Fusion Product of Positive Level Representations and Lie Algebra Homology

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

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 and a finite-dimensional irreducible representation V=V(ν) of 𝔤 with highest weight νP+. Recall the parabolic BGG resolution for Kac-Moody Lie algebras (cf. [RCW1982] and [Kum1990, Theorem (3.27)]): FpF1 F0L(V,) 0, (2.1) where FpwW,(w)=pM(V((w*ν)|𝔥),), W denotes the set of those wW such that w is the smallest element in its coset Ww, (w) denotes the length of w, and ν𝔥* is defined by ν|𝔥=ν, ν(K)=. (Observe that (w*ν)|𝔥P+ for any wW.)

Take any μP+, realize V(μ) as a module for Ω(𝔤) via evaluation at 1, and consider it as a module for 𝔤 (by letting K act trivially on V(μ)) via the Lie algebra homomorphism π (cf. (1.1)). We denote V(μ) with this 𝔤-module structure by V(μ;1).

Tensoring (2.1) with V(μ;1), we get a resolution: FpV(μ;1) F0V(μ;1) L(V,)V(μ;1) 0. (2.2) Tensoring the complex (2.2) with over U(𝔲-) and using the Hopf principle (cf. [GLe1976, Proposition 1.7]) we obtain a complex of 𝔤-modules and 𝔤-module maps: Fˆpδp δ1Fˆ0 0, (2.3) where FˆpwW,(w)=p[V((w*ν)|𝔥)V(μ)].

The maps δp are quite non-trivial, e.g., the map δ:Fˆ1Fˆ0 can be explicitly described as below:

First of all, Fˆ1=V(ν+mθ)V(μ), where m=+1-(ν,θ), and of course Fˆ0=V(ν)V(μ).

The map δ1:V(ν+mθ)V(μ)V(ν)V(μ) is the composite map η(jI) given as follows: (observe that V(θ) is the adjoint representation of 𝔤) V(ν+mθ) V(μ)jI V(ν)𝔤m V(μ)η V(ν)V(μ), where j:V(ν+mθ)V(ν)𝔤m is the canonical inclusion and η(v(x1xm)w)=vxmx1w, for vV(ν), wV(μ) and xi𝔤.

Observe that, since M(V((w*ν)|𝔥),) are (𝔲-)-free, H(Fˆ) is isomorphic to the Lie algebra homology H(𝔲-,L(V(ν),)V(μ;1)). Moreover, if a 𝔤-module V(λ) is a component of H0(Fˆ), then λP+.

We make the following conjecture.

Assume that μP+ (and of course νP+). Then for any λP+, the 𝔤-module V(λ) does not occur as a component of the homology Hp(𝔲-,L(V(ν),)V(μ;1))=Hp(Fˆ) of the complex (2.3), for any p1.

If we take μ=0, this conjecture follows immediately from Kostant's result on 𝔫-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-modules) H0(𝔲-L(V(ν),)V(μ;1)) =θP+mθ V(θ), where mθ=mθ(ν,μ) is the multiplicity of V(θ) in the left-hand side. Then for any i0 (and any such 𝔤, i.e., for which Conjecture 2.3 is true), Hi(𝔲-L(V(ν),)V(μ;1)) =θmθ wW(w)=i V((w*θ)|𝔥), as 𝔤-modules.

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