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

Stefan Dawydiak

TL;DR

This work develops explicit positivity properties for the based ring of the lowest two-sided cell in the extended affine Weyl group by expressing coefficients of the asymptotic basis elements $t_w$ in terms of generalized exponents of the Langlands dual group. Using a blend of the asymptotic Bernstein presentation, Demazure–Lusztig operators, and a K-theoretic model via $K( ext{St}/G^ abla imesoldsymbol{G}_m)$, the authors derive a concrete formula for $t_w(oldsymbol{ u})$ in dominant directions and show positivity (up to a sign) for the corresponding coefficients in a second positive basis of $t_d J t_{d'}$. They further establish a second positive basis $igl"{f_ u}igr"$ related to spherical Kazhdan–Lusztig polynomials, and prove recurrence relations for the two-sided cell data via Presburger constructibility, yielding computationally tractable structure constants. For $ ext{GL}_n$, they extend the method to certain other cells under a Plancherel-coincidence trick and relate the results to spherical KL polynomials, Hall–Littlewood data, and Levi-subgroup combinatorics. Overall, the paper connects the representation-theoretic and geometric aspects of affine Hecke algebras with Langlands dual geometry, providing explicit, positive, and computable descriptions of the lowest-cell coefficients and their recurrences.

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.

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

TL;DR

This work develops explicit positivity properties for the based ring of the lowest two-sided cell in the extended affine Weyl group by expressing coefficients of the asymptotic basis elements in terms of generalized exponents of the Langlands dual group. Using a blend of the asymptotic Bernstein presentation, Demazure–Lusztig operators, and a K-theoretic model via , the authors derive a concrete formula for in dominant directions and show positivity (up to a sign) for the corresponding coefficients in a second positive basis of . They further establish a second positive basis related to spherical Kazhdan–Lusztig polynomials, and prove recurrence relations for the two-sided cell data via Presburger constructibility, yielding computationally tractable structure constants. For , they extend the method to certain other cells under a Plancherel-coincidence trick and relate the results to spherical KL polynomials, Hall–Littlewood data, and Levi-subgroup combinatorics. Overall, the paper connects the representation-theoretic and geometric aspects of affine Hecke algebras with Langlands dual geometry, providing explicit, positive, and computable descriptions of the lowest-cell coefficients and their recurrences.

Abstract

Let be an extended affine Weyl group and and be the corresponding affine and asymptotic Hecke algebras with standard bases and , respectively. Viewing as a subalgebra of the -adic completion of , we give formulas for the coefficient of in for various and in the lowest two-sided cell, in terms of generalized exponents of the Langlands dual group, under a hypothesis on the left cell containing . In particular our results hold for the canonical left cell. For such we also define a seemingly new positive basis for the corresponding subring of . For , we give partial results for some other cells.

Paper Structure

This paper contains 14 sections, 10 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. A second positive basis of $t_d Jt_{d'}$
  3. Other cells
  4. Acknowledgements
  5. Notation and conventions
  6. The affine and asymptotic Hecke algebras
  7. The Schwartz space of the basic affine space
  8. Demazure-Lusztig operators and the Bernstein subalgebra
  9. Formulas for the lowest two-sided cell
  10. Proofs of Theorem \ref{['thm K theory formulas']} and Corollary \ref{['cor symm and pos dc0']}
  11. Proof of Corollary \ref{['cor tw0 Jtdc0']}
  12. Proof of Corollary \ref{['cor KL']}: Spherical Kazhdan-Lusztig polynomials
  13. Recurrence relations
  14. Other two-sided cells

Key Result

Theorem 1

Let $d, d'$ be distinguished involutions in the lowest two-sided cell $\mathbf{c}_0$ corresponding to $u,u'\in W$ via Shi's parametrization Shi, let $t_w\in t_d Jt_{d'}$ correspond to a dominant weight $\lambda$ under the same. Let $\left\{\mathcal{O}(x_u)\right\}_{u\in W}$ be Steinberg's $K(\mathrm where $\mathcal{O}(\mathcal{N}^\vee)_i$ and $\mathcal{O}(\mathfrak{g}^\vee)_i$ are the space of hom

Theorems & Definitions (27)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Example 1
  • Example 2
  • Lemma 1
  • proof
  • ...and 17 more