The singular integral block and its dot stabilizer

Assume the Axiom of Choice (The Axiom of Choice), inherited from the published block theory. Work in category $\mathcal{O}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with Cartan $\mathfrak{h}$, positive system ${\mathrm{\Phi }}^{+}$, simple roots ${\alpha }_1,\dots ,{\alpha }_r$, Weyl group $W$, Weyl vector $\rho$ and dot action $w\cdot \lambda =w(\lambda +\rho )-\rho$, as in The classical BGG category O and Dot-Weyl facets and single-wall translation data.

(1) Dot-antidominant integral weights. A weight $\mu$ is dot-antidominant if $⟨\mu +\rho ,{\alpha }^{\vee }⟩\le 0$ for every $\alpha \in {\mathrm{\Phi }}^{+}$, and integral if $⟨\mu ,{\alpha }_i^{\vee }⟩\in \mathbb{Z}$ for all $i$ (The integral Weyl group of a weight, Integral, dominant, and strictly dominant weights).

(2) The dot stabilizer. Fix such a weight $\mu$ and put $J=J(\mu ):=\{i:⟨\mu +\rho ,{\alpha }_i^{\vee }⟩=0\},W_{\mu }:=W_J=⟨s_i:i\in J⟩\mathrm{.}$ Then the dot stabilizer $\{v\in W:v\cdot \mu =\mu \}$ equals $W_J$. Indeed, for $i\in J$ the reflection $s_i$ fixes $\mu +\rho$, so $W_J$ lies in the stabilizer; conversely the stabilizer of an integral dot-antidominant weight is generated by the reflections it contains and is a standard parabolic (Finite Weyl closed chambers and stabilizers, applied to the dominant vector $-\mu -\rho$), and by Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups a standard parabolic subgroup is $W_I$ for the set $I$ of simple reflections it contains, which is exactly $J$. In particular $w^{\mathrm{\prime }}\cdot \mu =w\cdot \mu$ if and only if $w^{-1}{w}^{\mathrm{\prime }}\in W_J$, that is, $w^{\mathrm{\prime }}\in w{W}_J$.

(3) The singular integral block. The singular integral block ${\mathcal{O}}_{\mu }$ is the block of Central-character summands refine into linkage blocks of the central character ${\chi }_{\mu }$ of $\mu$; by Central characters are dot-Weyl orbits two central characters agree exactly when the highest weights lie in one dot orbit, and by Generalized central-character decomposition of O the block is a summand of $\mathcal{O}$. Its simple objects $L(w\cdot \mu )$, standard objects $\mathrm{\Delta }(w\cdot \mu )=M(w\cdot \mu )$, costandard objects $\mathrm{\nabla }(w\cdot \mu )$ and projective covers $P(w\cdot \mu )$ are indexed by $w\in W$ (Standard and costandard objects, Verma modules); by (2) the label $w\cdot \mu$ depends only on the coset $w{W}_J$, so the labels can be normalized by the minimal right coset representatives

$W^J=\{w\in W:\mathrm{\ell }(ws)>\mathrm{\ell }(w)\text{ for all }s\in J\},$

which are exactly the elements of minimal length in the cosets $w{W}_J$ (Support, intrinsic parabolic presentations, minimal coset representatives and length additivity, with the type-A identification (3)); inversion carries $W^J$ onto the minimal left coset representatives ${}^{J}W$.

(4) Highest-weight structure and multiplicities. The block is a highest-weight category: it has enough projectives (Category O has enough projectives), projective covers have finite Verma flags (Projectives in category O have finite Verma flags), the standard classes and simple classes are bases of the Grothendieck group (The Grothendieck group and character of O, Simple and standard bases of K0(O)), and it satisfies BGG reciprocity $(P(w\cdot \mu ):\mathrm{\Delta }(y\cdot \mu ))=[M(y\cdot \mu ):L(w\cdot \mu )]$ (BGG reciprocity, Finite Verma flags and their multiplicities). The multiplicity matrix is unitriangular for the refinement of the dominance order on the dot orbit: $(P(w\cdot \mu ):\mathrm{\Delta }(w\cdot \mu ))=1$, and $(P(w\cdot \mu ):\mathrm{\Delta }(y\cdot \mu ))\mathrm{\ne }0$ only if $y\cdot \mu \ge w\cdot \mu$ (The triangular restriction on projective Verma flags); the ordinary standard/costandard Hom and ${\mathrm{E}\mathrm{x}\mathrm{t}}^1$ vanishing used on this page is Standard-costandard Hom and Ext-one orthogonality.

(5) Regular versus singular. The block is regular when $J=\mathrm{\varnothing }$ (the weight lies on no wall) and singular when $J\mathrm{\ne }\mathrm{\varnothing }$; the singular translation statements of this page use this notation, and $J=\mathrm{\varnothing }$ recovers the regular case. The genuine regular-parabolic multiplicity statements below have their own separate explicit dominant regular base and are not consequences of the notation here.

• Type-$A$ comparison. When $W=S_n$ with its type-$A$ simple reflections, ${}^{J}W$ is the common standard-basis label set of the two parabolic Hecke modules of The parabolic Hecke module and the antispherical standard basis; the singular labels $W^J$ correspond to that set by inversion. This comparison is not a prerequisite of the general block definition.

• Choice. The Axiom of Choice is inherited with the published block theory: restricted duality, enough projectives, Verma flags of projectives and the Grothendieck-basis theorem are quoted from items that assume it (The Axiom of Choice, Standard and costandard objects, Category O has enough projectives, Projectives in category O have finite Verma flags, Simple and standard bases of K0(O)), and no new choice principle is introduced here.

• Highest-weight projective kernels. In an extension $0\to M(\beta )\to X\to M(\alpha )\to 0$ with $\alpha ≮\beta$, lift the highest vector of weight $\alpha$. Every positive-root image lands in the submodule at a weight strictly above $\alpha$, impossible because that would imply $\alpha <\beta$. Thus the lift is highest, and The universal property of Verma modules splits the extension. The projective flag in (4) has its labelled standard once and all other labels strictly higher. The preceding splitting swaps a higher section followed immediately by the labelled submodule section, moving the labelled section to the final quotient. Its kernel therefore has a flag of strictly higher standards, giving the projective-kernel axiom.

• Stabilizer notation. Here $W_{\mu }$ denotes the dot stabilizer $W_J$. The integral-reflection group called $W_{\mu }$ in The integral Weyl group of a weight is instead all of $W$ for integral $\mu$. In particular its linkage class is the entire dot orbit, so the integral central-character summand in (3) is one linkage block.

• No new multiplicity assertion. Everything in (4) is quoted structure of the published block theory; the first genuinely new multiplicity statements are the translation, singular and parabolic multiplicities proved later on this page. In particular no claim is made here that the singular matrix equals an ordinary Kazhdan--Lusztig matrix or a parabolic one.

• Label conventions. $W^J$ (right minima) is used for the singular block, ${}^{J}W$ (left minima) for the genuine regular parabolic category later; the two families are exchanged by inversion (Standard parabolic subgroups, descent-free one- and two-sided representatives, parabolic and reflection subgroups (2)), and the small $S_3\mathrm{/}{S}_2$ matrices of the companion examples exhibit both label orders.