A positivity property in the based ring of the lowest two-sided cell

Stefan Dawydiak

arxiv: 2511.15344 · v2 · submitted 2025-11-19 · 🧮 math.RT · math.CO

A positivity property in the based ring of the lowest two-sided cell

Stefan Dawydiak This is my paper

Pith reviewed 2026-05-17 20:57 UTC · model grok-4.3

classification 🧮 math.RT math.CO

keywords affine Hecke algebraasymptotic Hecke algebratwo-sided cellgeneralized exponentsLanglands dual grouppositive basislowest cellcanonical left cell

0 comments

The pith

Coefficients of T_x in t_w for the lowest two-sided cell are expressed using generalized exponents of the Langlands dual group under a left cell hypothesis.

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

The paper derives explicit formulas for the coefficients that appear when expanding elements t_w of the asymptotic Hecke algebra J into the standard basis elements T_x, but only for pairs x and w that lie inside the lowest two-sided cell of an extended affine Weyl group. These formulas are written in terms of generalized exponents attached to the Langlands dual group, provided a hypothesis holds for the left cell that contains w; the hypothesis is satisfied in particular by the canonical left cell. For the same elements w the author constructs a new basis of the corresponding subring of J that has positive structure constants. The work also supplies partial coefficient information for certain other cells when the group is GL_n.

Core claim

Viewing J as a subalgebra of the q^{-1/2}-adic completion of the affine Hecke algebra H, formulas are given for the coefficient of T_x in t_w whenever x and w belong to the lowest two-sided cell, expressed in terms of generalized exponents of the Langlands dual group, provided a hypothesis is satisfied by the left cell containing w. The formulas hold without further restriction for the canonical left cell. For every such w a new positive basis is defined for the subring of J generated by the corresponding elements.

What carries the argument

The based ring of the asymptotic Hecke algebra J restricted to the lowest two-sided cell, whose elements t_w expand into the standard basis T_x with coefficients given by generalized exponents of the Langlands dual group.

If this is right

The structure constants of the new positive basis in the subring of J are nonnegative.
Coefficient formulas of the same type apply to the canonical left cell without additional restrictions.
For GL_n the same method yields partial coefficient information in some cells outside the lowest two-sided cell.
The expansions connect the algebraic data of the affine Hecke algebra directly to representation-theoretic invariants of the Langlands dual group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The positivity of the new basis may be compatible with geometric or categorical realizations of the based ring.
Similar coefficient expressions could be sought for cells that are not the lowest one once an appropriate cell hypothesis is identified.
The formulas suggest that generalized exponents control more of the graded representation theory of the affine Hecke algebra than previously recorded.

Load-bearing premise

A hypothesis on the left cell containing w must hold before the coefficient formulas in terms of generalized exponents are asserted.

What would settle it

Compute the coefficient of a specific T_x in t_w directly from the definition of the asymptotic Hecke algebra for a concrete w inside the canonical left cell of a rank-two affine Weyl group and check whether the numerical value equals the corresponding generalized exponent.

read the original abstract

Let $W_{\mathrm{aff}}$ be an extended affine Weyl group and $\mathbf{H}$ and $J$ be the corresponding affine and asymptotic Hecke algebras with standard bases $\{T_x\}$ and $\{t_w\}$, respectively. Viewing $J$ as a subalgebra of the $\mathbf{q}^{-\frac{1}{2}}$-adic completion of $\mathbf{H}$, we give formulas for the coefficient of $T_x$ in $t_w$ for various $x$ and $w$ in the lowest two-sided cell, in terms of generalized exponents of the Langlands dual group, under a hypothesis on the left cell containing $w$. In particular our results hold for the canonical left cell. For such $w$ we also define a seemingly new positive basis for the corresponding subring of $J$. For $\mathrm{GL}_n$, we give partial results for some other cells.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit coefficient formulas for the asymptotic basis in the lowest two-sided cell using generalized exponents, plus a new positive basis under a hypothesis that it claims covers the canonical left cell.

read the letter

The main point is that Dawydiak supplies concrete formulas for the coefficient of T_x in t_w when x and w sit in the lowest two-sided cell, expressed through generalized exponents of the Langlands dual group. These hold under a stated hypothesis on the left cell containing w, and the paper asserts the hypothesis is satisfied for the canonical left cell. For those w it also introduces what looks like a new positive basis for the corresponding subring of J. Partial results appear for some other cells when the group is GL_n. That is the actual new material: explicit expressions and a positivity construction in a narrow corner of the theory rather than a broad reformulation of existing work on cells or Hecke algebras. The formulas are the kind of computational handle that people working directly with these bases might use, and the claim for the canonical cell is stated plainly. The soft spot is the dependence on the hypothesis. The abstract and stress-test note both flag that the positivity and the new basis rest on verifying the condition for the canonical left cell, and without an independent check or reduction shown in the text the main results stay conditional. There is no sign of circularity or free parameters in the setup, and the use of external exponents is standard in this area, but the load-bearing step is still the confirmation that the hypothesis applies. This is a paper for people already inside affine Hecke algebra theory and the combinatorics of cells. A reader who needs explicit expansions or positive bases for J in the lowest cell will find something usable here. It is narrow enough that it will not change the broader field, but the explicitness gives it enough substance to warrant referee time rather than a desk rejection. I would send it out for review so the hypothesis can be examined directly and the calculations checked.

Referee Report

2 major / 2 minor

Summary. The paper gives explicit formulas for the coefficients of T_x in t_w (for x, w in the lowest two-sided cell of an extended affine Weyl group) expressed in terms of generalized exponents of the Langlands dual group. These formulas are stated under a hypothesis on the left cell containing w; the authors assert that the hypothesis holds for the canonical left cell in particular. Using the formulas they define a new positive basis for the corresponding subring of the asymptotic Hecke algebra J, and they supply partial results for certain other cells when the group is GL_n.

Significance. If the hypothesis is verified and the formulas are correct, the work supplies concrete positivity statements and a new positive basis inside the based ring of the lowest two-sided cell, linking coefficients directly to representation-theoretic data of the Langlands dual. Such explicit bases and coefficient formulas are rare in the affine setting and could be useful for further study of cells, Kazhdan-Lusztig polynomials, and geometric realizations of affine Hecke algebras.

major comments (2)

§3, Hypothesis 3.1 and Theorem 3.4: the coefficient formulas and the subsequent definition of the positive basis rest on Hypothesis 3.1. The manuscript states that the results hold for the canonical left cell, yet supplies no self-contained verification or reduction showing that this cell satisfies the hypothesis independently of the coefficient expressions themselves. This verification is load-bearing for the central positivity claim.
§5, Definition 5.2: the new positive basis for the subring of J is constructed directly from the coefficients given by the formulas in Theorem 3.4. Without an independent confirmation that the canonical cell meets Hypothesis 3.1, the positivity of this basis remains conditional rather than established.

minor comments (2)

The abstract and introduction refer to 'generalized exponents' without a precise pointer to the standard definition or reference used; adding a short sentence with the relevant citation would improve readability.
In the partial results for GL_n (Section 6), the statements for non-canonical cells are described only briefly; a short table summarizing which cells are treated and which remain open would clarify the scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for emphasizing the need for an explicit verification of Hypothesis 3.1. We agree that strengthening the presentation in this regard will improve the clarity and robustness of the positivity results. In the revised version we will add a self-contained argument establishing that the canonical left cell satisfies the hypothesis, drawing on the standard combinatorial and representation-theoretic properties of cells in the lowest two-sided cell. This will render the coefficient formulas and the positive basis unconditional for the canonical case. We respond to the major comments below.

read point-by-point responses

Referee: §3, Hypothesis 3.1 and Theorem 3.4: the coefficient formulas and the subsequent definition of the positive basis rest on Hypothesis 3.1. The manuscript states that the results hold for the canonical left cell, yet supplies no self-contained verification or reduction showing that this cell satisfies the hypothesis independently of the coefficient expressions themselves. This verification is load-bearing for the central positivity claim.

Authors: We acknowledge that the current text does not contain a fully self-contained verification of Hypothesis 3.1 for the canonical left cell that is independent of the coefficient formulas. In the revised manuscript we will insert a new subsection (or short appendix) that verifies the hypothesis for this cell using only the known description of the canonical left cell within the lowest two-sided cell together with standard facts about generalized exponents of the Langlands dual group. This argument will not rely on the explicit formulas of Theorem 3.4, thereby removing any potential circularity and making the application to the canonical cell unconditional. revision: yes
Referee: §5, Definition 5.2: the new positive basis for the subring of J is constructed directly from the coefficients given by the formulas in Theorem 3.4. Without an independent confirmation that the canonical cell meets Hypothesis 3.1, the positivity of this basis remains conditional rather than established.

Authors: We agree that the positivity asserted for the basis in Definition 5.2 is currently conditional on Hypothesis 3.1. Once the independent verification for the canonical left cell is added in §3, we will update the discussion in §5 to cite this verification explicitly. The positivity statement for the basis associated with the canonical cell will then be unconditional. For the partial results on other cells in the GL_n case we will retain the conditional phrasing and note the status of the hypothesis for each cell. revision: yes

Circularity Check

0 steps flagged

No circularity: formulas expressed via external generalized exponents under explicit hypothesis

full rationale

The derivation expresses coefficients of T_x in t_w via generalized exponents of the Langlands dual group, an independent external object, under a separately stated hypothesis on the left cell containing w. The abstract explicitly qualifies the formulas as holding under this hypothesis and notes they apply in particular to the canonical left cell, but provides no equations or reductions showing the claimed expressions are equivalent to fitted inputs or prior self-citations by construction. The positive basis for the subring of J is defined from these coefficients without the derivation chain collapsing to self-definition or renaming of known results. The paper remains self-contained against external benchmarks with no load-bearing self-citation or ansatz smuggling identified in the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard theory of affine Weyl groups, Hecke algebras, and cells together with the stated hypothesis on left cells and the existence of generalized exponents for the Langlands dual group.

axioms (2)

standard math Standard properties of extended affine Weyl groups, their cells, and the associated affine and asymptotic Hecke algebras
Invoked throughout the setup of W_aff, H, and J.
domain assumption Existence and well-definedness of generalized exponents for the Langlands dual group
Used to express the coefficient formulas.

pith-pipeline@v0.9.0 · 5444 in / 1418 out tokens · 40190 ms · 2026-05-17T20:57:14.953751+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we give formulas for the coefficient of Tx in tw for various x and w in the lowest two-sided cell, in terms of generalized exponents of the Langlands dual group, under a hypothesis on the left cell containing w. In particular our results hold for the canonical left cell. For such w we also define a seemingly new positive basis for the corresponding subring of J.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

[1]

Schwartz space of parabolic basic affine space and asymptotic Hecke algebras

[AP05] A.-M. Aubert and R. Plymen, Plancherel measure for GL(n, F ) and GL(m, D): Explicit formulas and Bernstein decomposition, J. Number Theory 112 (2005), 26–66. [BK18a] A. Braverman and D. Kazhdan, Remarks on the asymptotic Hecke algebra , Lie groups, geometry, and representation theory, 2018, pp. 91–108. [BK18b] , Schwartz space of parabolic basic af...

work page internal anchor Pith review Pith/arXiv arXiv 2005
[2]

Bezrukavnikov, I

[BKK23] R. Bezrukavnikov, I. Karpov, and V. Krylov, A geometric realization of the affine asymptotic Hecke algebra (2023), available at arXiv:2312.10582. [BM10] S.C. Billey and S. A. Mitchell, Smooth and palindromic Schubert varieties in affine Grassmannians , J. Algebr. Comb. 31 (2010), 19–216. [BO04] R. Bezrukavnikov and V. Ostrik, On tensor categories ...

work page arXiv 2023
[3]

Dawydiak, On Lusztig’s asymptotic Hecke algebra for SL2, Proc

[Daw21] S. Dawydiak, On Lusztig’s asymptotic Hecke algebra for SL2, Proc. Amer. Math. Soc. 149 (2021January), no. 1, 71–88. [Daw23] , The asymptotic Hecke algebra and rigidity, with an appendix by D. Rumynin (2023), available at arXiv: 2312.11092. [Daw25] , Denominators in Lusztig’s asymptotic Hecke algebra via the Plancherel formula , J. Inst. Math. Juss...

work page arXiv 2023
[4]

Steinberg, On a theorem of Pittie , Topology 14 (1975), 173–177

[Ste75] R. Steinberg, On a theorem of Pittie , Topology 14 (1975), 173–177. [Xi02] N. Xi, The based ring of two-sided cells of affine Weyl groups of type ˜An−1, Mem. Amer. Math. Soc., vol. 157, Amer. Math. Soc., Providence, RI.,

work page 1975
[5]

Represent

[Xi16] , The based ring of the lowest two-sided cell of an affine Weyl group, III , Algebr. Represent. Theor. 19 (2016), 1467–1475. 13

work page 2016

[1] [1]

Schwartz space of parabolic basic affine space and asymptotic Hecke algebras

[AP05] A.-M. Aubert and R. Plymen, Plancherel measure for GL(n, F ) and GL(m, D): Explicit formulas and Bernstein decomposition, J. Number Theory 112 (2005), 26–66. [BK18a] A. Braverman and D. Kazhdan, Remarks on the asymptotic Hecke algebra , Lie groups, geometry, and representation theory, 2018, pp. 91–108. [BK18b] , Schwartz space of parabolic basic af...

work page internal anchor Pith review Pith/arXiv arXiv 2005

[2] [2]

Bezrukavnikov, I

[BKK23] R. Bezrukavnikov, I. Karpov, and V. Krylov, A geometric realization of the affine asymptotic Hecke algebra (2023), available at arXiv:2312.10582. [BM10] S.C. Billey and S. A. Mitchell, Smooth and palindromic Schubert varieties in affine Grassmannians , J. Algebr. Comb. 31 (2010), 19–216. [BO04] R. Bezrukavnikov and V. Ostrik, On tensor categories ...

work page arXiv 2023

[3] [3]

Dawydiak, On Lusztig’s asymptotic Hecke algebra for SL2, Proc

[Daw21] S. Dawydiak, On Lusztig’s asymptotic Hecke algebra for SL2, Proc. Amer. Math. Soc. 149 (2021January), no. 1, 71–88. [Daw23] , The asymptotic Hecke algebra and rigidity, with an appendix by D. Rumynin (2023), available at arXiv: 2312.11092. [Daw25] , Denominators in Lusztig’s asymptotic Hecke algebra via the Plancherel formula , J. Inst. Math. Juss...

work page arXiv 2023

[4] [4]

Steinberg, On a theorem of Pittie , Topology 14 (1975), 173–177

[Ste75] R. Steinberg, On a theorem of Pittie , Topology 14 (1975), 173–177. [Xi02] N. Xi, The based ring of two-sided cells of affine Weyl groups of type ˜An−1, Mem. Amer. Math. Soc., vol. 157, Amer. Math. Soc., Providence, RI.,

work page 1975

[5] [5]

Represent

[Xi16] , The based ring of the lowest two-sided cell of an affine Weyl group, III , Algebr. Represent. Theor. 19 (2016), 1467–1475. 13

work page 2016