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A Note On the Orbits of a Symmetric Subgroup in the Flag Variety

Leticia Barchini, Peter E. Trapa

Abstract

Motivated by relating the representation theory of the split real and $p$-adic forms of a connected reductive algebraic group $G$, we describe a subset of $2^r$ orbits on the complex flag variety for a certain symmetric subgroup. (Here $r$ is the semisimple rank of $G$.) This set of orbits has the property that, while the closure of individual orbits are generally singular, they are always smooth along other orbits in the set. This, in turn, implies consequences for the representation theory of the split real group.

A Note On the Orbits of a Symmetric Subgroup in the Flag Variety

Abstract

Motivated by relating the representation theory of the split real and -adic forms of a connected reductive algebraic group , we describe a subset of orbits on the complex flag variety for a certain symmetric subgroup. (Here is the semisimple rank of .) This set of orbits has the property that, while the closure of individual orbits are generally singular, they are always smooth along other orbits in the set. This, in turn, implies consequences for the representation theory of the split real group.
Paper Structure (6 sections, 10 theorems, 48 equations, 2 figures)

This paper contains 6 sections, 10 theorems, 48 equations, 2 figures.

Table of Contents

  1. introduction
  2. matching of orbits
  3. matching of geometric multiplicities
  4. Preliminaries.
  5. Induced bundles
  6. Matching

Key Result

Lemma 2.2

$\mathcal{X}^S$ is a smooth variety of dimension $\text{dim}(K\cdot \mathfrak{b}) + |S|.$ If $K$ is connected, $\mathcal{X}^S$ is irreducible.

Figures (2)

  • Figure 1: $G=\mathrm{GL}(4,\mathbb{C})$; see Example \ref{['e:gl4']}.
  • Figure 2: $G=\mathrm{Sp}(4,\mathbb{C})$; see Example \ref{['e:sp4']}.

Theorems & Definitions (24)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 14 more