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.
(Definition of a Complex). Fix a positive integer and a finite-dimensional irreducible representation of with highest weight Recall the parabolic BGG resolution for Kac-Moody Lie algebras (cf. [RCW1982] and [Kum1990, Theorem (3.27)]): where denotes the set of those such that is the smallest element in its coset denotes the length of and is defined by (Observe that for any
Take any realize as a module for via evaluation at 1, and consider it as a module for (by letting act trivially on via the Lie algebra homomorphism (cf. (1.1)). We denote with this structure by
Tensoring (2.1) with we get a resolution: Tensoring the complex (2.2) with over and using the Hopf principle (cf. [GLe1976, Proposition 1.7]) we obtain a complex of and maps: where
The maps are quite non-trivial, e.g., the map can be explicitly described as below:
First of all, where and of course
The map is the composite map given as follows: (observe that is the adjoint representation of where is the canonical inclusion and for and
Observe that, since are is isomorphic to the Lie algebra homology Moreover, if a is a component of then
We make the following conjecture.
Assume that (and of course Then for any the does not occur as a component of the homology of the complex (2.3), for any
If we take this conjecture follows immediately from Kostant's result on (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 be any (finite-dimensional simple) Lie algebra, for which Conjecture 2.3 is true. Decompose the Lie algebra homology (as where is the multiplicity of in the left-hand side. Then for any (and any such i.e., for which Conjecture 2.3 is true), as