Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the associated variety of a highest weight Harish-Chandra module

Zhanqiang Bai, Markus Hunziker, Xun Xie, Roger Zierau

Abstract

We prove a simple formula that calculates the associated variety of a highest weight Harish-Chandra module directly from its highest weight. We also give a formula for the Gelfand--Kirillov dimension of highest weight Harish-Chandra module which is uniform across Cartan types and is valid for arbitrary infinitesimal character.

On the associated variety of a highest weight Harish-Chandra module

Abstract

We prove a simple formula that calculates the associated variety of a highest weight Harish-Chandra module directly from its highest weight. We also give a formula for the Gelfand--Kirillov dimension of highest weight Harish-Chandra module which is uniform across Cartan types and is valid for arbitrary infinitesimal character.
Paper Structure (37 sections, 21 theorems, 108 equations, 2 tables)

This paper contains 37 sections, 21 theorems, 108 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Highest weight Harish-Chandra modules
  3. Associated varieties
  4. Statement of the main result
  5. Gelfand--Kirillov dimension
  6. About the proofs
  7. Connections with other work
  8. Posets of positive noncompact roots
  9. Lower-order ideals
  10. Antichains and width of lower-order ideals
  11. The number of lower-order ideals of a given width
  12. Proof for ${\mathfrak g}_\mathbb R={\mathfrak s}{\mathfrak u}(p,q)$
  13. Proof for ${\mathfrak g}_\mathbb R={\mathfrak s}{\mathfrak o}^*(2n)$
  14. Proof for ${\mathfrak g}_\mathbb R={\mathfrak s}{\mathfrak p}(2n,\mathbb R)$
  15. Proof for ${\mathfrak g}_\mathbb R={\mathfrak s}{\mathfrak o}(2,2n-1)$
  16. ...and 22 more sections

Key Result

Theorem 1.2

Suppose $L(\lambda)$ is a highest weight Harish-Chandra module with highest weight $\lambda$ and $\operatorname{AV}(L(\lambda))=\overline{\mathcal{O}_{k(\lambda)}}$. Let $m=\operatorname{width}(Y_\lambda)$. Then $k(\lambda)$ is given as follows.

Theorems & Definitions (45)

  • Definition 1.1
  • Theorem 1.2
  • Example 1.3
  • Theorem 1.4
  • Lemma 2.1: Jakobsen Jak:83
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Example 2.4
  • ...and 35 more