Asymptotic Hecke Category

• The asymptotic Hecke category is a monoidal truncation of the Hecke category that focuses on a two-sided Kazhdan–Lusztig cell and its leading asymptotic data.
• It categorifies Lusztig’s asymptotic algebra by isolating top a-degree contributions through Soergel bimodule techniques and H-reduction strategies.
• Its structure connects idempotent constructions and modular centers with unipotent character sheaves, and extends to frameworks like complex reflection groups.

The asymptotic Hecke category is the cellwise monoidal truncation of the Hecke category associated with a two-sided Kazhdan–Lusztig cell. In Lusztig’s framework, it is designed so that its Grothendieck ring recovers the corresponding summand $J_c$ of the asymptotic Hecke algebra, while its categorical center captures character-sheaf-type phenomena. In the Soergel-bimodule realization, this category is obtained by discarding lower two-sided cells and retaining only the $a(c)$-controlled leading piece of convolution; for finite cells it is a multifusion category, and in the Weyl-group case its center is identified with unipotent character sheaves supported on the same cell (Rogel et al., 2023).

1. Cellwise truncation and basic construction

Let $H_W$ be the equal-parameter Hecke algebra over $A=\mathbb Z[v^{\pm1}]$ with Kazhdan–Lusztig basis $\{b_w\}_{w\in W}$, and write

$b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$

The $a$-function is

$$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$

and the leading coefficients

$\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$

define the asymptotic Hecke algebra

$$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$

For a two-sided cell $a(c)$0, one has $a(c)$1, where $a(c)$2 is spanned by $a(c)$3 (Rogel et al., 2023).

In the Hecke category $a(c)$4 of Soergel bimodules, the asymptotic Hecke category attached to a two-sided cell $a(c)$5 of $a(c)$6-value $a(c)$7 is constructed by first quotienting by the tensor ideal $a(c)$8 generated by lower cells, then restricting to the full graded additive subcategory generated by indecomposables $a(c)$9 with $H_W$0, and finally defining the truncated monoidal product by

$H_W$1

This produces a monoidal category $H_W$2 categorifying $H_W$3 (Rogel et al., 2023).

A complementary computational formulation starts with a two-sided cell $H_W$4 of Lusztig $H_W$5-value $H_W$6. One first kills lower cells, then uses the fact that in $H_W$7 only summands $H_W$8 with $H_W$9 can occur for $A=\mathbb Z[v^{\pm1}]$0, defines the new tensor product by taking the degree-$A=\mathbb Z[v^{\pm1}]$1 summand and shifting by $A=\mathbb Z[v^{\pm1}]$2, and finally restricts to degree-zero morphisms. The resulting category is semisimple monoidal and is again called the asymptotic Hecke category; its objects are denoted $A=\mathbb Z[v^{\pm1}]$3 when descended from $A=\mathbb Z[v^{\pm1}]$4 (Elias et al., 14 Jul 2025).

2. Decategorification, based rings, and geometric meaning

The decisive decategorification statement is

$A=\mathbb Z[v^{\pm1}]$5

For finite cells, $A=\mathbb Z[v^{\pm1}]$6 is a multifusion category, and its unit is the direct sum over the Duflo involutions in the cell,

$A=\mathbb Z[v^{\pm1}]$7

Thus the asymptotic Hecke category is not an auxiliary enhancement of $A=\mathbb Z[v^{\pm1}]$8; it is its monoidal categorification in the precise sense encoded by the KL cell $A=\mathbb Z[v^{\pm1}]$9 (Rogel et al., 2023).

A useful variant of notation writes the basis of $\{b_w\}_{w\in W}$0 as $\{b_w\}_{w\in W}$1 rather than $\{b_w\}_{w\in W}$2. In the affine setting relevant to $\{b_w\}_{w\in W}$3-adic groups, $\{b_w\}_{w\in W}$4 decomposes by two-sided cells, equivalently by unipotent conjugacy classes $\{b_w\}_{w\in W}$5 of the dual group, and one has

$\{b_w\}_{w\in W}$6

for a finite centrally extended $\{b_w\}_{w\in W}$7-set $\{b_w\}_{w\in W}$8, with convolution product on the $\{b_w\}_{w\in W}$9-theory side. This furnishes a geometric shadow of the asymptotic category: each cell summand of the asymptotic algebra appears as the Grothendieck ring of a convolution-theoretic category of equivariant coherent sheaves (Dawydiak, 2023).

This relationship clarifies the role of the asymptotic Hecke category within Hecke-theoretic categorification. The ordinary Hecke category categorifies the full Hecke algebra, whereas the asymptotic Hecke category categorifies a cell summand of the leading-term algebra $b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$0. A plausible implication is that cellwise truncation is the categorical mechanism that separates genuine asymptotic data from the unreduced monoidal structure.

3. Center, $b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$1-reduction, and character-sheaf interpretation

For a multifusion category $b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$2, the Drinfeld center $b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$3 is the natural receptacle for character-sheaf-type structures. In the Weyl-group case, Lusztig showed geometrically that the center of the asymptotic Hecke category attached to a two-sided cell $b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$4 is equivalent to the category $b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$5 of unipotent character sheaves supported on that cell: $b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$6 This makes the center of the asymptotic Hecke category the categorical counterpart of unipotent character theory in the given cell (Rogel et al., 2023).

A central structural simplification is $b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$7-reduction. If the unit of $b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$8 decomposes according to Duflo involutions, the component fusion categories are indexed by diagonal $b_xb_y=\sum_{z\in W} h_{x,y,z}\,b_z.$9-cells $a$0. Writing

$a$1

one has

$a$2

Thus the center of the full asymptotic Hecke category is already determined by any diagonal $a$3-cell component. This reduction is the key technical device in explicit calculations of centers and modular data (Rogel et al., 2023).

In finite Weyl-group situations, non-exceptional cells admit tensor equivalences

$a$4

and then $a$5-reduction identifies the center with a finite-group-theoretic modular category. For exceptional cells in $a$6 and $a$7, the asymptotic category instead takes the form

$a$8

These descriptions explain why Fourier matrices of Lusztig–Malle–Broué type can appear as normalized $a$9-matrices of centers (Rogel et al., 2023).

The non-crystallographic case makes this especially explicit. For dihedral groups, the asymptotic Hecke category of the middle cell is reduced to

$$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$0

where $$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$1 is the type $$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$2 Verlinde category, and the resulting $$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$3-matrices agree with Lusztig’s dihedral Fourier matrices. In $$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$4 and $$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$5, many cells are similarly identified with small known fusion categories such as $$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$6, Fibonacci-type categories, or $$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$7, but the $$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$8 cell of $$a(z)=\min\{n\mid v^n h_{x,y,z}\in \mathbb Z[v]\text{ for all }x,y\},$$9-value $\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$0 remains unresolved; if the conjectural picture is correct, its center should have $\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$1 simple objects (Rogel et al., 2023).

4. Idempotents, top $\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$2-degree summands, and categorical dimensions

The asymptotic tensor product is defined by extracting the top $\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$3-degree contribution inside ordinary Hecke-category tensor products. The subtle point is that this degree-$\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$4 submodule is canonically a subobject but not canonically a direct summand. The computational program therefore hinges on constructing explicit projections onto these top summands, using the relative Hodge theory of Elias–Williamson and multiplication by $\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$5 (Elias et al., 14 Jul 2025).

This leads to a detailed idempotent technology. For a reduced expression $\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$6, the Bott–Samelson object $\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$7 contains the indecomposable $\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$8 as its top summand. A top idempotent is a primitive idempotent with image $\gamma_{x,y,z^{-1}}:=h_{x,y,z}v^{a(z)}(0)$9, and a clasp idempotent is a top idempotent satisfying orthogonality to lower terms. The coefficients appearing in recursive formulas for these idempotents are inverses of local intersection forms. For recursible triples $$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$0, these local intersection forms satisfy a recursion involving Demazure operators and admit closed expressions in terms of two-colored quantum numbers (Elias et al., 14 Jul 2025).

The same machinery computes categorical dimensions in asymptotic Hecke categories. For a diagonal $$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$1-cell $$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$2 with Duflo involution $$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$3, one defines scalars $$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$4 and $$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$5 from compositions of the form

$$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$6

modulo lower cells, and then

$$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$7

If $$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$8 is weak-right-above $$J=\bigoplus_{w\in W}\mathbb Z j_w,\qquad j_xj_y=\sum_{z\in W}\gamma_{x,y,z^{-1}}\,j_z.$$9, $a(c)$00 reduces to a partial trace computation, and for parabolic cells one has the further simplification

$a(c)$01

so dimensions become computable via Demazure operators applied to explicit partial traces (Elias et al., 14 Jul 2025).

These formulas are not merely local numerics. For finite Coxeter groups, asymptotic Hecke categories classify simple transitive $a(c)$02-representations of the Hecke category, and categorical $a(c)$03-reduction always holds: the category attached to a diagonal $a(c)$04-cell is $a(c)$05-Morita equivalent to the category attached to the full two-sided cell. The computational technology of idempotents, local intersection forms, and partial traces is therefore a structural tool for identifying the monoidal categories themselves, not only their decategorifications (Elias et al., 14 Jul 2025).

5. Extensions, analogues, and broader frameworks

A direct generalization beyond Coxeter groups appears for the complex reflection groups $a(c)$06. There the proposed asymptotic category is denoted $a(c)$07. It is defined as a monoidal additive $a(c)$08-linear matrix category built from the root-of-unity $a(c)$09 fusion category $a(c)$10: its indecomposable objects are matrices $a(c)$11 with $a(c)$12 simple in $a(c)$13 of color $a(c)$14, morphisms are entrywise, and tensor product is matrix multiplication. Its rank is

$a(c)$15

and its center is modular, with $a(c)$16- and $a(c)$17-matrices related to Malle’s Fourier matrices for $a(c)$18. A larger Morita equivalent category $a(c)$19 is also constructed, and its Grothendieck ring is identified with the asymptotic algebra of a Calogero–Moser cell (Lacabanne et al., 2024).

This complex-reflection-group construction is not a literal Soergel-cell quotient. It is instead a matrix-category analogue whose degree-zero behavior plays the role of a cellwise asymptotic truncation. That difference is substantive: the classical asymptotic Hecke category is cell-theoretic inside a Coxeter Hecke category, whereas the $a(c)$20 version is an analogue built from $a(c)$21 fusion data. Still, the persistence of KL-type bases, cells, modular centers, and Fourier matrices indicates that the asymptotic-Hecke pattern extends beyond Weyl and Coxeter groups (Lacabanne et al., 2024).

Two further structural directions are suggestive. First, the affine Hecke category can be reconstructed as a monoidal colimit of finite-type parahoric Hecke subcategories,

$a(c)$22

in the $a(c)$23-category of monoidal DG categories. This suggests that an affine asymptotic Hecke category, if formulated by cellwise or asymptotic truncation, may be approachable through compatible finite-type approximations (Tao et al., 2020).

Second, monodromic Hecke categories now admit algebraic, diagrammatic, and parity-sheaf categorifications of a monodromic Hecke algebroid, together with object-adapted cellular structures, lower-term ideals, and endoscopic reduction to ordinary Hecke categories of endoscopic Coxeter groups. This suggests a route to monodromic asymptotic Hecke categories by reducing blockwise to ordinary asymptotic categories and then regluing through endoscopic data (Sandvik, 16 Apr 2026).

6. Distinction from trace decategorification and related Hecke-categorical invariants

The asymptotic Hecke category should not be confused with the trace decategorification of the full Hecke category. For the diagrammatic Hecke category $a(c)$24, the trace is

$a(c)$25

and one has

$a(c)$26

Thus the trace of the full Hecke category recovers the ordinary Hecke algebra, or in the all-degree version a crossed product with the polynomial ring, but not Lusztig’s asymptotic algebra $a(c)$27 and not a cellwise asymptotic quotient (Elias et al., 2015).

This distinction matters conceptually. The word “cell” in the theory of strictly object-adapted cellular categories refers to cellular bases and factorization through distinguished objects; it is not the same notion as Kazhdan–Lusztig left, right, or two-sided cells. In particular, the trace computation for the full Hecke category is controlled by cellular factorization and identities of indecomposables, whereas the asymptotic Hecke category is controlled by two-sided-cell truncation and the top $a(c)$28-degree part of tensor products (Elias et al., 2015).

A plausible conclusion is that asymptotic information is not an automatic decategorified invariant of the full Hecke category. It emerges only after imposing cellwise structure—by quotienting lower cells, extracting the top $a(c)$29-degree piece, and passing to the monoidal category supported on a fixed two-sided cell. In that sense, the asymptotic Hecke category is not another decategorification of the Hecke category; it is a cell-localized recategorification of Lusztig’s leading-term algebra.