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

The ${\stackrel{\sim }{ℬ}}_k$ Modules ${ℳ}^{\lambda /\mu }$ and ${ℒ}^{\lambda /\mu }$

Let $\lambda$ be integrally dominant and let $\mu \in {𝔥}^{*}\text{.}$ Define ${\stackrel{\sim }{ℬ}}_k$ modules $$\begin{array}{cc}{ℳ}^{\lambda /\mu }=F_{\lambda }\left(M\left(\mu \right)\right)\text{and}{ℒ}^{\lambda /\mu }=F_{\lambda }\left(L\left(\mu \right)\right)\text{.}& \text{(4.1)}\end{array}$$ The following lemma is the main tool for studying the structure of these ${\stackrel{\sim }{ℬ}}_k$ modules.

([Jan1980, Theorem 2.2], [Dix1996, Lemma 7.6.14]) Let $E$ be a finite dimensional $U_h𝔤$ module and let $\left\{e_i\right\}$ be a basis of $E$ consisting of weight vectors ordered so that $i<j$ if $\text{wt}\left(e_i\right)<\text{wt}\left(e_j\right)\text{.}$ Suppose $M$ is a $U_h𝔤$ module generated by a highest weight vector $v_{\mu }^{+}$ of weight $\mu \text{.}$ Set $$M_i=\sum _{j\ge i}{U}_h{𝔫}^{-}(v_{\mu }^{+}\otimes e_j)\text{.}$$ Then

(a)	$M\otimes E=M_1\supseteq M_2\supseteq \cdots$ is a filtration of $U_h𝔤$ modules such that $M_i/M_{i+1}$ is $0$ or is a highest weight module of highest weight $\mu +\text{wt}\left(e_i\right)\text{.}$
(b)	If $M=M\left(\mu \right)$ then $M_i/M_{i+1}\cong M(\mu +\text{wt}\left(e_i\right))\text{.}$

The braid group ${ℬ}_k$ is the subgroup of ${\stackrel{\sim }{ℬ}}_k$ generated by $T_1,\dots ,T_{k-1}\text{.}$ By restriction, both ${ℳ}^{\lambda /\mu }$ and $V^{\otimes k}=L\left(0\right)\otimes V^{\otimes k}$ are ${ℬ}_k$ modules.

There is a unique $U_h𝔤$ contravariant form ${⟨,⟩}_M$ on the Verma module $M(w\circ \mu )$ determined by ${⟨v_{w\circ \mu }^{+},v_{w\circ \mu }^{+}⟩}_M=1$ where $v_{w\circ \mu }^{+}$ is the generating highest weight vector of $M(w\circ \mu )\text{.}$ As in (2.15), this form together with a nondegenerate $U_h𝔤$ contravariant form ${⟨,⟩}_V$ on $V$ gives $U_h𝔤$ contravariant forms ${⟨,⟩}_{V^{\otimes k}}$ and $⟨,⟩$ on $V^{\otimes k}$ and $M(w\circ \mu )\otimes V^{\otimes k},$ respectively.

With these notations at hand we use Lemma 4.1 to prove the fundamental facts about the ${\stackrel{\sim }{ℬ}}_k$ modules ${ℳ}^{\lambda /\mu }$ and ${ℒ}^{\lambda /\mu }$ defined in (4.1).

Let $\lambda ,$ $\mu$ be integrally dominant weights and $w\in W^{\mu }\text{.}$

(a)	As ${ℬ}_k$ modules, ${ℳ}^{\lambda /w\circ \mu }\cong {\left(V^{\otimes k}\right)}_{\lambda -w\circ \mu }$
(b)	${ℳ}^{\lambda /w\circ \mu }\cong {ℳ}^{\lambda /y\circ \mu }$ if $W_{\lambda +\rho }w{W}_{\mu +\rho }=W_{\lambda +\rho }y{W}_{\mu +\rho }\text{.}$
(c)	Use the same notation $⟨,⟩$ for the $U_h𝔤$ contravariant form $⟨,⟩$ on $M(w\circ \mu )\otimes V^{\otimes k}$ and the ${\stackrel{\sim }{ℬ}}_k$ contravariant form on ${ℳ}^{\lambda /(w\circ \mu )}$ obtained by restriction of $⟨,⟩$ to the subspace ${(M(w\circ \mu )\otimes V^{\otimes k})}_{\lambda }^{\left[\lambda \right]}\text{.}$ Then $${ℒ}^{\lambda /w\circ \mu }\cong \frac{{ℳ}^{\lambda /(w\circ \mu )}}{\text{rad}⟨,⟩}\text{.}$$
(d)	Assume $w\in W^{\mu }$ is maximal length in the coset $w{W}_{\mu +\rho }$ in $W^{\mu }\text{.}$ If ${ℒ}^{\lambda (w\circ \mu )}\ne 0$ then (1) $\lambda -w\circ \mu$ is a weight of $V^{\otimes k},$ (2) $w$ is maximal length in $W_{\lambda +\rho }w{W}_{\mu +\rho }\text{.}$
(e)	If $\mu$ is a dominant integral weight then $${ℒ}^{\lambda /\mu }\cong \{v\in {\left(V^{\otimes k}\right)}_{\lambda -\mu }\hspace{0.17em}|\hspace{0.17em}{X}_i^{⟨\mu +\rho ,{\alpha }_i^{\vee }⟩}v=0,\hspace{0.17em}\text{for all}\hspace{0.17em}1\le i\le n\text{.}\}\text{.}$$

		Proof.
	(a) Let $v_{w\circ \mu }^{+}$ be the generating highest weight vector of $M(w\circ \mu )$ and, for $n\in V^{\otimes k}$ let $pr(v_{w\circ \mu }^{+}\otimes n)$ be the image of $v_{w\circ \mu }^{+}\otimes n$ in $(M\otimes V^{\otimes k})/{𝔫}^{-}(M\otimes V^{\otimes k})\text{.}$ Then, since $\lambda$ is integrally dominant, Lemma 4.1 shows that $$\begin{array}{cc}\begin{array}{ccc}{\left(V^{\otimes k}\right)}_{\lambda -w\circ \mu }& ⟶& {ℳ}^{\lambda /(w\circ \mu )}\\ n& ⟼& pr(v_{w\circ \mu }^{+}\otimes n)\end{array}& \text{(4.2)}\end{array}$$ is a vector space isomorphism. This is a ${ℬ}_k\text{-module}$ isomorphism since the ${ℬ}_k$ action on $M(w\circ \mu )\otimes V^{\otimes k}$ commutes with ${𝔫}^{-}$ and fixes $v_{w\circ \mu }^{+}\text{.}$ (b) By (a), $\lambda -w\circ \mu \in P$ and so $W^{\lambda }=W^{\mu }\text{.}$ It is sufficient to show that ${ℳ}^{\lambda /w\circ \mu }\cong {ℳ}^{\lambda /(s_{i}w\circ \mu )}$ for all simple reflections of $W^{\lambda }$ such that $s_i\in W_{\lambda +\rho }$ and $s_{i}w>w\text{.}$ Applying the exact functor $F_{\lambda }$ to the Verma module inclusion $$M(s_{i}w\circ \mu )\hookrightarrow M(w\circ \mu )\text{gives}{ℳ}^{\lambda /s_{i}w\circ \mu }\hookrightarrow {ℳ}^{\lambda /w\circ \mu },$$ an inclusion of ${\stackrel{\sim }{ℬ}}_k\text{-modules.}$ Since $s_i(\lambda -w\circ \mu )=s_i(\lambda +\rho )-s_{i}w(\mu +\rho )=\lambda +\rho -s_{i}w(\mu +\rho )=\lambda -\left(s_{i}w\right)\circ \mu$ there is a (vector space) isomorphism of weight spaces $${\left(V^{\otimes k}\right)}_{\lambda -w\circ \mu }\cong V_{\lambda -s_{i}w\circ \mu }^{\otimes k}\text{.}$$ (This isomorphism can be realized by Lusztig’s braid group action [CPr1994, §8.1-8.2], $T_i:{\left(V^{\otimes k}\right)}_{\lambda -w\circ \mu }\to {\left(V^{\otimes k}\right)}_{s_i(\lambda -w\circ \mu )}\text{).}$ Thus, by part (a), the ${\stackrel{\sim }{ℬ}}_k\text{-module}$ inclusion ${ℳ}^{\lambda /s_{i}w\circ \mu }\hookrightarrow {ℳ}^{\lambda /w\circ \mu }$ is an isomorphism. (c) Use the notations for the bilinear forms on $M(w\circ \mu )$ and $V^{\otimes k}$ as given in the paragraph before the statement of the proposition. Let $\left\{b_i\right\}$ be an orthonormal basis of $V^{\otimes k}$ with respect to ${⟨,⟩}_{V^{\otimes k}}\text{.}$ If $r\in \text{rad}{⟨,⟩}_M$ then $$⟨r\otimes b,s\otimes b\prime ⟩={⟨r,s⟩}_M{⟨b,b\prime ⟩}_{V^{\otimes k}}=0,\hspace{0.17em}\text{for all}\hspace{0.17em}s\in M(w\circ \mu ),\hspace{0.17em}b,b\prime \in V^{\otimes k},$$ and so $\left(\text{rad}{⟨,⟩}_M\right)V^{\otimes k}\subseteq \text{rad}⟨,⟩\text{.}$ Conversely, if $r_i\in M(w\circ \mu )$ such that $\sum r_i\otimes b_i\in \text{rad}⟨,⟩$ then $$0=⟨\sum _i{r}_i\otimes b_i,s\otimes b_j⟩=\sum _i{⟨r_i,s⟩}_M{\delta }_{ij}=⟨r_i,s⟩,\hspace{0.17em}\text{for all}\hspace{0.17em}s\in M(w\circ \mu )\text{.}$$ So $r_i\in \text{rad}{⟨,⟩}_M$ and thus $\text{rad}⟨,⟩\subseteq \text{rad}{⟨,⟩}_M\otimes V^{\otimes k}\text{.}$ By the $U_h𝔤$ contravariance of $⟨,⟩$ $${⟨M(w\circ \mu )\otimes V^{\otimes k}⟩}_{\lambda }^{\left[\lambda \right]}\perp {⟨M(w\circ \mu )\otimes V^{\otimes k}⟩}_{\mu }^{\left[\mu \right]}$$ for integrally dominant weights $\lambda ,$ $\mu$ with $\lambda \ne \mu \text{.}$ Thus $$\begin{array}{cc}\text{rad}⟨,⟩={(\text{rad}{⟨,⟩}_M\otimes V^{\otimes k})}_{\lambda }^{\left[\lambda \right]},& \text{(4.3)}\end{array}$$ where $⟨,⟩$ is the restriction of the form on $M(w\circ \mu )\otimes V^{\otimes k}$ to ${(M(w\circ \mu )\otimes V^{\otimes k})}_{\lambda }^{\left[\lambda \right]}\text{.}$ Thus $$\begin{array}{ccc}\frac{{(M(w\circ \mu )\otimes V^{\otimes k})}_{\lambda }^{\left[\lambda \right]}}{\text{rad}⟨,⟩}& =& \frac{{(M(w\circ \mu )\otimes V^{\otimes k})}_{\lambda }^{\left[\lambda \right]}}{{(\text{rad}{⟨,⟩}_M\otimes V^{\otimes k})}_{\lambda }^{\left[\lambda \right]}}\\ & \cong & {(\frac{M(w\circ \mu )}{\text{rad}{⟨,⟩}_M}\otimes V^{\otimes k})}_{\lambda }^{\left[\lambda \right]}\\ & =& {(L(w\circ \mu )\otimes V^{\otimes k})}_{\lambda }^{\left[\lambda \right]}\\ & =& {ℒ}^{\lambda /(w\circ \mu )},\end{array}$$ where the isomorphism is a consequence of the fact that, because $\lambda$ is an integrally dominant weight, the functor ${(·\otimes V^{\otimes k})}_{\lambda }^{\left[\lambda \right]}$ is exact (3.8). (d) If $\lambda -w\circ \mu$ is not a weight of $V^{\otimes k}$ then, by part (a), ${ℳ}^{\lambda /w\circ \mu }=0\text{.}$ Since the functor $F_{\lambda }$ is exact and $L(w\circ \mu )$ is a quotient of $M(w\circ \mu ),$ ${ℒ}^{\lambda /w\circ \mu }$ is a quotient of ${ℳ}^{\lambda /w\circ \mu }\text{.}$ Thus ${ℳ}^{\lambda /w\circ \mu }=0$ implies ${ℒ}^{\lambda /w\circ \mu }=0\text{.}$ Assume that $w\in W^{\mu }$ is not the longest element of the double coset $W_{\lambda +\rho }w{W}_{\mu +\rho }\in W^{\mu }\text{.}$ Then there is a positive root $\alpha >0$ such that $s_{\alpha }\in W_{\lambda +\rho }$ and $s_{\alpha }w>w\text{.}$ Since $s_{\alpha }w{W}_{\mu +\rho }\ne w{W}_{\mu +\rho }$ there is an inclusion of Verma modules $M(s_{\alpha }w\circ \mu )\subseteq M(w\circ \mu )$ and $F_{\lambda }\left(L(w\circ \mu )\right)$ is a quotient of $F_{\lambda }\left(M(w\circ \mu )\right)/F_{\lambda }\left(M(s_{\alpha }w\circ \mu )\right)\text{.}$ On the other hand, by part (b), $${ℳ}^{\lambda /s_{\alpha }w\circ \mu }\cong {ℳ}^{\lambda /w\circ \mu },\text{and so}\frac{{ℳ}^{\lambda /w\circ \mu }}{{ℳ}^{\lambda /s_{\alpha }w\circ \mu }}=\frac{F_{\lambda }\left(M(w\circ \mu )\right)}{F_{\lambda }\left(M(s_{\alpha }w\circ \mu )\right)}=0\text{.}$$ Thus $F_{\lambda }\left(L(w\circ \mu )\right)=0\text{.}$ (e) When $\mu$ is a dominant integral weight $${ℒ}^{\lambda /\mu }\cong {\left(\frac{\text{span-}\left\{pr(v_{\mu }^{+}\otimes n)\hspace{0.17em}\right|\hspace{0.17em}n\in V^{\otimes k}\}}{\text{span-}\left\{pr(Y_i^{⟨\mu +\rho ,{\alpha }_i^{\vee }⟩}{v}_{\mu }^{+}\otimes n)\hspace{0.17em}\right|\hspace{0.17em}n\in V^{\otimes k}\}}\right)}_{\lambda }\text{.}$$ see [Dix1996, 7.2.7]. Thus, by (c) and the vector space isomorphism (4.2) it follows that, as vector spaces, $${ℒ}^{\lambda /\mu }\cong {(\left(\text{span-}\left\{pr(v_{\mu }^{+}\otimes n)\hspace{0.17em}\right|\hspace{0.17em}n\in V^{\otimes k}\}\right)/\left(\text{span-}\left\{pr(Y_i^{⟨\mu +\rho ,{\alpha }_i^{\vee }⟩}{v}_{\mu }^{+}\otimes n)\hspace{0.17em}\right|\hspace{0.17em}n\in V^{\otimes k}\}\right))}_{\lambda }\text{.}$$ For any $k\ge 0,$ there is a nonzero constant $c$ such that $pr(Y_i^{k+1}{v}_{\mu }^{+}\otimes n)=c·pr(Y_i(Y_i^k{v}_{\mu }^{+}\otimes n)-Y_i^k{v}_{\mu }^{+}\otimes Y_{i}n)=-c\hspace{0.17em}pr(Y_i^k{v}_{\mu }^{+}\otimes Y_{i}n),$ and so, by induction, $pr(Y_i^{k+1}{v}_{\mu }^{+}\otimes n)=\xi ·pr(v_{\mu }^{+}\otimes Y_i^{k+1}n)$ for some constant $\xi \ne 0\text{.}$ Thus ${ℒ}^{\lambda /\mu }$ is isomorphic to the vector space $${(V^{\otimes k}/\left(\sum _i{Y}_i^{⟨\mu +\rho ,{\alpha }_i^{\vee }⟩}{V}^{\otimes k}\right))}_{\lambda -\mu }\cong {\left(\sum _i{Y}_i^{⟨\mu +\rho ,{\alpha }_i^{\vee }⟩}{V}^{\otimes k}\right)}_{\lambda -\mu }^{\perp }\text{.}$$ If $v\in {\left(Y_i^{⟨\mu +\rho ,{\alpha }_i^{\vee }⟩}{V}^{\otimes k}\right)}^{\perp }$ then the $U_h𝔤$ contravariance of ${⟨,⟩}_{V^{\otimes k}}$ gives that $$0={⟨Y_i^{⟨\mu +\rho ,{\alpha }_i^{\vee }⟩}n,b⟩}_{V^{\otimes k}}={⟨v,X_i^{⟨\mu +\rho ,{\alpha }_i^{\vee }⟩}b⟩}_{V^{\otimes k}},\text{for all}\hspace{0.17em}n\in V^{\otimes k}\text{.}$$ Thus, by the nondegeneracy of ${⟨,⟩}_{V^{\otimes k}},$ $${ℒ}^{\lambda /\mu }\cong {\left(\sum _i{Y}_i^{⟨\mu +\rho ,{\alpha }_i^{\vee }⟩}{V}^{\otimes k}\right)}_{\lambda -\mu }^{\perp }=\{b\in {\left(V^{\otimes k}\right)}_{\lambda -\mu }\hspace{0.17em}|\hspace{0.17em}{X}_i^{⟨\mu +\rho ,{\alpha }_i^{\vee }⟩}b=0\}\text{.}$$ $\square$

Proposition 4.2d gives a necessary condition on $\lambda /w\circ \mu$ for the ${\stackrel{\sim }{ℬ}}_k\text{-module}$ ${ℒ}^{\lambda /w\circ \mu }$ to be nonzero. The following will be useful for analyzing the combinatorics of the examples in Section 6. If $P$ denotes the weight lattice $\lambda$ is an integrally dominant then the action of $W_{\lambda +\rho }$ on $\lambda -P$ by the dot action has fundamental domain $$C_{\lambda +\rho }^{-}=\{\mu \in \lambda -P\hspace{0.17em}|\hspace{0.17em}⟨\mu +\rho ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{\le 0},\hspace{0.17em}\text{for all}\hspace{0.17em}\alpha >0\hspace{0.17em}\text{such that}\hspace{0.17em}⟨\lambda +\rho ,{\alpha }_i^{\vee }⟩=0\}\text{.}$$ and the following are equivalent:

(a)	$\mu \in C_{\lambda +\rho }^{-},$
(b)	$\mu =w^{\lambda }\circ \nu$ with $\nu$ integrally dominant and $w^{\lambda }\in W^{\lambda }$ longest in the coset $W_{\lambda +\rho }{w}^{\lambda }$ in $W^{\lambda },$
(c)	$\mu =w_{\stackrel{\sim }{\mu }}^{\lambda }\circ \stackrel{\sim }{\mu }$ with $w_{\stackrel{\sim }{\mu }}^{\lambda }\in W^{\lambda }$ longest in the double coset $W_{\lambda +\rho }{w}_{\stackrel{\sim }{\mu }}^{\lambda }{W}_{\mu +\rho }$ in $W^{\lambda }\text{.}$

In the classical case, when $𝔤$ is type $A_n$ and $V=L\left({\omega }_1\right)$ is the $n+1$ dimensional fundamental representation the ${\stackrel{\sim }{ℬ}}_k\text{-module}$ ${ℒ}^{\lambda /w\circ \mu }$ is a simple ${\stackrel{\sim }{ℬ}}_k\text{-module}$ whenever it is nonzero (see [Suz1998]). As the following Proposition shows, this is a very special phenomenon.

Assume that $V=L\left(\nu \right)$ for a dominant integral weight $\nu \text{.}$ If the ${\stackrel{\sim }{ℬ}}_k\text{-module}$ ${ℒ}^{\lambda /\mu }$ is irreducible (or $0\text{)}$ for all $k,$ all dominant integral weights $\mu ,$ and all integrally dominant weights $\lambda$ then

(a)	$𝔤$ is type $A_n,$ $B_n,$ $C_n$ or $G_2$ and $V=L\left({\omega }_1\right),$ and
(b)	the action of the subgroup ${ℬ}_k$ of ${\stackrel{\sim }{ℬ}}_k$ generates ${\text{End}}_{U_h𝔤}\left(V^{\otimes k}\right)\text{.}$

		Proof.
	(a) If $\mu$ is large dominant integral weight (for example, we may take $\mu =n\rho ,$ $n\gg 0\text{)}$ then, as a $U_h𝔤\text{-module,}$ $$L\left(\mu \right)\otimes V\cong \underset{b}{⨁}L(\mu +\text{wt}\left(b\right)),$$ where the sum is over a basis of $V$ consisting of weight vectors and $\text{wt}\left(b\right)$ is the weight of the vector $b\text{.}$ The group ${\stackrel{\sim }{ℬ}}_1$ is generated by the element $X^{{\epsilon }_1}$ which acts on a summand $L\left(\lambda \right)$ in $L\left(\mu \right)\otimes V$ by the constant $q^{⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩}\text{.}$ Then $F_{\lambda }\left(L\left(\mu \right)\right)$ is the $L\left(\lambda \right)\text{-isotypic}$ component of $L\left(\mu \right)\otimes V$ and these are simple ${\stackrel{\sim }{ℬ}}_1$ modules (for the various $\lambda \text{)}$ only if all the values $$\begin{array}{cc}⟨\mu +\text{wt}\left(b\right),\mu +\text{wt}\left(b\right)+2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩=2⟨\mu +\rho ,\text{wt}\left(b\right)⟩+⟨\text{wt}\left(b\right),\text{wt}\left(b\right)⟩-⟨\nu ,\nu +2\rho ⟩,& \text{(4.4)}\end{array}$$ as $b$ ranges over a weight basis of $V,$ are distinct. It follows that all weight spaces of $V$ must be one dimensional. This means that (a) $𝔤$ is type $A_n,$ $B_n,$ $C_n,$ $D_n,$ $E_6,$ $E_7$ or $G_2$ and $V=L\left({\omega }_1\right),$ or (b) $𝔤$ is type $A_n$ and $V=L\left(k{\omega }_1\right)$ or $V=\left(k{\omega }_n\right)$ for some $k,$ or (c) $𝔤$ is type $B_n$ and $V=L\left({\omega }_n\right),$ or (d) $𝔤$ is type $D_n$ and $V=L\left({\omega }_{n-1}\right)$ or $V=L\left({\omega }_n\right)\text{.}$ Most of the weights of these representations lie in a single $W\text{-orbit.}$ If $\gamma$ and $\gamma \prime$ are two distinct weights of $V$ which are in the same $W\text{-orbit}$ then $⟨\gamma ,\gamma ⟩=⟨\gamma \prime ,\gamma \prime ⟩\text{.}$ If $\mu =n\rho$ with $n\gg 0$ then the condition that all the values in (4.4) be distinct forces that $$2(n+1)⟨\rho ,\gamma ⟩=2⟨\mu +\rho ,\gamma ⟩\hspace{0.17em}\hspace{0.17em}\ne \hspace{0.17em}\hspace{0.17em}2⟨\mu +\rho ,\gamma \prime ⟩=2(n+1)⟨\rho ,\gamma \prime ⟩\text{.}$$ Writing $\gamma =\nu -\sum _i{c}_i{\alpha }_i$ and $\gamma \prime =\nu -\sum _i{c}_i^{\prime }{\alpha }_i$ with $c_i,c_i^{\prime }\in {\mathbb{Z}}_{>0}$ the last equation becomes $$2(n+1)·\sum _i{c}_i\ne 2(n+1)·\sum _i{c}_i^{\prime }\text{.}$$ Finally, an easy case by case check verifies that the only choices of $V$ in (a-d) above which satisfy this last condition for all weights in the $W\text{-orbit}$ of the highest weight are those listed in the statement of the proposition. (b) Let ${𝒵}_k={\text{End}}_{U_h𝔤}\left(V^{\otimes k}\right)\text{.}$ As a $(U_h𝔤,{𝒵}_k)\text{-bimodule}$ $$V^{\otimes k}\cong \underset{\lambda }{⨁}L\left(\lambda \right)\otimes {𝒵}_k^{\lambda },$$ where ${𝒵}_k^{\lambda }$ is an irreducible ${𝒵}_k\text{-module}$ and the sum is over all dominant integral weights for which the irreducible $U_h𝔤\text{-module}$ $L\left(\lambda \right)$ appears in $V^{\otimes k}\text{.}$ By restriction ${𝒵}_k^{\lambda }$ is an ${ℬ}_k\text{-module}$ and this is the ${\stackrel{\sim }{ℬ}}_k\text{-module}$ $F_{\lambda }\left(L\left(0\right)\right)$ which, by assumption, is simple. Since $L\left(0\right)$ is the trivial module $X^{{\epsilon }_1}$ acts on $F_{\lambda }\left(L\left(0\right)\right)$ by the identity and so $F_{\lambda }\left(L\left(0\right)\right)$ is simple as a ${ℬ}_k\text{-module}$ Thus the simple ${𝒵}_k\text{-modules}$ in $V^{\otimes k}$ coincide exactly with the simple ${ℬ}_k\text{-modules}$ in $V^{\otimes k}$ and it follows that ${ℬ}_k$ generates ${𝒵}_k={\text{End}}_{U_h𝔤}\left(V^{\otimes k}\right)\text{.}$ $\square$

Jantzen filtrations for affine braid group representations

Applying the functor $F_{\lambda }$ to the Jantzen filtration of $M\left(\mu \right)$ produces a filtration of ${ℳ}^{\lambda /\mu },$ $$\begin{array}{cc}{ℳ}^{\lambda /\mu }=F_{\lambda }\left(M\left(\mu \right)\right)=F_{\lambda }\left({M\left(\mu \right)}^{\left(0\right)}\right)\supseteq F_{\lambda }\left({M\left(\mu \right)}^{\left(1\right)}\right)\supseteq \cdots \text{.}& \text{(4.5)}\end{array}$$ An argument of Suzuki [Suz1998, Thm. 4.3.5], shows that this filtration can be obtained directly from the ${\stackrel{\sim }{ℬ}}_k\text{-contravariant}$ form ${⟨,⟩}_t$ on $${ℳ}^{\lambda +t\delta /\mu +t\delta }=F_{\lambda +t\delta }\left(M(\mu +t\delta )\right)={(M(\mu +t\delta )\otimes V^{\otimes k})}_{\lambda +t\delta }^{[\lambda +t\delta ]}$$ which is the restriction of the $U_h𝔤$ contravariant form ${⟨,⟩}_t$ on $(M(\mu +t\delta )\otimes V^{\otimes k}),$ see (2.5) and (3.5)). To do this define $${ℳ}^{\lambda +t\delta /\mu +t\delta }\left(j\right)=\{m\in {ℳ}^{(\lambda +t\delta )/\mu +t\delta }\hspace{0.17em}|\hspace{0.17em}{⟨m,n⟩}_t\in t^jℂ\left[t\right]\hspace{0.17em}\text{for all}\hspace{0.17em}n\in {ℳ}^{\lambda +t\delta /\mu +t\delta }\}$$ and $${\left({ℳ}^{\lambda /\mu }\right)}^{\left(j\right)}=\text{image of}\hspace{0.17em}{ℳ}^{\lambda +t\delta /\mu +t\delta }\left(j\right)\hspace{0.17em}\text{in}\hspace{0.17em}{ℳ}^{\lambda +t\delta /\mu +t\delta }{\otimes }_{ℂ\left[t\right]}ℂ\left[t\right]/tℂ\left[t\right]$$ to obtain a filtration $$\begin{array}{cc}{ℳ}^{\lambda /\mu }={\left({ℳ}^{\lambda /\mu }\right)}^{\left(0\right)}\supseteq {\left({ℳ}^{\lambda /\mu }\right)}^{\left(1\right)}\supseteq \cdots & \text{(4.6)}\end{array}$$ such that the quotients ${\left({ℳ}^{\lambda /\mu }\right)}^{\left(j\right)}/{\left({ℳ}^{\lambda /\mu }\right)}^{(j+1)}$ carry nondegenerate ${\stackrel{\sim }{ℬ}}_k$ contravariant forms. Since, for different $\lambda ,$ the subspaces ${(M(\mu +t\delta )\otimes V^{\otimes k})}_{\lambda +t\delta }^{[\lambda +t\delta ]}$ are mutually orthogonal with respect to the $U_h𝔤$ contravariant form ${⟨,⟩}_t$ on $(M(\mu +t\delta )\otimes V^{\otimes k}),$ $${(M(\mu +t\delta )\left(j\right)\otimes V^{\otimes k})}_{\lambda +t\delta }^{[\lambda +t\delta ]}\subseteq {(M(\mu +t\delta )\otimes V^{\otimes k})}_{\lambda +t\delta }^{[\lambda +t\delta ]}\left(j\right)={ℳ}^{(\lambda +t\delta )/(\mu +t\delta )}\left(j\right)\text{.}$$ On the other hand, if $u\in {(M(\mu +t\delta )\otimes V^{\otimes k})}_{\lambda +t\delta }^{[\lambda +t\delta ]}\left(j\right)$ then write $u=\sum _i{a}_i\otimes b_i$ where $a_i\in M(\mu +t\delta )$ and $b_i$ is an orthonormal basis of $V^{\otimes k}\text{.}$ Then, for all $v\in M(\mu +t\delta ),$ and all $k,$ ${⟨a_k,v⟩}_t={⟨u,v\otimes b_k⟩}_t\in t^jℂ\left[t\right]$ and so $u\in {(M(\mu +t\delta )\left(j\right)\otimes V^{\otimes k})}_{\lambda +t\delta }^{[\lambda +t\delta ]}\text{.}$ So $$F_{\lambda +t\delta }{\left(M(\mu +t\delta )\right)}^{\left(j\right)}={ℳ}^{(\lambda +t\delta )/(\mu +t\delta )}\left(j\right)$$ and the filtrations in (4.6) and (4.5) are identical.

Let $\lambda$ and $\mu$ be integrally dominant weights and let $w,y\in W^{\mu }$ be elements of maximal length in $W_{\lambda +\rho }w{W}_{\mu +\rho }$ and $W_{\lambda +\rho }y{W}_{\mu +\rho }$ respectively. Assume that the ${\stackrel{\sim }{ℬ}}_k$ modules ${ℒ}^{\lambda /v\circ \mu },$ $v\in W^{\mu },$ are simple. Then multiplicities of ${ℒ}^{\lambda /y\circ \mu }$ in the filtration (4.5) are given by $$\sum _{j\ge 0}[\frac{{\left({ℳ}^{\lambda /w\circ \mu }\right)}^{\left(j\right)}}{{\left({ℳ}^{\lambda /w\circ \mu }\right)}^{(j+1)}}:{ℒ}^{\lambda /(y\circ \mu )}]v^{\frac{1}{2}(\ell \left(y\right)-\ell \left(w\right)-j)}=P_{wy}\left(v\right)\text{.}$$ where $P_{wy}\left(v\right)$ is the Kazhdan-Lusztig polynomial for the Weyl group $W^{\mu }\text{.}$

		Proof.
	Since the functor $F_{\lambda }$ is exact this result follows from the Beilinson-Bernstein theorem (2.6). The condition on $y$ is necessary for the module ${ℒ}^{\lambda /y\circ \mu }$ to be nonzero. $\square$

The BGG resolution for affine braid groups

Let $\mu \in {𝔥}^{*}$ be such that $-(\mu +\rho )$ is dominant and regular and let $W_J^{\mu }$ be a parabolic subgroup of the integral Weyl group $W^{\mu }\text{.}$ Let $w_0$ be the longest element of $W_J^{\mu }$ and fix $\nu =w_0\circ \mu \text{.}$ Following the method of [Che1987-2], applying the exact functor $F_{\lambda }$ to the BGG resolution in (2.7) produces an exact sequence of ${\stackrel{\sim }{ℬ}}_k\text{-modules}$ $$\begin{array}{cc}0⟶{𝒞}_N⟶\cdots ⟶{𝒞}_1⟶{𝒞}_0⟶{ℒ}^{\lambda /\nu }⟶0& \text{(4.7)}\end{array}$$ where ${𝒞}_k=\underset{\ell \left(w\right)=j}{⨁}{ℳ}^{\lambda /w\circ \nu },$ and the sum is over all $w\in W_J^{\mu }$ of length $j$ (in $W_J^{\mu }\text{).}$ Thus, in the Grothendieck group of the category ${\stackrel{\sim }{𝒪}}_k$ of finite dimensional ${\stackrel{\sim }{ℬ}}_k\text{-modules,}$ $$\begin{array}{cc}\left[{ℒ}^{\text{<mi>λ</mi><mo>/</mo><mi>ν</mi>}}\right]=\sum _{w\in W_J^{\mu }}{(-1)}^{\ell \left(w\right)}\left[{ℳ}^{\lambda /(w\circ \nu )}\right]& \text{(4.8)}\end{array}$$ where $\nu =w_0\circ \mu$ and $w_0$ is the longest element of $W_J^{\mu }\text{.}$ This identity is a generalization of the classical Jacobi-Trudi identity [Mac1995, I (5.4)] for expanding Schur functions in terms of homogeneous symmetric functions, $$\begin{array}{cc}{s}_{\lambda /\nu }=\sum _{w\in S_n}{(-1)}^{\ell \left(w\right)}{h}_{\lambda +\delta -w(\nu +\delta )}\text{.}& \text{(4.9)}\end{array}$$

Restriction of ${ℒ}^{\lambda /\mu }$ to the braid group

The braid group ${ℬ}_k$ is the quotient of the affine braid group by the relation $X^{{\epsilon }_1}=1$ and so the modules ${ℒ}^{\nu /0}$ are ${ℬ}_k\text{-modules.}$ The following proposition determines the structure of $F_{\lambda }\left(L\left(\mu \right)\right)$ as a ${ℬ}_k$ module when $L\left(\mu \right)$ is finite dimensional. This is a generalization of the Littlewood-Richardson rule.

Let $P^{+}$ be the set of dominant integral weights. Define the tensor product multiplicities $c_{\mu \nu }^{\lambda },$ $\lambda ,\mu ,\nu \in P^{+},$ by the $U_h𝔤\text{-module}$ decompositions $$L\left(\mu \right)\otimes L\left(\nu \right)\cong \underset{\lambda \in P^{+}}{⨁}{L\left(\lambda \right)}^{\oplus c_{\mu \nu }^{\lambda }}\text{.}$$ Then $${\text{Res}}_{{ℬ}_k}^{{\stackrel{\sim }{ℬ}}_k}\left({ℒ}^{\lambda /\mu }\right)=\underset{\nu \in P^{+}}{⨁}{\left({ℒ}^{\nu /0}\right)}^{\oplus c_{\mu \nu }^{\lambda }}\text{.}$$

		Proof.
	Let us abuse notation slightly and write sums instead of direct sums. Then, as a $(U_h𝔤,{ℬ}_k)$ bimodule $$L\left(\mu \right)\otimes V^{\otimes k}=\sum _{\lambda }L\left(\lambda \right)\otimes {ℒ}^{\lambda /\mu },$$ where ${ℒ}^{\lambda /\mu }=F_{\lambda }\left(L\left(\mu \right)\right)\text{.}$ As a $(U_h𝔤,{ℬ}_k)$ bimodule $$L\left(\mu \right)\otimes V^{\otimes k}=L\left(\mu \right)\otimes (\sum _{\nu }L\left(\nu \right)\otimes {ℒ}^{\nu /0})=\sum _{\lambda ,\nu }{c}_{\mu \nu }^{\lambda }L\left(\lambda \right)\otimes {ℒ}^{\nu /0}\text{.}$$ Comparing coefficients of $L\left(\lambda \right)$ in these two identities yields the formula in the statement. $\square$

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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