1. Category $O$
Let ${\germ g}\subset {\germ b}\subset {\germ h}$ be a semisimple complex Lie algebra, a Borel and a Cartan. Let ${\scr O}={\scr O}({\germ g},{\germ b})$ be the category of all finitely generated ${\germ g}$-modules that are locally finite over ${\germ b}$ and semisimple over ${\germ h}$, [BG80].
Let $U=U({\germ g})$ be the enveloping algebra, $Z\subset U$ its
center, $Z^+={\rm Ann}_Z{\bf C}$ the central annihilator of trivial
one-dimensional ${\germ g}$-module ${\bf C}$. To any $\lambda\in{\germ
h}^*$ we associate the Verma module $M(\lambda)=U\otimes_{U({\germ
b})}{\bf C}_\lambda$, its unique simple quotient $L(\lambda)$ and the
(indecomposable) projective cover $P(\lambda)$ of $L(\lambda)$ in
${\scr O}$. Let $L$ be the direct sum of all $L(\lambda)$ such that
$Z^+L(\lambda)=0$. There are finitely many of those, parametrized by
the Weyl group ${\scr W}$. Let $P$ be the direct sum of their
respective projective covers.
Theorem 1. [Soe90, BGS91] There exists an
isomorphism of finite dimensional ${\bf C}$-algebras ${\rm End}_{\germ
g}P\cong {\rm Ext}^*_{\scr O}(L,L)$. Furthermore the graded ring on
the right is Koszul.
Remarks:
(1) By ${\rm Ext}^*$ we mean the direct sum of all ${\rm
Ext}^i$, made into a ring via the cup product. Since $P$ is only
unique up to nonunique isomorphism, the isomorphism of the theorem
cannot be canonical.
(2) A "Koszul ring" is by definition a positively graded ring $A=\oplus_{i\geq 0}A^i$ such that (1) $A^0$ is semisimple and (2) considered as a left $A$-module $A^0$ admits a graded projective resolution $\cdots\to P_2\to P_1\to P_0\to A^0$ such that $P_i$ is generated by its elements that are homogeneous of degree $i$, $P_i=AP^i_i$. We can reformulate (2) as follows: Let $A$-gr denote the category of all ${\bf Z}$-graded $A$-modules $M=\oplus M^i$. (2)$'$ If $M,N\in A$-gr are concentrated in one degree, say $M=M^m$, $N=N^n$, then ${\rm Ext}^i_{A\text{-gr}}(M,N)=0$ unless $i=n-m$. Thus Koszulity is an analogon of semisimplicity for ${\bf Z}$-graded rings.
(3) Put ${\scr O}_0=\{M\in{\scr O}|(Z^+)^nM=0$ for $n\gg 0\}$. Since $P\in{\scr O}_0$ is a projective generator, the functor ${\rm Hom}_{\germ g}(P,)\colon {\scr O}_0\to$ Mof-End$_{\germ g}P$ from ${\scr O}_0$ to the category of finitely generated right ${\rm End}_{\germ g}P$-modules is an equivalence of categories. The theorem says that ${\rm End}_{\germ g}P$ admits a ${\bf Z}$-grading. Thus in some sense (which can be made precise) the category ${\scr O}_0$ admits a ${\bf Z}$-grading and is even "graded semisimle". The same is true for all other blocks of ${\scr O}$.
2. Positive characteristic
Let $R$ be a root system, $p$ a prime, $G$ the corresponding simply
connected semisimple algebraic group over ${\bf F}_p$, and ${\germ
g}={\rm Lie}\,G$ its Lie algebra. This comes equipped with a formal
$p$-power map $X\mapsto X^{[p]}$, $X\in{\germ g}$. For example, for
$R=A_n$ we get ${\germ g}={\rm sl}(n+1,{\overline {\bf F}}_p)$ and
$X^{[p]}=X^p$ is just the $p$-th power in the matrix ring. Let
$U^{[p]}=U({\germ g})/(X^p-X^{[p]})$ be the restricted enveloping
algebra of ${\germ g}$. (Be careful that here $X^p$ stands for the
$p$-th power in $U({\germ g})!) This is a finite dimensional
$\overline{\bf F}_p$ algebra.
Conjecture 1. [AJS94] Suppose $p$ is at least the
Coxeter number. Then $U^[[p]}$ admits a Koszul grading compatible with
its natural ${\bf Z}R$-grading.
Remark. By a Koszul grading on a ring we mean a ${\bf Z}$-grading that
makes the ring to a Koszul ring. The conjecture says that the category
of $G_1T$-modules should be "graded semisimple". In [AJS94] we
construct a ${\bf Z}$-grading on $U^[[p]}$ and prove that it gives a
Koszul grading on the regular blocks of $U^{[p]}$ if Lusztig's
conjecture holds, i.e. if $p\gg 0$ (for $R$ fixed). So the problem is
to treat singular blocks and small $p$.
3. Real groups
Let $X$ be a complex algebraic variety and $H$ a complex algebraic group acting on $X$ with finitely many orbits. Then we can form a positivity graded ring ${\rm Ext}^*_HX$, the "geometric extension algebra", as follows:
Let ${\scr D}_H(X)$ be the equivariant derived category, see [BL92].
Let ${\scr L\in{\scr D}_H(X)$ be the direct sum of all simple
$H$-equivariant perverse sheavers on $X$ (i.e. take on from each
isomorphism class). Then put ${\rm Ext}^*_HX=\oplus_i{\rm Hom}_{{\scr
D}_H(X)}({\scr L},{\scr L}[i])$. If $H\subset K$ is a closed subgroup,
then ${\rm Ext}^*_HX={\rm Ext}^*_K(Kx_HX)$. Using localization,
theorem 1 can be rewrtten as ${\scr O}_0\cong {\rm Mof-Ext}^*_N(G/B)$.
Let $\tilde{\scr O}_0$ be the category of all finite length ${\germ
g}$-modules which hall all composition factors in ${\scr O}_0$. For a
graded ${\bf C}$-algebra $A^\bullet$ let Nil-$A$ denote the category
of all right $A$-modules $M$ such that $\dim_{\bf C}M<\infty$,
$MA^i=0$ for $i\gg 0$. As a variant of the above, one shows [Soe92a]
$\tilde{\scr O}_0\cong {\rm Nil-Ext}^\bullet_B(G/B)$. Now one can prove that
the Ext-algebra on the right does not change when we replaced $G<B$ by
their Langlands-duals $G^\vee, B^\vee$. Furthermore $\tilde{\scr O}_0$
is equivalent to the category ${\scr H}$ of Harish-Chandra-modules
with trivial central character for the real Lie group $G({\bf C})$.
Finally $G^\vee/B^\vee\times V^\vee/B^\vee\cong
G^\vee\times_{B^\vee}G^\vee/B^\vee$ and using the induction
equivalence we arrive at
Theorem 2. ${\scr H}\cong {\rm
Nil-Ext}^\bullet_{G^\vee}(G^\vee/B^\vee\times F^\vee/B^\vee)$
Now $G^\vee/B^\vee\times G^\vee/B^\vee$ is just the modified Langlands parameter space of [ABV92] corresponding to ${\scr H}$, and from there it is evident how the theorem should generalize to arbitrary real reductive Lie groups [Soe92b].
Bibliography: