Fusion Product of Positive Level Representations and Lie Algebra Homology

Last updated: 26 March 2015

This is an excerpt of the paper Fusion Product of Positive Level Representations and Lie Algebra Homology by Shrawan Kumar, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA.

A Certain Complex and Lie Algebra Homology

(Definition of a Complex). Fix a positive integer $\ell$ and a finite-dimensional irreducible representation $V=V\left(\nu \right)$ of $𝔤$ with highest weight $\nu \in {P}_{\ell }^{+}\text{.}$ Recall the parabolic BGG resolution for Kac-Moody Lie algebras (cf. [RCW1982] and [Kum1990, Theorem (3.27)]): $⋯⟶Fp⟶⋯⟶F1 ⟶F0⟶L(V,ℓ) ⟶0, (2.1)$ where ${F}_{p}≔\underset{w\in \stackrel{\sim }{W}\prime ,\ell \left(w\right)=p}{⨁}M\left(V\left(\left(w*{\nu }_{\ell }\right){|}_{𝔥}\right),\ell \right),$ $\stackrel{\sim }{W}\prime$ denotes the set of those $w\in \stackrel{\sim }{W}$ such that $w$ is the smallest element in its coset $Ww,$ $\ell \left(w\right)$ denotes the length of $w,$ and ${\nu }_{\ell }\in {\stackrel{\sim }{𝔥}}^{*}$ is defined by ${\nu }_{\ell }{|}_{𝔥}=\nu ,$ ${\nu }_{\ell }\left(K\right)=\ell \text{.}$ (Observe that $\left(w*{\nu }_{\ell }\right){|}_{𝔥}\in {P}^{+}$ for any $w\in \stackrel{\sim }{W}\prime \text{.)}$

Take any $\mu \in {P}^{+},$ realize $V\left(\mu \right)$ as a module for $\mathrm{\Omega }\left(𝔤\right)$ via evaluation at 1, and consider it as a module for $\stackrel{\sim }{𝔤}$ (by letting $K$ act trivially on $V\left(\mu \right)\text{)}$ via the Lie algebra homomorphism $\pi$ (cf. (1.1)). We denote $V\left(\mu \right)$ with this $\stackrel{\sim }{𝔤}\text{-module}$ structure by $V\left(\mu ;1\right)\text{.}$

Tensoring (2.1) with $V\left(\mu ;1\right),$ we get a resolution: $⋯⟶ Fp⊗V(μ;1)⟶ ⋯⟶ F0⊗V(μ;1)⟶ L(V,ℓ)⊗V(μ;1)⟶ 0. (2.2)$ Tensoring the complex (2.2) with $ℂ$ over $U\left({\stackrel{\sim }{𝔲}}^{-}\right)$ and using the Hopf principle (cf. [GLe1976, Proposition 1.7]) we obtain a complex of $𝔤\text{-modules}$ and $𝔤\text{-module}$ maps: $⋯⟶Fˆp⟶δp ⋯⟶δ1Fˆ0 ⟶0, (2.3)$ where ${\stackrel{ˆ}{F}}_{p}≔\underset{w\in \stackrel{\sim }{W}\prime ,\ell \left(w\right)=p}{⨁}\left[V\left(\left(w*{\nu }_{\ell }\right){|}_{𝔥}\right)\otimes V\left(\mu \right)\right]\text{.}$

The maps ${\delta }_{p}$ are quite non-trivial, e.g., the map ${\delta }_{\prime }:{\stackrel{ˆ}{F}}_{1}\to {\stackrel{ˆ}{F}}_{0}$ can be explicitly described as below:

First of all, ${\stackrel{ˆ}{F}}_{1}=V\left(\nu +m\theta \right)\otimes V\left(\mu \right),$ where $m=\ell +1-\left(\nu ,{\theta }^{\vee }\right),$ and of course ${\stackrel{ˆ}{F}}_{0}=V\left(\nu \right)\otimes V\left(\mu \right)\text{.}$

The map ${\delta }_{1}:V\left(\nu +m\theta \right)\otimes V\left(\mu \right)\to V\left(\nu \right)\otimes V\left(\mu \right)$ is the composite map $\eta \circ \left(j\otimes I\right)$ given as follows: (observe that $V\left(\theta \right)$ is the adjoint representation of $𝔤\text{)}$ $V(ν+mθ)⊗ V(μ)↪j⊗I V(ν)⊗𝔤⊗m ⊗V(μ)⟶η V(ν)⊗V(μ),$ where $j:V\left(\nu +m\theta \right)↪V\left(\nu \right)\otimes {𝔤}^{\otimes m}$ is the canonical inclusion and $\eta \left(v\otimes \left({x}_{1}\otimes \cdots \otimes {x}_{m}\right)\otimes w\right)=v\otimes {x}_{m}\cdots {x}_{1}w,$ for $v\in V\left(\nu \right),$ $w\in V\left(\mu \right)$ and ${x}_{i}\in 𝔤\text{.}$

Observe that, since $M\left(V\left(\left(w*{\nu }_{\ell }\right){|}_{𝔥}\right),\ell \right)$ are $\left({\stackrel{\sim }{𝔲}}^{-}\right)\text{-free,}$ ${H}_{•}\left(\stackrel{ˆ}{F}\right)$ is isomorphic to the Lie algebra homology ${H}_{•}\left({\stackrel{\sim }{𝔲}}^{-},L\left(V\left(\nu \right),\ell \right)\otimes V\left(\mu ;1\right)\right)\text{.}$ Moreover, if a $𝔤\text{-module}$ $V\left(\lambda \right)$ is a component of ${H}_{0}\left(\stackrel{ˆ}{F}\right),$ then $\lambda \in {P}_{\ell }^{+}\text{.}$

We make the following conjecture.

Assume that $\mu \in {P}_{\ell }^{+}$ (and of course $\nu \in {P}_{\ell }^{+}\text{).}$ Then for any $\lambda \in {P}_{\ell }^{+},$ the $𝔤\text{-module}$ $V\left(\lambda \right)$ does not occur as a component of the homology ${H}_{p}\left({\stackrel{\sim }{𝔲}}^{-},L\left(V\left(\nu \right),\ell \right)\otimes V\left(\mu ;1\right)\right)={H}_{p}\left(\stackrel{ˆ}{F}\right)$ of the complex (2.3), for any $p\ge 1\text{.}$

If we take $\mu =0,$ this conjecture follows immediately from Kostant's result on $𝔫\text{-homology}$ (for affine Kac-Moody Lie algebras, as proved by Garland-Lepowsky [GLe1976]).

Assuming the validity of the above conjecture, and using the Hochschild-Serre spectral sequence for the Lie algebra homology, we obtain the following:

Let $𝔤$ be any (finite-dimensional simple) Lie algebra, for which Conjecture 2.3 is true. Decompose the Lie algebra homology (as $𝔤\text{-modules)}$ $H0(𝔲∼-L(V(ν),ℓ)⊗V(μ;1)) =∑θ∈Pℓ+mθ V(θ),$ where ${m}_{\theta }={m}_{\theta }\left(\nu ,\mu \right)$ is the multiplicity of $V\left(\theta \right)$ in the left-hand side. Then for any $i\ge 0$ (and any such $𝔤,$ i.e., for which Conjecture 2.3 is true), $Hi(𝔲∼-L(V(ν),ℓ)⊗V(μ;1)) =∑θmθ ∑w∈W∼′ℓ(w)=i V((w*θℓ)|𝔥),$ as $𝔤\text{-modules.}$

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