Affine Braids, Markov Traces and the Category $𝒪$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 22 December 2013

Examples

Affine and cyclotomic Hecke algebras

Let $q\in {ℂ}^{*}$ be transcendental (so that we may view it as a variable when necessary). The affine Hecke algebra ${\stackrel{\sim }{H}}_k$ is the quotient of the group algebra $ℂ{\stackrel{\sim }{ℬ}}_k$ of the affine braid group by the relations $$\begin{array}{cc}{T}_i^2=(q-q^{-1})T_i+1,1\le i\le k-1\text{.}& \text{(6.1)}\end{array}$$ The affine Hecke algebra ${\stackrel{\sim }{H}}_k$ is an infinite dimensional algebra with a very interesting representation theory (see [KLu0862716] and [CGi1997]). With $X$ as in (3.4) the subalgebra $$ℂ\left[X\right]=ℂ[X^{\pm {\epsilon }_1},\dots ,X^{\pm {\epsilon }_k}]=\text{span}\hspace{0.17em}\left\{X^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in L\}$$ is a commutative subalgebra of ${\stackrel{\sim }{H}}_k\text{.}$ It is a theorem of Bernstein and Zelevinsky (see [RRa0401322, Theorem 4.12]), that the center of ${\stackrel{\sim }{H}}_k$ is the ring of symmetric (Laurent) polynomials in $X^{\pm {\epsilon }_1},\dots ,X^{\pm {\epsilon }_k},$ $$Z\left({\stackrel{\sim }{H}}_k\right)={ℂ\left[X\right]}^{S_k}={ℂ[X^{\pm {\epsilon }_1},\dots ,X^{\pm {\epsilon }_k}]}^{S_k}\text{.}$$ If $w\in S_k$ define $T_w=T_{i_1}\cdots T_{i_p}$ if $w=s_{i_1}\cdots s_{i_p}$ is a reduced word for $w$ in terms of the generating reflections $s_i=(i,i+1),$ $1\le i\le k-1,$ Then, with $X^{\lambda }$ as in (3.4) $$\left\{X^{\lambda }{T}_w\hspace{0.17em}\right|\hspace{0.17em}\lambda \in L,w\in S_k\}\text{is a basis of}\hspace{0.17em}{\stackrel{\sim }{H}}_k\text{.}$$

Let $u_1,\dots ,u_r\in ℂ\text{.}$ The cyclotomic Hecke algebra $H_{r,1,n}$ with parameters $u_1,\dots ,u_r,q$ is the quotient of the affine Hecke algebra by the relation $$\begin{array}{cc}(X^{{\epsilon }_1}-u_1)(X^{{\epsilon }_1}-u_2)\cdots (X^{{\epsilon }_1}-u_r)=0\text{.}& \text{(6.2)}\end{array}$$ The algebra $H_{r,1,n}$ is a deformation of the group algebra of the complex reflection group $G(r,1,n)=(\mathbb{Z}/r\mathbb{Z})\wr S_n$ and is of dimension $r^{n}n!\text{.}$ It was introduced by Ariki and Koike [Ari1996] and its representations and its connection to the affine Hecke algebra have been well studied ([Ari1996],[Ari1996],[Gec1998]).

The affine and cyclotomic BMW algebras

Fix $q,z\in {ℂ}^{*}$ and an infinite number of values $Q_1,Q_2,\dots$ in $ℂ\text{.}$ The affine BMW (Birman-Murakami-Wenzl) algebra ${\stackrel{\sim }{𝒵}}_k$ is the quotient of the group algebra $ℂ{\stackrel{\sim }{ℬ}}_k$ of the affine braid group by the relations

(6.3a)	$(T_i-z^{-1})(T_i+q^{-1})(T_i-q)=0,$
(6.3b)	$E_i+T_i^{\pm 1}=T_i^{\pm 1}{E}_i=z^{\mp 1}{E}_i,$
(6.3c)	$E_i{T}_{i-1}^{\pm 1}{E}_i=z^{\pm 1}{E}_i$ and $E_i{T}_{i+1}^{\pm 1}{E}_i=z^{\pm 1}{E}_i,$
(6.3d)	$E_1{\left(X^{{\epsilon }_1}\right)}^r{E}_1=Q_r{E}_1,$
(63.e)	$E_1{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}=z^{-1}{E}_1,$

where the $E_i,$ $1\le i\le k-1,$ are defined by the equations $$\begin{array}{cc}\frac{T_i-T_i^{-1}}{q-q^{-1}}=1-E_i,1\le i\le k-1\text{.}& \text{(6.4)}\end{array}$$ It follows that $$\begin{array}{cc}{E}_i^2=x{E}_i,\text{where}x=\frac{z-z^{-1}}{q-q^{-1}}+1\text{.}& \text{(6.5)}\end{array}$$ The classical BMW algebra is the subalgebra ${𝒵}_k$ of the affine BMW algebra which is generated by $T_1,\dots ,T_{k-1},$ and $E_1,\dots ,E_{k-1}\text{.}$ Fix $u_1,\dots ,u_r\in ℂ\text{.}$ The cyclotomic BMW algebra ${𝒵}_{r,1,k}$ is the quotient of the affine BMW algebra by the relation $$\begin{array}{cc}(X^{{\epsilon }_1}-u_1)(X^{{\epsilon }_1}-u_2)\cdots (X^{{\epsilon }_1}-u_r)=0\text{.}& \text{(6.6)}\end{array}$$

The cyclotomic BMW algebras have been defined and studied by [Här1673464], [Här9712030], [Här1834081] and, for all practical purposes the affine BMW algebras appear in these papers. The cyclotomic BMW algebras are quotients of the affine BMW algebras in the same way that cyclotomic Hecke algebras are quotients of affine Hecke algebras. The classical BMW algebras ${𝒵}_k={𝒵}_{1,1,k}$ have been studied in [Wen1988], [HRa1995], [Mur1990], [LRa1977] and many other works. The “degenerate” version of the affine BMW algebras was defined by Nazarov [Naz1996] who called them “degenerate affine Wenzl algebras”. The relation between his algebras and the affine BMW algebras ${\stackrel{\sim }{𝒵}}_k$ is analogous to the relation between the graded Hecke algebras (sometimes called the degenerate affine Hecke algebras) and the affine Hecke algebras (see [Lus1989]).

Goodman and Hauschild [GHa0411155] have proved that elements of the affine BMW algebra can be viewed as linear combinations of affine tangles. An affine tangle has $k$ strands and a flagpole just as in the case of an affine braid, but there is no restriction that a strand must connect an upper vertex to a lower vertex. Let $X^{{\epsilon }_1}$ and Ti be the affine braids given in (3.1) and let $$\begin{array}{cc}{E}_i=\begin{array}{c} \end{array}& \text{(6.7)}\end{array}$$ Then ${\stackrel{\sim }{𝒵}}_k$ is the algebra of linear combinations of tangles generated by $X^{{\epsilon }_1},T_1,\dots ,T_{k-1},E_1,\dots ,E_{k-1}$ and the relations in (6.3) expressed in the form $$\begin{array}{cc}\begin{array}{c} \end{array}-\begin{array}{c} \end{array}=(q-q^{-1})(\begin{array}{c} \end{array}-\begin{array}{c} \end{array})& \text{(6.8)}\\ \begin{array}{c} \end{array}=z\begin{array}{c} \end{array}\text{and}\begin{array}{c} \end{array}=z^{-1}\begin{array}{c} \end{array},& \text{(6.9)}\\ r\hspace{0.17em}\text{loops}\hspace{0.17em}\{\begin{array}{c} \end{array}=Q_r\begin{array}{c} \end{array}\text{and}\begin{array}{c} \end{array}=z^{-1}·\begin{array}{c} \end{array}& \text{(6.10)}\\ \begin{array}{c} \end{array}=\frac{z-z^{-1}}{q-q^{-1}}+1=x\text{.}& \text{(6.11)}\end{array}$$ When working with this algebra it is useful to note that $$T_{i-1}{X}^{{\epsilon }_{i-1}}{T}_{i-1}{X}^{{\epsilon }_{i-1}}=X^{{\epsilon }_i}{X}^{{\epsilon }_{i-1}}=X^{{\epsilon }_{i-1}}{T}_{i-1}{X}^{{\epsilon }_{i-1}}{T}_{i-1},$$ and, by induction, $$\begin{array}{cc}\begin{array}{ccc}{E}_{i-1}{X}^{{\epsilon }_{i-1}}{T}_{i-1}{X}^{{\epsilon }_{i-1}}& =& E_{i-1}{T}_{i-2}{T}_{i-1}{T}_{i-1}^{-1}{X}^{{\epsilon }_{i-2}}{T}_{i-2}{T}_{i-1}{X}^{{\epsilon }_{i-1}}\\ & =& E_{i-1}{E}_{i-2}{X}^{{\epsilon }_{i-2}}{T}_{i-2}{T}_{i-1}{T}_{i-2}^{-1}{X}^{{\epsilon }_{i-1}}\\ & =& E_{i-1}{E}_{i-2}{X}^{{\epsilon }_{i-2}}{T}_{i-2}{X}^{{\epsilon }_{i-2}}{T}_{i-1}{T}_{i-2}\\ & =& z^{-1}{E}_{i-1}{E}_{i-2}{T}_{i-1}{T}_{i-2}\\ & =& z^{-1}{E}_{i-1}{E}_{i-2}{E}_{i-1}=z^{-1}{E}_{i-1}\text{.}\end{array}& \text{(6.12)}\end{array}$$

Schur-Weyl duality for affine and cyclotomic Hecke and BMW algebras

In order to explicitly compute the representations of affine and cyclotomic Hecke algebras and BMW algebras which are obtained by applying the functors $F_{\lambda }$ we need to fix notations for working with the representations of finite dimensional complex semisimple L ie algebras of classical type.

Let $𝔤$ be a complex semisimple Lie algebra of type $A_n,$ $B_n,$ $C_n$ or $D_n$ and let $U_h𝔤$ be the corresponding Drinfeld-Jimbo quantum group. Use the notations in [Bou1968, p. 252-258], for the root systems of types $A_n,$ $B_n,$ $C_n$ and $D_n$ so that ${\epsilon }_1,\dots ,{\epsilon }_n$ are orthonormal (in type $A_n$ also include ${\epsilon }_{n+1}\text{),}$ $$\begin{array}{ccc}{𝔥}^{*}& =& \{{\lambda }_1{\epsilon }_1+\cdots {\lambda }_{n+1}{\epsilon }_{n+1}\hspace{0.17em}|\hspace{0.17em}{\lambda }_i\in ℝ,\hspace{0.17em}\sum _i{\lambda }_i=0\},\hspace{0.17em}\text{in type}\hspace{0.17em}{A}_n\text{, and}\\ {𝔥}^{*}& =& \{{\lambda }_1{\epsilon }_1+\cdots {\lambda }_n{\epsilon }_n\hspace{0.17em}|\hspace{0.17em}{\lambda }_i\in ℝ\},\hspace{0.17em}\text{in types}\hspace{0.17em}{B}_n,\hspace{0.17em}{C}_n\hspace{0.17em}\text{and}\hspace{0.17em}{D}_n,\end{array}$$ the fundamental weights are given by $$\begin{array}{cccc}{\omega }_i& =& {\epsilon }_1+\cdots +{\epsilon }_i-\frac{i}{n+1}({\epsilon }_1+\cdots +{\epsilon }_{n+1}),1\le i\le n,& \text{in Type}\hspace{0.17em}{A}_n,\\ {\omega }_i& =& {\epsilon }_1+\cdots +{\epsilon }_i,1\le i\le n-1,& \text{in Type}\hspace{0.17em}{B}_n,\\ {\omega }_n& =& \frac{1}{2}({\epsilon }_1+\cdots +{\epsilon }_n),\\ {\omega }_i& =& {\epsilon }_1+\cdots +{\epsilon }_i,1\le i\le n,& \text{in Type}\hspace{0.17em}{C}_n,\\ {\omega }_i& =& {\epsilon }_1+\cdots +{\epsilon }_i,1\le i\le n-2,\\ {\omega }_{n-1}& =& \frac{1}{2}({\epsilon }_1+\cdots +{\epsilon }_{n-1}-{\epsilon }_n),& \text{in Type}\hspace{0.17em}{D}_n,\\ {\omega }_n& =& \frac{1}{2}({\epsilon }_1+\cdots +{\epsilon }_{n-1}+{\epsilon }_n),\end{array}$$ and the finite dimensional $U_h𝔤$ modules $L\left(\lambda \right)$ are indexed by dominant integral weights $$\begin{array}{c}\begin{array}{ccc}\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n& {\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n\ge 0,& \text{in Type}\hspace{0.17em}{A}_n,\\ -\frac{\left|\lambda \right|}{n+1}({\epsilon }_1+\cdots +{\epsilon }_{n+1}),& {\lambda }_1,\dots ,{\lambda }_n\in \mathbb{Z},\end{array}\\ \begin{array}{cc}\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n,& {\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n\ge 0,\\ & {\lambda }_1,\dots ,{\lambda }_n\in \mathbb{Z}\text{, or}& \text{in Type}\hspace{0.17em}{B}_n,\\ & {\lambda }_1,\dots ,{\lambda }_n\in \frac{1}{2}+\mathbb{Z},\end{array}\\ \begin{array}{ccc}\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n& {\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_n\ge 0,& \text{in Type}\hspace{0.17em}{C}_n,\\ & {\lambda }_1,\dots ,{\lambda }_n\in \mathbb{Z},\end{array}\\ \begin{array}{cc}\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n,& {\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_{n-1}\ge \left|{\lambda }_n\right|\ge 0,\\ & {\lambda }_1,\dots ,{\lambda }_n\in \mathbb{Z}\text{, or}& \text{in Type}\hspace{0.17em}{D}_n,\\ & {\lambda }_1,\dots ,{\lambda }_n\in \frac{1}{2}+\mathbb{Z},\end{array}\end{array}$$ where $\left|\lambda \right|={\lambda }_1+\cdots +{\lambda }_n\text{.}$ $$\begin{array}{cc}2\rho =\sum _{i=1}^n(y-2i+1){\epsilon }_i,\text{where}y=\{\begin{array}{cc}n+1,& \text{in type}\hspace{0.17em}{A}_n,\\ 2n,& \text{in type}\hspace{0.17em}{B}_n,\\ 2n+1,& \text{in type}\hspace{0.17em}{C}_n,\\ 2n-1,& \text{in type}\hspace{0.17em}{D}_n,\end{array}& \text{(6.13)}\end{array}$$ and, in type $A_n$ the sum is over $1\le i\le n+1$ instead of $1\le i\le n\text{.}$

Identify $\lambda$ with the configuration of boxes which has ${\lambda }_i$ boxes in row $i$ $(1\le i\le n)\text{.}$ If ${\lambda }_i\le 0$ put $\left|{\lambda }_i\right|$ boxes in row $i$ and mark them with $-$ signs. For example $$\begin{array}{ccc}\lambda & =& \begin{array}{c} \end{array}=\{\begin{array}{c}5{\epsilon }_1+5{\epsilon }_2+3{\epsilon }_3+3{\epsilon }_4+{\epsilon }_5+{\epsilon }_6\\ -\frac{18}{n+1}({\epsilon }_1+\cdots +{\epsilon }_{n+1})\text{, in type}\hspace{0.17em}{A}_n,\\ 5{\epsilon }_1+5{\epsilon }_2+3{\epsilon }_3+3{\epsilon }_4+{\epsilon }_5+{\epsilon }_6,\hspace{0.17em}\text{in types}\hspace{0.17em}{B}_n,\hspace{0.17em}{C}_n,\hspace{0.17em}{D}_n,\end{array}\\ \lambda & =& \begin{array}{c} \end{array}=\frac{11}{2}{\epsilon }_1+\frac{11}{2}{\epsilon }_2+\frac{7}{2}{\epsilon }_3+\frac{7}{2}{\epsilon }_4+\frac{3}{2}{\epsilon }_5+\frac{3}{2}{\epsilon }_6,\hspace{0.17em}\text{in Types}\hspace{0.17em}{B}_n,\hspace{0.17em}{D}_n,\end{array}$$ and $$\lambda =\begin{array}{c} - - \end{array}=6{\epsilon }_1+6{\epsilon }_2+4{\epsilon }_3+4{\epsilon }_4+2{\epsilon }_5-2{\epsilon }_6,\text{in Type}\hspace{0.17em}{D}_6,$$

If $b$ is a box in position $(i,j)$ of $\lambda$ the content of $b$ is $$\begin{array}{cc}c\left(b\right)=j-i=\left(\text{the diagonal number of}\hspace{0.17em}b\text{.}\right)& \text{(6.14)}\end{array}$$ If $\lambda ={\lambda }_1{\epsilon }_1+\cdots {\lambda }_n{\epsilon }_n,$ then $$⟨\lambda ,\lambda +2\rho ⟩-⟨\lambda -{\epsilon }_i,\lambda -{\epsilon }_i+2\rho ⟩=2{\lambda }_i+2{\rho }_i-1=y+2{\lambda }_i-2i=y+2c(\lambda /{\lambda }^{-}),$$ where $\lambda /{\lambda }^{-}$ is the box at the end of row $i$ in $\lambda \text{.}$ Note that $c(\lambda /{\lambda }^{-})$ may be a $\frac{1}{2}\text{-integer.}$ Also, in types $B_n$ and $D_n,$ $$\begin{array}{ccc}⟨{\omega }_n,{\omega }_n+2\rho ⟩=\frac{n}{4}+\frac{1}{2}\sum _{i=1}^n(y-2i+1)& =& \frac{n}{4}+\frac{n}{2}·y-\frac{n^2}{2}\\ & =& \{\begin{array}{cc}\frac{n^2}{2}+\frac{n}{4},& \text{in type}\hspace{0.17em}{B}_n,\\ \frac{n^2}{2}-\frac{n}{4},& \text{in type}\hspace{0.17em}{D}_n\text{.}\end{array}\end{array}$$ Using these formulas $⟨\lambda ,\lambda +2\rho ⟩$ can easily be computed for all dominant integral weights $\lambda \text{.}$ For example $$\begin{array}{cc}⟨\lambda ,\lambda +2\rho ⟩=y\left|\lambda \right|+2\sum _{b\in \lambda }c\left(b\right)+\{\begin{array}{cc}-\frac{{\left|\lambda \right|}^2}{n+1},& \text{in type}\hspace{0.17em}{A}_n,\\ 0,& \text{in type}\hspace{0.17em}{C}_n\hspace{0.17em}\text{or in type}\hspace{0.17em}{B}_n\hspace{0.17em}\text{with}\hspace{0.17em}{\lambda }_i\in \mathbb{Z},\\ \frac{n}{4}+\frac{n^2}{2},& \text{in type}\hspace{0.17em}{B}_n\hspace{0.17em}\text{with}\hspace{0.17em}{\lambda }_i\in \frac{1}{2}+\mathbb{Z}\text{.}\end{array}& \text{(6.15)}\end{array}$$

For all dominant integral weights $\lambda$ in type $A_n,$ $B_n$ and $C_n$ we have $$\begin{array}{cc}L\left(\lambda \right)\otimes L\left({\omega }_1\right)=\{\begin{array}{cc}\underset{{\lambda }^{+}}{⨁}L\left({\lambda }^{+}\right),& \text{in type}\hspace{0.17em}{A}_n,\\ L\left(\lambda \right)⨁\left(\underset{{\lambda }^{\pm }}{⨁}L\left({\lambda }^{\pm }\right)\right),& \text{in Type}\hspace{0.17em}{B}_n\hspace{0.17em}\text{with}\hspace{0.17em}{\lambda }_n>0,\\ \left(\underset{{\lambda }^{\pm }}{⨁}L\left({\lambda }^{\pm }\right)\right),& \text{in types}\hspace{0.17em}{C}_n\hspace{0.17em}\text{and}\hspace{0.17em}{D}_n\text{, and}\\ & \text{in type}\hspace{0.17em}{B}_n\hspace{0.17em}\text{with}\hspace{0.17em}{\lambda }_n=0,\end{array}& \text{(6.16)}\end{array}$$ where the sum over ${\lambda }^{+}$ is a sum over all partitions (of length $\le n\text{)}$ obtained by adding a box to $\lambda ,$ and the sum over ${\lambda }^{\pm }$ denotes a sum over all dominant weights obtained by adding or removing a box from $\lambda \text{.}$ In type $D_n$ 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 $n$ when ${\lambda }_n=\frac{1}{2}$ changes ${\lambda }_n$ to $-\frac{1}{2}\text{.}$

Let $𝔤$ be the simple complex Lie algebra of classical type, $U=U_h𝔤$ the corresponding quantum group and let $V=L\left({\omega }_1\right)$ be the irreducible $U_h𝔤$ module of highest weight ${\omega }_1\text{.}$ For each $M\in 𝒪$ let ${\Phi }_k:{\stackrel{\sim }{ℬ}}_k\to {\text{End}}_U(M\otimes V^{\otimes k})$ be the affine braid group representation defined in Proposition 3.1.

(a)	If $𝔤$ is type $A_n$ then ${\Phi }_k$ is a representation of the affine Hecke algebra ${\stackrel{\sim }{H}}_k$ with $q=e^{h/2}\text{.}$ (In this Type $A_n$ case use a different normalization of the map ${\Phi }_k$ and set ${\Phi }_k\left(T_i\right)=q^{1/(n+1)}{Ř}_i\text{.)}$
(b)	If $𝔤$ is type $A_n$ and if $M=L\left(\mu \right)$ where $\mu$ is a dominant integral weight then ${\Phi }_k$ is a representation of the cyclotomic Hecke algebra $H_{r,1,n}(u_1,\dots ,u_r)$ for any (multi)set of parameters $u_1,\dots ,u_r$ containing the (multi)set of values $q^{2c\left(b\right)}$ as $b$ runs over the addable boxes of $\mu \text{.}$
(c)	If $𝔤$ is type $B_n,$ $C_n$ or $D_n$ and $M$ is a highest weight module then there are unique values $Q_1,Q_2,\dots \in ℂ,$ depending only on the central character of $M,$ such that ${\Phi }_k$ is a representation of the affine BMW algebra ${\stackrel{\sim }{𝒵}}_k$ with parameters $Q_1,Q_2,\dots ,$ $$q=e^{h/2},\text{and}z=\{\begin{array}{cc}{q}^{2n},& \text{in Type}\hspace{0.17em}{B}_n,\\ -q^{2n+1},& \text{in Type}\hspace{0.17em}{C}_n,\\ q^{2n-1},& \text{in Type}\hspace{0.17em}{D}_n\text{.}\end{array}$$
(d)	If $𝔤$ is type $B_n,$ $C_n$ or $D_n,$ and $M=L\left(\mu \right)$ where $\mu$ is a dominant integral weight then ${\Phi }_k$ is a representation of the cyclotomic BMW algebra ${\stackrel{\sim }{𝒵}}_{r,1,k}$ with $q$ and $z$ as in (c), $$Q_r=\sum _{{\mu }^{\pm }}{q}^{r\stackrel{\sim }{c}({\mu }^{\pm },\mu )}\frac{{\text{dim}}_q\left(L\left({\mu }^{\pm }\right)\right)}{{\text{dim}}_q\left(L\left(\mu \right)\right){\text{dim}}_q\left(L\left({\omega }_1\right)\right)},r\in {\mathbb{Z}}_{>0},$$ and any (multi)set of parameters $u_1,\dots ,u_r$ containing the (multi)set of values $q^{\stackrel{\sim }{c}({\mu }^{\pm },\mu )}$ as ${\mu }^{\pm }$ runs over the dominant integral weights appearing in the decomposition (6.16) of $L\left(\mu \right)\otimes L\left({\omega }_1\right)\text{.}$ Here $$\stackrel{\sim }{c}({\mu }^{\pm },\mu )=\{\begin{array}{cc}-y,& \text{if}\hspace{0.17em}{\mu }^{\pm }=\mu ,\\ 2c({\mu }^{\pm }/\mu ),& \text{if}\hspace{0.17em}{\mu }^{\pm }\supseteq \mu ,\\ -2(c(\mu /{\mu }^{\pm })+y),& \text{if}\hspace{0.17em}{\mu }^{\pm }\subseteq \mu ,\end{array}$$ where $y$ and $c\left(b\right)$ are as defined in (6.13) and (6.14), respectively.

		Proof.
	(a) It is only necessary to show that ${\Phi }_k\left(T_i\right)=q^{1/(n+1)}{Ř}_i$ satisfies ${\left(q^{1/(n+1)}{Ř}_i\right)}^2=(q-q^{-1})\left(q^{1/(n+1)}{Ř}_i\right)+1$ for $2\le i\le n\text{.}$ This is proved in [LRa1977, Prop. 4.4]. (c) The arguments establishing the relations (6.3a–c) in the definition of the affine BMW algebra are exactly as in [LRa1977, Prop. 5.10]. It remains to establish (6.3d–e). The element $E_1$ in the affine BMW algebra acts on $V^{\otimes 2}$ as $x·{\text{pr}}_0$ where ${\text{pr}}_0$ is the unique $U_h𝔤\text{-invariant}$ projection onto the invariants in $V^{\otimes 2}$ and $x$ is as in (6.5). Using the identity (6.9) and the pictorial equalities $$\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=z^{-1}·\begin{array}{c} \end{array}$$ it follows that ${\Phi }_2\left(E_1{X}^{{\epsilon }_1}{T}_1{X}^{{\epsilon }_1}\right)$ acts as $x{z}^{-1}·{Ř}_{L\left(0\right),M}{Ř}_{M,L\left(0\right)}({\text{id}}_M\otimes {\text{pr}}_0)\text{.}$ By (2.11), this is equal to $$\begin{array}{ccc}{z}^{-1}·(C_M\otimes C_{L\left(0\right)})C_{M\otimes L\left(0\right)}^{-1}{\Phi }_2({\text{id}}_M\otimes E_1)& =& z^{-1}·C_M{C}_M^{-1}{\Phi }_2({\text{id}}_M\otimes E_1)\\ & =& z^{-1}·{\Phi }_2\left(E_1\right),\end{array}$$ establishing the relation in (6.3e). Since ${\Phi }_2\left(E_1\right)$ acts as $x·({\text{id}}_M\otimes {\text{pr}}_0)$ on $M\otimes V^{\otimes 2}$ the morphism ${\Phi }_2\left(E_1{X}^{r{\epsilon }_1}{E}_1\right)$ is a morphism from $M\otimes L\left(0\right)\to M\otimes L\left(0\right)\text{.}$ Since $M=M\otimes L\left(0\right)$ is a highest weight module this morphism is $Q_r·{\text{id}}_M,$ for some $Q_r\in ℂ\text{.}$ By the results of Drinfeld [Dri1990] and Reshetikhin [Res1990] (see [Bau1620662, p. 250]), the action of the morphism ${\Phi }_2\left(E_1{X}^{r{\epsilon }_1}{E}_1\right)$ corresponds to the action of a central element of $U_h𝔤$ on $M\text{.}$ Thus the constant $Q_r$ depends only on the central character of $M\text{.}$ (b) Let $b_1,\dots ,b_r$ be the addable boxes of $\mu$ and consider the action of $X^{{\epsilon }_1}$ on $M\otimes V=L\left(\mu \right)L\left({\omega }_1\right)\text{.}$ We will show that ${\Phi }_k\left(X^{{\epsilon }_1}\right)={Ř}_0^2$ satisfies the relation $({Ř}_0^2-u_1)\cdots ({Ř}_0^2-u_r)=0,$ where $u_i=q^{2c\left(b_i\right)}\text{.}$ By (2.11) and (2.14) it follows that $${Ř}_0^2=\sum _{{\mu }^{+}}{q}^{⟨{\mu }^{+},{\mu }^{+}+2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨{\omega }_1,{\omega }_1+2\rho ⟩}{P}_{\mu ,{\omega }_1}^{{\mu }^{+}}=\sum _{{\mu }^{+}}{q}^{2c({\mu }^{+}/\mu )}{P}_{\mu ,{\omega }_1}^{{\mu }^{+}},$$ where the sum is over all partitions ${\mu }^{+}$ obtained by adding a box to $\mu ,$ $P_{\mu ,{\omega }_1}^{{\mu }^{+}}$ is the projection onto $L\left({\mu }^{+}\right)$ in the tensor product $M\otimes V=L\left(\mu \right)\otimes L\left({\omega }_1\right),$ and $c({\mu }^{+}/\mu )$ is the content of the box ${\mu }^{+}/\mu$ which is added to $\mu$ to get ${\mu }^{+}\text{.}$ Thus ${Ř}_0^2$ is a diagonal operator with eigenvalues $q^{2c({\mu }^{+}/\mu )}$ and so it satisfies the equation (6.2). (d) Using the appropriate case of the decomposition rule for $L\left(\mu \right)\otimes L\left({\omega }_1\right),$ the proof of the relation $(X^{{\epsilon }_1}-u_1)\cdots (X^{{\epsilon }_1}-u_r)=0$ is as in (b). The values of $\stackrel{\sim }{c}({\mu }^{\pm },\mu )$ are determined from (6.16). To compute the value of $Q_r$ note that ${\Phi }_2\left(E_1{X}^{r{\epsilon }_1}{E}_1\right)={\Phi }_2\left({\epsilon }_1\left(X^{r{\epsilon }_1}\right)E_1\right),$ in the notations of the proof of Theorem 5.1. Thus $Q_r$ is determined by the formula in Theorem 5.1 and the decomposition of $L\left(\mu \right)\otimes L\left({\omega }_1\right)$ in (6.14). $\square$

Remark. The parameters in $Q_1,Q_2,\dots \in ℂ$ needed in Theorem 6.1c can be determined by using the formula of Baumann [Bau1620662, Theorem 1], which characterizes $Q_r$ in terms of the values $Q_1$ given in (5.6). To do this it is necessary to use formula (5.6) for $Q_1$ several times: $\mu$ is always the highest weight of $M,$ but many different $\nu$ will be needed. Note that the proof of the formula (5.6) for $Q_1$ in [TWe1217386] does not require $\mu$ to be dominant integral.

The following theorem provides an analogue of Schur-Weyl duality for the affine Hecke algebras, cyclotomic Hecke algebras, affine BMW algebras and cyclotomic BMW algebras. Alternative Schur-Weyl dualities have been given by Chari-Pressley [CPr1996] for the case of affine Hecke algebras and by Sakamoto and Shoji [SSh1999] for cyclotomic Hecke algebras. Cherednik [Che1987] also used a Schur-Weyl duality for the affine Hecke algebra which is different from the Schur-Weyl duality given here.

Assume that $𝔤$ is not of type $D_n\text{.}$ Let $\mu$ be a dominant integral weight and let $M=L\left(\mu \right)\text{.}$ In each of the cases given in Theorem 6.1 the representation ${\Phi }_k$ is surjective.

		Proof.
	Part (a) is a consequence of (b) since the representation of ${\stackrel{\sim }{H}}_k$ in (a) is the composition of the representation ${\Phi }_k:H_{r,1,k}\to {\text{End}}_{U_h𝔤}(L\left(\mu \right)\otimes V^{\otimes k})$ from (b) with the surjective algebra homomorphism ${\stackrel{\sim }{H}}_k\to H_{r,1,k}$ coming from the definition of $H_{r,1,k}\text{.}$ Similarly part (c) is a consequence of part (d). The proof of the surjectivity of the representation in Theorem 6.1b and Theorem 6.1d are exactly the same as the proofs of [LRa1977, Cor. 4.15], and [LRa1977, Cor. 5.22], respectively. The case considered there is the $\mu =0$ case but all the arguments there generalize verbatim to the case when $\mu$ is an arbitrary dominant integral weight. In [LRa1977, §4], the elements $X^{{\epsilon }_i}$ in the affine braid group are denoted $D_i\text{.}$ The assumption $n\gg k$ in [LRa1977] is unecessary for this theorem if the full decomposition rule given in (6.14) is used. The main point is that the eigenvalues of $X^{{\epsilon }_1},\dots ,X^{{\epsilon }_i}$ separate the components of the decomposition of $L\left(\mu \right)\otimes V^{\otimes i}\text{.}$ By induction it is sufficient to check that the eigenvalues of $X^{{\epsilon }_1}$ distinguish the components of $L\left(\lambda \right)\otimes V$ for all $\lambda \text{.}$ By (2.10), (2.11) and (2.14), the eigenvalues of $X^{{\epsilon }_i}$ are of the form $q^{2\stackrel{\sim }{c}({\lambda }^{\pm },\lambda )}$ where $\lambda$ is a dominant integral weight $\stackrel{\sim }{c}({\lambda }^{\pm },\lambda )$ is as in Theorem 6.1d and ${\lambda }^{\pm }$ runs over the components in the decomposition (6.16) of $L\left(\lambda \right)\otimes V\text{.}$ Different addable boxes for $\lambda$ can never have the same content since they cannot be in the same diagonal. Similarly for two different removable boxes. Let $b$ be an addable box and $b\prime$ a removable box for $\lambda \text{.}$ Unless $𝔤$ is type $D_n$ and $b$ and $b\prime$ are in row $n,$ we have $c\left(b\right),c\left(b\prime \right)\ge -n-1\text{.}$ Thus, when $𝔤$ is not of type $D_n,$ $c\left(b\right)\ne -c\left(b\prime \right)=y$ and so the two eigenvalues coming from these boxes are different. $\square$

Let ${\stackrel{\sim }{𝒵}}_k$ denote the affine Hecke algebra, the cyclotomic Hecke algebra, the affine BMW algebra or the cyclotomic BMW algebra corresponding to the case of Theorem 6.1 which is being considered. Then, as in the classical Schur-Weyl duality setting, Theorem 6.2 implies that as $(U_h𝔤,{\stackrel{\sim }{𝒵}}_k)$ bimodules $$\begin{array}{cc}L\left(\mu \right)\otimes V^{\otimes k}\cong \underset{\lambda }{⨁}L\left(\lambda \right)\otimes {ℒ}^{\lambda /\mu },& \text{(6.17)}\end{array}$$ where $L\left(\lambda \right)$ is the irreducible $U_h𝔤\text{-module}$ of highest weight $\lambda$ and ${ℒ}^{\lambda /\mu }$ is the irreducible ${\stackrel{\sim }{𝒵}}_k$ module defined by (4.1).

The irreducible ${\stackrel{\sim }{𝒵}}_k$ modules ${ℒ}^{\lambda /\mu }$ appearing in (6.17) can be constructed quite explicitly. All the necessary computations for doing this have already been done in [LRa1977, §4 and 5], which does the case $\mu =0\text{.}$ All the arguments in [LRa1977, §4 and 5], generalize directly to the case when $\mu$ is an arbitrary dominant integral weight. The final result is Theorem 6.3 below. The result in part (a) of Theorem 6.3 is due to Cherednik [Che1987].

If $\lambda$ and $\mu$ are partitions such that $\lambda \supseteq \mu$ the skew shape $\lambda /\mu$ is the configuration of boxes of in $\lambda$ which are not in $\mu \text{.}$ Let $\lambda /\mu$ be a skew shape with $k$ boxes. A standard tableau of shape $\lambda /\mu$ is a filling $T$ of the boxes of $\lambda /\mu$ with $1,2,\dots ,k$ such that

(a)	the rows of $T$ are increasing (left to right), and
(b)	the columns of $T$ are increasing (top to bottom).

For example, $$\begin{array}{c} 3 4 9 12 1 5 10 7 13 14 2 6 8 11 \end{array}$$ is a standard tableau of shape $\lambda /\mu =\left(977421\right)/\left(5443\right)\text{.}$

For any two partitions $\mu$ and $\lambda$ an up down tableau of length $k$ from $\mu$ to $\lambda$ is a sequence of partitions $T=(\mu ={\tau }^{\left(0\right)},{\tau }^{\left(1\right)},\dots ,{\tau }^{(k-1)},{\tau }^{\left(k\right)}=\lambda )$ such that

(a)	${\tau }^{\left(i\right)}\supseteq {\tau }^{(i-1)}$ and ${\tau }^{\left(i\right)}/{\tau }^{(i-1)}=▫,$ or
(b)	${\tau }^{(i-1)}\supseteq {\tau }^{\left(i\right)}$ and ${\tau }^{(i-1)}/{\tau }^{\left(i\right)}=▫,$

and, in type $B_n$ the situation ${\tau }^{(i-1)}={\tau }^{\left(i\right)}$ with $\ell \left({\tau }^{(i-1)}\right)=n$ is also allowed. Note that a standard tableau $\lambda /\mu$ with $k$ boxes is exactly an up down tableau of length $k$ from $\mu$ to $\lambda$ where all steps in the sequence satisfy condition (a).

(a)

Let $\lambda /\mu$ be a skew shape with $k$ boxes. Then the module ${ℒ}^{\lambda /\mu }=F_{\lambda }\left(L\left(\mu \right)\right)$ for the affine Hecke algebra ${\stackrel{\sim }{H}}_k$ is irreducible and is given by $${ℒ}^{\lambda /\mu }=\text{span}\left\{v_T\hspace{0.17em}\right|\hspace{0.17em}T\hspace{0.17em}\text{standard tableaux of shape}\hspace{0.17em}\lambda /\mu \}$$ (so that the symbols $v_T$ are a $ℂ\text{-basis}$ of ${ℒ}^{\lambda /\mu }\text{)}$ with ${\stackrel{\sim }{H}}_k$ action given by $$\begin{array}{ccc}{X}^{{\epsilon }_i}{v}_T& =& q^{2c\left(T\left(i\right)\right)}{v}_T,\hspace{0.17em}1\le i\le k,\\ T_i{v}_T& =& {\left(T_i\right)}_{TT}{v}_T+\sqrt{(q^{-1}+{\left(T_i\right)}_{TT})(q^{-1}+{\left(T_i\right)}_{s_{i}T,s_{i}T})}{v}_{s_{i}T},\end{array}$$ for $1\le i\le k-1,$ where ${\left(T_i\right)}_{TT}\hspace{0.17em}$ is the constant $\frac{q-q^{-1}}{1-q^{2c\left(T\left(i\right)\right)-c\left(T(i+1)\right)}},$ $c\left(b\right)$ denotes the content of the box $b,$ $T\left(i\right)$ is the box containing $i$ in $T,$ $s_{i}T$ is the same filling as $T$ except $i$ and $i+1$ are switched, and $v_{s_{i}T}=0$ if $s_{i}T$ is not a standard tableau.

(b)

Let $\lambda /\mu$ be a pair of partitions. Then the module ${ℒ}^{\lambda /\mu }=F_{\lambda }\left(L\left(\mu \right)\right)$ for the affine BMW algebra ${\stackrel{\sim }{𝒵}}_k$ is irreducible and is given by $${ℒ}^{\lambda /\mu }=\text{span}\left\{v_T\hspace{0.17em}\right|\hspace{0.17em}\begin{array}{c}T=(\mu ={\tau }^{\left(0\right)},\dots ,{\tau }^{\left(k\right)}=\lambda )\hspace{0.17em}\text{an}\\ \text{up down tableau of length}\hspace{0.17em}k\hspace{0.17em}\text{from}\hspace{0.17em}\mu \hspace{0.17em}\text{to}\hspace{0.17em}\lambda \end{array}\}$$ (so that the symbols $v_T$ are a $ℂ\text{-basis}$ of ${ℒ}^{\lambda /\mu }\text{)}$ with ${\stackrel{\sim }{𝒵}}_k$ action given by $$\begin{array}{ccc}{X}^{{\epsilon }_i}{v}_T& =& \stackrel{\sim }{c}({\tau }^{\left(i\right)},{\tau }^{(i-1)})v_T,1\le i\le k,\\ E_i{v}_T& =& {\delta }_{{\tau }^{(i+1)},{\tau }^{(i-1)}}·\sum _S{\left(E_i\right)}_{ST}{v}_S,\hspace{0.17em}\text{and}\\ T_i{v}_T& =& \sum _S{\left(T_i\right)}_{ST}{v}_S,\hspace{0.17em}1\le i\le k-1,\end{array}$$ where both sums are over up-down tableaux $$S=(\mu ={\tau }^{\left(0\right)},\dots ,{\tau }^{(i-1)},{\sigma }^{\left(i\right)},{\tau }^{(i+1)},\dots ,{\tau }^{\left(k\right)}=\lambda )$$ that are the same as $T$ except possibly at the $i\text{th}$ step and $$\begin{array}{ccc}{\left(E_i\right)}_{ST}& =& ϵ·\frac{\sqrt{{\text{dim}}_q\left(L\left({\tau }^{\left(i\right)}\right)\right){\text{dim}}_q\left(L\left({\sigma }^{\left(i\right)}\right)\right)}}{{\text{dim}}_q\left({\tau }^{(i-1)}\right)},\\ {\left(T_i\right)}_{ST}& =& \{\begin{array}{cc}\sqrt{(q^{-1}+{\left(T_i\right)}_{TT})(q^{-1}+{\left(T_j\right)}_{SS})},& \text{if}\hspace{0.17em}{\tau }^{(i-1)}\ne {\tau }^{(i+1)}\hspace{0.17em}\text{and}\hspace{0.17em}S\ne T,\\ \left(\frac{q-q^{-1}}{1-\stackrel{\sim }{c}{({\tau }^{(i+1)},{\sigma }^{\left(i\right)})}^{-1}\stackrel{\sim }{c}{({\tau }^{\left(i\right)},{\tau }^{(i-1)})}^{-1}}\right)({\delta }_{ST}-{\left(E_i\right)}_{ST}),& \text{otherwise,}\end{array}\\ \stackrel{\sim }{c}({\tau }^{\left(i\right)},{\tau }^{(i-1)})& =& \{\begin{array}{cc}{z}^{-1},& \text{if}\hspace{0.17em}{\tau }^{\left(i\right)}={\tau }^{(i-1)},\\ q^{2c({\tau }^{\left(i\right)}/{\tau }^{(i-1)})},& \text{if}\hspace{0.17em}{\tau }^{\left(i\right)}\supseteq {\tau }^{(i-1)},\\ z^{-2}{q}^{-2c({\tau }^{(i-1)}/{\tau }^{\left(i\right)})},& \text{if}\hspace{0.17em}{\tau }^{\left(i\right)}\subseteq {\tau }^{(i-1)},\end{array}\end{array}$$ and $ϵ=1,$ in type $B_n$ and $D_n,$ and $ϵ=-1$ in type $C_n\text{.}$

Markov traces on affine and cyclotomic Hecke and BMW algebras

If $M=L\left(\mu \right)$ where $\mu$ is a dominant integral weight and $V=L\left({\omega }_1\right)$ then each of the representations ${\Phi }_k:{\stackrel{\sim }{𝒵}}_k\to {\text{End}}_{U_h𝔤}(M\otimes V^{\otimes k})$ (where ${\stackrel{\sim }{𝒵}}_k$ is the affine Hecke algebra, the cyclotomic Hecke algebra, the affine BMW algebra, or the cyclotomic BMW algebra) gives rise to a Markov trace via Theorem 5.1. The parameters and the weights of these Markov traces are given by Theorems 5.1 and 5.3.

In type A case $\lambda /\mu$ is a skew shape with $k$ boxes and the parameters and the weights of (most of) these traces have been given in terms of partitions in [GIM2000]. In [GIM2000], $\mu$ is a partition of a special form ([GIM2000, 2.2(*)]), and so, in their case, the skew shape $\lambda /\mu$ can be viewed as an $r\text{-tuple}$ of partitions. Their formulas can be recovered from ours by rewriting the quantum dimension ${\text{dim}}_q\left(L\left(\lambda \right)\right)$ from (5.3) in terms of the partition as in [Mac1995, I §3 Ex. 1]: $$\begin{array}{cc}{\text{dim}}_q\left(L\left(\lambda \right)\right)=\prod _{b\in \lambda }\frac{[n+1+c\left(b\right)]}{\left[h\left(b\right)\right]},\text{in the type}\hspace{0.17em}{A}_n\hspace{0.17em}\text{case,}& \text{(6.18)}\end{array}$$ where, if $b$ is the box in position $(i,j)$ of $\lambda ,$ then $h\left(b\right)={\lambda }_i-i+{\lambda }_j^{\prime }-j+1$ is the hook length at $b,$ and $\left[d\right]=(q^d-q^{-d})/(q-q^{-1})$ for a positive integer $d\text{.}$ Thus, the first formula in [GIM2000, §2.3], coincides with ${\text{dim}}_q\left(L\left(\lambda \right)\right)/{\left({\text{dim}}_q\left(V\right)\right)}^{\left|\lambda \right|}$ and so the formula for the weights of the Markov trace on cyclotomic Hecke algebras which is given in [GIM2000, Prop. 2.3], coincides exactly with the formula in Theorem 5.3. To remove the constants that come from the difference between ${𝔤𝔩}_n$ and ${𝔰𝔩}_n$ the affine braid group action in Theorem 6.1a should be normalized so that ${\Phi }_k\left(X^{{\epsilon }_1}\right)=q^{2\left|\mu \right|/(n+1)}{Ř}_0^2$ and ${\Phi }_k\left(T_i\right)=q^{1/(n+1)}{Ř}_i\text{.}$ Then, from Theorem 5.1, (6.16) and (6.18) it follows that the parameters of the Markov trace are $z=q^{n+1}[n+1]$ and $$\begin{array}{ccc}{Q}_r& =& \sum _{{\mu }^{+}}{q}^{2rc({\mu }^{+}/\mu )}\frac{{\text{dim}}_q\left(L\left({\mu }^{+}\right)\right)}{{\text{dim}}_q\left(L\left(\mu \right)\right){\text{dim}}_q\left(V\right)}\\ & =& \sum _{{\mu }^{+}}{q}^{2rc({\mu }^{+}/\mu )}\left(\prod _{b\in {\mu }^{+}}\frac{[n+1+c\left(b\right)]}{\left[h\left(b\right)\right]}\right)\left(\prod _{b\in \mu }\frac{\left[h\left(b\right)\right]}{[n+1+c\left(b\right)]}\right)\frac{1}{[n+1]}\\ & =& \sum _{{\mu }^{+}}{q}^{2rc({\mu }^{+}/\mu )}\left(\frac{\prod _{b\in \mu }\left[h\left(b\right)\right]}{\prod _{b\in {\mu }^{+}}\left[h\left(b\right)\right]}\right)\frac{[n+1+c({\mu }^{+}/\mu )]}{[n+1]}\\ & =& \sum _{{\mu }^{+}}{q}^{2rc({\mu }^{+}/\mu )}\left(\prod _{b\prime }\frac{\left[h\left(b\prime \right)\right]}{[h\left(b\prime \right)+1]}\right)\left(\prod _{b″}\frac{\left[h\left(b″\right)\right]}{[h\left(b″\right)+1]}\right)\frac{[n+1+c({\mu }^{+}/\mu )]}{[n+1]},\end{array}$$ where, in the last expression, the first product is over boxes $b\prime \in \mu$ which are in the same row as the added box ${\mu }^{+}/\mu$ and the second product is over $b″\in \mu$ which are in the same column as ${\mu }^{+}/\mu \text{.}$ Then cancellation of the common terms in the numerator and denominators of each product yields the combinatorial formulas for the parameters of the Markov traces on cyclotomic Hecke algebras which are given in [GIM2000, Thm. 2.4].

Lambropoulou [Lam1999, §4], has proved that there is a unique Markov trace on the affine Hecke algebra with a given choice of parameters $z,Q_1,\dots ,Q_r\in ℂ\text{.}$ A similar result is true for the affine BMW algebra.

For each fixed choice of parameters $q,$ $z$ and $Q_1,Q_2,\dots$ there is a unique Markov trace on the affine BMW algebra ${𝒵}_k\text{.}$

		Proof (Sketch).
	Consider the image of an affine braid $b$ in the affine BMW algebra. The Markov trace of this braid can be viewed pictorially as the closure of the braid $b\text{.}$ $${\text{mt}}_k\left(b\right)={\text{mt}}_1\left(\begin{array}{c} b \end{array}\right)$$ Consider a string in the closure as it winds around the other strings and the pole. If the string crosses another string twice without going around the pole between these two crossings then we can use the relation $$\begin{array}{c} \end{array}-\begin{array}{c} \end{array}=(q-q^{-1})(\begin{array}{c} \end{array}-\begin{array}{c} \end{array})$$ to rewrite the closed braid as a linear combination of closed braids with fewer crossings between strings. By successive steps of this type we can reduce the computation of the Markov trace of a braid to a linear combination of $$\begin{array}{c}{r}_1\hspace{0.17em}\text{loops}\hspace{0.17em}\{\begin{array}{c} \end{array}\\ ⋮\\ r_k\hspace{0.17em}\text{loops}\hspace{0.17em}\{\begin{array}{c} \end{array}\end{array}=\frac{Q_{r_1}\cdots Q_{r_k}}{{\text{dim}}_q{\left(V\right)}^k}·\text{mt}\left(\begin{array}{c} \end{array}\right)=\frac{Q_{r_1}\cdots Q_{r_k}}{{\text{dim}}_q{\left(V\right)}^k}\text{.}$$ $\square$

Remark. For computations it is helpful to note that $$\begin{array}{cc}{\text{dim}}_q\left(L\left({\omega }_1\right)\right)=\{\begin{array}{cc}[n+1],& \text{in type}\hspace{0.17em}{A}_n,\\ \left[2n\right]+1,& \text{in type}\hspace{0.17em}{B}_n,\\ [2n+1]-1,& \text{in type}\hspace{0.17em}{C}_n,\\ [2n-1]+1,& \text{in type}\hspace{0.17em}{D}_n\text{.}\end{array}& \text{(6.19)}\end{array}$$

Standard and simple modules for affine Hecke algebras

The original construction of the irreducible representations of the affine Hecke algebra of type A is due to Zelevinsky [Zel1980] and is an analogue of the Langlands construction of admissible representations of real reductive Lie groups. Zelevinsky used the combinatorics of multisegments which is easily seen to be equivalent to the combinatorics of unipotent-semisimple pairs used later in [KLu0862716] (see [Ari1996]). Here we show how the construction of affine Hecke algebra representations via the functors $F_{\lambda }$ naturally matches up with Zelevinsky’s indexings by multisegments. Using the multisegment indexing of representations, Theorem 6.6 below explicitly matches up the decomposition numbers for affine Hecke algebras with Kazhdan-Lusztig polynomials. Recall that the functor $F_{\lambda }$ gives representations of the affine Hecke algebra in the setting of Theorem 6.1a when $𝔤$ is of type $A_n$ and $V=L\left({\omega }_1\right)$ is the $n\text{-dimensional}$ fundamental representation.

Consider an (infinite) sheet of graph paper which has its diagonals labeled consecutively by $\dots ,-2,-1,0,1,2,\dots \text{.}$ The content $c\left(b\right)$ of a box $b$ on this sheet of graph paper is $$c\left(b\right)=\text{the diagonal number of the box}\hspace{0.17em}b$$ (a natural generalization of the definition of $c\left(b\right)$ in (6.15)). A multisegment is a collection of rows of boxes (segments) placed on graph paper. We can label this multisegment by a pair of weights $\lambda ={\lambda }_1{\epsilon }_1+\cdots {\lambda }_{n+1}{\epsilon }_{n+1}$ and $\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_{n+1}{\epsilon }_{n+1}$ by setting $$\begin{array}{ccc}{(\lambda +\rho )}_i& =& \text{content of the last box in row}\hspace{0.17em}i,\text{and}\\ {(\mu +\rho )}_i& =& \left(\text{content of the first box in row}\hspace{0.17em}i\right)-1\text{.}\end{array}$$ For example $$\begin{array}{c} 3 4 5 6 7 3 4 5 6 7 5 6 7 1 2 3 4 5 3 4 5 \end{array}\text{corresponds to}$$ $$\begin{array}{cc}\begin{array}{ccc}\lambda +\rho & =& (7,\hspace{0.17em}7,\hspace{0.17em}7,\hspace{0.17em}5,\hspace{0.17em}5)\text{and}\\ \mu +\rho & =& (2,\hspace{0.17em}2,\hspace{0.17em}4,\hspace{0.17em}0,\hspace{0.17em}2)\end{array}& \text{(6.20)}\end{array}$$ (the numbers in the boxes in the picture are the contents of the boxes). The construction forces the condition

(a)	${(\lambda +\rho )}_i-{(\mu +\rho )}_i\in {\mathbb{Z}}_{\ge 0}\text{.}$

and since we want to consider unordered collections of boxes it is natural to take the following pseudo-lexicographic ordering on the segments

(b)	${(\lambda +\rho )}_i\ge {(\lambda +\rho )}_{i+1},$
(c)	${(\mu +\rho )}_i\le {(\mu +\rho )}_{i+1}$ if ${(\lambda +\rho )}_i={(\lambda +\rho )}_{i+1},$

when we denote the multisegment $\lambda /\mu$ by a pair of weights $\lambda ,$ $\mu \text{.}$ In terms of weights the conditions (a), (b) and (c) can be restated as (note that in this case both $\lambda$ and $\mu$ are integral)

(a')	$\lambda -\mu$ is a weight of $V^{\otimes k},$ where $k$ is the number of boxes in $\lambda /\mu ,$
(b')	$\lambda$ is integrally dominant,
(c')	$\mu =w\circ \nu$ with $\nu$ integrally dominant and $w$ maximal length in the coset $W_{\lambda +\rho }w{W}_{\nu +\rho }\text{.}$

These conditions on the pair of weights $(\lambda ,\mu )$ arose previously in Propositions 4.2d and Remark 4.3.

Let $\lambda /\mu$ be a multisegment with $k$ boxes and number the boxes of $\lambda /\mu$ from left to right (like a book). Define ${\stackrel{\sim }{H}}_{\lambda /\mu }$ to be the subalgebra of ${\stackrel{\sim }{H}}_k$ generated by $$\{X^{\lambda },\hspace{0.17em}{T}_j\hspace{0.17em}|\hspace{0.17em}\lambda \in L,\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{is not at the end of its row}\},$$ so that ${\stackrel{\sim }{H}}_{\lambda /\mu }$ is the “parabolic” subalgebra of ${\stackrel{\sim }{H}}_k$ corresponding to the multisegment $\lambda /\mu \text{.}$ Define a one-dimensional ${\stackrel{\sim }{H}}_{\lambda /\mu }$ module ${ℂ}_{\lambda /\mu }=ℂv_{\lambda /\mu }$ by setting $$\begin{array}{cc}{X}^{{\epsilon }_i}{v}_{\lambda /\mu }=q^{2c\left({\text{box}}_i\right)}{v}_{\lambda /\mu },\text{and}{T}_j{v}_{\lambda /\mu }=q{v}_{\lambda /\mu },& \text{(6.21)}\end{array}$$ for $1\le i\le k$ and $j$ such that ${\text{box}}_j$ is not at the end of its row.

Let $𝔤$ be of type $A_n$ and let $F_{\lambda }$ be the functor ${\text{Hom}}_{U_h𝔤}(M\left(\lambda \right),·\otimes V^{\otimes k})$ from the setting of Theorem 6.1a, where $V=L\left({\omega }_1\right)\text{.}$ The standard module for the affine Hecke algebra ${\stackrel{\sim }{H}}_k$ is $$\begin{array}{cc}{ℳ}^{\lambda /\mu }=F_{\lambda }\left(M\left(\mu \right)\right)& \text{(6.22)}\end{array}$$ as defined in (4.1). It follows from the above discussion that these modules are naturally indexed by multisegments $\lambda /\mu \text{.}$ The following proposition shows that this standard module coincides with the usual standard module for the affine Hecke algebra as considered by Zelevinsky [Zel1980] (see also [Ari1996], [CGi1997] and [KLu0862716]).

Let $\lambda /\mu$ be a multisegment determined by a pair weights $(\lambda ,\mu )$ with $\lambda$ integrally dominant. Let ${ℂ}_{\lambda /\mu }$ be the one dimensional representation of the parabolic subalgebra ${\stackrel{\sim }{H}}_{\lambda /\mu }$ of the affine Hecke algebra ${\stackrel{\sim }{H}}_k$ defined in (6.21). Then $${ℳ}^{\lambda /\mu }\cong {\text{Ind}}_{{\stackrel{\sim }{H}}_{\lambda /\mu }}^{{\stackrel{\sim }{H}}_k}\left({ℂ}_{\lambda /\mu }\right)\text{.}$$

		Proof.
	To remove the constants that come from the difference between ${𝔤𝔩}_n$ and ${𝔰𝔩}_n$ the affine braid group action in Theorem 6.1a should be normalized so that ${\Phi }_k\left(X^{{\epsilon }_1}\right)=q^{2\left|\mu \right|/(n+1)}{Ř}_0^2$ and ${\Phi }_k\left(T_i\right)=q^{1/(n+1)}{Ř}_i\text{.}$ By Proposition 4.2a, ${ℳ}^{\lambda /\mu }\cong {\left(V^{\otimes k}\right)}_{\lambda -\mu }$ as a vector space. Let ${\left\{v_i\right\}}_{i=1}^{n+1}$ be the standard basis of $V=L\left({\omega }_1\right)$ with $\text{wt}\left(v_i\right)={\epsilon }_i\text{.}$ If we let the symmetric group $S_k$ act on $V^{\otimes k}$ by permuting the tensor factors then $${\left(V^{\otimes k}\right)}_{\lambda -\mu }=\text{span-}\{\pi ·v^{\otimes (\lambda -\mu )}\hspace{0.17em}|\hspace{0.17em}\pi \in S_k\}=\text{span-}\{\pi ·v^{\otimes (\lambda -\mu )}\hspace{0.17em}|\hspace{0.17em}\pi \in S_k/S_{\lambda -\mu }\},$$ where $$v^{\otimes (\lambda -\mu )}=\underset{\underset{{\lambda }_1-{\mu }_1}{⏟}}{v_1\otimes \cdots \otimes v_1}\otimes \cdots \otimes \underset{\underset{{\lambda }_n-{\mu }_n}{⏟}}{v_n\otimes \cdots \otimes v_n}$$ and $$S_{\lambda -\mu }=S_{{\lambda }_1-{\mu }_1}\times \cdots \times S_{{\lambda }_n-{\mu }_n}$$ is the parabolic subgroup of $S_k$ which stabilizes the vector $v^{\otimes (\lambda -\mu )}\in V^{\otimes k}\text{.}$ This shows that, as vector spaces, $$\begin{array}{cc}{ℳ}^{\lambda /\mu }\cong {\text{Ind}}_{{\stackrel{\sim }{H}}_{\lambda /\mu }}^{{\stackrel{\sim }{H}}_k}\left({ℂ}_{\lambda /\mu }\right)=\text{span-}\{T_{\pi }\otimes v_{\lambda /\mu }\hspace{0.17em}|\hspace{0.17em}\pi \in S_k/S_{\lambda -\mu }\}& \text{(6.23)}\end{array}$$ are isomorphic. For notational purposes let $$b_{\lambda /\mu }=v_{\mu }^{+}\otimes v^{\otimes (\lambda -\mu )}=v_{\mu }^{+}\otimes v_{i_1}\otimes \cdots \otimes v_{i_k}$$ and let ${\bar{b}}_{\lambda /\mu }$ be the image of $b_{\lambda /\mu }$ in ${(M\otimes V^{\otimes k})}^{\left[\lambda \right]}\text{.}$ Since $\lambda$ is integrally dominant and ${\bar{b}}_{\lambda /\mu }$ has weight $\lambda$ it must be a highest weight vector. We will show that $X^{{\epsilon }_{\ell }}$ acts on ${\bar{b}}_{\lambda /\mu }$ by the constant $q^{c\left({\text{box}}_{\ell }\right)},$ where $c\left({\text{box}}_{\ell }\right)$ is the content of the $\ell \text{th}$ box of the multisegment $\lambda /\mu$ (read left to right and top to bottom like a book). Consider the projections $${\text{pr}}_{\ell }:M\left(\mu \right)\otimes V^{\otimes k}\to {(M\left(\mu \right)\otimes V^{\otimes \ell })}^{\left[{\lambda }^{\left(\ell \right)}\right]}\otimes V^{\otimes (k-\ell )}$$ where $${\lambda }^{\left(\ell \right)}=\mu +\sum _{j\le \ell }\text{wt}\left(v_{i_{\ell }}\right)$$ and ${\text{pr}}_i$ acts as the identity on the last $k-i$ factors of $M\left(\mu \right)\otimes V^{\otimes k}\text{.}$ Then $${\bar{b}}_{\lambda /\mu }={\text{pr}}_k{\text{pr}}_{k-1}\cdots {\text{pr}}_1{b}_{\lambda /\mu },$$ and for each $1\le \ell \le k,$ (the first $\ell$ components of) ${\text{pr}}_{\ell -1}\cdots {\text{pr}}_1\left(b_{\lambda /\mu }\right)$ form a highest weight vector of weight ${\lambda }^{\left(\ell \right)}$ in $M\otimes V^{\otimes \ell }\text{.}$ It is the “highest” highest weight vector of $$\begin{array}{cc}{({(M\left(\mu \right)\otimes V^{\otimes (\ell -1)})}^{\left[{\lambda }^{(\ell -1)}\right]}\otimes V)}^{\left[{\lambda }^{\left(\ell \right)}\right]}& \text{(6.24)}\end{array}$$ with respect to the ordering in Lemma 4.1 and thus it is deepest in the filtration constructed there. Note that the quantum Casimir element acts on the space in (6.24) as the constant $q^{⟨{\lambda }^{\left(\ell \right)},{\lambda }^{\left(\ell \right)}+2\rho ⟩}$ times a unipotent transformation, and the unipotent transformation must preserve the filtration coming from Lemma 4.1. Since ${\text{pr}}_{\ell }\left(b_{\lambda /\mu }\right)$ is the highest weight vector of the smallest submodule of this filtration (which is isomorphic to a Verma module by Lemma 4.1b) it is an eigenvector for the action of the quantum Casimir. Thus, by (2.11) and (2.14), $X^{{\epsilon }_{\ell }}$ acts on ${\text{pr}}_{\ell }\left(b_{\lambda /\mu }\right)$ by the constant $$q^{⟨{\lambda }^{\left(\ell \right)},{\lambda }^{\left(\ell \right)}+2\rho ⟩-⟨{\lambda }^{(\ell -1)},{\lambda }^{(\ell -1)}+2\rho ⟩-⟨{\omega }_1,{\omega }_1+2\rho ⟩}=q^{2c\left({\text{box}}_{\ell }\right)}\text{.}$$ (see [LRa1977]). Since $X^{{\epsilon }_{\ell }}$ commutes with ${\text{pr}}_j$ for $j>\ell$ this also specifies the action of $X^{{\epsilon }_{\ell }}$ on ${\bar{b}}_{\lambda /\mu }={\text{pr}}_{\ell }\left(b_{\lambda /\mu }\right)\text{.}$ The explicit $R\text{-matrix}$ ${Ř}_{VV}:V\otimes V\to V\otimes V$ for this case $\text{(}𝔤$ of type $A$ and $V=L\left({\omega }_1\right)\text{)}$ is well known (see, for example, the proof of [LRa1977, Prop. 4.4]) and given by $$(v_i\otimes v_j)q^{1/(n+1)}{Ř}_{VV}=\{\begin{array}{cc}{v}_j\otimes v_i,& \text{if}\hspace{0.17em}i>j,\\ (q-q^{-1})v_i\otimes v_j+v_j\otimes v_i,& \text{if}\hspace{0.17em}i<j,\\ q{v}_i\otimes v_j,& \text{if}\hspace{0.17em}i=j\text{.}\end{array}$$ Since $T_i$ acts by ${Ř}_{VV}$ on the $i\text{th}$ and $(i+1)\text{st}$ tensor factors of $V^{\otimes k}$ and commutes with the projection ${\text{pr}}_{\lambda }$ it follows that $T_j\left({\bar{b}}_{\lambda /\mu }\right)=q{\bar{b}}_{\lambda /\mu },$ if ${\text{box}}_j$ is not a box at the end of a row of $\lambda /\mu \text{.}$ This analysis of the action of ${\stackrel{\sim }{H}}_{\lambda /\mu }$ on ${\bar{b}}_{\lambda /\mu }$ shows that there is an ${\stackrel{\sim }{H}}_k\text{-homomorphism}$ $$\begin{array}{ccc}{\text{Ind}}_{{\stackrel{\sim }{H}}_{\lambda /\mu }}^{{\stackrel{\sim }{H}}_k}\left(ℂv_{\lambda /\mu }\right)& ⟶& {ℳ}^{\lambda /\mu }\\ v_{\lambda /\mu }& ⟼& {\bar{b}}_{\lambda /\mu }\text{.}\end{array}$$ This map is surjective since ${ℳ}^{\lambda /\mu }$ is generated by ${\bar{b}}_{\lambda /\mu }$ (the ${ℬ}_k$ action on $v^{\lambda -\mu }$ generates all of ${\left(V^{\otimes k}\right)}_{\lambda -\mu }\text{).}$ Finally, (6.23) guarantees that it is an isomorphism. $\square$

In the same way that each weight $\mu \in {𝔥}^{*}$ has a normal form $$\mu =w\circ \stackrel{\sim }{\mu },\text{with}\begin{array}{c}\stackrel{\sim }{\mu }\hspace{0.17em}\text{integrally dominant,}\text{and}\\ w\hspace{0.17em}\text{maximal length in the coset}\hspace{0.17em}w{W}_{\stackrel{\sim }{\mu }+\rho },\end{array}$$ every multisegment $\lambda /\mu$ has a normal form $$\lambda /\mu =\nu /(w\circ \stackrel{\sim }{\nu }),\text{with}\begin{array}{c}\nu +\rho \hspace{0.17em}\text{the sequence of contents of boxes of}\\ \lambda /\mu ,\hspace{0.17em}\stackrel{\sim }{\nu }=\nu -(1,1,\dots ,1),\hspace{0.17em}\text{and}\hspace{0.17em}w\\ \text{maximal length in}\hspace{0.17em}{W}_{\nu +\rho }w{W}_{\nu +\rho }\text{.}\end{array}$$ The element $w$ in the normal form $\nu /(w\circ \stackrel{\sim }{\nu })$ of $\lambda /\mu$ can be constructed combinatorially by the following scheme. We number (order) the boxes of $\lambda /\mu$ in two different ways.

First ordering: To each box $b$ of $\lambda /\mu$ associate the following triple $$(\text{content of the box to the left of}\hspace{0.17em}b,-\left(\text{content of}\hspace{0.17em}b\right),-\left(\text{row number of}\hspace{0.17em}b\right))$$ where, if a box is the leftmost box in a row “the box to its left” is the rightmost box in the same row. The lexicographic ordering on these triples induces an ordering on the boxes of $\lambda /\mu \text{.}$

Second ordering: To each box $b$ of $\lambda /\mu$ associate the following pair $$(\text{content of}\hspace{0.17em}b,-\left(\text{the number of box}\hspace{0.17em}b\hspace{0.17em}\text{in the first ordering}\right))$$ The lexicographic ordering of these pairs induces a second ordering on the boxes of $\lambda /\mu \text{.}$

If $v$ is the permutation defined by these two numberings of the boxes then $w=w_{0}v{w}_0\text{.}$ For example, for the multisegment $\lambda /\mu$ displayed in (6.20) the numberings of the boxes are given by $$\begin{array}{ccc}\begin{array}{c} 21 6 10 13 18 20 5 9 12 17 19 11 16 15 1 2 4 8 14 3 7 \end{array}& \text{and}& \begin{array}{c} 3 7 12 16 19 4 8 13 17 20 11 18 21 1 2 6 9 14 5 10 15 \end{array}\\ \text{first ordering of boxes}& & \text{second ordering of boxes}\end{array}$$ and the normal form of $\lambda /\mu$ is $$\begin{array}{ccc}\nu & =& (7,7,7,6,6,6,5,5,5,5,5,4,4,4,4,3,3,3,3,2,1),\\ \stackrel{\sim }{\nu }& =& (6,6,6,5,5,5,4,4,4,4,4,3,3,3,3,2,2,2,2,1,0),\end{array}$$ and $w=w_{0}v{w}_0$ where $$v=\left(\begin{array}{ccccccccccccccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15& 16& 17& 18& 19& 20& 21\\ 15& 1& 21& 20& 14& 2& 6& 5& 4& 3& 19& 10& 9& 8& 7& 13& 12& 11& 18& 17& 16\end{array}\right)$$ Let $𝔤$ be of type $A_n$ and $V=L\left({\omega }_1\right)$ and let $$\begin{array}{cc}{ℒ}^{\lambda /\mu }=F_{\lambda }\left(L\left(\mu \right)\right),& \text{(6.25)}\end{array}$$ as defined in (4.1). It is known (a consequence of Proposition 6.5 and Proposition 4.2c) that ${ℒ}^{\lambda /\mu }$ is always a simple ${\stackrel{\sim }{H}}_k\text{-module}$ or $0\text{.}$ Furthermore, all simple ${\stackrel{\sim }{H}}_k$ modules are obtained by this construction. See [Suz1998] for proofs of these statements. The following theorem is a reformulation of Proposition 4.5 in terms of the combinatorics of our present setting.

Let $\lambda /\mu$ and $\rho /\tau$ be multisegments with $k$ boxes (with $\mu$ and $\tau$ assumed to be integral) and let $$\lambda /\mu =\nu /(w\circ \stackrel{\sim }{\nu })\text{and}\rho /\tau =\gamma /(v\circ \stackrel{\sim }{\gamma })$$ be their normal forms. Then the multiplicities of ${ℒ}^{\rho /\tau }$ in a Jantzen filtration of ${ℳ}^{\lambda /\mu }$ are given by $$\sum _{j\ge 0}[\frac{{\left({ℳ}^{\lambda /\mu }\right)}^{\left(j\right)}}{{\left({ℳ}^{\lambda /\mu }\right)}^{(j+1)}}:{ℒ}^{\rho /\tau }]v^{\frac{1}{2}(\ell \left(y\right)-\ell \left(w\right)+j)}=\{\begin{array}{cc}{P}_{wv}\left(v\right),& \text{if}\hspace{0.17em}\nu =\gamma ,\\ 0,& \text{if}\hspace{0.17em}\nu \ne \gamma ,\end{array}$$ where $P_{wv}\left(v\right)$ is the Kazhdan-Lusztig polynomial for the symmetric group $S_k\text{.}$

Theorem 6.6 says that every decomposition number for affine Hecke algebra representations is a Kazhdan-Lusztig polynomial. The following is a converse statement which says that every Kazhdan-Lusztig polynomial for the symmetric group is a decomposition number for affine Hecke algebra representations. This statement is interesting in that Polo [Pol1999] has shown that every polynomial in $1+v{\mathbb{Z}}_{\ge 0}\left[v\right]$ is a Kazhdan-Lusztig polynomial for some choice of $n$ and permutations $v,w\in S_n\text{.}$ Thus, the following proposition also shows that every polynomial arises as a generalized decomposition number for an appropriate pair of affine Hecke algebra modules.

Let $\lambda =(r,r,\dots ,r)=\left(r^r\right)$ and $\mu =(0,0,\dots ,0)=\left(0^r\right)\text{.}$ Then, for each pair of permutations $v,w\in S_r,$ the Kazhdan-Lusztig polynomial $P_{vw}\left(v\right)$ for the symmetric group $S_r$ is equal to $$P_{vw}\left(v\right)=\sum _{j\ge 0}[\frac{{\left({ℳ}^{\lambda /w\circ \mu }\right)}^{\left(j\right)}}{{\left({ℳ}^{\lambda /w\circ \mu }\right)}^{(j+1)}}:{ℒ}^{\lambda /v\circ \mu }]v^{\frac{1}{2}(\ell \left(y\right)-\ell \left(w\right)+j)}\text{.}$$

		Proof.
	Since $\mu +\rho$ and $\lambda +\rho$ are both regular, $W_{\lambda +\rho }=W_{\mu +\rho }=1$ and the standard and irreducible modules ${ℒ}^{\lambda /(w\circ \mu )}$ and ${ℳ}^{\lambda /(v\circ \mu )}$ range over all $v,w\in S_k\text{.}$ Thus, this statement is a corollary of Proposition 4.5. $\square$

Notes and references

This is a typed version of the paper Affine Braids, Markov Traces and the Category $𝒪$ by Rosa Orellana and Arun Ram*.

*Research supported in part by National Science Foundation grant DMS-9971099, the National Security Agency and EPSRC grant GR K99015.

This paper is a slightly revised version of a preprint of 2001. We thank F. Goodman, A. Henderson and an anonymous referee for their very helpful comments on the original preprint.

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