The antidominant projective and the Soergel functor

Assume the Axiom of Choice, inherited from the published block theory (The Axiom of Choice, Central-character summands refine into linkage blocks).

In the regular integral block ${\mathcal{O}}_{\lambda }$ of The regular integral block and its antidominant weight fix the antidominant weight $w_0\cdot \lambda$ and let $P:=P(w_0\cdot \lambda )$ be a projective cover of $L(w_0\cdot \lambda )$ (Category O has enough projectives); it is unique up to isomorphism and indecomposable (Projective covers in O are indecomposable and unique, Projective object), and it is an injective object of ${\mathcal{O}}_{\lambda }$. The Soergel functor is $V:={\mathrm{Hom}}_{\mathcal{O}}(P,-)\text{\hspace{0.17em}⁣}:{\mathcal{O}}_{\lambda }⟶{\mathrm{End}}_{\mathcal{O}}(P{)}^{\mathrm{o}\mathrm{p}}\text{-}\mathbf{m}\mathbf{o}\mathbf{d},M⟼{\mathrm{Hom}}_{\mathcal{O}}(P,M),$ with the left action of ${\mathrm{End}}_{\mathcal{O}}(P{)}^{\mathrm{o}\mathrm{p}}$ on ${\mathrm{Hom}}_{\mathcal{O}}(P,M)$ induced by composition on the right of the source: $a\cdot f=f\circ a$ (Projective functors on a regular integral block). Its values on the standard objects and on the indecomposable projectives are written $V\mathrm{\Delta }(w\cdot \lambda ):=V(\mathrm{\Delta }(w\cdot \lambda ))$ and $VP(w\cdot \lambda ):=V(P(w\cdot \lambda ))$. The functor $V$ is exact, because $P$ is projective, and hence left exact; the identity $VP\cong {\mathrm{End}}_{\mathcal{O}}(P)$ with its regular module structure is the Yoneda identification. Replacing the chosen projective cover $P$ by an isomorphic one changes $V$ by a natural equivalence of functors compatible with the endomorphism-ring identification, so the construction depends only on the isomorphism class of $P$.

Well-definedness: $P$ is injective. We prove that $P$ is an injective object of ${\mathcal{O}}_{\lambda }$ using only the listed suppliers, following the argument of [Soe90, Sections 1.2 and 2.1] in its algebraic form. Put $\eta :=-\rho$; then $W\cdot \eta =\{\eta \}$, because $w\cdot \eta =w(\eta +\rho )-\rho =-\rho$ for every $w$ (Central characters are dot-Weyl orbits). By The simple objects of O the simple objects are the $L(\mu )$ with $L(\mu )$ determined by $\mu$, and the composition factors of $M(\eta )$ have highest weights in $W\cdot \eta \cap (\eta -Q^{+})=\{\eta \}$; hence $M(\eta )=L(\eta )$ is simple. By BGG reciprocity (BGG reciprocity) applied to the single-label block, and the existence of Verma flags for projectives (Projectives in category O have finite Verma flags), the projective cover $P(\eta )$ has a Verma flag with the single factor $\mathrm{\Delta }(\eta )=M(\eta )=L(\eta )$, so $P(\eta )\cong L(\eta )$; the block ${\mathcal{O}}_{\eta }$ has a single simple object, which is projective, hence is semisimple by enough projectives (Category O has enough projectives), and therefore every object of ${\mathcal{O}}_{\eta }$, in particular $L(\eta )$, is injective as well as projective in ${\mathcal{O}}_{\eta }$ (Injective object). A projective object of a block is projective in $\mathcal{O}$ (Exact projections onto linkage blocks preserve projectives), and an injective object of a block is injective in $\mathcal{O}$ because the block decomposition is a direct sum of abelian subcategories and the exact projections restrict split exactness blockwise. Second, we identify a projective-injective object in the block ${\mathcal{O}}_{\lambda }$: let $E=L(\lambda +\rho )$ be the finite-dimensional simple module of highest weight $\lambda +\rho$ (Finite-dimensional simple modules are classified by dominant highest weights) and set $Q:={\mathrm{pr}}_{{\chi }_{\lambda }}(E\otimes L(\eta ))\in {\mathcal{O}}_{\lambda },$ the exact central projection of Translation functors by tensoring and projection. Since $L(\eta )$ is projective and injective in $\mathcal{O}$, the tensor product $E\otimes L(\eta )$ is projective and injective (Finite-dimensional tensoring preserves projectives in category O), and its direct summand $Q$ is projective and injective in $\mathcal{O}$; the general exact tensor-block biadjunction of Translation functors are exact and biadjoint gives the same conclusion with all adjoints and, by Finite-dimensional tensoring preserves projectives in category O, injectivity (equivalently, $Q$ is a direct summand of a projective-injective object, and injectivity passes to summands). Third, we compute a one-dimensional Hom space: since $M(a)$ with $a=w_0\cdot \lambda$ is simple by the same top-weight argument as for $\eta$ (every other highest weight in $W\cdot a$ lies strictly above $a$ in the root order), the tensor product $E^{\ast }\otimes M(a)$ has a finite Verma flag whose factors are the $\mathrm{\Delta }(a+\nu )$ with $\nu$ running through the weights of the dual module $E^{\ast }$, by Finite-dimensional tensoring preserves Verma flags; here $E^{\ast }=L(-w_0(\lambda +\rho ))$ (Highest weight of the dual representation). Projecting to the central character ${\chi }_{\eta }$ retains exactly the factors with $a+\nu \in W\cdot \eta =\{\eta \}$, that is $\nu =-w_0(\lambda +\rho )$, the highest weight of $E^{\ast }$ of multiplicity one, so ${\mathrm{pr}}_{{\chi }_{\eta }}(E^{\ast }\otimes M(a))=L(\eta )\mathrm{.}$ The tensor-hom adjunction and the tensor-block adjunction now give ${\mathrm{Hom}}_{\mathcal{O}}(Q,L(a))\cong {\mathrm{Hom}}_{\mathcal{O}}(L(\eta ),E^{\ast }\otimes M(a))\cong {\mathrm{Hom}}_{\mathcal{O}}(L(\eta ),L(\eta ))\cong \mathbb{C},$ because every homomorphism from the block-$\eta$ simple lands in the ${\chi }_{\eta }$-summand and $L(\eta )$ is simple. Choose the nonzero, hence surjective, element $q:Q\to L(a)$ and lift it through the projective cover $P(a)\to L(a)$ to a map $Q\to P(a)$; its image is a submodule of $P(a)$ surjecting onto the head $L(a)$, hence is all of $P(a)$ by essentiality of a projective cover, so the lift is surjective and splits by projectivity of $P(a)$. Therefore $P(a)$ is a direct summand of the injective object $Q$ (Injective object), and direct summands of injective objects are injective, so $P=P(w_0\cdot \lambda )$ is injective. This argument uses The BGG criterion for homomorphisms between Verma modules only through the standard embedding criterion implicit in the top-weight computations, and it does not assume the Endomorphismensatz.

The once-each flag. The tensor flag of $E\otimes M(-\rho )$ has labels $-\rho +\gamma$ for weights $\gamma$ of $E=L(\lambda +\rho )$. Projection to ${\chi }_{\lambda }$ keeps precisely $\gamma \in W(\lambda +\rho )$, each of multiplicity one by Weights of a finite-dimensional simple module lie in the norm ball. Thus $Q$ has each $\mathrm{\Delta }(x\cdot \lambda )$ once. Every such Verma contains $M(w_0\cdot \lambda )=L(a)$ by Dominant integral dot translates embed canonically in the Verma module. BGG reciprocity gives $(P(a):\mathrm{\Delta }(x\cdot \lambda ))\ge 1$; the splitting $Q=P(a)\oplus Q^{\mathrm{\prime }}$ and Direct summands of Verma-filtered objects are Verma-filtered give the opposite bound one. Hence all these multiplicities equal one, $Q^{\mathrm{\prime }}$ has an empty flag and is zero, and $Q=P(a)$. Since $P(a)$ is projective, $V$ is exact and $\dim VX=[X:L(a)]$ for finite-length $X$: on simples its values are $\mathbb{C}$ at the head label and zero otherwise. Consequently $\dim V\mathrm{\Delta }(x\cdot \lambda )=1$ for every $x$. Every simple submodule of a Verma is itself a simple Verma: its universal Verma map into that Verma is injective. Thus the unique simple standard $L(a)$ is its only possible socle label, and Homomorphism spaces between Verma modules have dimension at most one makes this socle a single copy of $L(a)$.