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Micro-packets for real groups of type $G_2$

Leticia Barchini, Nicolas Arancibia Robert, Paul Mezo

Abstract

In their study of Arthur's conjectures for real groups, Adams, Barbasch, and Vogan introduced the notion of micro-packets. Micro-packets are finite sets of irreducible representations defined using microlocal geometric methods and characteristic cycles. We explore an action of the Weyl group on characteristic cycles to compute all micro-packets of real groups of type $G_2$.

Micro-packets for real groups of type $G_2$

Abstract

In their study of Arthur's conjectures for real groups, Adams, Barbasch, and Vogan introduced the notion of micro-packets. Micro-packets are finite sets of irreducible representations defined using microlocal geometric methods and characteristic cycles. We explore an action of the Weyl group on characteristic cycles to compute all micro-packets of real groups of type .

Paper Structure

This paper contains 17 sections, 16 theorems, 234 equations, 7 tables.

Table of Contents

  1. Introduction
  2. ${^\vee}K$-orbits of the flag variety of ${^\vee}\mathrm{G}_{2}$
  3. The group ${^\vee}K$
  4. ${^\vee}K$-orbits of ${^\vee}\mathrm{G}_{2}/ \,{^\vee}B$
  5. Langlands Component groups
  6. Orbits on flag varieties for regular non-integral infinitesimal character
  7. Complete geometric parameters and the moment map
  8. Weyl group actions and the characteristic cycles map
  9. The coherent continuation representation
  10. The $W$-action on $\mathscr{L}(X, {^\vee}K)$
  11. The $W$-equivariance of the characteristic cycle map
  12. Interim progress on the $W$-action on $\mathscr{L}(X, {^\vee}K)$
  13. Computing characteristic cycles for ${}^{\vee}\mathrm{G}_{2}$
  14. Characteristic cycles in the case of a regular non-integral infinitesimal character
  15. Computing Micro-Packets
  16. ...and 2 more sections

Key Result

Lemma 2.1

There are exactly two ${^\vee}\mathrm{G}_{2}$-orbits of $\mathcal{I}$, and the two orbits are represented by the elements $1$ and $e(\rho/2) = \exp(\uppi i \rho)$.

Theorems & Definitions (25)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 3.1: Lemma 19.14 ABV
  • Proposition 4.1
  • ...and 15 more