## Affine and degenerate affine BMW algebras: The center

Last update: 18 October 2013

## Introduction

The degenerate affine BMW algebras ${𝒲}_{k}$ and the affine BMW algebras ${W}_{k}$ arise naturally in the context of Schur-Weyl duality and the application of Schur functors to modules in category $𝒪$ for orthogonal and symplectic Lie algebras and quantum groups (using the Schur functors of [Zel1987], [ASu9710037], and [ORa0401317]). The degenerate algebras ${𝒲}_{k}$ were introduced in [Naz1996] and the affine versions ${W}_{k}$ appeared in [ORa0401317], following foundational work of [Här1673464], [Här9712030] and [Här1834081]. The representation theory of ${𝒲}_{k}$ and ${W}_{k}$ contains the representation theory of any quotient: in particular, the degenerate cyclotomic BMW algebras ${𝒲}_{r,k},$ the cyclotomic BMW algebras ${W}_{r,k},$ the degenerate affine Hecke algebras ${ℋ}_{k},$ the affine Hecke algebras ${H}_{k},$ the degenerate cyclotomic Hecke algebras ${ℋ}_{r,k},$ and the cyclotomic Hecke algebras ${H}_{r,k}$ as quotients. The representation theory of the affine BMW algebras is an image of the representation theory of category $𝒪$ for orthogonal and symplectic Lie algebras and their quantum groups in the same way that the affine Hecke algebras arise in Schur-Weyl duality with the enveloping algebra of ${𝔤𝔩}_{n}$ and its Drinfeld-Jimbo quantum group.

In the literature, the algebras ${𝒲}_{k}$ and ${W}_{k}$ have often been treated separately. One of the goals of this paper is to unify the theory. To do this we have begun by adjusting the definitions of the algebras carefully to make the presentations match, relation by relation. In the same way that the affine BMW algebra is a quotient of the group algebra of the affine braid group, we have defined a new algebra, the degenerate affine braid algebra which has the degenerate affine BMW algebra and the degenerate affine Hecke algebras as quotients. We have done this carefully, to ensure that the Schur-Weyl duality framework is completely analogous for both the degenerate affine and the affine cases. We have also added a parameter $\epsilon$ (which takes values $±1\text{)}$ so that both the orthogonal and symplectic cases can be treated simultaneously. Our new presentations of the algebras ${𝒲}_{k}$ and ${W}_{k}$ are given in section 2.

In section 3 we consider some remarkable recursions for generating central elements in the algebras ${𝒲}_{k}$ and ${W}_{k}\text{.}$ These recursions were given by Nazarov [Naz1996] in the degenerate case, and then extended to the affine BMW algebra by Beliakova-Blanchet [BBl1866492]. Another proof in the affine cyclotomic case appears in [RXu0801.0465, Lemma 4.21] and, in the degenerate case, in [AMR0506467, Lemma 4.15]. In all of these proofs, the recursion is obtained by a rather mysterious and tedious computation. We show that there is an “intertwiner” like identity in the full algebra which, when “projected to the center” produces the Nazarov recursions. Our approach dramatically simplifies the proof and provides insight into where these recursions are coming from. Moreover, the proof is exactly analogous in both the degenerate and the affine cases, and includes the parameter $\epsilon ,$ so that both the orthogonal and symplectic cases are treated simultaneously.

In section 4 we identify the center of the degenerate and affine BMW algebras. In the degenerate case this has been done in [Naz1996]. Nazarov stated that the center of the degenerate affine BMW algebra is the subring of the ring of symmetric functions generated by the odd power sums. We identify the ring in a different way, as the subring of symmetric functions with the Q-cancellation property, in the language of Pragacz [Pra1180989]. This is a fascinating ring. Pragacz identifies it as the cohomology ring of orthogonal and symplectic Grassmannians; the same ring appears again as the cohomology of the loop Grassmannian for the symplectic group in [LSS0710.2720, Lam0906.0385]; and references for the relationship of this ring to the projective representation theory of the symmetric group, the BKP hierarchy of differential equations, representations of Lie superalgebras, and twisted Gelfand pairs are found in [Mac1354144, Ch. II §8]. For the affine BMW algebra, the Q-cancellation property can be generalized well to provide a suitable description of the center. From our perspective, one would expect that the ring which appears as the center of the affine BMW algebra should also appear as the K-theory of the orthogonal and symplectic Grassmannians and as the K-theory of the loop Grassmannian for the symplectic group, but we are not aware that these identifications have yet been made in the literature.

This paper is part of a more comprehensive work on affine and degenerate affine BMW algebras. In future work [DRV1205.1852v1] we may:

 (a) set up the commuting actions between the algebras ${𝒲}_{k}$ and ${W}_{k}$ and the enveloping algebras of orthogonal and symplectic Lie algebras and their quantum groups, (b) show how the central elements which arise in the Nazarov recursions coincide with central elements studied in Baumann [Bau1620662], (c) provide a new approach to admissibility conditions by providing “universal admissible parameters” in an appropriate ground ring (arising naturally, from Schur-Weyl duality, as the center of the enveloping algebra, or quantum group), (d) classify and construct the irreducible representations of ${𝒲}_{k}$ and ${W}_{k}$ by multisegments, and (e) define Khovanov-Lauda-Rouquier analogues of the affine BMW algebras.
Many parts of this program are already available in the works of Goodman, Rui, Wilcox-Yu, and others (see, for example, [RSi2418167], [RSi0607266], [RXu0702633], [RXu0801.0465], [Goo0801.0306], [Goo0905.4258], [Goo0905.4253], [GHa0411155], [GMo0612064], [GMo0612065], [WYu0611518], [WYu0911.5284], [Yu0810.0069]). Some parts of our work are also available at [RamNotes].

Acknowledgements: Significant work on this paper was done while the authors were in residence at the Mathematical Sciences Research Institute (MSRI) in 2008, and the writing was completed when A. Ram was in residence at the Hausdorff Insitute for Mathematics (HIM) in 2011. We thank MSRI and HIM for hospitality, support, and a wonderful working environment during these stays. This research has been partially supported by the National Science Foundation (DMS-0353038) and the Australian Research Council (DP-0986774). We thank S. Fomin for providing the reference [Pra1180989] and Fred Goodman for providing the reference [BBl1866492], many informative discussions, detailed proofreading, and for much help in processing the theory around admissibility conditions. We thank J. Enyang for his helpful comments on the manuscript.

## Notes and references

This is a typed exert of the paper Affine and degenerate affine BMW algebras: The center by Zajj Daugherty, Arun Ram and Rahbar Virk.