The deformed category O family over the Cartan symmetric algebra

Assume the Axiom of Choice, inherited from the published block theory (The Axiom of Choice). Let $S=S(\mathfrak{h})$ be the symmetric algebra of the Cartan subalgebra of The classical BGG category O, graded by $\mathrm{deg}\mathfrak{h}=2$ as fixed on this page (Associative graded algebras, bimodules, and internal shifts, Polynomial rings in finitely many commuting indeterminates by iteration). Let $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$ act on $S\otimes {\mathbb{C}}_{\lambda }$ by $x\cdot (s\otimes 1)=xs\otimes 1+\lambda (x)s\otimes 1(x\in \mathfrak{h}),x\cdot (s\otimes 1)=0(x\in {\mathfrak{n}}^{+}),$ where for $x\in \mathfrak{h}$, $xs$ means multiplication by the linear polynomial $x\in S(\mathfrak{h})$; ${\mathfrak{n}}^{+}$ acts trivially. Put $M_S(\lambda ):=U(\mathfrak{g}){\otimes }_{U(\mathfrak{b})}(S\otimes {\mathbb{C}}_{\lambda }),$ a module over $\mathfrak{g}\otimes S$: the left $\mathfrak{g}$-action is induced, and $S$ acts on the right by multiplication on the last factor, which commutes with $\mathfrak{g}$. Specializing at a closed point $ϵ$ of $\mathrm{Spec}S$ gives $M_S(\lambda ){\otimes }_S{\mathbb{C}}_{ϵ}\cong M(\lambda +ϵ),$ the Verma module of Verma modules; this is the universal family of Verma modules with deformation parameter $ϵ$.

The family. For a localization $T$ of $S$ put $M_T(\lambda ):=M_S(\lambda ){\otimes }_{S}T$, and let ${\mathcal{D}}_T(\lambda )$ be the full subcategory of $\mathfrak{g}\otimes T$-modules that are direct summands of a finite direct sum of modules $E\otimes M_T(\lambda )$ for finite-dimensional $\mathfrak{g}$-modules $E$. Write ${\mathcal{D}}_R(\lambda )$ for $R=S_{(0)}$, the localization of $S$ at the origin of $\mathrm{Spec}S$. The deformed family is the assignment $T\mapsto {\mathcal{D}}_T(\lambda )$ together with the specialization functors $-{\otimes }_T\kappa (p)$ at closed points $p$ of $\mathrm{Spec}T$, where $\kappa (p)=T\mathrm{/}p$ is the residue field. The image of the tautological map $\mathfrak{h}\to T\to \kappa (p)$ is the parameter ${ϵ}_p$, and $M_T(\lambda ){\otimes }_T\kappa (p)$ is the Verma module over $\kappa (p)$ with highest weight $\lambda +{ϵ}_p$ for $\mathfrak{g}{\otimes }_{\mathbb{C}}\kappa (p)$. For a complex-valued point $ϵ:T\to \mathbb{C}$, this is precisely $-{\otimes }_T{\mathbb{C}}_{ϵ}$ and the ordinary complex Verma module $M(\lambda +ϵ)$. An arbitrary localization can have closed points whose residue fields are larger than $\mathbb{C}$; for example, $T=\mathbb{C}(t)$ has only the residue field $\mathbb{C}(t)$ and has no complex-valued point. By the tensor identity for finite-dimensional modules and the Verma flags of tensor products (Finite-dimensional tensoring preserves Verma flags) the specializations have Verma constituents over their residue fields of highest weights $\lambda +\nu +{ϵ}_p$, where $\nu$ runs through the weights of the finite-dimensional tensor factors. At complex-valued points they can meet several central-character blocks, whose labels are the dot-Weyl orbits through these highest weights (Central characters are dot-Weyl orbits); a generic parameter is generally nonintegral, and the distinct constituent labels have distinct central characters at the generic parameter: an equality $\lambda +\gamma +\tau +\rho =w(\lambda +{\gamma }^{\mathrm{\prime }}+\tau +\rho )$ as affine functions of $\tau$ forces $w=1$ by comparing linear parts, and then $\gamma ={\gamma }^{\mathrm{\prime }}$ (Central characters are dot-Weyl orbits).

Selected components. For a regular dominant integral source $\lambda$ as in The regular integral block and its antidominant weight and a dot-dominant integral target $\nu$, so that $\nu +\rho$ lies in the closed positive chamber, denote by ${\mathcal{D}}_R^{[\nu ]}(\lambda )$ the component of ${\mathcal{D}}_R(\lambda )$ whose closed fiber at the origin lies in the selected target block ${\mathcal{O}}_{\nu }$ of Translation functors by tensoring and projection. Finite Verma flags of the deformed tensor products and pairwise-unit resultants of their central-character polynomials define this component by actual central idempotents over $R$, obtained from Chinese remainder theorem for pairwise comaximal ideals. Indeed choose a linear combination $z$ of central elements separating the finitely many distinct special characters. On each deformed Verma factor, $z$ acts by a polynomial $c_i(\tau )$ by The Harish-Chandra projection computes the highest-weight scalar, and a flag is annihilated by ${\prod }_i(z-c_i(\tau ))$. Group these factors by their constant values at the origin; differences from different groups are units in $R$, so their product polynomials are comaximal in $R[X]$. CRT gives mutually orthogonal polynomial projectors, whose images are the selected components and whose sum is the identity. Other central-character components remain part of the full tensor family; thus whole-family statements and selected-block statements are distinguished.

Weight structure. Write $\tau :\mathfrak{h}\to T$ for the tautological linear map. Modules in ${\mathcal{D}}_T(\lambda )$ carry a deformed weight decomposition $X={⨁}_{\xi }{X}^{\xi }$, where $X^{\xi }=\{m:(h-\tau (h))m=\xi (h)m\text{ for every }h\in \mathfrak{h}\}\mathrm{.}$ These are eigenmodules for $h-\tau (h)$, not ordinary complex weight spaces for $h$. They are finite projective, hence locally free of finite rank over $T$, as follows from the PBW model of Verma modules and the flatness of the localization (PBW gives an ordered monomial basis for the enveloping algebra). The conventions are those of [Soe90, Section 2.1]; no choice beyond the inherited Axiom of Choice of the block theory is used.