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: 21 December 2013

Introduction

This paper provides a unified approach to results on representations of affine Hecke algebras, cyclotomic Hecke algebras, affine BMW algebras, cyclotomic BMW algebras, Markov traces, Jacobi-Trudi type identities, dual pairs [Zel1987], and link invariants [Tur1990]. The key observation in the genesis of this paper was that the technical tools used to obtain the results in Orellana [Ore1999] and Suzuki [Suz1998], two a priori unrelated papers, are really the same. Here we develop this method and explain how to apply it to obtain results similar to those in [Ore1999] and [Suz1998] in more general settings. Some specific new results which are obtained are the following:

(a)	A generalization of the results on Markov traces obtained by Orellana [Ore1999] to centralizer algebras coming from quantum groups of all Lie types.
(b)	A generalization of the results of Suzuki [Suz1998] to show that Kazhdan-Lusztig polynomials of all finite Weyl groups occur as decomposition numbers in the representation theory of affine braid groups of type $A\text{.}$
(c)	A generalization of the functors used by Zelevinsky [Zel1987] to representations of affine braid groups of type $A\text{.}$
(d)	We define the affine BMW algebra (Birman-Murakami-Wenzl) and show that it has a representation theory analogous to that of affine Hecke algebras. In particular there are “standard modules” for these algebras which have composition series where multiplicities of the factors are given by Kazhdan-Lusztig polynomials for Weyl groups of types $A,$ $B,$ and $C\text{.}$
(e)	We generalize the results of Leduc and Ram [LRa1977] to affine centralizer algebras.

Let $U_h𝔤$ be the Drinfel’d-Jimbo quantum group associated to a finite dimensional complex semisimple Lie algebra $𝔤\text{.}$ If $M$ is a (possibly infinite dimensional) $U_h𝔤\text{-module}$ in the category $𝒪$ and $V$ is a finite dimensional $U_h𝔤\text{-module}$ then we show that the affine braid group ${\stackrel{\sim }{ℬ}}_k$ acts on the $U_h𝔤\text{-module}$ $M\otimes V^{\otimes k}\text{.}$ Fix $V$ and define $$\begin{array}{ccc}{F}_{\lambda }\left(M\right)& =& (\text{the vector space of highest weight vectors of weight}\hspace{0.17em}\lambda \hspace{0.17em}\text{in}\hspace{0.17em}M\otimes V^{\otimes k}\text{.})\text{.}\end{array}$$ Then $F_{\lambda }$ is a functor from $U_h𝔤$ modules in category $𝒪$ to finite dimensional modules for the affine braid group ${\stackrel{\sim }{ℬ}}_k$ which takes

(1)	finite dimensional $U_h𝔤$ modules to “calibrated” ${\stackrel{\sim }{ℬ}}_k$ modules,
(2)	Verma modules to “standard” modules, and
(3)	under appropriate conditions, irreducible $U_h𝔤$ modules to irreducible ${\stackrel{\sim }{ℬ}}_k$ modules.

Applying the functor $F_{\lambda }$ to a Jantzen filtration of Verma modules of $U_h𝔤$ provides a “Jantzen filtration” of the standard modules of ${\stackrel{\sim }{ℬ}}_k$ and shows that the irreducible ${\stackrel{\sim }{ℬ}}_k$ modules appear in a composition series of the standard module with multiplicities given by the Kazhdan-Lusztig polynomials of the Weyl group of $𝔤\text{.}$ Though ${\stackrel{\sim }{ℬ}}_k$ is always the affine braid group of type A, the Weyl group of $𝔤$ is not usually of type A.

Applying the functor $F_{\lambda }$ to the BGG resolution of an irreducible highest weight module provides a BGG resolution for the corresponding ${\stackrel{\sim }{ℬ}}_k\text{-modules}$ and a corresponding “Jacobi-Trudi” identity for the characters of ${\stackrel{\sim }{ℬ}}_k$ modules. Once again, it is interesting to note that, though ${\stackrel{\sim }{ℬ}}_k$ is the affine braid group of type A, it is the Weyl group of a different type which appears in this Jacobi-Trudi identity.

Using the general formulation for constructing Markov traces on braid groups, given for example in [Tur1988], we obtain a Markov trace on the affine braid group ${\stackrel{\sim }{ℬ}}_k$ for every choice of $𝔤$ and $U_h𝔤$ modules $M$ and $V\text{.}$

(a)	If $𝔤={𝔰𝔩}_{n+1},$ $M=L\left(0\right)$ and $V=L\left({\omega }_1\right)$ this gives the Markov trace on the Hecke algebra studied in [Jon1987] and [Wen1988].
(b)	If $𝔤={𝔰𝔩}_2,$ $M=L\left(0\right)$ and $V=L\left({\omega }_1\right)$ this gives the Markov trace on the Temperley-Lieb algebra used by Jones [Jon1983].
(c)	If $𝔤={𝔰𝔩}_{n+1},$ $M=L\left(k{\omega }_{\ell }\right)$ with $k$ and $\ell$ large and $n$ very large, and $V=L\left({\omega }_1\right)$ this gives the Markov traces on the Hecke algebra of type B studied by [GLa1997], [Lam1999], [Ian2001] and [Ore1999].
(d)	If $𝔤={𝔰𝔩}_{n+1},$ $M=L\left(\lambda \right),$ where $\lambda$ is “large”, and $V=L\left({\omega }_1\right)$ this gives the Markov traces on the cyclotomic Hecke algebras introduced by Lambropoulou [Lam1999] and studied in [GIM2000].
(e)	If $𝔤={𝔰𝔬}_n$ or $𝔤={𝔰𝔭}_{2n},$ $M=L\left(0\right)$ and $V=L\left({\omega }_1\right)$ this gives the Markov traces used to construct Kauffman polynomials.

For general $𝔤,$ general $V,$ and $M=L\left(0\right),$ this mechanism gives the traces necessary to compute the Reshetikhin-Turaev link invariants [RTu1991]. In some sense, this paper is a study of the representation theory behind the generalization of the Reshetikhin-Turaev method given in [Tur1990].

In the final section of this paper we describe precisely the combinatorics of the representations $F_{\lambda }\left(M\right)$ in the cases when $𝔤$ is type $A_n,B_n,C_n$ or $D_n$ and $V$ is the fundamental representation. In these cases the representations can be constructed with partitions, standard tableaux, up-down tableaux, multisegments and the combinatorics of Young diagrams. In particular, in type A, the functor $F_{\lambda }$ naturally constructs the standard modules and irreducible modules of affine Hecke algebras of type A in terms of multisegments (a classification originally obtained by Zelevinsky [Zel1980] by different methods). We then specify explicitly the correspondence between the decomposition numbers of the affine Hecke algebra and Kazhdan-Lusztig polynomials for the symmetric group.

Acknowledgements. A. Ram thanks P. Littelmann, the Department of Mathematics at the University of Strasbourg and the Isaac Newton Institute for the Mathematical Sciences at Cambridge University for hospitality and support during residencies when this paper was written.

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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