Common use of Module Clause in Contracts

Module. For i, j ≥ 0 define the subset M (i,j)× of the (i, j)-level of M as follows: M (i,j)× := {v ∈ M (i,j)|En−1nv = 0} n−1n Remark 4.1.8. If En−1nv = 0 then of course E(r) v = 0 for all r > 0. Therefore by Proposition 2.4.10 M (i,j)× is the fixed point space of the GL(n − 2)-module M (i,j) under the root subgroup Uєn−1−єn . In particular it is a GL(n − 2) module. Furthermore if j = 0 then En−1nM (i,j) = 0 since otherwise it would be contained in a weight space for M corresponding to a weight with a negative entry.