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Conjectures on reductive homogeneous spaces

Toshiyuki Kobayashi

Abstract

We address some conjectures and open problems in "analysis of symmetries" which include the study of non-commutative harmonic analysis and discontinuous groups for reductive homogeneous spaces beyond the classical framework: (1) discrete series for non-symmetric homogeneous spaces $G/H$; (2) discontinuous group $Γ$ for $G/H$ beyond the Riemannian setting; (3) analysis on pseudo-Riemannian locally homogeneous spaces.

Conjectures on reductive homogeneous spaces

Abstract

We address some conjectures and open problems in "analysis of symmetries" which include the study of non-commutative harmonic analysis and discontinuous groups for reductive homogeneous spaces beyond the classical framework: (1) discrete series for non-symmetric homogeneous spaces ; (2) discontinuous group for beyond the Riemannian setting; (3) analysis on pseudo-Riemannian locally homogeneous spaces.
Paper Structure (5 sections, 12 equations, 2 figures)

This paper contains 5 sections, 12 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic settings
  3. Problems on discrete series for $G/H$
  4. Problems on discontinuous groups for $G/H$
  5. Spectral analysis for pseudo-Riemannian locally homogeneous spaces $\Gamma \backslash G/H$

Theorems & Definitions (23)

  • Example 2.2: semisimple coadjoint orbits
  • Example 2.3: symmetric spaces, real spherical spaces
  • Example 2.4: pseudo-Riemannian space form, see W11
  • Definition 3.2
  • Example 3.3
  • Example 3.5: real forms of $SL(2n+1, {\mathbb{C}})/Sp(n, {\mathbb{C}})$, KInvent94
  • Conjecture 3.6: xkrons94
  • Conjecture 3.7
  • Remark 3.8
  • Conjecture 3.9
  • ...and 13 more