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Dual pairs in $PGL(n,\mathbb{C})$

Marisa Gaetz

Abstract

In Roger Howe's seminal 1989 paper "Remarks on classical invariant theory," he introduces the notion of Lie algebra dual pairs, and its natural analog in the groups context: a pair $(G_1,G_2)$ of reductive subgroups of an algebraic group $G$ is a dual pair in $G$ if $G_1$ and $G_2$ equal each other's centralizers in $G$. While reductive dual pairs in the complex reductive Lie algebras have been classified, much less is known about algebraic group dual pairs, which were only fully classified in the context of certain classical matrix groups. In this paper, we classify the reductive dual pairs in $PGL(n,\mathbb{C})$.

Dual pairs in $PGL(n,\mathbb{C})$

Abstract

In Roger Howe's seminal 1989 paper "Remarks on classical invariant theory," he introduces the notion of Lie algebra dual pairs, and its natural analog in the groups context: a pair of reductive subgroups of an algebraic group is a dual pair in if and equal each other's centralizers in . While reductive dual pairs in the complex reductive Lie algebras have been classified, much less is known about algebraic group dual pairs, which were only fully classified in the context of certain classical matrix groups. In this paper, we classify the reductive dual pairs in .

Paper Structure

This paper contains 41 sections, 43 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Prior work
  3. Further motivation
  4. Constructing "single-orbit" dual pairs in PGL(U)
  5. Connected dual pairs in $PGL(U)$
  6. Disconnected dual pairs in $PGL(U)$
  7. Preliminaries for showing we've found all "single-orbit" dual pairs
  8. Orbits of $G$-, $G^{\circ}$-, $H$-, and $H^{\circ}$-irreducible representations
  9. Choosing $\Gamma$-representatives (resp. $\widehat{\Gamma}$-representatives) in $G$ (resp. $H$)
  10. Defining a $\widehat{\Gamma}$-action on the $G$-irreducibles
  11. Choosing representatives for the $G$-irreducibles
  12. Defining a $\Gamma$-action on the $G^{\circ}$-irreducibles
  13. Identifying the embedding of $G^{\circ}$ with $\bigoplus_{k \in \mathcal{K}} \prescript{\widetilde{\gamma_k}}{}{\beta} \otimes \mathop{\mathrm{Hom}}\nolimits_{G^{\circ}} ( \prescript{\widetilde{\gamma_k}}{}{B}, U )$
  14. Defining the corresponding $H^{\circ}$-irreducibles
  15. Defining the corresponding $H$-irreducibles
  16. ...and 26 more sections

Key Result

Theorem 4

The dual pairs of $GL(U)$ are exactly the pairs of groups of the form where $U = \bigoplus_{i=1}^{r} V_i \otimes W_i$ is a vector space decomposition of $U$. In particular, any member of a $GL(U)$ dual pair is connected.

Theorems & Definitions (74)

  • Definition 1: Howe HoweRemarks
  • Remark 3
  • Theorem 4: G. myarxiv
  • Proposition 5: G. myarxiv
  • Proposition 6: G. myarxiv
  • Proposition 7
  • proof
  • Lemma 8
  • Theorem 9
  • proof
  • ...and 64 more