## Affine Braids, Markov Traces and the Category $𝒪$

Last update: 21 December 2013

## Preliminaries on Quantum Groups

Let ${U}_{h}𝔤$ be the Drinfel’d-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra $𝔤\text{.}$ Let us fix some notations. In particular, fix a triangular decomposition $𝔤=𝔫-⊕𝔥⊕𝔫+, 𝔫+=⨁α>0𝔤α, 𝔟+=𝔥⊕𝔫+,$ and let $W$ be the Weyl group of $𝔤\text{.}$ Let $⟨,⟩$ be the usual inner product on ${𝔥}^{*}$ so that, if $\alpha$ is a root, the corresponding reflection ${s}_{\alpha }$ in $W$ is given by $sαλ=λ- ⟨λ,α∨⟩ α,$ where ${\alpha }^{\vee }=\frac{2\alpha }{⟨\alpha ,\alpha ⟩}\text{.}$ The element $\rho =\frac{1}{2}\sum _{\alpha >0}\alpha$ is often viewed as an element of $𝔥$ by using the form $⟨,⟩$ to identify $𝔥$ and ${𝔥}^{*}\text{.}$ We shall use the conventions for quantum groups as in [Dri1990] and [LRa1977] so that $q={e}^{h/2},$ $𝔥\subseteq {U}_{h}𝔤,$ and ${U}_{h}𝔤\cong U𝔤\left[\left[h\right]\right],$ as algebras. The quantum group has a triangular decomposition corresponding to that of $𝔤,$ $Uh𝔤=Uh𝔫-⊗ Uh𝔥⊗Uh𝔫+ andUh𝔟+ =Uh𝔥⊗Uh𝔫+.$

### The category $𝒪$

If $M$ is a ${U}_{h}𝔤$ module and $\lambda \in {𝔥}^{*}$ the $\lambda$ weight space of $M$ is $Mλ = { m∈M | am=λ (a)m, for all a∈𝔥 } .$ The category $𝒪$ is the category of ${U}_{h}𝔤$ modules $M$ such that

 (a) $M=\underset{\lambda \in {𝔥}^{*}}{⨁}{M}_{\lambda },$ (b) For all $m\in M,$ $\text{dim}\left({U}_{h}{𝔫}^{+}m\right)$ is finite, (c) $M$ is finitely generated as a ${U}_{h}𝔤$ module.
For $\mu \in {𝔥}^{*}$ let $M\left(\mu \right)$ be the Verma module of highest weight $\mu ,$ and let
$L\left(\mu \right)$ be the irreducible module of highest weight $\mu \text{.}$
The irreducible module $L\left(\mu \right)$ is the quotient of $M\left(\mu \right)$ by a maximal proper submodule and $M\left(\mu \right)={U}_{h}𝔤{\otimes }_{{U}_{h}{𝔟}^{+}}ℂ{v}_{\mu }^{+}$ where $ℂ{v}_{\mu }^{+}$ is the one dimensional ${U}_{h}{𝔟}^{+}$ module spanned by a vector ${v}_{\mu }^{+}$ such that $a{v}_{\mu }^{+}=\mu \left(a\right){v}_{\mu }^{+}$ for $a\in 𝔥$ and ${U}_{h}{𝔫}^{+}{v}_{\mu }^{+}=0\text{.}$ Every module $M\in 𝒪$ has a finite composition series with factors $L\left(\mu \right),$ $\mu \in {𝔥}^{*}\text{.}$ Each of the sets ${[L(λ)] | λ∈𝔥*} and{[M(λ)] | λ∈𝔥*}$ (where $\left[M\right]$ denotes the isomorphism class of the module $M\text{)}$ are bases of the Grothendieck group of the category $𝒪\text{.}$

If $M$ is a ${U}_{h}𝔤$ module generated by a highest weight vector of weight $\lambda$ (i.e., a vector ${v}^{+}$ such that $a{v}^{+}=\lambda \left(a\right){v}^{+}$ for $a\in 𝔥$ and ${U}_{h}{𝔫}^{+}{v}^{+}=0\text{)}$ then any element of the center $Z\left({U}_{h}𝔤\right)$ acts on $M$ by a constant, $zm=χλ(z)m, for z∈Z(Uh𝔤), m∈M, χλ(z) ∈ℂ.$ For each ${U}_{h}𝔤$ module $M\in 𝒪$ let $M[λ]=⨁ν∈Q Mλ+ν[λ], whereQ=∑i=1n ℤαi,$ ${\alpha }_{1},\dots ,{\alpha }_{n}$ are the simple roots and $Mλ+ν[λ] = { m∈Mλ+ν | there is k∈ℤ>0 such that (z-χλ(z))k m=0 for all z∈Z(Uh𝔤) } .$ Then $M=⨁λM[λ],$ where the sum is over all integrally dominant weights $\lambda \in {𝔥}^{*}$ i.e., $\lambda \in {𝔥}^{*}$ such that $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\notin {ℤ}_{<0}$ for all $\alpha \in {R}^{+}\text{.}$ To summarize, there is a decomposition of the category $𝒪,$ $𝒪=⨁λ𝒪[λ], (2.1)$ where the sum is over all integrally dominant weights $\lambda \in {𝔥}^{*}$ and ${𝒪}^{\left[\lambda \right]}$ is the full subcategory of modules $M\in 𝒪$ such that $M={M}^{\left[\lambda \right]}\text{.}$ The Grothendieck group of the category ${𝒪}^{\left[\lambda \right]}$ has bases ${[L(μ)] | μ∈Wλ∘λ} and {[M(μ)]μ∈Wλ∘λ}. (2.2)$ where the integral Weyl group corresponding to $\lambda$ is $Wλ=⟨sα | ⟨λ+ρ,α∨⟩∈ℤ⟩ andw∘λ=w(λ+ρ) -ρ, w∈W,λ∈𝔥*. (2.3)$ defines the dot action of $W$ on ${𝔥}^{*}\text{.}$

### Jantzen filtrations

Following the notations for the quantum group used in [LRa1977, §2], let $𝔥,$ ${X}_{1},\dots ,{X}_{r}$ and ${Y}_{1},\dots ,{Y}_{r}$ be the standard generators of the quantum group ${U}_{h}𝔤$ which satisfy the quantum Serre relations. The Cartan involution $\theta :{U}_{h}𝔤\to {U}_{h}𝔤$ is the algebra anti-involution defined by $θ(Xi)=Yi, θ(Yi)=Xi, and θ(a)=a,for a∈𝔥. (2.4)$ The Cartan involution $\theta$ is a coalgebra homomomorphism. A contravariant form on a ${U}_{h}𝔤$ module $M$ is a symmetric bilinear form $⟨,⟩:M×M\to ℂ$ such that $⟨um1,m2⟩= ⟨m1,θ(u)m2⟩, u∈Uh𝔤, m1,m2∈M.$

Fix $\lambda \in {𝔥}^{*}$ and $\delta \in {𝔥}^{*}$ such that $\lambda +t\delta$ is integrally dominant for all small positive real numbers $t\text{.}$ Consider $t$ as an indeterminate and consider the Verma module $M(λ+tδ)=Uh 𝔤[t]⊗Uh𝔟+[t] ℂvλ+tδ$ as the module for ${U}_{h}𝔤\left[t\right]=ℂ\left[t\right]{\otimes }_{ℂ}{U}_{h}𝔤$ generated by a vector ${v}^{+}$ such that $a{v}^{+}=\left(\lambda +t\delta \right)\left(a\right){v}^{+}$ for $a\in 𝔥$ and ${U}_{h}{𝔫}^{+}\left[t\right]{v}^{+}=0\text{.}$ There is a unique contravariant form ${⟨,⟩}_{t}:M\left(\lambda +t\delta \right)×M\left(\lambda +t\delta \right)\to ℂ\left[t\right]$ such that ${⟨{v}^{+},{v}^{+}⟩}_{t}=1\text{.}$ Define $M(λ+tδ)(j)= { m∈M(λ+tδ) | ⟨m,n⟩t∈ tjℂ[t] for all n∈M(λ+tδ) } .$ The “specialization of $M\left(\lambda +t\delta \right)\left(j\right)$ at $t=0\text{”}$ is $M(λ)(j)= image of M(λ+tδ) (j) in M(λ+tδ) ⊕ℂ[t]ℂ[t] /tℂ[t]$ and the Jantzen filtration of $M\left(\lambda \right)$ is $M(λ)= M(λ)(0)⊇ M(λ)(1)⊇ ⋯. (2.5)$ By [Jan1980, Theorem 5.3], the Jantzen filtration is a filtration of $M\left(\lambda \right)$ by ${U}_{h}𝔤$ modules, the module $M{\left(\lambda \right)}^{\left(1\right)}$ is a maximal proper submodule of $M\left(\lambda \right)$ and each quotient $M{\left(\lambda \right)}^{\left(i\right)}M{\left(\lambda \right)}^{\left(i+1\right)}$ has a nondegenerate contravariant form. It is known [Bar1983] that the Jantzen filtration does not depend on the choice of $\delta \text{.}$ It is a deep theorem [BBe1993] that the quotients $M{\left(\lambda \right)}^{\left(i\right)}/M{\left(\lambda \right)}^{\left(i+1\right)}$ are semisimple and that if $w\in {W}^{\mu }$ and $y\in {W}^{\mu }$ are maximal length in their cosets $w{W}_{\mu +\rho }$ and $y{W}_{\mu +\rho },$ respectively, then the Kazhdan-Lusztig polynomial for ${W}^{\mu }$ is $∑j≥0 [ M(w∘μ)(j)/ M(w∘μ)(j+1) :L(y∘μ) ] v12(ℓ(y)-ℓ(w)-j) =Pwy(v), (2.6)$ where $\ell$ is the length function on ${W}^{\mu }$ and $\left[M{\left(w\circ \mu \right)}^{\left(j\right)}/M{\left(w\circ \mu \right)}^{\left(j+1\right)}:L\left(y\circ \mu \right)\right]$ is the multiplicity of the simple module $L\left(y\circ \mu \right)$ in the $j\text{th}$ factor of the Jantzen filtration of $M\left(\lambda \right)\text{.}$

### The BGG resolution

Not all simple modules $L\left(\lambda \right)$ 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 $\mu \in {𝔥}^{*}$ be such that $-\left(\mu +\rho \right)$ 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 $\nu ={w}_{0}\circ \mu \text{.}$ Define a resolution $0⟶Cℓ(w0) ⟶⋯⟶C2⟶d2 C1⟶d1C0 ⟶L(ν)⟶0 (2.7)$ of the simple module $L\left(\nu \right)$ by Verma modules by setting $Cj=⨁ℓ(w)=j M(w∘ν),$ where the sum is over all $w\in {W}_{J}^{\mu }$ of length $j,$ and defining the map $dj:Cj→Cj-1, by the matrix(dj)v,w = { εv,w ιv,w, if v→w, 0, otherwise,$ where $v\to w$ means that there is a (not necessarily simple) root $\alpha$ such that $w={s}_{\alpha }v$ and $\ell \left(w\right)=\ell \left(v\right)-1,$ the maps ${\iota }_{v,w}$ are fixed choices of inclusions $ιv,w:M(v∘ν) ↪M(w∘ν),and εv,w=±1,$ are fixed choices of signs such that $εu,vεv,w=- εu,v′εv′,w ifu→v→w, u→v′→wandv≠ v′.$ 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 $-\left(\mu +\rho \right)$ is dominant and regular then, in the Grothendieck group of the category $𝒪,$ $[L(ν)]= ∑w∈WJμ (-1)ℓ(w) [M(w∘ν)], (2.8)$ where $\nu ={w}_{0}\circ \mu$ and ${w}_{0}$ is the longest element of ${W}_{J}^{\mu }\text{.}$

### ${Ř}_{MN}$ matrices and the quantum Casimir ${C}_{M}$

Let ${U}_{h}𝔤$ be the Drinfeld-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra $𝔤\text{.}$ There is an invertible element $ℛ=\sum {a}_{i}\otimes {b}_{i}$ in (a suitable completion of) ${U}_{h}𝔤\otimes {U}_{h}𝔤$ such that, for any two ${U}_{h}𝔤$ modules $M$ and $N,$ the map $ŘMN: M⊗N ⟶ N⊗M m⊗n ⟼ ∑bin⊗aim M ⊗ N N ⊗ M$ is a ${U}_{h}𝔤$ module isomorphism. There is also a quantum Casimir element ${e}^{-h\rho }u$ in the center of ${U}_{h}𝔤$ and, for a ${U}_{h}𝔤$ 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 $ℛ$ and ${e}^{-h\rho }u$ satisfy relations (see [LRa1977, (2.1-2.12)]), which imply that, for ${U}_{h}𝔤$ modules $M,N,P$ and a ${U}_{h}𝔤$ module isomorphism ${\tau }_{M}:M\to M,$ $M ⊗ N N ⊗ M τM = M ⊗ N N ⊗ M τM ŘMN (idN⊗τM) = (τM⊗idN) ŘMN, (2.9) M⊗(N⊗P) (N⊗P)⊗M = M ⊗ N ⊗ P P ⊗ N ⊗ M (M⊗N)⊗P P⊗(M⊗N) = M ⊗ N ⊗ P P ⊗ N ⊗ M ŘM,N⊗P = (ŘMN⊗idP) (idN⊗ŘMP) ŘM⊗N,P = (idM⊗ŘNP) (ŘMP⊗idN), (2.10) CM⊗N = (ŘMNŘNM)-1 (CM⊗CN). (2.11)$ The relations (2.9) and (2.10) together imply the braid relation $M ⊗ N ⊗ P P ⊗ N ⊗ M = M ⊗ N ⊗ P P ⊗ N ⊗ M (ŘMN⊗idP) (idN⊗ŘMP) (ŘNP⊗idM) = (idM⊗ŘNP) (ŘMP⊗idN) (idP⊗ŘMN), (2.12)$ If $M$ is a ${U}_{h}𝔤$ module generated by a highest weight vector ${v}^{+}$ of weight $\lambda$ then, by [Dri1990, Prop. 3.2], $CM= q-⟨λ,λ+2ρ⟩ idM. (2.13)$ Note that $⟨\lambda ,\lambda +2\rho ⟩=⟨\lambda +\rho ,\lambda +\rho ⟩-⟨\rho ,\rho ⟩$ are the eigenvalues of the classical Casimir operator [Dix1996, 7.8.5]. If $M$ is a finite dimensional ${U}_{h}𝔤$ module then $M$ is a direct sum of the irreducible modules $L\left(\lambda \right),$ $\lambda \in {P}^{+},$ and $CM=⨁λ∈P+ q-⟨λ,λ+2ρ⟩ Pλ,$ where ${P}_{\lambda }:M\to M$ is the projection onto ${M}^{\left[\lambda \right]}$ in $M\text{.}$ From the relation (2.11) it follows that if $M=L\left(\mu \right),$ $N=L\left(\nu \right)$ are finite dimensional irreducible ${U}_{h}𝔤$ modules then ${Ř}_{MN}{Ř}_{NM}$ acts on the $\lambda$ isotypic component ${L\left(\lambda \right)}^{\oplus {c}_{\mu \nu }^{\lambda }}$ of the decomposition $L(μ)⊗L(ν)= ⨁λ L(λ)⊕cμνλ (2.14)$ by the constant ${q}^{⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩}$ where ${c}_{\mu \nu }^{\lambda }$ are positive integers. Suppose that $M$ and $N$ are ${U}_{h}𝔤$ modules with contravariant forms ${⟨,⟩}_{M}$ and ${⟨,⟩}_{N},$ respectively. Since the Cartan involution is a coalgebra homomorphism the form on $M\otimes N$ defined by $⟨m1⊗n1,m2⊗n2⟩ =⟨m1,m2⟩M ⟨n1,n2⟩N, (2.15)$ for ${m}_{1},{m}_{2}\in M,$ ${n}_{1},{n}_{2}\in N,$ is also contravariant. If $\theta$ is the Cartan involution defined in (2.4) then a formula of Drinfeld [Dri1990, Prop. 4.2], states $(θ⊗θ)(ℛ)= ∑ibi⊗ai,$ from which it follows that $⟨ (m1⊗n1)ŘMN, n2⊗m2 ⟩ = ∑i ⟨ (bi⊗ai) (n1⊗m1), n2⊗m2 ⟩ = ∑i ⟨ n1⊗m1, (θ(bi)⊗θ(ai)) (n2⊗m2) ⟩ = ∑i ⟨ n1⊗m1, (ai⊗bi) (n2⊗m2) ⟩ = ∑i ⟨ m1⊗n1,bi m2⊗ain2 ⟩ .$ Thus $⟨ (m1⊗n1) ŘMN,n2⊗m2 ⟩ = ⟨ m1⊗n1, (n2⊗m2) ŘNM ⟩ . (2.16)$

## 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.