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Two-sided cells of Weyl groups and certain splitting Whittaker polynomials

Fan Gao, Yannan Qiu

Abstract

Consider the subset of a Weyl group with a fixed descent set. For Weyl groups of classical types, we determine the number of two-sided cells this subset intersect. Moreover, we apply this result to prove that certain rational Whittaker polynomials associated with covering groups split over the field of rational numbers.

Two-sided cells of Weyl groups and certain splitting Whittaker polynomials

Abstract

Consider the subset of a Weyl group with a fixed descent set. For Weyl groups of classical types, we determine the number of two-sided cells this subset intersect. Moreover, we apply this result to prove that certain rational Whittaker polynomials associated with covering groups split over the field of rational numbers.
Paper Structure (20 sections, 16 theorems, 132 equations, 7 tables)

This paper contains 20 sections, 16 theorems, 132 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Main result
  3. Acknowledgement
  4. Two-sided cells intersecting $\mathcal{C}_S$
  5. Cells of $W$
  6. The representation $\sigma_S$
  7. More explicit formulas for $\sigma_S$
  8. $\bf{a}$-function and upper bound of $\left| \mathfrak{C}^{\rm LR}(S) \right|$
  9. Proof of Theorem \ref{['T:ABD']}
  10. Further result for type $A_r$
  11. The Whittaker polynomial $\mathcal{P}_{G,S}(X)$
  12. Whittaker models for oasitic covers
  13. Regular principal series
  14. The polynomial $\mathcal{P}_{G,S}(X)$
  15. Splitting of $\mathcal{P}_{G,S}(X)$ over $\mathbbm{Q}$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $\Delta$ be of type $A_r, B_r$ or $D_r$ and let $S_j \in \mathscr{P}(\Delta), 1\leqslant j \leqslant r-1$. Then the following data are determined explicitly: More precise results are tabulated in Tables T:A, T:B, T:D-odd and T:D-even.

Theorems & Definitions (31)

  • Theorem 1.1: Theorem \ref{['T:ABD']}
  • Theorem 1.2: Theorem \ref{['T:poly']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • ...and 21 more