Asymptotic Character Map

• The asymptotic character map is a representation-theoretic tool that scales weight characters to yield rational functions reflecting equivariant multiplicities and geometric invariants.
• It refines the ordinary weight character in category O via a normalization process that isolates the leading order behavior in terms of the Gelfand–Kirillov dimension.
• Its applications to truncated shifted Yangians and KLR algebra representations bridge algebraic representation theory with the geometry of characteristic cycles and MV basis comparisons.

The asymptotic character map is a representation-theoretic morphism from the Grothendieck group of a category $\mathcal O$ attached to an integral filtered quantization to rational functions on the Lie algebra of a torus. It is defined by taking the usual weight character of a module, scaling weights by $1/n$, normalizing by $n^d$, and passing to a limit, where $d$ is the Gelfand–Kirillov dimension / dimension of the support. In the framework developed for filtered quantizations and then applied to truncated shifted Yangians, the asymptotic character is shown to compute the equivariant multiplicity of the characteristic cycle; this places asymptotic character theory at the intersection of category $\mathcal O$, Hamiltonian torus actions, equivariant Borel–Moore homology, Mirković–Vilonen geometry, and KLR character theory (Leroux-Lapierre, 22 Jul 2025).

1. Definition and ambient framework

The general setup begins with a filtered algebra $A$ with filtration $F^\bullet$. The hypotheses are that $A$ has a separated increasing filtration $F^\bullet$, that $A$ is finitely generated, that

$1/n$0

is commutative and finitely generated, and that the filtration admits an expansion

$1/n$1

as filtered vector spaces (Leroux-Lapierre, 22 Jul 2025).

The relevant category $1/n$2 is attached to a cocharacter $1/n$3 and is defined by

$1/n$4

The asymptotic character is first defined on the Gelfand–Kirillov truncated pieces

$1/n$5

If $1/n$6, then for $1/n$7 the asymptotic character is

$1/n$8

This definition extracts the leading asymptotic behavior of the ordinary weight character near the identity in the torus, with the normalization chosen to isolate the contribution in support dimension $1/n$9 (Leroux-Lapierre, 22 Jul 2025).

The ordinary character itself is regarded as a map into a formal character space. For a weight module $n^d$0,

$n^d$1

The target is the $n^d$2-vector space of formal sums

$n^d$3

with multiplication defined whenever one factor has finite support, and more generally for sums supported in suitable cones. The asymptotic character map refines this formal character by replacing it with a scaling limit valued in rational functions on $n^d$4 (Leroux-Lapierre, 22 Jul 2025).

2. Rationality and the commutative model

A key point is that the limit in the definition is not merely formal. Under the stated hypotheses, the asymptotic character becomes a rational function on $n^d$5, giving a group homomorphism

$n^d$6

and the kernel contains $n^d$7 (Leroux-Lapierre, 22 Jul 2025).

The mechanism for rationality passes through the associated graded algebra and then to a polynomial model. One introduces the repelling set

$n^d$8

a reduced quotient $n^d$9, and a polynomial algebra

$d$0

with a torus action and a surjection

$d$1

For a finitely generated $d$2-equivariant $d$3-module $d$4, the character takes the form

$d$5

From this expression the paper constructs the equivariant Hilbert polynomial $d$6, and the crucial asymptotic identity is

$d$7

This is the formal step that converts the asymptotic character from a scaled exponential sum into a rational function (Leroux-Lapierre, 22 Jul 2025).

This rationality statement is not secondary. It identifies asymptotic character as an invariant that is simultaneously character-theoretic and geometric: it is built from weight multiplicities, but its final form is governed by the same denominator data that appear in equivariant multiplicity theory.

3. Characteristic cycles and equivariant multiplicity

The central theorem identifies the asymptotic character with the equivariant multiplicity of the characteristic cycle. For a good filtration on a finitely generated $d$8-module $d$9, one has $\mathcal O$0, and the support cycle defines

$\mathcal O$1

The characteristic cycle map is then

$\mathcal O$2

and, in the supported-on-$\mathcal O$3 version,

$\mathcal O$4

(Leroux-Lapierre, 22 Jul 2025).

If

$\mathcal O$5

then the asymptotic character satisfies

$\mathcal O$6

Here $\mathcal O$7 is the equivariant multiplicity map on $\mathcal O$8-equivariant Borel–Moore homology, and the $\mathcal O$9 are the $A$0-dimensional irreducible components of the support of the associated graded module (Leroux-Lapierre, 22 Jul 2025).

This theorem gives the asymptotic character map its defining significance. It is not merely a renormalized character, and not merely a large-$A$1 extraction device. It is an exact bridge between representation theory and the geometry of characteristic cycles. In this sense, the asymptotic character computes a localized equivariant invariant of singular support rather than a generic growth rate.

4. Truncated shifted Yangians and category $A$2

The abstract formalism is applied to truncated shifted Yangians

$A$3

associated to a simply-laced Lie algebra $A$4, a shift $A$5, a dominant coweight $A$6, and integral parameters $A$7. The paper verifies that these algebras satisfy the hypotheses required for the asymptotic character formalism: filtered algebra hypotheses, Hamiltonian torus action, $A$8-finiteness, $A$9-integrality, and local fixed-point algebra conditions (Leroux-Lapierre, 22 Jul 2025).

The relevant category is

$F^\bullet$0

For $F^\bullet$1, with

$F^\bullet$2

the asymptotic character is related to Mirković–Vilonen geometry by

$F^\bullet$3

where the sum runs over MV cycles $F^\bullet$4 appearing in the characteristic cycle of $F^\bullet$5, and

$F^\bullet$6

is the sign-twist arising from an anti-$F^\bullet$7-equivariance convention (Leroux-Lapierre, 22 Jul 2025).

This application makes the asymptotic character map a concrete tool in geometric representation theory. The category $F^\bullet$8 of a filtered quantization, the characteristic cycle, and the equivariant multiplicity of MV cycles become different realizations of the same invariant. A plausible implication is that the asymptotic character serves as a uniform interface between algebraic and geometric incarnations of highest-weight-type representation theory for these quantized Coulomb-branch-related objects.

5. KLR characters, the $F^\bullet$9-map, and basis comparisons

The paper next connects the Yangian-side asymptotic character to KLR and cyclotomic KLR representation theory. It recalls the KLR algebra

$A$0

the cyclotomic quotient $A$1, the KLRW algebra and its steadied quotient, and the cyclotomic idempotent $A$2, together with the isomorphism

$A$3

An equivalence of categories identifies the Yangian category $A$4 with a parity KLRW category,

$A$5

and composition with the cyclotomic idempotent functor gives

$A$6

(Leroux-Lapierre, 22 Jul 2025).

The asymptotic character then matches the bar-character on the cyclotomic KLR side: $A$7 At the same time, the paper connects the construction to Baumann’s map

$A$8

which on MV basis elements computes equivariant multiplicity at the bottom fixed point: $A$9 Thus asymptotic characters on the Yangian side correspond to $F^\bullet$0-values on the KLR side (Leroux-Lapierre, 22 Jul 2025).

The broader consequence is formulated as evidence for a change-of-basis statement between the dual canonical basis and the MV basis of $F^\bullet$1. The paper does not prove the full change-of-basis theorem, but it interprets the commutative diagram involving characteristic cycles, asymptotic characters, KLR characters, and MV geometry as evidence that the change-of-basis matrix is computed by characteristic-cycle multiplicities and therefore has non-negative integer coefficients (Leroux-Lapierre, 22 Jul 2025).

6. Terminological scope and nearby constructions

The expression “asymptotic character map” is specific in the filtered-quantization and category $F^\bullet$2 setting just described, but nearby areas use related language for structurally different objects. In $F^\bullet$3-adic representation theory, one paper describes a semisimple-centered asymptotic expansion

$F^\bullet$4

and explains that the resulting “asymptotic character map” is conceptually the assignment

$F^\bullet$5

namely the coefficient vector of a local character expansion near a semisimple element rather than a morphism from $F^\bullet$6 to rational functions (Spice, 2017). A second $F^\bullet$7-adic construction makes the passage

$F^\bullet$8

precise, and calls this an “asymptotic character” viewpoint for positive-depth local character expansions (Ciubotaru et al., 2023).

In noncommutative geometry, the asymptotic Connes–Moscovici characteristic map extends the classical Connes–Moscovici characteristic map by replacing the cyclic-cohomology target with asymptotic cyclic cohomology. There the map is built from a cup product, a diagonal morphism, and a characteristic homomorphism

$F^\bullet$9

and it is used to recover index and spectral-flow cocycles rather than weight-multiplicity asymptotics (Kaygun et al., 2017).

A further nearby but distinct usage arises in $A$0 recoupling theory, where asymptotic analysis of Wigner matrix elements yields the character formula

$A$1

and the remarkable point is that an “asymptotic” derivation reproduces the exact $A$2 character formula through a localization-like phenomenon for a discrete sum. This concerns the asymptotics of characters, not an asymptotic character map in the category-$A$3 sense (Geloun et al., 2010).

These distinctions remove a common source of ambiguity. The filtered-quantization asymptotic character map is a morphism on a Grothendieck group whose values are rational functions and whose meaning is fixed by characteristic cycles and equivariant multiplicities; the similarly named constructions in $A$4-adic harmonic analysis, cyclic cohomology, and $A$5 asymptotics address different invariants, different targets, and different geometric mechanisms.