Lecture 20: Kac-moody Algebra Actions on Categories, Ii

1.1. Recap. In the previous lecture we have considered the category CF := ⊕ n>0 FSn -mod. We have equipped it with two endofunctors, E = ⊕ n Res n n−1 and F = ⊕ n Ind n n+1 that are biadjoint. We have decomposed E into the direct sum of eigenfunctors, E = ⊕ i∈ZF Ei, for the endomorphism X that is given by XMm = Lnm for M ∈ FSn -mod, where Ln is the Jucys-Murphy element ∑n−1 i=1 (in). We have also considered the corresponding decomposition F = ⊕ i∈ZF Fi. Besides, we have introduced the decomposition FSn -mod = ⊕ A FSn -modA, where the summation is taken over all cardinality n multi-subsets in ZF, and FSn -modA consists of all M ∈ FSn -mod such that P (L1, . . . , Ln) acts on M with a single eigenvalue P (A), for every P ∈ Z[x1, . . . , xn] . This decomposition is related to the functors Ei, Fi as follows. Let πA denote the projection FSn -mod FSn -modA. Then, for M ∈ FSn -modA, we have EiM = πA\{i}(EM), FiM = πA∪{i}(FM). Below, we will write CF,A = FS|A| -modA. So we get the direct sum decomposition CF = ⊕ A CF,A, where the sum is taken over all multi-subsets A of ZF. Finally, we have also introduced an endomorphism T of E: TMm = (n− 1, n)m for m ∈ M,M ∈ FSn -mod. We have seen that the assignment Xi 7→ 1i−1X1d−i, Ti 7→ 1i−1T1d−i−1 extends to an algebra homomorphism H(d) → End(E).