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The Atiyah-Schmid formula for reductive groups

Jun Yang

TL;DR

The paper extends the Atiyah-Schmid formula from discrete series of semisimple groups to projective tempered representations and to real reductive groups by introducing von Neumann densities and ω-projective representations. It proves a general density formula d_{Γ,ω}(H_π) = μ(Γ/G) dν_{G,ω}(π) for lattices Γ in unimodular type I groups and their ω-projective duals, unifying with known special cases. It then derives a reductive-group version that factors through the center, yielding d_{Γ}(H_π) = μ_{overline{G}}(overline{Γ}/overline{G}) / |Z∩Γ| · dν_G(π) for finite Plancherel-measure slices, using a central-Mackey decomposition. These results provide a cohesive framework linking representation-theoretic invariants to lattice covolumes and central quotients, with implications for arithmetic subgroups like G(ℤ).

Abstract

We give the generalized Atiyah-Schmid formula for projective tempered representations. Then we prove the Atiyah-Schmid formula for arithmetic subgroups of real reductive groups.

The Atiyah-Schmid formula for reductive groups

TL;DR

The paper extends the Atiyah-Schmid formula from discrete series of semisimple groups to projective tempered representations and to real reductive groups by introducing von Neumann densities and ω-projective representations. It proves a general density formula d_{Γ,ω}(H_π) = μ(Γ/G) dν_{G,ω}(π) for lattices Γ in unimodular type I groups and their ω-projective duals, unifying with known special cases. It then derives a reductive-group version that factors through the center, yielding d_{Γ}(H_π) = μ_{overline{G}}(overline{Γ}/overline{G}) / |Z∩Γ| · dν_G(π) for finite Plancherel-measure slices, using a central-Mackey decomposition. These results provide a cohesive framework linking representation-theoretic invariants to lattice covolumes and central quotients, with implications for arithmetic subgroups like G(ℤ).

Abstract

We give the generalized Atiyah-Schmid formula for projective tempered representations. Then we prove the Atiyah-Schmid formula for arithmetic subgroups of real reductive groups.

Paper Structure

This paper contains 4 sections, 12 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. The density over a von Neumann algebra
  3. The Atiyah-Schmid formula for projective representations
  4. The Atiyah-Schmid formula for reductive groups

Key Result

Lemma 1

Let $\Gamma$ be a lattice in a unimodular type I locally compact group $G$. Let $\nu_{G,\omega}$ be the Plancherel measure on the $\omega$-projective dual $\Pi(G,\omega)$ of $G$ for a 2-cocycle $\omega$. We have

Theorems & Definitions (23)

  • Lemma
  • Theorem
  • Lemma 2.1
  • Definition 2.1: von Neumann densities
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 13 more