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Residue Formulae for the Trace on Affine Hecke Algebras

Paul Mammen

TL;DR

This work develops residue-based formulas for the trace on extended affine Hecke algebras by combining Opdam’s trace generating function with the Szenes–Vergne total-residue framework. The authors formulate a general mechanism that expresses trace coefficients as sums of residues over residual points in big chambers of the positive root cone, and apply it to Type $A_{n-1}$ to obtain explicit product formulas for translations in several chamber families. In particular, they derive concrete trace expressions for chambers $\mathfrak{a}_{n-1}^{1}$, $\mathfrak{a}_{n-1}^{n-1}$, and the general $\mathfrak{a}_{n-1}^{m}$, linking Kostant partitions, Tesler-matrix weights, and permutation data. The results extend known finite-type traces to the affine setting, providing a concrete computational bridge between representation-theoretic traces and combinatorial objects like Tesler matrices and Kostant partitions, with potential connections to Plancherel theory for affine Hecke algebras. This framework paves the way for systematic residue-based evaluations of traces in broader affine settings and types.

Abstract

Motivated by recent advances in Catalan combinatorics, we study special values of the standard trace on affine Hecke algebras. Starting from a generating function for this trace calculated by Opdam, we use the theory of Szenes and Vergne to obtain residue formulae for the trace. This allows us to derive a product formula for the trace of translation elements corresponding to weights in certain ``Big Chambers'' of the positive root cone.

Residue Formulae for the Trace on Affine Hecke Algebras

TL;DR

This work develops residue-based formulas for the trace on extended affine Hecke algebras by combining Opdam’s trace generating function with the Szenes–Vergne total-residue framework. The authors formulate a general mechanism that expresses trace coefficients as sums of residues over residual points in big chambers of the positive root cone, and apply it to Type to obtain explicit product formulas for translations in several chamber families. In particular, they derive concrete trace expressions for chambers , , and the general , linking Kostant partitions, Tesler-matrix weights, and permutation data. The results extend known finite-type traces to the affine setting, providing a concrete computational bridge between representation-theoretic traces and combinatorial objects like Tesler matrices and Kostant partitions, with potential connections to Plancherel theory for affine Hecke algebras. This framework paves the way for systematic residue-based evaluations of traces in broader affine settings and types.

Abstract

Motivated by recent advances in Catalan combinatorics, we study special values of the standard trace on affine Hecke algebras. Starting from a generating function for this trace calculated by Opdam, we use the theory of Szenes and Vergne to obtain residue formulae for the trace. This allows us to derive a product formula for the trace of translation elements corresponding to weights in certain ``Big Chambers'' of the positive root cone.
Paper Structure (15 sections, 18 theorems, 96 equations)

This paper contains 15 sections, 18 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Affine Hecke Algebras
  3. Trace formula and residual points
  4. Link with Tesler Matrices
  5. Preliminaries on Total Residues and Coefficient formulas
  6. Total Residue
  7. The ring $M^{\Phi^+}$ and Reduced Set of Poles
  8. The cone $C(\Phi^+)$ and the main theorem
  9. Computing the Trace using Residues
  10. Type A
  11. Calculating $c(\lambda,\eta)$
  12. Trace formula for $\mathfrak{a}_{n-1}^{1}$
  13. Trace formula for $\mathfrak{a}_{n-1}^{n-1}$
  14. Trace formula for $\mathfrak{a}_{n-1}^m$
  15. Concluding Remarks

Key Result

Theorem 1.1

Let $\tilde{H} = H(\tilde{A}_{n-1})$ be the extended affine Hecke algebra. Let $\lambda = \sum_{i=1}^{n-1} a_i \alpha_{i}$, where $\alpha_{i} = e_{i} - e_{i+1}$ denote the simple roots. Suppose Then where $\theta_{-\lambda}$ is the basis element corresponding to the translation $t_\lambda \in \tilde{A}_{n-1}$ (def2.2) and $[k]_{q} := \frac{q^{k}-1}{q-1}$ is the $k^{\text{th}}$$q$-integer.

Theorems & Definitions (49)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • Definition 2.5
  • Theorem 2.2: op1
  • Example
  • Definition 2.6
  • ...and 39 more