Cherednik category O has a standard-filtered projective generator

Write $H=H_{1,c}$, $X=S({\mathfrak{h}}^{\ast })$, $Y=S(\mathfrak{h})$, and $B=Y⋊\mathbb{C}W$. Let $a_E$ be the scalar of $z=\dim (\mathfrak{h})\mathrm{/}2-{\sum }_s{c}_{s}s$ on $E$, and put $\mathrm{\Sigma }=\{a_E:E\in \mathrm{Irr}W\}$. The Euler element is $\mathrm{e}\mathrm{u}={\sum }_i{x}_i{y}_i+z$. The Euler and finite-length argument in [F1] gives locally finite Euler action, finite generalized eigenspaces, and spectra contained in $\mathrm{\Sigma }+{\mathbb{Z}}_{\ge 0}$: the spectral assertion holds on standards and their simple quotients, then on every finite composition series. Fixed generalized eigenspaces are exact, since one can decompose a preimage into its finitely many generalized eigencomponents and retain the desired component. Put $b={\min }_E\mathrm{Re}(a_E)$ and choose $n$ such that $\mathrm{Re}(\beta )-(n+1)<b$ for every $\beta \in \mathrm{\Sigma }$. Every degree-$(n+1)$ monomial in $Y$ sends $M[\beta ]$ to $M[\beta -(n+1)]=0$, for every object $M$. Higher monomials also vanish by factoring them. Thus $Y_{>n}M[\beta ]=0$ uniformly over all objects and all $\beta \in \mathrm{\Sigma }$.

Let $E_n=\mathbb{C}W\otimes (Y\mathrm{/}{Y}_{>n})$, with diagonal $W$-action and $Y$ multiplying the second factor, and put $G_n=H{\otimes }_B{E}_n$. Filtering the truncated $Y$-factor by descending degree gives a finite standard filtration: PBW makes induction exact, and each layer $S^d(\mathfrak{h})\otimes \mathbb{C}W$ is a finite semisimple $W$-module, since averaging a projection over $W$ splits each submodule. Hence $G_n\in {\mathcal{O}}_c$. PBW identifies its vector space with $X\otimes \mathbb{C}W\otimes (Y\mathrm{/}{Y}_{>n})$. Give $x$ degree $1$, $y$ degree $-1$, and $W$ degree $0$. Each total-degree space is finite-dimensional because the $Y$-degree is at most $n$. The operator $Z=\mathrm{e}\mathrm{u}-\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}$ commutes with every homogeneous generator, since both commutators equal its degree times that generator. It preserves total degree. The finite degree-zero space contains the finite generating set $\mathbb{C}W\otimes 1$, so $Z$ is locally finite with a finite set of generalized eigenvalues on all of $G_n$. Therefore the generalized $Z$-eigenspaces $G_n^{\beta }$ are genuine $H$-direct summands.

Induction adjunction is explicit: an $H$-map $G_n\to M$ is determined by $m=f(1\otimes 1\otimes 1)$, and this may be any vector killed by $Y_{>n}$. The value on $w\otimes p(y)$ is $p(y)wm$. This is $Y$-linear, and its $W$-linearity follows from the diagonal action; equivalently, $w\otimes p$ corresponds to $pw$ in the regular $B$-quotient. On the degree-zero generator, evaluation intertwines precomposition by $Z$ with ${\mathrm{e}\mathrm{u}}_M$, because $f(Zv)={\mathrm{e}\mathrm{u}}_{M}f(v)$. Consequently ${\mathrm{Hom}}_H(G_n^{\beta },M)=M[\beta ]\cap {\mathrm{Ann}}_M(Y_{>n})$. For $\beta \in \mathrm{\Sigma }$, step 1.1 makes this exactly $M[\beta ]$. Thus $P_{\beta }=G_n^{\beta }$ represents an exact functor and is projective. The finite direct sum $P_{\mathrm{\Sigma }}={⨁}_{\beta \in \mathrm{\Sigma }}{P}_{\beta }$ detects every nonzero object: a nonzero vector has a finite $Y$-orbit, whose commuting nilpotent operators have a nonzero common kernel; that singular space contains an irreducible $W$-module, on which $\mathrm{e}\mathrm{u}$ acts by one of the scalars in $\mathrm{\Sigma }$.

Hom spaces are finite-dimensional: choose finitely many generators of the source decomposed into generalized Euler eigenvectors; their images lie in finitely many finite-dimensional target eigenspaces. The evaluation map $P_{\mathrm{\Sigma }}\otimes \mathrm{Hom}(P_{\mathrm{\Sigma }},M)\to M$ is therefore a map from a finite direct sum. If its cokernel were nonzero, a nonzero map from $P_{\mathrm{\Sigma }}$ to that cokernel would lift by projectivity and contradict the definition of the evaluation image. Thus evaluation is surjective. Its kernel belongs to ${\mathcal{O}}_c$, so each object has a presentation $P_{\mathrm{\Sigma }}^b\to P_{\mathrm{\Sigma }}^a\to M\to 0$. Put $\mathrm{\Gamma }=\mathrm{End}(P_{\mathrm{\Sigma }}{)}^{\mathrm{o}\mathrm{p}}$, a finite-dimensional algebra. Applying $\mathrm{Hom}(P_{\mathrm{\Sigma }},-)$ to these presentations proves equivalence with finite-dimensional $\mathrm{\Gamma }$-modules: a presentation by finite free $\mathrm{\Gamma }$-modules lifts to a map between finite sums of $P_{\mathrm{\Sigma }}$, and its cokernel is the inverse construction; the same presentations identify morphisms. This argument requires only finite presentations, not cocompleteness of ${\mathcal{O}}_c$.

The standard filtration of $G_n$ in step 2.1 is graded. A section $\mathrm{\Delta }(F)$ arising from $Y$-degree $d$ has its generating copy of $F$ in total degree $-d$. On that entire section, $Z$ acts by the scalar $a_F+d$, since $\mathrm{e}\mathrm{u}$ increases by one with each multiplication by $x$ and the grading does the same. Applying the exact generalized $Z$-projection to the filtration keeps a whole standard section if $a_F+d=\beta$ and kills it otherwise. Hence $P_{\beta }$ and $P_{\mathrm{\Sigma }}$ have finite standard flags. Since the $\beta$ are distinct, their sum is itself a direct summand of the single $G_n$.

If ${\mathrm{Ext}}_{\mathcal{O}}^1(\mathrm{\Delta }(E),\mathrm{\Delta }(F))\mathrm{\ne }0$, then $a_E-a_F\in {\mathbb{Z}}_{>0}$. Indeed, in an extension lift the generating $E$ into its generalized Euler space of eigenvalue $a_E$ and split the resulting $W$-surjection by averaging. Each $y$ sends this lift into $\mathrm{\Delta }(F)[a_E-1]$. Unless $a_E-a_F$ is a positive integer, that space is zero. The lift is then singular and induction adjunction gives a splitting from $\mathrm{\Delta }(E)$. This includes the case $a_E=a_F$. Also $\mathrm{End}(\mathrm{\Delta }(E))=\mathbb{C}$, because an endomorphism is determined on the irreducible generating copy of $E$. A map $\mathrm{\Delta }(E)\to \mathrm{\Delta }(F)$ vanishes when $\mathrm{Re}(a_E)<\mathrm{Re}(a_F)$, or when those real parts agree and $E\mathrm{\ne }F$: the target either has no space of weight $a_E$ or has only its degree-zero $F$ there.

Let $B$ be the subobject formed by the factors of minimal real part in a standard filtration of $M$. By step 6.1, adjacent inversions split, so the filtration can be sorted by increasing $\mathrm{Re}(a)$; extensions among equal-real-part factors split as well. Hence $B$ is a direct sum of standards and $M\mathrm{/}B$ has only larger-real-part factors. Every map $\mathrm{\Delta }(E)\to M$ from a minimal label factors through $B$, since its map to $M\mathrm{/}B$ vanishes on each higher-real-part standard section by step 6.1 and induction along that filtration. Conversely, the summand inclusions in $B$ are such maps, so their images sum to $B$. For minimal labels, step 6.1 gives $\mathrm{Hom}(\mathrm{\Delta }(E),\mathrm{\Delta }(F))=0$ when $E\mathrm{\ne }F$ and $\mathrm{End}(\mathrm{\Delta }(E))=\mathbb{C}$. Thus $\dim \mathrm{Hom}(\mathrm{\Delta }(E),M)$ is the multiplicity of $\mathrm{\Delta }(E)$ in $B$. The evaluation map from ${⨁}_E\mathrm{\Delta }(E{)}^{\dim \mathrm{Hom}(\mathrm{\Delta }(E),M)}$ onto $B$ is therefore surjective between finite-length modules of equal length, so it is an isomorphism. This identifies $B$ canonically as the sum of all images from minimal standards. Every idempotent $e:M\to M$ preserves $B$; on $B$, it is a block matrix over $\mathrm{End}(\mathrm{\Delta }(E))=\mathbb{C}$ with no maps between distinct labels, so $eB$ is a direct sum of standards. By induction on the number of real-part levels, the image of the induced idempotent on $M\mathrm{/}B$ is standard-filtered. Since $eM\cap B=eB$, the sequence $0\to eB\to eM\to e(M\mathrm{/}B)\to 0$ is exact. Concatenating the flags of its standard-filtered ends gives a standard filtration of $eM$.

The finite-algebra model from step 4.1 has projective covers by [F2]. Transporting them back gives $P(E)\twoheadrightarrow L(E)$ with simple head $L(E)$ once. Each is a summand of a finite sum of $P_{\mathrm{\Sigma }}$, hence is standard-filtered by step 7.1. Sort its flag as there. Its top real-part group is a quotient which is a direct sum of standards, so its simple heads are quotients of $P(E)$. The simple head condition forces this group to be exactly one $\mathrm{\Delta }(E)$. Thus $0\to K(E)\to P(E)\to \mathrm{\Delta }(E)\to 0$, where all standard labels $F$ in $K(E)$ have $\mathrm{Re}(a_F)<\mathrm{Re}(a_E)$. Euler spectra split every object into congruence classes modulo $\mathbb{Z}$, because generators shift Euler eigenvalues by integers. The indecomposable $P(E)$ lies in its head's class, so every such $F$ also has $a_E-a_F\in \mathbb{Z}$ and hence $a_E-a_F\in {\mathbb{Z}}_{>0}$. Other composition factors of $\mathrm{\Delta }(E)$ have $a_F-a_E\in {\mathbb{Z}}_{>0}$, since a proper submodule misses degree zero. These are precisely the standard-support and projective-kernel conditions for the stated finite highest-weight order.

The simple labels are distinct and have scalar endomorphisms. If $L(E)\cong L(F)$, the two lower Euler bounds force both $a_E-a_F$ and $a_F-a_E$ to be nonnegative integers, so they are zero; the common lowest $W$-space then identifies $E=F$. An endomorphism of $L(E)$ is determined on its irreducible generating lowest space and is scalar. Every nonzero map from a projective cover $P(E)$ to a simple has its unique maximal kernel, so $\mathrm{Hom}(P(E),L(F))={\delta }_{EF}\mathbb{C}$. Exactness of $\mathrm{Hom}(P(E),-)$ along composition series gives $\dim \mathrm{Hom}(P(E),M)=[M:L(E)]$. This proves the projective-generator and highest-weight assertions with the multiplicity formula used below.