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The parabolic Kazhdan–Lusztig character formula

Statement

Assume the Axiom of Choice (The Axiom of Choice) and the dominant regular integral conventions of Parabolic category O multiplicities are parabolic Kazhdan–Lusztig values. Put $N = (n_{y, x} (1))_{y, x \in^{J} W}$ , with rows indexed by generalized standards and columns by simples in a Bruhat-compatible order (The parabolic Kazhdan–Lusztig bases and polynomials). Then $N$ is upper unitriangular and, for each $x$ , $ch L (x \cdot λ) = \sum_{y \geq x} (N^{- 1})_{x, y} ch Δ^{J} (y \cdot λ) .$ Its inverse entries are integers computed by the finite signed chain sum of products of $n_{a, b} (1)$ ; they are weighted incidence inverses, not generally ordinary Bruhat Mobius values. In the antidominant base $λ_{a n t i} = w_{0} \cdot λ$ , set $a = l f (y) = w_{J} y w_{0}$ and encode the same weight as $(w_{J} a) \cdot λ_{a n t i}$ . The reindexed matrix is ${\tilde{N}}_{a, b} = N_{{l f}^{- 1} (a), {l f}^{- 1} (b)}$ and its inverse is reindexed by the same permutation; thus the same formula has reversed support in the transported order. This does not replace the entries by $n_{a, b} (1)$ . This is the ungraded character formula for the regular parabolic block, and constructs no Koszul or Ext grading.

Facts & Assumptions

Given: The regular parabolic block with generalized standards $Δ^{J} (y \cdot λ)$ , simples $L (x \cdot λ)$ and projective covers $P^{J} (x \cdot λ)$ indexed by $^{J} W$ , the multiplicity matrix $N = (n_{y, x} (1))$ of Parabolic category O multiplicities are parabolic Kazhdan–Lusztig values, and the Grothendieck group with its standard and simple bases (The Grothendieck group and character of O, Simple and standard bases of K0(O)).

[F1]

Multiplicity theorem: $[M_{p} (y \cdot λ) : L (x \cdot λ)] = n_{y, x} (1)$ with support $y \leq x$ and diagonal one, and $(P^{J} (x \cdot λ) : M_{p} (y \cdot λ)) = n_{y, x} (1)$ (Parabolic category O multiplicities are parabolic Kazhdan–Lusztig values).

[F2]

The simple and generalized-standard classes are bases of the finite block Grothendieck group, characters are injective on it, and the standard and simple matrices are unitriangular; the blocks are finite (Parabolic category O is a highest weight category (4), Simple and standard bases of K0(O), BGG reciprocity).

[F3]

The Bruhat order on the finite set $^{J} W$ is a partial order, and the index range is finite, so the strictly upper triangular matrix $U = N - I$ relative to such an order is nilpotent (Parabolic category O multiplicities are parabolic Kazhdan–Lusztig values, The parabolic Kazhdan–Lusztig bases and polynomials); the inverse entries are the finite alternating sums of products of the $n_{a, b} (1)$ along strictly increasing chains (Inverse Kazhdan–Lusztig polynomials, The Kazhdan–Lusztig inversion formula provide the ordinary inversion-formula comparison).

Proof

technique · direct finite linear algebra on the unitriangular multiplicity matrix

1.1F1F2algebra

The matrix equation. In the Grothendieck group write $M_{y} : = [Δ^{J} (y \cdot λ)]$ and $L_{x} : = [L (x \cdot λ)]$ ; by [F1] and the Jordan-Holder theorem in the finite-length parabolic block, $M_{y} = \sum_{x} [Δ^{J} (y \cdot λ) : L (x \cdot λ)] L_{x} = \sum_{x \geq y} n_{y, x} (1) L_{x}$ . Since the simple classes form a basis by [F2], this is the matrix equation $M = N L$ with $N = (n_{y, x} (1))$ .

1.2F1F3algebra

Unitriangularity and inversion. By [F1], $n_{x, x} (1) = 1$ and $n_{y, x} (1) = 0$ unless $y \leq x$ ; the diagonal of $N$ is one and $U : = N - I$ is strictly upper triangular for the finite Bruhat-compatible order on $^{J} W$ . Hence $U$ is nilpotent ( $U^{m} = 0$ for $m > #^{J} W$ ) and $N^{- 1} = I - U + U^{2} - \dots$ is a finite sum of integer matrices; the $(x, y)$ -entry of $N^{- 1}$ is supported on $y \geq x$ and equals the finite signed chain sum $\sum_{m \geq 0} (- 1)^{m} \sum_{x = z_{0} < z_{1} < \dots < z_{m} = y} \prod_{i} n_{z_{i}, z_{i + 1}} (1)$ over strictly increasing chains. These are weighted incidence inverses of the finite poset, not generally ordinary Mobius values.

2.1F1F2step 1.1step 1.2algebra

The character formula. Solving step 1.1 by step 1.2 gives $L = N^{- 1} M$ with $N^{- 1}$ read as a matrix whose rows are indexed by the simples and whose columns are indexed by the generalized standards, that is, $ch L (x \cdot λ) = \sum_{y} (N^{- 1})_{x, y} ch Δ^{J} (y \cdot λ)$ ; since $(N^{- 1})_{x, y}$ is supported on $y \geq x$ , this is exactly the displayed formula.

3.1F1step 1.2step 2.1algebra∎

Reindexing and conclusion. Use $a = l f (y) = w_{J} y w_{0}$ as the shortest-left encoding of the actual longest-left label $y w_{0}$ in the antidominant base, so the same weight is $(w_{J} a) \cdot λ_{a n t i}$ . The corrected dictionary in Parabolic category O multiplicities are parabolic Kazhdan–Lusztig values gives ${\tilde{N}}_{a, b} = N_{{l f}^{- 1} (a), {l f}^{- 1} (b)}$ . Reindexing both coordinates is conjugation by a permutation matrix, hence ${\tilde{N}}_{a, b}^{- 1} = N_{{l f}^{- 1} (a), {l f}^{- 1} (b)}^{- 1}$ ; therefore $ch L ((w_{J} a) \cdot λ_{a n t i}) = \sum_{b} {\tilde{N}}_{a, b}^{- 1} ch Δ^{J} ((w_{J} b) \cdot λ_{a n t i})$ , with the original support transported by $l f$ . Choosing the transported reversed order makes this matrix lower triangular. This preserves every character formula and constructs no Koszul or Ext grading.

Depends on

Used by

A small parabolic category O character formula Example

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Sources

Lin Chen, lecture notes (Spring 2024), Lecture 9: standard filtrations, BGG reciprocity and translation functors (complete 7-page lecture) (standard reference, not scraped)
Volodymyr Mazorchuk and Catharina Stroppel, Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module, Trans. Amer. Math. Soc. 357 (2005) 2939-2973 (author-hosted published version) (standard reference, not scraped)