Indecomposable projective functors are indexed by the Weyl group

Let $\mathcal{A}$ be the class of projective functors and evaluate a natural transformation $\alpha$ at $\mathrm{\Delta }(\lambda )$. For ambient tensor functors, lift the finite equivariant coefficient map in [F1] to $U$. Its tensor coefficients act on every module by the same formula, yielding a natural $\mathfrak{g}$-map. For projected functors, extend a map between projected evaluations by inclusion and projection to the ambient tensors, lift it there, then project on the output and include on the input. The same inclusion/retraction procedure handles functor summands. Thus evaluation is surjective on Hom spaces.

The evaluation kernel is nilpotent. If $\alpha$ vanishes on $\mathrm{\Delta }(\lambda )$, then by exactness and naturality it vanishes on every standard, since each standard embeds in $\mathrm{\Delta }(\lambda )$ by [F2] and exactness preserves the embedding; on a fixed projective generator with a finite Verma flag of length $k$, vanishing on the standard factors strictly lowers the flag, so a composition of $k$ such maps is zero. Hence $\ker$ is a nilpotent ideal on every finite additive functor sum.

Idempotents lift across a nilpotent ideal (if $e^2-e=d$ lies in a nilpotent ideal, replace $e$ by $e+(1-2e)d$; its idempotent error lies in the square of the previous error ideal, so finitely many repetitions give an idempotent) and an element $1+$nilpotent is invertible by a finite geometric series; hence decompositions and isomorphism classes of projective functors coincide with those of their evaluations at $\mathrm{\Delta }(\lambda )$, and the endomorphism algebra of an indecomposable is local exactly when its evaluation is. Hom spaces are finite-dimensional for a separate reason: a natural transformation between right-exact functors is determined by its value on a finite projective generator $H={⨁}_{x}P(x\cdot \lambda )$, since every object has a finite presentation by sums of $H$. Its component is a member of the finite-dimensional space $\mathrm{Hom}(FH,F^{\mathrm{\prime }}H)$ (maps are determined by images of finitely many weight-homogeneous generators into finite-dimensional weight spaces). Thus the functor category has finite-dimensional endomorphism algebras and split idempotents. Nilpotent-kernel lifting matches its indecomposable decompositions to those of the evaluations, proving Krull–Schmidt.

By [F3] the wall word ${\theta }_{s_k}\cdots {\theta }_{s_1}$ evaluated at $\mathrm{\Delta }(\lambda )$ has the decompositions $P(w\cdot \lambda )\oplus {⨁}_{x<w}P(x\cdot \lambda {)}^{m_x}$; by step 3.1 this decomposition lifts uniquely to a decomposition of the functor, producing ${\mathrm{\Theta }}_w$ with ${\mathrm{\Theta }}_w\mathrm{\Delta }(\lambda )\cong P(w\cdot \lambda )$ and with all other summands labelled $x<w$; conversely every indecomposable projective functor is a summand of such a word, so the labels $w\in W$ exhaust the indecomposables. This proves (1) and (2).

If two projective functors have the same class operator on $K_0$, then their evaluations at $\mathrm{\Delta }(\lambda )$ are projectives with the same class; by the unitriangular BGG matrix [F4] their indecomposable decompositions agree, so the evaluations are isomorphic, and by step 3.1 the functors are naturally equivalent; the converse is formal. This proves (3).