As it is shown in ** Representations of semi-simple Lie algebras in the BGG category $ mathcal {O} $**, each non-zero module $ M in mathcal {O} ^ mathfrak {p} $ has finished filtration with nonzero quotients, each of these modules having the highest weight $ mathcal {O} ^ mathfrak {p} $.

So the action of $ Z (g) $ sure $ M $ is finished.

Let

$

M ^ chi = {v in M | (z – chi (z)) ^ nv = 0 text {for some $ n in mathbb {Z} _ {> 0} $ according to $ z $} },

$

then $ z – chi (z) $ acts locally nilpotently on $ M ^ chi $ for everyone $ z in Z ( mathfrak {g}) $ and $ M ^ chi $ is a $ U ( mathfrak {g}) $-module of $ M $.

Note by $ mathcal {O} ^ mathfrak {p} _ chi $ the full subcategory of $ mathcal {O} ^ mathfrak {p} $ whose objects are of the form $ M ^ chi $, then we have the decomposition of the next direct sum

$

mathcal {O} ^ mathfrak {p} = bigoplus _ { chi} mathcal {O} ^ mathfrak {p} _ chi,

$

or $ chi = chi_ lambda $ for some people $ lambda in mathfrak {h} ^ * $.

Because $ M $ is generated by a lot of weight

vectors, so it must act from the direct sum of many finite non-null sub-modules $ M ^ chi $.

We call $ lambda, mu in mathfrak {h} ^ * $ are linked if $ lambda = w cdot mu: = w ( mu + rho) – rho $ for some people $ w in W $ or $ rho $ is half of the positive roots and $ W $ is the Weyl group of $ Phi $.

The set of $ Phi ^ + _ I $dominant integral in $ mathfrak {h} ^ * $ is

$

Lambda ^ + _ I = { lambda in mathfrak {h} ^ *: langle lambda, alpha ^ lor row in mathbb {Z} ^ { ge 0} text { for all} alpha in Phi ^ + _ I },

$ there is a fact that $ L ( lambda) in mathcal {O} ^ mathfrak {p} $ Yes Yes $ lambda in Lambda ^ + _ I $.

So here's the question, leave $ mathfrak {g} = mathfrak {sl} (2, mathbb {C}) $, so note $ Phi $ the root system and $ Delta = { alpha, beta } $ the set of simple roots in $ Phi $. Consider $ t alpha + (1-t) ( alpha + beta) in Lambda_I ^ + $ with $ 0 the t $ 1. Then there should be an infinity of unbound weights (since $ t $ infinite and $ t alpha + (1-t) ( alpha + beta) $ are weights are between $ alpha $ and $ alpha + beta $) such as $ L (t alpha + (1-t) ( alpha + beta)) in mathcal {O} ^ mathfrak {p} _ { chi_ {t alpha + (1-t) ( alpha + beta)}} $,

who should then give an infinity of $ mathcal {O} ^ mathfrak {p} _ { chi} $.

Can any one give me some examples / explain why my intuition fails to convince me that there is indeed a lot $ Phi ^ + _ I $integral integral up to the link?