Two boundary Hecke Algebras and the combinatorics of type $C$

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

Last updated: 27 January 2015

Introduction

This paper explores a Schur-Weyl duality approach to the representations of the affine Hecke algebras of type C with unequal parameters. Following Kazhdan-Lusztig [KLu1987], the irreducible representations of the affine Hecke algebra are usually constructed via the K-theory of generalized Springer fibers. This method works well in the cases that an algebraic group is available, which is only for special cases of the three parameters $t,t_0,t_k$ of the affine Hecke algebra of type C.

G. Lusztig gave a general approach to the unequal parameter case using Kazhdan-Lusztig bases and cells. In [Lus2003], the challenges for pushing this method through in type C are outlined in a set of conjectures, many of which have now been settled in work of Geck, Bonnafé, and others (see [Gec2004, Bon2772408, Gui2008] and references there). Another analytic approach, closer to the original classification and construction of Kazhdan-Lusztig, is given by Opdam and Solleveld (see [OSo2010] and [Sol2012] and the references there). In the type C case, Kato [Kat2009] explained that the "exotic nilpotent cone" can be used to replace the Kazhdan-Lusztig geometry and obtain a complete geometric classification of the irreducible representations of the affine Hecke algebra (with mild restrictions on parameters).

In the type A case, there is a powerful alternative to the geometric method via Schur-Weyl duality (see for example [ASu9710037, ORa0401317, VVa1996]). In this paper we provide an analogue of this Schur-Weyl duality approach for the type C case, with unequal parameters. This is a generalization of the degenerate case studied by Daugherty [Dau2012].

The method is the following: Let $U_q{𝔤𝔩}_n$ be the Drinfeld-Jimbo quantum group corresponding to the general linear Lie algebra, and let $V={ℂ}^n$ be the standard representation of $U_q{𝔤𝔩}_n\text{.}$ Write $L\left(\lambda \right)$ for the irreducible polynomial representation of $U_q{𝔤𝔩}_n$ indexed by the partition $\lambda \text{,}$ let $M=L\left(\left(a^c\right)\right)$ and $N=L\left(\left(b^d\right)\right)$ be irreducible representations of $U_q{𝔤𝔩}_n$ indexed by $a\times c$ and $b\times d$ rectangles. There is an action of the affine Hecke algebra (more precisely, a slightly extended affine Hecke algebra) $H_k^{\text{ext}}$ of type $C_k$ with parameters $$t^{\frac{1}{2}}=q,t_0^{\frac{1}{2}}=-i{q}^{b+d},\text{and}{t}_k^{\frac{1}{2}}=-i{q}^{a+c}\text{(where}\hspace{0.17em}i=\sqrt{-1}\text{),}$$ such that $$M\otimes N\otimes V^{\otimes k}\text{is a}\hspace{0.17em}(U_q{𝔤𝔩}_n,H_k^{\text{ext}})\text{-bimodule.}$$ We show that these commuting actions of $U_q{𝔤𝔩}_n$ and $H_k^{\text{ext}}$ provide a Schur-Weyl duality, which can be used to derive the representation theory of $H_k^{\text{ext}}$ from the quantum group $U_q{𝔤𝔩}_n\text{.}$ We work out the combinatorics of this correspondence, relating the natural indexing of $H_k^{\text{ext}}\text{-modules}$ coming from the Schur-Weyl duality to the other indexings, by describing the weights for the action of the polynomial part (Bernstein generators) on each irreducible module.

A significant portion of the work in identifying the centralizer of the $U_q{𝔤𝔩}_n$ action on $M\otimes N\otimes V^{\otimes k}$ as an extended affine Hecke algebra of type C is in relating the Coxeter and Bernstein presentations and getting the parameter conversions into focus. The relationships between these presentations are given in Theorem 2.1 for the affine braid group of type C, and in Theorem 2.2 for the affine Hecke algebra of type C. Sections 3, 4 and 5 could, perhaps have stood as papers on their own. In Section 3, we give the combinatorics of local regions and standard tableaux for the case of type C with unequal parameters (following the equal parameter case done in [Ram2003]). The main result of Section 3, Theorem 3.3, provides a classification and construction of all irreducible calibrated $H_k^{\text{ext}}\text{-modules.}$ As in [Ram2003] this classification is via skew local regions and it is important to get the definition of skew local region exactly right in order to achieve the result. In [Ram2003] this depends on the careful analysis of the structure of the irreducible representations of rank two affine Hecke algebras which was done, in the single parameter case, in [Ram2002]. Since the corresponding analysis for three distinct parameters in the type $C_2$ case is, to our knowledge, not available in the literature, we have provided this analysis in Section 4. This will ensure that our classification of calibrated irreducible representations for $H_k^{\text{ext}}$ with distinct parameters, as given in Theorem 3.3, is on firm footing. The construction of the action of $H_k^{\text{ext}}$ on tensor space is completed in Theorems 5.1 and 5.4. Finally, armed with these tools we prove the main result, Theorem 5.5, which determines exactly which representations of $H_k^{\text{ext}}$ appear in tensor space, comparing the indexing that arises naturally from the highest weight theory for ${𝔤𝔩}_n$ to the combinatorics of the weights of the action of the polynomial part of $H_k^{\text{ext}}\text{.}$

Following the schematic from [ORa0401317], one would like to generalize the analysis in this paper by replacing finite-dimensional $M$ and $N$ with, for example, other modules from category $𝒪\text{.}$ Forcing the $R\text{-matrices}$ for $M\otimes V$ and $N\otimes V$ to have only 2 eigenvalues severely restricts the choices of $M$ and $N\text{.}$ Non-finite-dimensional choices of modules $M$ and $N$ that satisfy these conditions exist in category $𝒪\text{,}$ but additional work understanding the combinatorics of $M\otimes N\otimes V^{\otimes k}$ in these cases is needed. This understanding would yield an interesting generalization of the work in this paper.

The seeds of the idea for this paper were sown during conversations of A. Ram with P. Pyatov and V. Rittenberg in Bonn in 2005. They suggested that one should analyze two-boundary spin chains by $R\text{-matrices,}$ thus implying the possibility for Schur-Weyl duality approach to representations of affine braid groups of type C. This idea was completed in the degenerate case in [Dau2012], and significant information was obtained in the Temperley-Lieb case in [dAl2009] (see also references there). In [DRaPREP] we shall complete the connection between this paper and the statistical mechanics by using the results of this paper to identify the representations of the two-boundary Temperley-Lieb algebra given, in a diagrammatic form, by de Gier and Nichols in [dAl2009].

Acknowledgements. We thank the Australian Research Council and the National Science Foundation for support of our research under grants DP130100674 and DMS-1162010. Much of the research for this paper was completed during residency at the special semester on "Automorphic forms, Combinatorial representation theory, and Multiple Dirichlet series" at ICERM in 2013. We thank ICERM, all the ICERM staff and the organizers of the special semester for providing a wonderful and stimulating working environment.

Notes and References

This is an excerpt from the paper Two boundary Hecke Algebras and the combinatorics of type $C$ Zajj Daugherty (Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY 10031) and Arun Ram (Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia).

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