## Actions of tantalizers

Let $𝔤$ be a complex semisimple Lie algebra and let $\mathrm{U𝔤}$ be the universal enveloping algebra. Let $C$ be the center of $\mathrm{U𝔤}$ and let $\kappa$ be the Casimir in $C$. Let $M$ and $V$ be $𝔤$-modules.

1. Let ${𝔹}_{k}$ be the degenerate affine braid group. Then $M\otimes {V}^{\otimes k}$ is a ${𝔹}_{k}$-module with action given by $Φ : 𝔹k → EndU M⊗V⊗k , tsi ↦ idM ⊗ idV⊗i-1 ⊗ s1 ⊗ idV⊗k-i-1 , ci ↦ κM⊗ V⊗i ⊗ idV⊗k-i z ↦ z⊗ idV⊗k,$ for $z\in C$ and where ${s}_{1}\cdot \left(u\otimes v\right)=v\otimes u,$ for $u,v\in V$ and ${\kappa }_{M\otimes {V}^{\otimes i}}$ is $\kappa$ acting on $M$ and the first $i$ factors of $V$. This action commutes with the $𝔤$-action on $M\otimes {V}^{\otimes k}$.
2. Let $𝔤$ be ${𝔰𝔬}_{2r+1}$, ${𝔰𝔬}_{2r}$ or ${𝔰𝔭}_{2r}$. Using notations for irreducible representations as in (5.27), let $ε= 1, if𝔤is 𝔰𝔬2r+1 or 𝔰𝔬2r, -1, if𝔤 is 𝔰𝔭2r, V=Lϵ1 , y= ⟨ϵ1, ϵ1+2ρ⟩,$ and $zV 𝓁 = ε id⊗ trV y+t 𝓁 , for 𝓁∈ℤ≥0.$ Then the algebra homomorphism $\Phi :{𝔹}_{k}\to {End}_{U}\left(M\otimes {V}^{\otimes k}\right)$ is a representation of the degenerate affine BMW algebra ${𝒲}_{k}$.
3. Let $𝔤=𝔰{𝔩}_{r+1}$ and $V=L\left({ϵ}_{1}\right)$. Then $\Phi :{𝔹}_{k}\to {End}_{U}\left(M\otimes {V}^{\otimes k}\right)$ is a representation of the graded Hecke algebra.

Proof.
1. Since the ${t}_{{s}_{i}}$ act as simple transpositions, they generate an action of ${S}_{k}$ on $M\otimes {V}^{\otimes k}$. This action commutes with the $𝔤$-action. Since $\kappa \in C$, ${c}_{i}\in {End}_{𝔤}\left(M\otimes {V}^{\otimes i}\right)\subseteq {End}_{𝔤}\left(M\otimes {V}^{\otimes j}\right)$ for $i\le j$. Then, since $\kappa \in U𝔤$, ${c}_{i}$ commutes with ${c}_{j}$ for $i\le j$. So ${c}_{0},\dots ,{c}_{k}$ all commute with each other. By (ctoy), ${y}_{1},\dots ,{y}_{k}$ commute with each other.

Since ${c}_{1},\dots ,{c}_{i-1}$ act as the identity on the $i$th and $\left(i+1\right)$st components of $M\otimes {V}^{\otimes \left(i+1\right)}$, ${t}_{{s}_{i}}{c}_{j}={c}_{j}{t}_{{s}_{i}}$ for $j. Since ${t}_{{s}_{i}}$ commutes with the action of $U𝔤$ on $M\otimes {V}^{\otimes j}$, it follows that ${t}_{{s}_{i}}{c}_{j}={c}_{j}{t}_{{s}_{i}}$ for $j>i$. Hence the relations in (gbp3), and thus in (gbp1), are satisfied.

Let $t=\sum b\otimes {b}^{*}$ as in (tdefn). If

 (ktop)
(so that ${\kappa }_{0}$ is $\kappa$ acting on $M$) then the relations in (gbp4) hold. Applying the coproduct to compute the action of $\kappa$ on $m\otimes {v}_{1}\otimes \cdots \otimes {v}_{j}$ gives the first identity in (cytokt) and so the relations in (gbp2) hold.

2. By (5.20), the computations in (5.30) determine the action of $t$ on the components of $V\otimes V$. The decompositions in (5.31) and (5.32) determine the action of ${t}_{{s}_{1}}$ on $V\otimes V$. The operator $\Phi \left({e}_{1}\right)$ is determined from $\Phi \left({t}_{{s}_{1}}\right)$ and $\Phi \left(t\right)$ via (gbp6), $Φ(t) Φ(ts1) = 1- Φ(e1).$ Then $\Phi \left({t}_{{s}_{1}}\right)$, $\Phi \left({e}_{1}\right)$, and $\Phi \left(t\right)$ act on the components of $V\otimes V$ by $L(0) L(2ϵ1) L(ϵ1 +ϵ2) Φ(t) -y 1 -1 Φ(ts1) ε 1 -1 Φ(e1) 1+εy 0 0$ where $y$ and $t$ are as in (5.24) and (5.17) respectively. The first relation in (dbw1) follows. Since ${dim}_{q}\left(V\right)=\epsilon +y$, the first identity in (5.15) gives that $Φ(e1) = εEV.$ By (5.12), the second identity in (5.15), and (5.4), $Φ(ei tsi-1 ei) = ε(1⊗EV )( RVV⊗1 )( 1⊗EV)ε = (id⊗trV) (RVV) ⊗EV = CV-1 ⊗EV = id⊗EV= εΦ(ei),$ which establishes the second relation in (dbw1). By (cytokt), $\Phi \left({y}_{1}\right)=\frac{1}{2}{\kappa }_{1}+{t}_{0,1}=\frac{1}{2}y+{t}_{0,1},$ and by (5.12), $Φ e1y1𝓁 e1 = ε id⊗EV 12 y+t 𝓁 ε id⊗EV = id⊗trV 12 y+t 𝓁 ⊗EV = ε id⊗trV 12 y+t 𝓁 Φ e1 = zV(𝓁) Φ(e1),$ which gives the first relation in (dbw2). Since the ${y}_{i}$ commute and ${t}_{{s}_{i}}\left({y}_{i}+{y}_{i+1}\right)=\left({y}_{i}+{y}_{i+1}\right){t}_{{s}_{i}},$ $ei ( yi+yi+1 )=( tsi yi - yi+1 tsi +1 ) ( yi+yi+1 ) = ( yi+yi+1 ) ( tsi yi - yi+1 tsi +1 ) = ( yi+yi+1 ) ei.$ Use notations similar to that in (ktop), so that, for an element $b\in U𝔤$ or $U𝔤\otimes U𝔤$, ${b}_{i}$ and ${b}_{i,i+1}$ denote the action of an element $b$ on the $i$th, respectively $i$th and $\left(i+1\right)$st, factors of $V$ in $M\otimes {V}^{\otimes \left(i+1\right)}$. Then $(yi+yi+1) ei = 12 κi+ ∑ i-1 r=0 tr,i + 12 κi+1 + ∑ i r=0 tr,i+1 ei = 12 κi,i+1 + ∑ r=0 i-1 ( tr,i + tr,i+1 ) ei = 12 κi,i+1 + ∑ r=0 i-1 ∑b b⊗Δ b* i,i+1 ei =0,$ because the action of ${b}^{*}$ and $\kappa$ on $L\left(0\right)$ is $0$.
3. In this case, $V⊗V = L(2ε1) ⊕ L(ε1 +ε2) with Λ2(V) = L(ε1 +ε2)$ so that $L(2ε1) L(ε1 +ε2) Φ(t) 1 -1 Φ ts1 1 -1 and Φ(e1) = Φ(t) - Φ(ts1) =0.$
$\square$

Say I wish to discuss irreducible representations of SU(3). I think it should be perfectly acceptable to point out that: $⊗ Fundamental Representation = Adjoint Representation ⊕ Rank-3 Symmetric Tensor Representation$.

Remark. In Theorem 1.1, if $\Phi \left({y}_{1}\right)$ has eigenvalues ${u}_{1},\dots ,{u}_{r}$, then $\Phi$ is a representation of the degenerate cyclotomic BMW algebra ${𝒲}_{r,k}\left({u}_{1},\dots ,{u}_{r}\right)$.

Pictorially,

$Adjoint Representation$

STUFF   STUFF

$PICTURE GOES HERE.$ By (5.20) and (2.3), the eigenvalues of ${y}_{j}$ are related to the eigenvalues of the Casimir.

The Schur functors are the functors $FVλ : U𝔤-modules → 𝔹k-modules M ↦ HomU𝔤 M(λ),M⊗V⊗k$ where ${Hom}_{U𝔤}\left(M\left(\lambda \right),M\otimes {V}^{\otimes k}\right)$ is the vector space of highest weight $\lambda$ in $M\otimes {V}^{\otimes k}$.

Let $𝔤$ be a complex semisimple Lie algebra and let $U={U}_{h}𝔤$ be the corresponding Drinfel'd-Jimbo quantum group. Let $M$ and $V$ be $U$-modules.

Proof.
1. The relations (2.26) and (2.29) are consequences of the definition of the action of ${T}_{i}$ and ${X}^{{\epsilon }_{1}}$. The relations (2.27) and (2.28) follow from the following computations: $R ˇ i R ˇ i+1 R ˇ i =PICTURE=PICTURE= R ˇ i+1 R ˇ i R ˇ i+1$ and $R ˇ 02 R ˇ 1 R ˇ 02 R ˇ 1 =PICTURE=PICTURE= R ˇ 1 R ˇ 02 R ˇ 1 R ˇ 02 .$
2. By (5.23), the computations in (5.30) determine the action of ${\stackrel{ˇ}{R}}_{VV}^{2}$ on the components of $V\otimes V$. The operator $\Phi \left({T}_{1}\right)={\stackrel{ˇ}{R}}_{VV}$ is the square root of ${\stackrel{ˇ}{R}}_{VV}^{2}$ and, at $q=1$, specializes to ${t}_{{s}_{1}}$, the operator that switches the factors in $V\otimes V$. Thus equations (5.31) and (5.32) 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 (2.39), $ΦE1 = 1 - ΦT1 - ΦT1-1 q-q-1 .$ Then ${\stackrel{ˇ}{R}}_{VV}^{2},$ $\Phi \left({T}_{1}\right)$ and $\Phi \left({E}_{1}\right)$ act on the components of $V\otimes V$ by $L 0 L 2ε1 L ε1+ε2 R ˇ VV 2 q-2ε q2 q-2 ΦT1 εq-y q -q-1 ΦE1 1+εy 0 0 where y = qy-q-y q-q-1 .$ The first relation in (phidefn) follows from $Φ E1 T1 = ε y-y Φ E1 = z-1 Φ E1 .$ Since ${dim}_{q}\left(V\right)=\epsilon +\left[y\right]$, the first identity in (5.15) gives that
 $Φ E1 = εEV.$ LABEL
By (5.12), the second identity in (5.15), and (5.22),
 $Φ Ei Ti-1 Ei = ε 1⊗EV R ˇ VV ⊗1 1⊗EV ε = id⊗qtrV R ˇ VV ⊗EV = CV-1 ⊗EV = q⟨ ε1, ε1+2ρ ⟩ id⊗EV =qy Φ Ei ,$ LABEL
which establishes the second relation in (2.36). By (5.12),
 $Φ E1 Y1l E1 = ε id⊗EV zℛ21ℛ l ε id⊗EV = id⊗qtrV zℛ21ℛ l ⊗EV = ε id⊗qtrV zℛ21ℛ l Φ E1 = ZV(l) Φ E1 ,$ LABEL
which gives the first relation in (2.37). Since the ${Y}_{i}$ commute, and ${T}_{i}{Y}_{i}{Y}_{i+1}={Y}_{i}{Y}_{i+1}{T}_{i},$ $EiYiYi+1 = 1- Ti - Ti-1 q-q-1 YiYi+1 = YiYi+1 1- Ti - Ti-1 q-q-1 = YiYi+1Ei.$ The proof that ${E}_{i}{Y}_{i}{Y}_{i+1}={E}_{i}$ is exactly as in the proof of [
OR, Thm. 6.1(c)]: Since $\Phi \left({E}_{1}\right)=\epsilon {E}_{V},$ using ${E}_{1}{T}_{1}={z}^{-1}{E}_{1},$ the pictorial equalites $εz2· =εz2· =εz2z-1·$ 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 $\epsilon {z}^{2}{z}^{-1}\cdot {\stackrel{ˇ}{R}}_{L\left(0\right),M}{\stackrel{ˇ}{R}}_{M,L\left(0\right)}\left({id}_{M}\otimes {E}_{V}\right).$ By (5.10), this is equal to $ε z CM ⊗ C L 0 C M⊗L0 -1 idM ⊗ EV = εz⋅ CM CM-1 idM ⊗ EV = z⋅ ΦD1 = ΦE1 T1-1 ,$ so that $\left({E}_{1}{Y}_{1}{T}_{1}^{-1}\right)=\Phi \left({E}_{1}{T}_{1}^{-1}\right).$ This establishes the second relation in (2.37).
3. When $𝔤=𝔰{𝔩}_{r}$, $V\otimes V=L\left(2{\epsilon }_{1}\right)\oplus L\left({\epsilon }_{1}+{\epsilon }_{2}\right)$ with ${S}^{2}\left(V\right)=L\left(2{\epsilon }_{1}\right)$ and ${\wedge }^{2}\left(V\right)=L\left({\epsilon }_{1}+{\epsilon }_{2}\right)$, and $L2ε1 Lε1+ε2 Φ ( R ˇ VV 2 ) q2 q-2 Φ(T1) q -q-1 so that Φ (E1) = Φ(T1) - Φ( T1-1 ) q-q-1 =0.$ Thus the relations (2.44), (2.36) and (2.37) are satisfied.
$\square$

Remark. In Theorem 1.2, if $\Phi \left({Y}_{1}\right)$ has eigenvalues ${u}_{1},\dots ,{u}_{r}$, then $\Phi$ is a representation of the cyclotomic BMW algebra ${𝒲}_{r,k}\left({u}_{1},\dots ,{u}_{r}\right)$.

By the definition of ${Y}_{i}$ in (2.33), $Φ Yi = z R ˇ M⊗V⊗(i-1) , V R ˇ V , M⊗V⊗(i-1) = zPICTURE.$

The Schur functors are the functors $FVλ : U𝔤-modules → ℬ ˜ k-modules M ↦ HomU𝔤 M(λ),M⊗V⊗k$ where ${Hom}_{U𝔤}\left(M\left(\lambda \right),M\otimes {V}^{\otimes k}\right)$ is the vector space of highest weight $\lambda$ in $M\otimes {V}^{\otimes k}$.

## References

[AMR] S. Ariki, A. Mathas and H. Rui Cyclotomic Nazarov Wenzl algebras, Nagoya Math. J. 182, (2006), 47-134. MR2235339 (2007d:20005)

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

[OR] R. Orellana and A. Ram, Affine braids, Markov traces and the category $𝒪$, Algebraic groups and homogeneous spaces, 423-473, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007. MR2348913 (2008m:17034)