\in C{B}_{k}$ by $TiYi=yi+1 Ti-(q-q-1) Yi+1(1-Ei). (2.38)$ The affine BMW algebra ${W}_{k}$ is the quotient of the group algebra $C{B}_{k}$ by the relations $EiTi±1= Ti±1Ei= Z∓1Ei, EiTi-1±1 Ei=ei Ti+1±1Ei =z±1Ei, (2.39) E1Y1ℓE1= Z0(ℓ)E1, EiYiYi+1 =Ei=Yi Yi+1Ei. (2.40)$ Left multiplying (2.38) by ${Y}_{i+1}^{-1}$ and using the second identity in (2.36) shows that (2.38) is equivalent to ${T}_{i}-{T}_{i}^{-1}=\left(q-{q}^{-1}\right)\left(1-{E}_{i}\right)\text{.}$ So $Ei=1- Ti-Ti-1 q-q-1 ,andEi2= (1+z-z-1q-q-1) Ei (2.41)$ follows by multiplying the first equation in (2.41) by ${E}_{i}$ and using (2.39).

The affine Hecke algebra ${H}_{k}$ is the affine BMW algebra ${W}_{k}$ with the additional relations $Ei=0,for i= 1,…,k-1. (2.41)$

Let $\Phi :C{B}_{k}\to {\text{End}}_{{U}_{h}𝔤}\left(M\otimes {V}^{\otimes k}\right)$ be the representation defined in Theorem 1.3.

 (a) Let $𝔤$ be ${𝔰𝔬}_{2r+1},$ ${𝔰𝔭}_{2r}$ or ${𝔰𝔬}_{2r}$ and $\gamma ={\sum }_{b}b\otimes {b}^{*}$ as in (2.11). 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 $Z0(ℓ)=ε (id⊗qtrV) ((zℛ21ℛ)ℓ), for ℓ∈ℤ.$ Then $\Phi :C{B}_{k}\to {\text{End}}_{U}\left(M\otimes {V}^{\otimes k}\right)$ is a representation of the degenerate affine BMW algebra ${W}_{k}\text{.}$ (b) If $𝔤={𝔤𝔩}_{r},$ $\gamma ={\sum }_{b}b\otimes {b}^{*}$ is as in (2.2), and $V=L\left({\epsilon }_{1}\right)$ then $\Phi :C{B}_{k}\to {\text{End}}_{U𝔤}\left(M\otimes {V}^{\otimes k}\right)$ is a representation of the degenerate affine Hecke algebra. If $𝔤={𝔰𝔩}_{r},$ $\gamma ={\sum }_{b}b\otimes {b}^{*}$ is as in (2.3), and $V=L\left(\stackrel{‾}{{\epsilon }_{1}}\right)$ then $Φ′:CBk→EndU(M⊗V⊗k) given by Φ′(Ti)= q1/rΦ(Ti) andΦ′ (Xei)=Φ (Xei),$ extends to a representation of the degenerate affine Hecke algebra.

 Proof. (a) By (1.28), the computations in (2.18) determine the action of ${Ř}_{VV}^{2}$ on the components of $V\otimes V\text{.}$ The operator $\Phi \left({T}_{1}\right)={Ř}_{VV}$ is the square root of ${Ř}_{VV}^{2}$ and, at $q=1,$ specializes to ${t}_{{s}_{1}},$ the operator that switches the factors in $V\otimes V\text{.}$ Thus equations (2.19) and (2.20) determine the sign of $\Phi \left({T}_{1}\right)$ on each component. The operator $\Phi \left({E}_{1}\right)$ is determined from $\Phi \left({T}_{1}\right)$ via the first identity in (2.41), $Φ(Ei)=1- Φ(Ti)-Φ(Ti-1) q-q-1 .$ Then ${Ř}_{VV}^{2},$ $\Phi \left({T}_{1}\right)$ and $\Phi \left({E}_{1}\right)$ act on the components of $V\otimes V$ by $L(0) L(2ε1) L(ε1+ε2) ŘVV2 q-2y q2 q-2 Φ(T1) εq-y q -q-1 Φ(E1) 1+ε[y] 0 0 where[y]= qy-q-y q-q-1 .$ The first relation in (2.39) follows from $Φ(E1T1)= εq-yΦ(E1) =z-1Φ(E1).$ Since ${\text{dim}}_{q}\left(V\right)=\epsilon +\left[y\right],$ (2.25) gives $Φ(E1)=ε EV. (2.43)$ By (2.23), (2.26), (1.27), and (2.13), $Φ(EiTi-1Ei) = ε(1⊗EV) (ŘVV⊗1) (1⊗EV)ε= (id⊗qtrV) (ŘVV)⊗EV = CV-1⊗EV= q⟨ε1,ε1+2ρ⟩ (id⊗EV)=qy εΦ(Ei)= zΦ(Ei).$ This establishes the second relation in (2.39). By (2.23), $Φ(E1Y1ℓE1) = ε(id⊗EV) (zℛ21ℛ)ℓ ε(id⊗EV)= (id⊗qtrV) ((zℛ21ℛ)ℓ) ⊗EV = ε(id⊗qtrV) ((zℛ21ℛ)ℓ) Φ(E1)= Z0(ℓ)Φ (E1), (2.44)$ which gives the first relation in (2.40). Since the ${Y}_{i}$ commute and ${T}_{i}{Y}_{i}{Y}_{i+1}={Y}_{i}{Y}_{i+1}{T}_{i},$ $EiYiYi+1= (1-Ti-Ti-1q-q-1) YiYi+1=Yi Yi+1 (1-Ti-Ti-1q-q-1) =YiYi+1Ei.$ The proof that ${E}_{i}{Y}_{i}{Y}_{i+1}={E}_{i}$ is exactly as in the proof of [ORa0401317, Thm. 6.1(c)]: Since $\Phi \left({E}_{1}\right)=\epsilon {E}_{V},$ using ${E}_{1}{T}_{1}={z}^{-1}{E}_{1}$ and the pictorial equalities $εz2· =εz2· =εz2z-1·$ it follows that $\Phi \left({E}_{1}{Y}_{1}{Y}_{2}{T}_{1}^{-1}\right)=\epsilon \left(1\otimes {E}_{V}\right)\Phi \left(z{X}^{{\epsilon }_{1}}\right)\Phi \left(z{T}_{1}{X}^{{\epsilon }_{1}}\right)$ acts as $\epsilon {z}^{2}{z}^{-1}·{Ř}_{L\left(0\right),M}{Ř}_{M,L\left(0\right)}\left({\text{id}}_{M}\otimes {E}_{V}\right)\text{.}$ By (1.26), this is equal to $εz(CM⊗CL(0)) CM⊗L(0)-1 (idM⊗EV)=εz ·CMCM-1 (idM⊗EV)=z· Φ(E1)=Φ (E1T1-1),$ so that $\Phi \left({E}_{1}{Y}_{1}{Y}_{2}{T}_{1}^{-1}\right)=\Phi \left({E}_{1}{T}_{1}^{-1}\right)\text{.}$ This establishes the second relation in (2.40). (b) In the case where $𝔤={𝔤𝔩}_{r}$ and $V=L\left({\epsilon }_{1}\right),$ $V⊗V=L(2ε1)⊕ L(ε1+ε2) with S2(V)=L (2ε1)and Λ2(V)=L (ε1+ε2).$ So by (1.28), $L(2ε1) L(ε1+ε2) Φ(ŘVV2) q2 q-2 Φ(T1) q -q-1 so thatΦ(E1) =1- Φ(T1)-Φ(T1)-1 q-q-1 =0. (2.45)$ In the case where $𝔤={𝔰𝔩}_{r}$ and $V=L\left({\stackrel{‾}{\epsilon }}_{1}\right),$ $V⊗V=L(2ε1‾) ⊕L(ε1+ε2‾), with Λ2(V) =L(ε1+ε2‾) andS2(V)= L(2ε1‾).$ Since the map $\varphi :{B}_{k}\to {B}_{k}$ given by $Ti↦aTi, Xεi↦Xεi, for invertible a∈C$ is an automorphism, the result then follows from (2.45) and (2.4) (also see [LRa1977, Prop. 4.4]). $\square$

Fix ${b}_{1},\dots ,{b}_{r}\in C\text{.}$ The cyclotomic BMW algebra ${W}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$ is the affine BMW algebra ${W}_{k}$ with the additional relation $(Y1-b1)⋯ (Y1-br)=0. (2.46)$ The cyclotomic Hecke algebra ${H}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$ is the affine Hecke algebra ${H}_{k}$ with the additional relation (2.46). In Theorem 2.5, if $\Phi \left({Y}_{1}\right)$ has eigenvalues ${u}_{1},\dots ,{u}_{r},$ then $\Phi$ is a representation of ${W}_{r,k}\left({u}_{1},\dots ,{u}_{r}\right)$ or ${H}_{r,k}\left({u}_{1},\dots ,{u}_{r}\right)\text{.}$

## 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.