# Wednesday April 15th

Section 7.6, proved theorem about translation functors on Verma modules.
We fixed an antidominant weight, since we can apply elements of $W$ to obtain the rest.
We proved that translation functors take Verma modules to Verma modules.

Corollary (Translations Have Standard Filtrations)
: If $M\in \OO_\chi$ has a standard filtration, then so does $T_\lambda^\mu M \in \OO_\mu$.

Proof
:   By induction on the length of the filtration, where the length 1 case is handled by the theorem.
    In general we have $0 \to N \to M \to M(w\cdot \lambda) \to 0$ and since $T_\lambda^\mu$ is exact, we can apply it to get another exact sequence.
    ??? See notes.

## Translation Functors and Simple Modules

Proposition (Translation Functors Applied to L for Antidominant Weights)
:   Let $\lambda, \mu \in \Lambda$ be antidominant with a facet $F$ such that $\lambda \in F$ and $\mu \in \bar F$.
    Then for all $w\in W$, $T_\lambda^\mu L(w\cdot \lambda)$ is either $L(w\cdot \mu)$ or 0.

Idea: we're pushing $\lambda$ to something more singular.

Proof
:   Apply the exact functor $T_\lambda^\mu$ to the surjection $M(w\cdot \lambda) \surjects L(w\cdot \lambda)$ so obtain $M(w\cdot \mu) \surjects M$ for some $M$.
    Since $M$ is a quotient of a Verma module, it is a highest weight module of highest weight $w\cdot \mu$.
    Suppose $M\neq 0$, we can apply $T_\lambda^\mu$ to $L(w\cdot \lambda) \injects M(w\cdot \lambda) \dual$ to obtain $M(w\cdot \mu) \surjects M \injects M(w\cdot \mu) \dual$, a nonzero map.
    By Theorem 3.3c, the image is the socle, so we obtain $M \cong L(w\cdot \mu)$.

It turns out that $T_\lambda^\mu \cong L(w\cdot \mu)$ precisely when $w\cdot \mu \in \hat{w\cdot F}$ (the upper closure, see Theorem 7.9 and example 7.7 for $\liesl(2, \CC)$).
Example: take $\rho = -1$.

So if we can figure out $L(w\cdot \lambda)$ for $\lambda$ $\rho\dash$regular, we can determine the composition factors of all Verma modules by "pushing to walls".
There's also a need to "cross walls", and there's a nice rule for what happens for Verma modules in this case.
Going "off the wall" on the other side is more complicated.

## 7.8: Category Equivalences

We saw in the case of $\liesl(2, \CC)$ and $\liesl(3, \CC)$ that the composition factors depended more on the Weyl group than the highest weight.
We want to show that $T_\lambda^\mu$ gives an equivalence of categories between $\OO_\lambda$ and $\OO_\mu$. when  $\lambda, \mu$ are antidominant and lie in the same facet.
We'll first show that it induces an isomorphism on the Grothendieck groups.

Proposition (Isomorphism of Grothendieck Groups for Weights in a Common Facet)
:   Suppose there is a single facet $F$ containing $\lambda, \mu$. 
    Then the obvious map is an isomorphism:
    \begin{align*}
    T_\lambda^\mu: K(\OO_\lambda) &\mapsvia\cong K(\OO_\mu)\\
    [M(w\cdot \lambda)] &\mapsto [M(w\cdot \mu)] \\
    [L(w\cdot \lambda)] &\mapsto [L(w\cdot \mu)]
    .\end{align*}

Proof
:   Recall that $\theset{[M(w\cdot \lambda)] \suchthat w\in W}$ (and/or replacing by $L$) forms $\ZZ\dash$basis for $K(\OO_\lambda)$ and similarly for $\mu$.
    So when $[M]\in K(\OO_\lambda)$ is written was a $\ZZ\dash$linear combination of $[M(w\cdot \lambda)]$, it's clear that $T_\mu^\lambda T_\lambda^\mu [M] = [M]$.
    So these functors are mutually inverse.
    \
    By the previous proposition, if we take $L$ instead, the result is either the identity or zero.
    But zero is impossible by what we just proved, so we must have $T_\lambda^\mu [ L(w\cdot \lambda)] = [L(w\cdot \mu)]$ forcing $T_\lambda^\mu L(w\cdot \lambda) = L(w\cdot \mu)$.

Since $K(\OO) \cong \chi_0$, the group of formal characters of modules in $\OO$, we in fact get
\begin{align*}
[M: L(w\cdot \lambda)] = [T_\lambda^\mu M: L(w\cdot \mu)] \quad \forall M\in \OO
.\end{align*}

Thus the $\lambda, \mu$ don't matter as much (as long as they're in the same facet), and the $w$ is what's important.
There is a theorem (2005) that for any artinian abelian category, and isomorphism of Grothendieck groups implies an equivalence of categories.