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Translation to a singular wall in type A_2

Example

Let $g = {s l}_{2}$ and use the coordinate $l = ⟨ λ, α^{\lor} ⟩$ , so the dot action is $s \cdot l = - l - 2$ and the facets are $(- \infty, - 1)$ , ${- 1}$ , $(- 1, \infty)$ (Dot-Weyl facets and single-wall translation data, Example 3.11 of the source). Take $λ = - 2$ , $μ = - 1$ . Then the singular block $O_{- 1}$ has the single simple $L (- 1) = M (- 1) = P (- 1)$ , and the translation functor $T = T_{- 2}^{- 1}$ satisfies $T M (- 2) = M (- 1)$ , $T M (0) = M (- 1)$ , $T L (- 2) = L (- 1) (the label 1 \in W^{J}), T L (0) = 0 (the label s \notin W^{J}),$ and $T P (- 2) = 2 P (- 1)$ , because the projective cover $P (- 2)$ of $L (- 2)$ has the Verma flag $Δ (- 2), Δ (0)$ and both factors are sent to $Δ (- 1)$ .

For $g = {s l}_{3}$ take $λ = - 2 ρ$ , $J = {s_{1}}$ and the dot-antidominant $μ$ with $μ + ρ = - ω_{2}$ ; then $W^{J} = {1, s_{2}, s_{1} s_{2}}$ and by Singular category O multiplicities are parabolic Kazhdan–Lusztig values the composition and Verma-flag matrices of the singular block are $[M (y \cdot μ) : L (w \cdot μ)] = (P (w \cdot μ) : Δ (y \cdot μ)) = (\begin{array}{ccc} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array})$ in the row/column order $(1, s_{2}, s_{1} s_{2})$ : explicitly $M (μ)$ is irreducible, $M (s_{2} \cdot μ)$ has composition factors $L (μ), L (s_{2} \cdot μ)$ once each, and $M (s_{1} s_{2} \cdot μ)$ has all three simples once; dually $(P (μ) : Δ (y \cdot μ)) = 1$ for all three $y$ , $(P (s_{2} \cdot μ) : Δ (s_{2} \cdot μ)) = (P (s_{2} \cdot μ) : Δ (s_{1} s_{2} \cdot μ)) = 1$ , with its $Δ (μ)$ multiplicity zero, and $P (s_{1} s_{2} \cdot μ) = Δ (s_{1} s_{2} \cdot μ)$ has the single top-label standard. The surviving simple labels are exactly the minimal coset representatives, so translation onto the wall kills $L (w \cdot λ)$ for $w \notin W^{J}$ (for instance $L (s_{1} \cdot λ)$ and $L (w_{0} \cdot λ)$ in the ${s l}_{3}$ case).

Facts & Assumptions

Given: The Axiom of Choice; for ${s l}_{2}$ the facet datum $λ = - 2$ , $μ = - 1$ with $J = {s}$ , $W^{J} = {1}$ ; for ${s l}_{3}$ take $λ = - 2 ρ$ and the standard parabolic $J = {s_{1}}$ , $W_{J} = {1, s_{1}}$ , $W^{J} = {1, s_{2}, s_{1} s_{2}}$ and the dot-antidominant weight with $μ + ρ = - ω_{2}$ (Dot-Weyl facets and single-wall translation data, The singular integral block and its dot stabilizer).

[F1]

Wall translation: $T Δ (w \cdot λ) ≅ Δ (w \cdot μ)$ , and $T L (w \cdot λ) ≅ L (w \cdot μ)$ for $w \in W^{J}$ , $T L (w \cdot λ) = 0$ otherwise (Translation to the wall on Verma, simple and projective objects (1),(2)); $T$ is exact and preserves finite Verma flags.

[F2]

The most singular block: for $μ = - ρ$ (that is $μ + ρ = 0$ ) the Verma $M (μ) = L (μ)$ is simple and its block has the single simple; the projective cover has flag $[Δ (μ)]$ and equals $L (μ)$ , so it is projective and injective (Translation to the wall on Verma, simple and projective objects (3)).

[F3]

The singular multiplicity formula is $[M (y \cdot μ) : L (w \cdot μ)] = P_{y w_{0}, w w_{0}} (1)$ (Singular category O multiplicities are parabolic Kazhdan–Lusztig values). Every supported pair needed here has length gap at most two, so its value is one by the ordinary degree bound and constant term (Kazhdan–Lusztig polynomials in the classical $q$ -normalization); unsupported pairs have value zero. The reduced-subword test is Basic properties of the Bruhat order on $S_{n}$ .

[F4]

BGG reciprocity gives $(P (w \cdot μ) : Δ (y \cdot μ)) = [M (y \cdot μ) : L (w \cdot μ)]$ (BGG reciprocity). For the rank-one computation, PBW gives the basis $f^{k} v_{l}$ of $M (l)$ (The PBW model of a Verma module); the ${s l}_{2}$ relations are The root sl_2 triple, and $M (- 2)$ is simple by Antidominant regular Verma modules are simple.

[F5]

The sl_3 weights: $μ = - (4 α_{1} + 5 α_{2}) / 3$ , $s_{2} \cdot μ = - (4 α_{1} + 2 α_{2}) / 3$ , $s_{1} s_{2} \cdot μ = - (α_{1} + 2 α_{2}) / 3$ , with $s_{2} \cdot μ - μ = α_{2}$ , $s_{1} s_{2} \cdot μ - s_{2} \cdot μ = α_{1}$ , $s_{1} s_{2} \cdot μ - μ = α_{1} + α_{2}$ in $Q^{+}$ , and no other pairs of the three labels are dominance-linked (The singular integral block and its dot stabilizer, Dot-Weyl facets and single-wall translation data).

Verification

technique · direct: evaluate the wall-translation theorem on the two smallest wall data and read the matrices off the regular Kazhdan-Lusztig values at one

1.1F2algebra

The sl_2 singular block. For $μ = - 1 = - ρ$ the dot orbit is ${- 1}$ , so $O_{- 1}$ has the single simple $L (- 1)$ , and by [F2] $M (- 1) = L (- 1) = P (- 1) = \nabla (- 1)$ is projective and injective.

1.2F1algebra

The sl_2 translation. Here $W^{J} = {1}$ . By [F1], $T M (- 2) = M (- 1)$ and $T M (0) = M (- 1)$ ; and on simples $T L (- 2) = L (- 1)$ , while $T L (0) = 0$ because $s \notin W^{J}$ .

1.3F3F4F5algebra

The sl_3 matrices. For $y = (1, s_{2}, s_{1} s_{2})$ the products $y w_{0} = (w_{0}, s_{1} s_{2}, s_{2})$ form the descending Bruhat chain $w_{0} > s_{1} s_{2} > s_{2}$ by reduced subwords. Their supported length gaps are zero, one or two. Thus [F3] gives the displayed LOWER unitriangular matrix $A$ : its row $y$ , column $w$ is one exactly when $y \geq w$ . BGG [F4] gives the same entries for $(P (w \cdot μ) : Δ (y \cdot μ))$ . Thus the column at $w = 1$ consists of three ones, the column at $w = s_{2}$ has zeros at $y = 1$ and ones at $y = s_{2}, s_{1} s_{2}$ , and the column at $w = s_{1} s_{2}$ has only the diagonal one. These are precisely the corrected projective flags displayed above; reading rows gives the unchanged simple composition factors of the three Vermas.

2.1F1F4step 1.1step 1.2algebra

The sl_2 projective. The relations in [F4] give $e f^{k} v_{l} = k (l - k + 1) f^{k - 1} v_{l}$ by induction using $[e, f] = h$ and $[h, f] = - 2 f$ . For $l = 0$ , the span of $f^{k} v_{0}$ , $k \geq 1$ , is the copy of $M (- 2)$ generated by $f v_{0}$ , and the quotient is the one-dimensional $L (0)$ . Since $M (- 2)$ is simple, BGG [F4] gives $(P (- 2) : Δ (- 2)) = (P (- 2) : Δ (0)) = 1$ ; the rank-one dot orbit has no further labels. Applying exact $T$ to this Verma flag gives two factors $Δ (- 1) = P (- 1)$ by [F1] and step 1.1. Every extension of these projectives splits, so $T P (- 2) ≅ P (- 1) \oplus P (- 1)$ .

3.1F1step 1.3algebra∎

Surviving labels. Applying [F1] to the simple labels: $L (w \cdot λ)$ survives translation iff $w \in W^{J} = {1, s_{2}, s_{1} s_{2}}$ ; hence $L (s_{1} \cdot λ)$ and $L (w_{0} \cdot λ) = L (s_{1} s_{2} s_{1} \cdot λ)$ are sent to zero, as are all other non-minimal labels. This proves every displayed assertion.

Depends on

Used by

A small parabolic category O character formula Example

Dependency tree · two levels

88 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lin Chen, lecture notes (Spring 2024), Lecture 9: standard filtrations, BGG reciprocity and translation functors (complete 7-page lecture) (standard reference, not scraped)
Anders Bjoerner and Francesco Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231, Springer 2005 (author-hosted complete 370-page PDF) (standard reference, not scraped)