Projective Functors and the Soergel Functor in Category O

Assume the Axiom of Choice inherited from the cited invariant-theory results (The Axiom of Choice). Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Weyl group $W$, acting on ${\mathfrak{h}}^{\ast }$ by the root reflections of Root reflections and the Weyl group action (Finite Weyl root system, lattice and chamber conventions), hence on the symmetric algebra $S=S({\mathfrak{h}}^{\ast })$ by graded algebra automorphisms; a nondegenerate $W$-invariant bilinear form identifies $\mathfrak{h}$ with ${\mathfrak{h}}^{\ast }$ and thus identifies $S$ with $S(\mathfrak{h})$ as a graded $W$-algebra. Such a form is the complexification of a positive definite form on the real Cartan, obtained by averaging any positive definite form over finite $W$. Fix this identification; the page writes $S=S(\mathfrak{h})$ for either presentation. Write $S_{+}^W={⨁}_{d>0}(S^W{)}_d,I:=S{S}_{+}^W,$ so that $I$ is the homogeneous ideal of $S$ generated by the invariants of positive degree; this is the specialization to the reflection group case of the quotient construction of Finite linear invariant and coinvariant polynomial algebras, because $S_{+}^W$ is exactly the space of positive-degree elements of the invariant algebra $S^W$.

The coinvariant algebra of the pair $(\mathfrak{h},W)$ is the graded quotient $C=C(\mathfrak{h},W):=S\mathrm{/}I\mathrm{.}$ It is a commutative graded $\mathbb{C}$-algebra with $C_0=\mathbb{C}$: the ideal $I$ is homogeneous of positive degree, so constants survive and every homogeneous piece of $S$ maps onto its image. The quotient map $\rho :S⟶C,\rho (f)=f+I,$ is a surjective graded algebra homomorphism, and the action of $W$ on $S$ descends to a $\mathbb{C}$-linear action on $C$ by graded algebra automorphisms, because $I$ is $W$-stable. The algebra $C$ is finite dimensional: $W$ is generated by its root reflections, so Chevalley shephard todd for finite weyl groups writes $S$ as a free module of finite rank over the polynomial algebra $S^W$, and Weyl coinvariant hilbert series has order w dimension gives ${\dim }_{\mathbb{C}}C=\mathrm{\mid }W\mathrm{\mid }<\mathrm{\infty }$ with the graded Hilbert series recorded there.

Degree normalization. Throughout the Soergel material on this page the Cartan generators are given degree $2$, that is $\mathrm{deg}\mathfrak{h}=2$; the symmetric algebra $S$ is then graded with $\mathrm{deg}{S}_d=2d$ for homogeneous polynomials of ordinary degree $d$, and $C$ inherits this doubled grading. In particular a basic invariant of ordinary degree $d_i$ has page degree $2d_i$, and the top nonzero degree of $C$ is $2N$ with $N={\sum }_i(d_i-1)$. The separate degree-sum lemma on this page proves that this number equals the positive-root count.

Auxiliary permutation model for type $A$. For $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_n$ one may instead work with the place-permuting $S_n$-action on the auxiliary polynomial ring $S_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}}=\mathbb{C}[x_1,\dots ,x_n]$, where a permutation $\sigma$ sends $x_i$ to $x_{\sigma (i)}$, and with the quotient $C_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}}:=\mathbb{C}[x_1,\dots ,x_n]\mathrm{/}(e_1,\dots ,e_n),$ where $e_1,\dots ,e_n$ are the elementary symmetric polynomials. Splitting the permutation representation ${\mathbb{C}}^n=\mathfrak{h}\oplus \mathbb{C}\cdot (1,\dots ,1)$ into its sum-zero Cartan part and its fixed average coordinate exhibits $\mathbb{C}[x_1,\dots ,x_n]\cong S(\mathfrak{h})\otimes \mathbb{C}[e_1]$ with $e_1=x_1+\cdots +x_n$ the invariant coordinate, and restriction to the hyperplane $e_1=0$ has kernel $(e_1)$. By Fundamental theorem of symmetric polynomials: unique expression as a polynomial in $e_1,\dots ,e_n$, $e_1,\dots ,e_n$ generate the invariant algebra and its positive-degree ideal of the permutation action, while $e_2,\dots ,e_n$ restrict to basic invariants of $S(\mathfrak{h}{)}^W$, dividing out by the positive-degree invariants commutes with this presentation and yields a canonical isomorphism of graded algebras $C(\mathfrak{h},W)\cong C_{\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}}=\mathbb{C}[x_1,\dots ,x_n]\mathrm{/}(e_1,\dots ,e_n),$ in which $x_i$ has degree $2$ and $e_i$ has degree $2i$. Elements of $C$ are written as classes $\rho (f)$; the auxiliary model is used only for explicit rank computations and never replaces the intrinsic definition.

The quotient and finite coordinate constructions require no additional choice; the freeness and Hilbert-series conclusions inherit AC from their suppliers.

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

Work in category $\mathcal{O}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with the conventions of The classical BGG category O: a Cartan subalgebra $\mathfrak{h}$, a positive system with simple roots ${\alpha }_i$, the Weyl group $W$ of Root reflections and the Weyl group action, the Weyl vector $\rho$, and the dot action $w\cdot \lambda =w(\lambda +\rho )-\rho$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$ and $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$.

The fixed weight. Fix a weight $\lambda$ which is integral, dot-regular and dominant: $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\in \mathbb{Z}$ for every root $\alpha$ (so the integral Weyl group $W_{\lambda }$ of The integral Weyl group of a weight is all of $W$), and $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\mathrm{\ne }0$, respectively $>0$, for every root, respectively every positive root (Dot-Weyl facets and single-wall translation data). The set ${\mathrm{\Phi }}_{\lambda }$ of The integral Weyl group of a weight is therefore the full root system and $W_{\lambda }=W$.

The block. The regular point $\lambda +\rho$ lies in an open chamber. An element fixing it preserves that chamber, so Simple transitivity on Weyl chambers makes its stabilizer trivial; hence the linkage class $C=W_{\lambda }\cdot \lambda =W\cdot \lambda$ has exactly $\mathrm{\mid }W\mathrm{\mid }$ elements. The regular integral block is the full subcategory ${\mathcal{O}}_{\lambda }:={\mathcal{O}}_C$ of objects of $\mathcal{O}$ whose simple composition factors have labels in $C$, as in Central-character summands refine into linkage blocks; it is nonzero, abelian and extension-closed as a block, and $K_0({\mathcal{O}}_{\lambda })$ is the subgroup of $K_0(\mathcal{O})$ of The Grothendieck group and character of O spanned by the classes of objects of the block.

Its labelled objects. For $w\in W$ define the standard and costandard objects and the simple object by $\mathrm{\Delta }(w\cdot \lambda ):=M(w\cdot \lambda ),\mathrm{\nabla }(w\cdot \lambda ):=D(M(w\cdot \lambda )),L(w\cdot \lambda ):=M(w\cdot \lambda )\mathrm{/}J(w\cdot \lambda ),$ where $M$ is the Verma module of Verma modules, $D$ the restricted dual of Standard and costandard objects, and $J(w\cdot \lambda )$ the sum of all proper submodules, so that $L(w\cdot \lambda )$ is the unique simple quotient of $\mathrm{\Delta }(w\cdot \lambda )$ by A Verma module has a unique simple quotient. By The simple objects of O the simple objects of ${\mathcal{O}}_{\lambda }$ are exactly the $L(w\cdot \lambda )$ for $w\in W$, pairwise non-isomorphic. Let $P(w\cdot \lambda )$ denote a projective cover of $L(w\cdot \lambda )$, which may be chosen inside the linkage class of $w\cdot \lambda$ and is unique up to isomorphism and indecomposable (Category O has enough projectives); every such $P(w\cdot \lambda )$ carries a finite Verma flag (Projectives in category O have finite Verma flags, Finite Verma flags and their multiplicities).

Triangularity of the Verma-flag matrix. For all $v,w\in W$, $[\mathrm{\Delta }(w\cdot \lambda ):L(v\cdot \lambda )]\mathrm{\ne }0⟹v\cdot \lambda \le w\cdot \lambda ,[\mathrm{\Delta }(w\cdot \lambda ):L(w\cdot \lambda )]=1,$ where the multiplicity is the composition multiplicity of Standard and costandard objects. Indeed $(P(v\cdot \lambda ):\mathrm{\Delta }(w\cdot \lambda ))=[\mathrm{\Delta }(w\cdot \lambda ):L(v\cdot \lambda )]$ by BGG reciprocity and $(P(v\cdot \lambda ):\mathrm{\Delta }(w\cdot \lambda ))\mathrm{\ne }0$ forces $w\cdot \lambda -v\cdot \lambda \in Q^{+}$ while $(P(v\cdot \lambda ):\mathrm{\Delta }(v\cdot \lambda ))=1$ by The triangular restriction on projective Verma flags.

The antidominant weight. Let $w_0\in W$ be the longest element. Since $⟨w_0\cdot \lambda +\rho ,{\alpha }^{\vee }⟩=⟨\lambda +\rho ,w_0^{-1}{\alpha }^{\vee }⟩<0$ for every positive root $\alpha$, the antidominant Verma module $\mathrm{\Delta }(w_0\cdot \lambda )$ is simple (Antidominant regular Verma modules are simple), so $\mathrm{\Delta }(w_0\cdot \lambda )=L(w_0\cdot \lambda )$ and this is the unique standard object of the block that is simple: for $w\mathrm{\ne }{w}_0$ the regular weight $w\cdot \lambda$ is not antidominant, so some positive root $\alpha$ has $⟨w\cdot \lambda +\rho ,{\alpha }^{\vee }⟩\in {\mathbb{Z}}_{>0}$, and Verma embedding for an arbitrary positive root embeds $M(s_{\alpha }\cdot (w\cdot \lambda ))$ as a proper submodule of $\mathrm{\Delta }(w\cdot \lambda )$. The weight $w_0\cdot \lambda$ is called the antidominant weight of the block and $\mathrm{\Delta }(w_0\cdot \lambda )$ its antidominant Verma module.

Grothendieck group. By Simple and standard bases of K0(O) applied to the finite label set $C=W\cdot \lambda$, both the classes $[\mathrm{\Delta }(w\cdot \lambda )]$ and the classes $[L(w\cdot \lambda )]$, $w\in W$, form $\mathbb{Z}$-bases of $K_0({\mathcal{O}}_{\lambda })$, and the transition matrix between them is unitriangular for any linear order of $W$ extending the order on the dot orbit induced by $\le$.

Every statement above uses the published block, Verma, duality and projective-cover theory named in the citations, and that theory assumes the Axiom of Choice; the definition itself selects nothing further.

Assume the Axiom of Choice, inherited from invariant theory. Let $\mathrm{\Phi }$ be a finite reduced crystallographic root system of rank $r$ in the real Cartan space $E$ with Weyl group $W$ (Finite Weyl root system, lattice and chamber conventions, Root reflections and the Weyl group action), and let $S=\mathbb{C}[E]$ carry the induced graded $W$-action with Cartan generators in ordinary degree $1$. Let $d_1,\dots ,d_r$ be the ordinary degrees of basic invariants, so that $S^W=\mathbb{C}[w_1,\dots ,w_r]$ with $w_i$ homogeneous of degree $d_i$ (Chevalley shephard todd for finite weyl groups). Then ${\sum }_{i=1}^r(d_i-1)=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }\mathrm{.}$ With the page degree normalization $\mathrm{deg}E=2$ the coinvariant algebra $S\mathrm{/}(S_{+}^{W}S)$ has top degree $2{\sum }_i(d_i-1)=2\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=\mathrm{\mid }\mathrm{\Phi }\mathrm{\mid }$.

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, $S=S(\mathfrak{g})$ the symmetric algebra of the adjoint representation, $R=S^{\mathfrak{g}}$ its invariants, $Z=Z(U(\mathfrak{g}))$ the center of the enveloping algebra, and let $\mathcal{H}\subseteq S$ be a Kostant harmonic subspace, so that multiplication is an isomorphism $R{\otimes }_{\mathbb{C}}\mathcal{H}\to S$ of graded $\mathfrak{g}$-modules with $R$ acting trivially (Kostant harmonic decomposition of the symmetric algebra).

For every finite-dimensional $\mathfrak{g}$-module $E$ with zero-weight space $E_0$, the module ${\mathrm{Hom}}_{\mathfrak{g}}(E,S)$ is a free graded $R$-module of rank $\dim E_0$, and the module ${\mathrm{Hom}}_{\mathfrak{g}}(E,U(\mathfrak{g}))$ is a free $Z$-module of rank $\dim E_0$; with the filtered lifting $U(\mathfrak{g})=Z\cdot \mathrm{sym}(\mathcal{H})$ of The enveloping algebra is free over its center these freenesses are compatible with the natural quotient maps. More precisely, ${\mathrm{Hom}}_{\mathfrak{g}}(E,S)\cong R{\otimes }_{\mathbb{C}}{\mathrm{Hom}}_{\mathfrak{g}}(E,\mathcal{H}),{\mathrm{Hom}}_{\mathfrak{g}}(E,U(\mathfrak{g}))\cong Z{\otimes }_{\mathbb{C}}{\mathrm{Hom}}_{\mathfrak{g}}(E,\mathcal{H}),$ where ${\mathrm{Hom}}_{\mathfrak{g}}(E,\mathcal{H})$ is a finite-dimensional graded vector space of dimension $\dim E_0$; consequently every central reduction has adjoint $E$-multiplicity $\dim E_0$, and the natural map ${\mathrm{Hom}}_{\mathfrak{g}}(E,S){\otimes }_{R}R\mathrm{/}{R}_{+}\to {\mathrm{Hom}}_{\mathfrak{g}}(E,S\mathrm{/}(R_{+}S))$ is an isomorphism onto a space of dimension $\dim E_0$.

Assume the Axiom of Choice, inherited from the published block theory (Translation functors by tensoring and projection, The Axiom of Choice).

Fix a regular integral block ${\mathcal{O}}_{\lambda }$ as in The regular integral block and its antidominant weight, with inclusion ${\mathrm{incl}}_{\lambda }:{\mathcal{O}}_{\lambda }\to \mathcal{O}$ and exact projection ${\mathrm{pr}}_{\lambda }:\mathcal{O}\to {\mathcal{O}}_{\lambda }$ of Translation functors by tensoring and projection, so that ${\mathrm{pr}}_{\lambda }\circ {\mathrm{incl}}_{\lambda }=\mathrm{id}$; the block is a direct summand of $\mathcal{O}$ and contains the projective covers $P(w\cdot \lambda )$ of Category O has enough projectives. For a finite-dimensional $\mathfrak{h}$-semisimple $\mathfrak{g}$-module $E$ define the endofunctor $F_E:={\mathrm{pr}}_{\lambda }\circ (E\otimes -)\circ {\mathrm{incl}}_{\lambda }\text{\hspace{0.17em}⁣}:{\mathcal{O}}_{\lambda }⟶{\mathcal{O}}_{\lambda }\mathrm{.}$

Basic properties. The functor $F_E$ is well defined and additive: $E\otimes -$ is an exact endofunctor of $\mathcal{O}$ and maps $\mathcal{O}$ into itself (Finite-dimensional tensoring preserves O), and ${\mathrm{pr}}_{\lambda }$ is exact. Hence $F_E$ is exact as a composite of exact functors. It preserves projective objects: if $P\in {\mathcal{O}}_{\lambda }$ is projective, then $E\otimes P$ is projective in $\mathcal{O}$ (Finite-dimensional tensoring preserves projectives in category O), and ${\mathrm{pr}}_{\lambda }$ preserves projectives (Exact projections onto linkage blocks preserve projectives). The zero module $E=0$ gives the zero functor, and $E=\mathbb{C}$ the identity functor, since the one-dimensional trivial module acts by the identity.

Projective functors. A projective functor on ${\mathcal{O}}_{\lambda }$ is an endofunctor of ${\mathcal{O}}_{\lambda }$ isomorphic to a direct summand of some $F_E$, $E$ finite dimensional and $\mathfrak{h}$-semisimple. Write ${\mathcal{F}}_{\lambda }$ for the full subcategory of the endofunctor category of ${\mathcal{O}}_{\lambda }$ whose objects are the projective functors, with arbitrary natural transformations as morphisms; it is additive with the pointwise biproducts of Additive category, and "direct summand" means natural maps $i:F\to F_E$, $p:F_E\to F$ with $pi=1_F$; then $ip$ is an idempotent natural transformation.

Closure properties. The class of projective functors is closed under finite direct sums, composition, direct summands and passage to adjoints:

1. $F_{E\oplus E^{\mathrm{\prime }}}\cong F_E\oplus F_{E^{\mathrm{\prime }}}$, since tensoring with a direct sum distributes over direct sums and ${\mathrm{pr}}_{\lambda }$ is additive.
2. $F_E\circ F_{E^{\mathrm{\prime }}}$ is a natural direct summand of $F_{E\otimes E^{\mathrm{\prime }}}$: the natural block decomposition of $E^{\mathrm{\prime }}\otimes X$ splits off its $\lambda$-component before applying $E\otimes -$ and the final projection. Its complementary blocks can return to ${\mathcal{O}}_{\lambda }$, so an isomorphism with the entire $F_{E\otimes E^{\mathrm{\prime }}}$ is not asserted.
3. $F_{E^{\ast }}$ is both a left and a right adjoint of $F_E$: this is Translation functors are exact and biadjoint applied to the two generalized central characters of the block, since $F_{E^{\ast }}={\mathrm{pr}}_{\lambda }\circ (E^{\ast }\otimes -)\circ {\mathrm{incl}}_{\lambda }$ is the translation functor $T_{{\chi }_{\lambda },E^{\ast },{\chi }_{\lambda }}$ and similarly for $F_E$.
4. A direct summand of an exact additive functor is exact, and a direct summand of a functor preserving projective objects again preserves projective objects: apply the splitting to the image of each object. Hence every projective functor is exact and preserves projective objects, and composition of summand splittings makes a composite of projective functors a summand of the composite of their ambient tensor functors. Clause 2 therefore proves closure under composition; clause 1 proves closure under sums. For an adjoint, take the mate of the summand idempotent under clause 3; the triangle identities carry its square to itself, and its image is the corresponding adjoint summand of $F_{E^{\ast }}$.

This is the definition of Bernstein-Gelfand [BGe, Section 3.1] and of [Soe90, Section 1.2]; unlike the functors $T_{\lambda }^{\mu }$ of Translation functors by tensoring and projection, projective functors are not required to come from a single finite-dimensional tensor functor, only from a direct summand of one.

Assume the Axiom of Choice, inherited from the published invariant-theory items. Let $W$ act on the Cartan space $\mathfrak{h}$ by reflections and on $S=S(\mathfrak{h})$ by graded algebra automorphisms, with coinvariant algebra $C=S\mathrm{/}I$ as in The coinvariant algebra of a finite Weyl group. Let $w_1,\dots ,w_r$ be homogeneous basic invariants of page degrees $2d_1,\dots ,2d_r$, where $d_1,\dots ,d_r$ are the classical degrees ($\mathrm{deg}\mathfrak{h}=1$; for the Cartan of ${\mathfrak{s}\mathfrak{l}}_n$ they are $2,3,\dots ,n$), so that $S^W=\mathbb{C}[w_1,\dots ,w_r]$ (Chevalley shephard todd for finite weyl groups), and let $\rho :S\to C$ be the quotient by the ideal $I=(w_1,\dots ,w_r)$. Put $N:={\sum }_{i=1}^r(d_i-1)$. Then: