Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Action of the relative Weyl group on partial Springer sheaf

Tamanna Chatterjee, Laura Rider

Abstract

In the context of the Springer correspondence, the Weyl group action on the Springer sheaf can be defined in two ways: via restriction or the Fourier transform. It is well-known that these two actions differ by the sign character. This was proven by Hotta for sheaves with characteristic 0 coefficients in 1981, and more recently extended to arbitrary coefficients by Achar, Henderson, Juteau, and Riche in 2014. In this short article, we study an extension of this problem to the partial Springer sheaf with arbitrary field coefficients. This involves an action of the so-called relative Weyl group $W(L)$.

Action of the relative Weyl group on partial Springer sheaf

Abstract

In the context of the Springer correspondence, the Weyl group action on the Springer sheaf can be defined in two ways: via restriction or the Fourier transform. It is well-known that these two actions differ by the sign character. This was proven by Hotta for sheaves with characteristic 0 coefficients in 1981, and more recently extended to arbitrary coefficients by Achar, Henderson, Juteau, and Riche in 2014. In this short article, we study an extension of this problem to the partial Springer sheaf with arbitrary field coefficients. This involves an action of the so-called relative Weyl group .
Paper Structure (14 sections, 21 theorems, 95 equations, 2 figures)

This paper contains 14 sections, 21 theorems, 95 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgement
  4. Preliminaries
  5. Reminder on Springer Theory
  6. Fourier--Sato transform
  7. The partial Springer resolution
  8. $W_L$-invariants
  9. The sheaf complex $\mathop{\mathrm{\underline{Spr}^U}}\nolimits$.
  10. Parabolic induction
  11. Two actions of $W(L)$
  12. Actions on $\mathop{\mathrm{H}}\nolimits^*(G/P)$
  13. Comparison of $W(L)$-actions
  14. The comparison of the Springer and the partial Springer.

Key Result

Theorem 1.1

There exists a map $\Lambda_U: \mathop{\mathrm{\mathit{k}}}\nolimits[W(L)]\to \mathop{\mathrm{\mathit{k}}}\nolimits[W(L)]$ such that,

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (32)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2: The Springer Correspondence by Restriction
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • Theorem 2.8
  • ...and 22 more