Category $\mathcal{O}$ and asymptotic characters

Alexis Leroux-Lapierre

REVIEW 3 major objections 4 minor 1 cited by

Category $\mathcal{O}$ and asymptotic characters

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves the D-image version of a conjectured non-negative change of basis between the dual canonical and Mirkovi\'c–Vilonen bases of $\mathbb{C}[N]$, with coefficients counting MV cycles in characteristic cycles of category…

desk verdict The asymptotic character formalism is a real contribution, but the proof of Theorem 5.38 has a genuine, load-bearing gap in the partition of GT-weights. read the letter →

arxiv 2507.16215 v1 pith:G77IJDUK submitted 2025-07-22 math.RT

Alexis Leroux-Lapierre This is my paper · ORCID

classification math.RT MSC 17B3717B1014M1517B67

keywords categoryOtruncatedshiftedYangiansasymptoticcharactersMirkovic-VilonencyclesdualcanonicalbasisKLRalgebrascharacteristicequivariantmultiplicities

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's aim is to compare two well-known bases of the coordinate ring $\mathbb{C}[N]$ of a unipotent group in the simply-laced case: the dual canonical basis, built from representation theory of KLR algebras, and the Mirkovi\'c–Vilonen (MV) basis, built from geometry of the affine Grassmannian. The proposed comparison is that each dual canonical basis vector $d_L$ is a non-negative integer combination of MV basis vectors $b_Z$, with the coefficient $n_{L,Z}$ equal to the multiplicity of the cycle $Z$ in the characteristic cycle of a category $\mathcal{O}$ module for a truncated shifted Yangian. The paper proves the weak form of this statement (Theorem 2.13): after applying the rational-function map $D^-$ to both sides, $D^-(d_L)=\sum_Z n_{L,Z}D^-(b_Z)$. A new asymptotic character $\chi^\infty$ carries the argument, since it turns ordinary weight multiplicities into rational functions and, by Theorem 3.55, equals the equivariant multiplicity of the characteristic cycle, while an existing equivalence of categories identifies it with a KLR bar-character. If the main theorem is right, the conjectured basis change is not a formal coincidence: its coefficients are computable geometric invariants of a concrete module-support construction.

What carries the argument

The load-bearing construction is the asymptotic character map $\chi^\infty_d$, defined for a module $M$ of Gelfand–Kirillov dimension $d$ by $$\chi^\infty_d(M)(h)=\lim_{n\to\infty}\frac{1}{n^d}\sum_\mu \dim M(\mu)\, $e^{{\langle\mu,h\rangle/n}}$.$$ The limit is not a formal device: by Theorem 3.55 it computes the equivariant multiplicity of the characteristic cycle of $M$, namely the sum $\sum_Z n_Z\,\varepsilon_T(Z)$ over the $d$-dimensional irreducible components $Z$ of the support of the associated graded module. Once the paper verifies that truncated shifted Yangians satisfy the required finiteness and integrality hypotheses, this same $\chi^\infty$ is rewritten, through the equivalence of Theorem 5.31, as the bar-character of a cyclotomic KLR module, yielding the main identity.

What would settle it

Pick a concrete case not forced by minuscule symmetry, e.g. $\mathfrak{g}=\mathfrak{sl}_4$, $\lambda=\varpi_1+\varpi_3$, $\mu=0$, and an integral parameter set $R$ in general position. For each simple cyclotomic KLR module $L$, compute independently the KLR character side $D^-(d_L)$ and the geometric side: construct the corresponding category $\mathcal{O}$ module, pass to its associated graded module, read off the multiplicities $n_{L,Z}$ of the top-dimensional components, and evaluate $\sum_Z n_{L,Z}D^-(b_Z)$. Any difference between the two rational functions at a generic point of the Cartan would refute Theorem 2.13.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is an equality of rational functions that ties two previously separate computational languages. For $\mathfrak{g}$ simple, simply-laced, with $\lambda$ dominant and $\mu$ a weight of $V(\lambda)$, every simple module $L$ over the cyclotomic KLR algebra $R^\lambda_{\lambda-\mu}$ gives a module in category $\mathcal{O}$ of the truncated shifted Yangian $Y^\lambda_\mu(R)$. The theorem asserts $$D^-(d_L)=\sum_Z n_{L,Z}D^-(b_Z),$$ where $d_L$ is the dual canonical basis vector attached to $L$, $b_Z$ is the MV basis vector attached to a stable MV cycle $Z$, and $n_{L,Z}\in\mathbb{Z}_{\geq 0}$ is the multiplicity of $Z$ in the top-dimensional characteristic cycle of that module. The equality is established through a commutative diagram whose three arrows are the characteristic cycle map, the category equivalence to parity KLRW modules followed by the cyclotomic idempotent, and the asymptotic character map; the latter is shown to compute equivariant multiplicities of supports. The paper therefore treats Conjecture 2.15, the unshadowed basis change $d_L=\sum_Z n_{L,Z}b_Z$, as a strictly stronger statement that its methods support but do not reach.

Load-bearing premise

The paper's chain of reasoning rests on a previously proved equivalence between category $\mathcal{O}$ for truncated shifted Yangians and modules over parity KLRW algebras, assumed as a black box for every integral set of parameters; if that equivalence fails in any needed case, the identification of $\chi^\infty$ with the KLR bar-character, and with it the main formula, would not follow.

Editorial extensions

If this is right

If Theorem 2.13 holds, the conjectured change of basis from the dual canonical basis to the MV basis has the explicit, computable coefficients $n_{L,Z}$, and these coefficients are non-negative integers because they are cycle multiplicities.
The asymptotic character map $\chi^\infty_d$ becomes a general invariant of filtered quantizations: on category $\mathcal{O}$ it is a group homomorphism from $K_0(\mathcal{O}_{\leq d})$ to rational functions that kills objects of lower GK dimension and equals the equivariant multiplicity of the characteristic cycle (Theorem 3.55).
In the minuscule case, the paper's setup recovers Nakada's colored hook formula and the Peterson–Proctor hook formula as special cases, showing that the new formalism reproduces known combinatorial identities (Example 2.16).
When $\lambda$ is a sum of minuscule coweights and the parameters avoid finitely many affine hyperplanes, the characteristic cycle map is an isomorphism between the top category $\mathcal{O}$ and the top Borel–Moore homology of the repelling set (Corollary 4.52).
In the symmetric Kac–Moody setting, the same commutative diagram shows that the top homology of the repelling set of the Coulomb branch is non-zero for every weight of $V(\lambda)$ (Corollary 6.8), evidence toward the conjectured Kac–Moody geometric Satake.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper only proves the $D^-$-shadow of the conjectured basis change; a natural next step that the paper does not take is to test whether the same coefficients $n_{L,Z}$ satisfy the stronger identity $d_L=\sum_Z n_{L,Z}b_Z$ inside $\mathbb{C}[N]$, which would fully resolve Conjecture 2.15.
Because Section 3's asymptotic character is developed for any filtered quantization satisfying hypotheses H1–H9, the same limit construction could be applied to other quantizations with a Hamiltonian torus action; the paper's checks for truncated shifted Yangians suggest a general recipe for turning characters into equivariant multiplicities.
The equivalence used is simply-laced; if the corresponding B-integrality and category equivalence were extended to non-simply-laced parameters, the main identity would presumably extend to the symmetrized Coulomb-branch setting, where the paper can currently only give evidence.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: the asymptotic character formalism is a real contribution, and the first half of the paper (sections 2–4) is careful and largely convincing. But the proof of Theorem 5.38, which is the bridge from χ∞ to KLR bar-characters and hence the main formula, has a genuine technical gap. As printed, the claimed partition of the GT-weights in the proof is not a partition: F′ and F″ overlap, and the singular non-cyclotomic weights are counted twice with different coefficients. Lemma 5.44 also invokes the regular parametrization (5.41) for singular sets of parameters, where it does not apply. And the application of Lemma 5.25 to all S∈E(i) requires ecyc e(S)=e(S), which only holds in the cyclotomic range. These are not cosmetic slips; they are load-bearing for the equality ς(χ∞d(M)) = ch(Θcyc(M)). So Theorem 5.38 and the main theorem are not proven as written.

The good news is that the core idea is worth taking seriously. The definition of the asymptotic character (Definition 3.47) and the theorem that it computes equivariant multiplicities of characteristic cycles (Theorem 3.55) are genuinely new and are proved from the hypotheses in a way that mostly checks out. The verification of H1–H11 for truncated shifted Yangians is careful, including the sign conventions which are flagged explicitly. The paper also does the right thing by stating clearly that the full change-of-basis conjecture remains open and positions Theorem 2.13 as evidence. The reliance on the Kamnitzer–Tingley–Webster–Weekes–Yacobi equivalence is a black box, but an honest one; that is deep prior work, not a defect of this paper.

The fixes for the proof might be local — redefine F″ as the singular cyclotomic weights, restrict the use of (5.41) to regular subsequences, and apply Lemma 5.25 only to Ecyc(i). But as of now, the central computation is not established, and a referee would need to verify those repairs carefully. I would send this to a serious referee, because the stakes and the novelty are high, but the revision should be substantial, not cosmetic.

Referee Report

3 major / 4 minor

Summary. The paper introduces an asymptotic character map $\chi^\infty_d$ on category $\mathcal{O}$ of filtered quantizations satisfying axioms H1--H11 and proves that it computes the equivariant multiplicity of the characteristic cycle (Theorem 3.55). It then verifies these hypotheses in detail for truncated shifted Yangians associated to simply-laced simple Lie algebras (Corollaries 4.23, 4.30, 4.41), and uses the equivalence of categories of [Kam+19b] to relate the asymptotic character to the KLR bar-character (Theorem 5.38). Assembling these results gives the main commutative diagram (Theorem 6.1), which is reformulated as the weak change-of-basis formula $D^-(d_L)=\sum_Z n_{L,Z}D^-(b_Z)$ (Theorem 2.13), with $n_{L,Z}$ the multiplicities of MV cycles in the characteristic cycle. The paper also discusses injectivity of the characteristic cycle map, gives worked examples including Nakada's hook formula, and sketches consequences for a conjecture of Nakajima in the Kac--Moody setting.

Significance. If the main theorem is established, this is a valuable contribution: it provides a new, geometrically meaningful mechanism for comparing Lusztig's dual canonical basis with the Mirkovi\'c--Vilonen basis of $\mathbb{C}[N]$, and it predicts integrality and non-negativity of the change-of-basis coefficients through characteristic cycles. The abstract framework of Section 3 is clearly formulated and reusable, the hypotheses H1--H11 are checked explicitly for truncated shifted Yangians, and the paper is careful to flag sign and convention issues (Remarks 4.20, 4.32, 5.33). The derivation of the asymptotic character from equivariant Hilbert polynomials is largely independent of the KLR side, which speaks against circularity. However, the proof of Theorem 5.38, which is the bridge between $\chi^\infty_d$ and the KLR bar-character, contains serious gaps as written; since Theorem 6.1 and hence Theorem 2.13 depend on it, the main claim is not fully proven in the present form. The flaws appear local and repairable, so the appropriate response is a major revision rather than rejection.

major comments (3)

[§5.5 (definitions before Theorem 5.38)] The claimed partition of $(\Lambda_\nu)_R$ into three disjoint subsets is not disjoint as stated. The displayed definitions give $F'=(\Lambda^{<R}_\nu)^\complement\cap \Lambda^{\mathrm{sing}}_\nu\cap (\Lambda_\nu)_R$ and $F''=\Lambda^{\mathrm{sing}}_\nu\cap (\Lambda_\nu)_R$, so $F'\subset F''$. Consequently, in the decomposition of $\chi(M)$ in the proof of Theorem 5.38, every singular GT-weight outside the cyclotomic range is counted twice: once in the $F'$-term with coefficient $\dim W_S(M)$ and once in the $F''$-term with coefficient $(1/\sigma(S))\dim W_S(M)$. If $F'$ was instead intended to consist of regular weights, then $F'$ overlaps $E(i)$ on the regular weights with $\max S\ge \min R$. Thus the triple $(E(i),F',F'')$ is not a partition in either reading, and the limit computation that follows cannot be valid as written.
[§5.5 (Lemma 5.44 and its proof)] Lemma 5.44 is stated for the singular set $F'(k,j)_t$, but its proof invokes the bijection $Q_i$ of (5.41), which parametrizes regular GT-weights by strictly increasing integers. For a singular $S$ with repeated longitudes, the summation over $q_1<\cdots<q_k<N$ does not enumerate the elements of $F'(k,j)_t$ bijectively, and the multiplicity factor $\sigma(S)$ is not taken into account. Hence the conclusion of Lemma 5.44 is not established for the singular locus it is stated for. Relatedly, the second case in the proof of Lemma 5.45 applies Lemma 5.44 to the regular set $\tilde S$ obtained by forgetting multiplicities, so the cited lemma does not actually cover that case. These gaps mean that Lemmas 5.43--5.45 do not, as written, prove the displayed formula for $\chi^\infty_d(M)$ used in Theorem 5.38.
[§5.5 (proof of Theorem 5.38)] In the proof of Theorem 5.38, Lemma 5.25 is applied to every $S\in E(i)$ to obtain $\dim e(S)N=\dim e(i)(e_{\mathrm{cyc}}N)$. The required idempotent identity $e_{\mathrm{cyc}}e(S)=e(S)$ holds only for $S\in E_{\mathrm{cyc}}(i)=E(i)\cap(\Lambda_\nu)^{<R}$. For regular $S$ with $\min R\le \max S<\max R$, the right-hand side is zero while $e(S)$ need not annihilate $N$, so the rewriting of the first sum in $\chi(M)$ is false. These regular outside-cyclotomic weights are not covered by Lemmas 5.44--5.45 (which are formulated for singular weights), so their contribution to $\chi^\infty_d(M)$ is not shown to vanish. A corrected proof must restrict the Lemma 5.25 step to $E_{\mathrm{cyc}}(i)$ and handle $E(i)\setminus E_{\mathrm{cyc}}(i)$ separately, for instance by showing that its asymptotic contribution is of order $O(n^{d-1})$ or by including it in a correctly partitioned singular/regular outside-cyclotomic case.

minor comments (4)

[§4.1 (Definition 4.1)] The sentence introducing Definition 4.1 contains a duplicated phrase: "It first appeard in first appeared in" should be corrected to "It first appeared in".
[§4.7 (Lemma 4.51)] In the proof of Lemma 4.51, the set on which $|\varpi^\vee_{i_1,\dots,i_N}(c_1,\dots,c_N)|=r$ is the complement of a union of finitely many affine hyperplanes, not itself a union of affine hyperplanes; the wording should be adjusted to say that the desired set is the complement of a finite union.
[§5.5 (Lemma 5.45)] The displayed denominator "$\rho_{i_1}!\cdot\!\cdot\!\cdot\!\cdot\!\cdot\rho_{i_q}!$" appears to be a typesetting artifact and should read $\rho_{i_1}!\cdots\rho_{i_q}!$.
[§5.5 (Lemma 5.44 proof)] The formula "$\lambda(S)=\frac12(s_{i_1}\alpha_{k+1}+\cdots+s_{i_k}\alpha_{i_d})+\mu_r$" in the proof of Lemma 5.44 appears to contain an indexing/notation error; it should refer to the longitudes of the $k$ left black strands in a way consistent with (5.42).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic character is defined independently, and the bridge to KLR bar-characters is an external equivalence plus explicit GT-weight limits.

full rationale

The paper's central derivation is self-contained against its cited inputs. The asymptotic character χ∞_d is defined (Definition 3.47) directly from weight space dimensions and a limit, with no reference to the D-map, MV cycles, or KLR characters. Theorem 3.55, identifying χ∞_d with the equivariant multiplicity of the characteristic cycle, is proved through equivariant Hilbert polynomials and Lemma 3.50, which is a calculus-level limit computation from [CG97]; it does not assume the target equality. The later identification with the KLR bar-character (Theorem 5.38) uses the category equivalence Θ from [Kam+19b] as an external black box and then performs an explicit decomposition of GT-weights into three families; the conclusion ς(χ∞_d(M)) = ch(Θcyc(M)) is not used as an input anywhere. The final Theorem 6.1 simply assembles Theorems 4.44 and 5.38 into a commutative diagram, and Theorem 2.13 reads off the equality of rational functions D^-(d_L) = Σ n_{L,Z}D^-(b_Z) from that diagram; the multiplicities n_{L,Z} come from the characteristic cycle, not from a fit to D^- values. The cited [Kam+19b] equivalence is independent prior work by different authors, and the only self-citation ([Kal+25], in preparation) is forward-looking and not load-bearing. The proof of Theorem 5.38 has a potential partition inconsistency (F' ⊂ F'' and the use of the regular parametrization (5.41) for singular weights), but that is a correctness gap in the written proof, not a circularity: a failed limit computation does not mean the claimed equality was assumed. Accordingly, no step reduces by definition or self-citation to its own inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results are built on a tower of deep prior theorems, but no ad hoc entities or fitted free parameters are introduced. The asymptotic character is a new invariant defined independently in the paper.

assumptions (4)

domain assumption Equivalence of categories between category O for truncated shifted Yangians and parity KLRW modules (Theorem 5.31, from [Kam+19b]).
Used as the bridge between the Yangian category O and KLRW modules; essential for Theorem 5.38 and Theorem 6.1.
domain assumption [BKK21, Theorem 1.4]: D^-(b_Z) equals the equivariant multiplicity of the MV cycle at its bottom fixed point.
Connects the asymptotic character to the D-map in Theorem 4.44 and Theorem 2.13.
domain assumption [Kam+19b, Corollary 5.22] bijection between maximal ideals of B(Y^lambda_mu(R)) and the product monomial crystal B(lambda,R)_mu.
Used in Lemma 4.40 to establish B-integrality (H9) and in Section 4.6 to describe simple objects of category O.
domain assumption KLR categorification theorems (Theorem 5.17 and 5.20) identifying Grothendieck groups with C[N] and V(lambda).
Used to translate the KLR bar-character into the D-map of the dual canonical basis vector.

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Cite this review

Pith. "Pith review of Category $\mathcal{O}$ and asymptotic characters." pith.science (2026). https://pith.science/paper/G77IJDUK

@misc{pith2026250716215,
  author       = {Pith},
  title        = {Pith review of: Category $\mathcalO$ and asymptotic characters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G77IJDUK}},
  note         = {Machine review of arXiv:2507.16215}
}

abstract

This paper defines an asymptotic character map which is a morphism from the Grothendieck group of category $\mathcal{O}$ of an integral filtered quantization to rational functions on the Lie algebra of a torus. We show that the asymptotic character of a module computes the equivariant multiplicity of its characteristic cycle. We then apply this construction to truncated shifted Yangians coming from simple, simply-laced Lie algebras and draw connections with characters of modules over KLR algebras using an equivalence of categories of arXiv:1806.07519. Our main theorem shows how this new formalism gives formulas relating equivariant multiplicities of Mirkovi\'{c}-Vilonen cycles and characters of modules over cyclotomic KLR algebras. We explain how this result provides evidence that the change-of-basis between Lusztig's dual canonical basis and the Mirkovi\'{c}-Vilonen basis of $\mathbb{C}[N]$ is computed by a characteristic cycle map whose domain is category $\mathcal{O}$ for truncated shifted Yangians, implying that the coefficients are non-negative integers.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Category $\mathcal{O}$ for truncated shifted Yangians and the bi-infinite Bott-Samelson variety
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The Grothendieck ring of integral category O for shifted Yangians equals the Cox ring of a new open bi-infinite Bott-Samelson pro-variety, proving the Hernandez-Zhang and Frenkel-Hernandez/GHL conjectures in simply-la...

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Reviewed August 6, 2026 · model on record in the stance chip above.

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