# representation theory – About the finite direct sum of the full subcategory of the \$ mathcal {O} ^ mathfrak {p} \$ category

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?