Affine Braids, Markov Traces and the Category 𝒪

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 21 December 2013

Preliminaries on Quantum Groups

Let Uh𝔤 be the Drinfel’d-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra 𝔤. Let us fix some notations. In particular, fix a triangular decomposition 𝔤=𝔫-𝔥𝔫+, 𝔫+=α>0𝔤α, 𝔟+=𝔥𝔫+, and let W be the Weyl group of 𝔤. Let , be the usual inner product on 𝔥* so that, if α is a root, the corresponding reflection sα in W is given by sαλ=λ- λ,α α, where α=2αα,α. The element ρ=12α>0α is often viewed as an element of 𝔥 by using the form , to identify 𝔥 and 𝔥*. We shall use the conventions for quantum groups as in [Dri1990] and [LRa1977] so that q=eh/2, 𝔥Uh𝔤, and Uh𝔤U𝔤[[h]], as algebras. The quantum group has a triangular decomposition corresponding to that of 𝔤, Uh𝔤=Uh𝔫- Uh𝔥Uh𝔫+ andUh𝔟+ =Uh𝔥Uh𝔫+.

The category 𝒪

If M is a Uh𝔤 module and λ𝔥* the λ weight space of M is Mλ = { mM|am=λ (a)m,for alla𝔥 } . The category 𝒪 is the category of Uh𝔤 modules M such that

(a) M=λ𝔥*Mλ,
(b) For all mM, dim(Uh𝔫+m) is finite,
(c) M is finitely generated as a Uh𝔤 module.
For μ𝔥* let M(μ) be the Verma module of highest weight μ, and let
L(μ) be the irreducible module of highest weight μ.
The irreducible module L(μ) is the quotient of M(μ) by a maximal proper submodule and M(μ)=Uh𝔤Uh𝔟+vμ+ where vμ+ is the one dimensional Uh𝔟+ module spanned by a vector vμ+ such that avμ+=μ(a)vμ+ for a𝔥 and Uh𝔫+vμ+=0. Every module M𝒪 has a finite composition series with factors L(μ), μ𝔥*. Each of the sets {[L(λ)]|λ𝔥*} and{[M(λ)]|λ𝔥*} (where [M] denotes the isomorphism class of the module M) are bases of the Grothendieck group of the category 𝒪.

If M is a Uh𝔤 module generated by a highest weight vector of weight λ (i.e., a vector v+ such that av+=λ(a)v+ for a𝔥 and Uh𝔫+v+=0) then any element of the center Z(Uh𝔤) acts on M by a constant, zm=χλ(z)m, forzZ(Uh𝔤), mM,χλ(z) . For each Uh𝔤 module M𝒪 let M[λ]=νQ Mλ+ν[λ], whereQ=i=1n αi, α1,,αn are the simple roots and Mλ+ν[λ] = { mMλ+ν| there isk>0 such that(z-χλ(z))k m=0for allzZ(Uh𝔤) } . Then M=λM[λ], where the sum is over all integrally dominant weights λ𝔥* i.e., λ𝔥* such that λ+ρ,α<0 for all αR+. To summarize, there is a decomposition of the category 𝒪, 𝒪=λ𝒪[λ], (2.1) where the sum is over all integrally dominant weights λ𝔥* and 𝒪[λ] is the full subcategory of modules M𝒪 such that M=M[λ]. The Grothendieck group of the category 𝒪[λ] has bases {[L(μ)]|μWλλ} and {[M(μ)]μWλλ}. (2.2) where the integral Weyl group corresponding to λ is Wλ=sα|λ+ρ,α andwλ=w(λ+ρ) -ρ,wW,λ𝔥*. (2.3) defines the dot action of W on 𝔥*.

Jantzen filtrations

Following the notations for the quantum group used in [LRa1977, §2], let 𝔥, X1,,Xr and Y1,,Yr be the standard generators of the quantum group Uh𝔤 which satisfy the quantum Serre relations. The Cartan involution θ:Uh𝔤Uh𝔤 is the algebra anti-involution defined by θ(Xi)=Yi, θ(Yi)=Xi, and θ(a)=a,fora𝔥. (2.4) The Cartan involution θ is a coalgebra homomomorphism. A contravariant form on a Uh𝔤 module M is a symmetric bilinear form ,:M×M such that um1,m2= m1,θ(u)m2, uUh𝔤,m1,m2M.

Fix λ𝔥* and δ𝔥* such that λ+tδ is integrally dominant for all small positive real numbers t. Consider t as an indeterminate and consider the Verma module M(λ+tδ)=Uh 𝔤[t]Uh𝔟+[t] vλ+tδ as the module for Uh𝔤[t]=[t]Uh𝔤 generated by a vector v+ such that av+=(λ+tδ)(a)v+ for a𝔥 and Uh𝔫+[t]v+=0. There is a unique contravariant form ,t:M(λ+tδ)×M(λ+tδ)[t] such that v+,v+t=1. Define M(λ+tδ)(j)= { mM(λ+tδ)| m,nt tj[t]for all nM(λ+tδ) } . The “specialization of M(λ+tδ)(j) at t=0 is M(λ)(j)= image ofM(λ+tδ) (j)inM(λ+tδ) [t][t] /t[t] and the Jantzen filtration of M(λ) is M(λ)= M(λ)(0) M(λ)(1) . (2.5) By [Jan1980, Theorem 5.3], the Jantzen filtration is a filtration of M(λ) by Uh𝔤 modules, the module M(λ)(1) is a maximal proper submodule of M(λ) and each quotient M(λ)(i)M(λ)(i+1) has a nondegenerate contravariant form. It is known [Bar1983] that the Jantzen filtration does not depend on the choice of δ. It is a deep theorem [BBe1993] that the quotients M(λ)(i)/M(λ)(i+1) are semisimple and that if wWμ and yWμ are maximal length in their cosets wWμ+ρ and yWμ+ρ, respectively, then the Kazhdan-Lusztig polynomial for Wμ is j0 [ M(wμ)(j)/ M(wμ)(j+1) :L(yμ) ] v12((y)-(w)-j) =Pwy(v), (2.6) where is the length function on Wμ and [ M(wμ)(j)/ M(wμ)(j+1) :L(yμ) ] is the multiplicity of the simple module L(yμ) in the jth factor of the Jantzen filtration of M(λ).

The BGG resolution

Not all simple modules L(λ) in the category 𝒪 have a BGG resolution. The general form of the BGG resolution given by Gabber and Joseph [GJo1981] is as follows.

Let μ𝔥* be such that -(μ+ρ) is dominant and regular and let WJμ be a parabolic subgroup of the integral Weyl group Wμ. Let w0 be the longest element of WJμ and fix ν=w0μ. Define a resolution 0C(w0) C2d2 C1d1C0 L(ν)0 (2.7) of the simple module L(ν) by Verma modules by setting Cj=(w)=j M(wν), where the sum is over all wWJμ of length j, and defining the map dj:CjCj-1, by the matrix(dj)v,w = { εv,w ιv,w, ifvw, 0, otherwise, where vw means that there is a (not necessarily simple) root α such that w=sαv and (w)=(v)-1, the maps ιv,w are fixed choices of inclusions ιv,w:M(vν) M(wν),and εv,w=±1, are fixed choices of signs such that εu,vεv,w=- εu,vεv,w ifuvw, uvwandv v. Gabber and Joseph [GJo1981] prove that the sequence (2.7) is exact in this general setting. See [BGG1975] and [Dix1996, 7.8.14], for the original form of the BGG resolution. From the exactness of (2.7) it follows that if -(μ+ρ) is dominant and regular then, in the Grothendieck group of the category 𝒪, [L(ν)]= wWJμ (-1)(w) [M(wν)], (2.8) where ν=w0μ and w0 is the longest element of WJμ.

ŘMN matrices and the quantum Casimir CM

Let Uh𝔤 be the Drinfeld-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra 𝔤. There is an invertible element =aibi in (a suitable completion of) Uh𝔤Uh𝔤 such that, for any two Uh𝔤 modules M and N, the map ŘMN: MN NM mn binaim M N N M is a Uh𝔤 module isomorphism. There is also a quantum Casimir element e-hρu in the center of Uh𝔤 and, for a Uh𝔤 module M we define CM: M M m (e-hρu)m M M CM In order to be consistent with the graphical calculus these operators should be written on the right. The elements and e-hρu satisfy relations (see [LRa1977, (2.1-2.12)]), which imply that, for Uh𝔤 modules M,N,P and a Uh𝔤 module isomorphism τM:MM, M N N M τM = M N N M τM ŘMN (idNτM) = (τMidN) ŘMN, (2.9) M(NP) (NP)M = M N P P N M (MN)P P(MN) = M N P P N M ŘM,NP = (ŘMNidP) (idNŘMP) ŘMN,P = (idMŘNP) (ŘMPidN), (2.10) CMN = (ŘMNŘNM)-1 (CMCN). (2.11) The relations (2.9) and (2.10) together imply the braid relation M N P P N M = M N P P N M (ŘMNidP) (idNŘMP) (ŘNPidM) = (idMŘNP) (ŘMPidN) (idPŘMN), (2.12) If M is a Uh𝔤 module generated by a highest weight vector v+ of weight λ then, by [Dri1990, Prop. 3.2], CM= q-λ,λ+2ρ idM. (2.13) Note that λ,λ+2ρ=λ+ρ,λ+ρ-ρ,ρ are the eigenvalues of the classical Casimir operator [Dix1996, 7.8.5]. If M is a finite dimensional Uh𝔤 module then M is a direct sum of the irreducible modules L(λ), λP+, and CM=λP+ q-λ,λ+2ρ Pλ, where Pλ:MM is the projection onto M[λ] in M. From the relation (2.11) it follows that if M=L(μ), N=L(ν) are finite dimensional irreducible Uh𝔤 modules then ŘMNŘNM acts on the λ isotypic component L(λ)cμνλ of the decomposition L(μ)L(ν)= λ L(λ)cμνλ (2.14) by the constant qλ,λ+2ρ-μ,μ+2ρ-ν,ν+2ρ where cμνλ are positive integers. Suppose that M and N are Uh𝔤 modules with contravariant forms ,M and ,N, respectively. Since the Cartan involution is a coalgebra homomorphism the form on MN defined by m1n1,m2n2 =m1,m2M n1,n2N, (2.15) for m1,m2M, n1,n2N, is also contravariant. If θ is the Cartan involution defined in (2.4) then a formula of Drinfeld [Dri1990, Prop. 4.2], states (θθ)()= ibiai, from which it follows that (m1n1)ŘMN, n2m2 = i (biai) (n1m1), n2m2 = i n1m1, (θ(bi)θ(ai)) (n2m2) = i n1m1, (aibi) (n2m2) = i m1n1,bi m2ain2 . Thus (m1n1) ŘMN,n2m2 = m1n1, (n2m2) ŘNM . (2.16)

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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