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Associated varieties of integral minimal highest weight modules

Zhanqiang Bai, Jing Jiang, Rui Wang

Abstract

Let $\mathfrak{g}$ be a complex simple Lie algebra and $L(λ)$ be a highest weight module of $\mathfrak{g}$ with highest weight $λ-ρ$, where $ρ$ is half the sum of positive roots. A simple $\mathfrak{g}$-module $L_w:=L(-wρ)$ is called integral minimal if the associated variety of its annihilator ideal equals the closure of the minimal special nilpotent orbit. In this paper, we find that the associated variety of any integral minimal module $L_w$ is irreducible and equal to the orbital variety corresponding to the minimal length element in the Kazhdan--Lusztig right cell containing $w$.

Associated varieties of integral minimal highest weight modules

Abstract

Let be a complex simple Lie algebra and be a highest weight module of with highest weight , where is half the sum of positive roots. A simple -module is called integral minimal if the associated variety of its annihilator ideal equals the closure of the minimal special nilpotent orbit. In this paper, we find that the associated variety of any integral minimal module is irreducible and equal to the orbital variety corresponding to the minimal length element in the Kazhdan--Lusztig right cell containing .

Paper Structure

This paper contains 18 sections, 34 theorems, 71 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schubert variety
  4. Pattern avoidance of type $A$
  5. The root subsystem pattern avoidance criterion
  6. Associated variety
  7. Kazhdan--Lusztig cells
  8. The case of $A_{n}$
  9. The case of $C_n$ and $B_n$
  10. The case of $C_n$
  11. The case of $B_n$
  12. The case of $D_n$
  13. Proof for even $n$
  14. Proof for odd $n$
  15. The case of $E$ and $F_4$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\mathcal{C}_i$ be the KL right cell containing the simple reflection $s_i=s_{\alpha_i}$. Then the smooth elements $S(w_0\mathcal{C}_i)$ of the KL right cell $w_0\mathcal{C}_i$ are given as follows:

Theorems & Definitions (46)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2: Ca-11
  • Definition 2.3
  • Proposition 2.4: LS-90
  • Definition 2.5: Robinson--Schensted insertion algorithm
  • Theorem 2.6: BP-05
  • Theorem 2.7: BP-05
  • Definition 2.8
  • ...and 36 more