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A spectral theorem for compact representations and non-unitary cusp forms

Anton Deitmar

Abstract

We show that a compact representation of a semisimple Lie group has an orthogonal decomposition into finite length representations. This generalises and simplifies a number of more special spectral theorems in the literature. We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms.

A spectral theorem for compact representations and non-unitary cusp forms

Abstract

We show that a compact representation of a semisimple Lie group has an orthogonal decomposition into finite length representations. This generalises and simplifies a number of more special spectral theorems in the literature. We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms.
Paper Structure (5 sections, 10 theorems, 39 equations)

This paper contains 5 sections, 10 theorems, 39 equations.

Table of Contents

  1. The Spectral Theorem for Lie groups
  2. Filtrations
  3. Totally disconnected groups
  4. Flat bundles and automorphic forms
  5. Cusp forms

Key Result

Theorem 1.10

. Let $G$ be a semisimple Lie group with finite center and finitely many components. Let $(R,H)$ be a $K$-unitary compact Hilbert representation of $G$. Then the representation $R$ is a direct sum of the subrepresentations $H_\lambda$, $\lambda\in{\mathbb C}^R$, i.e., the space $\bigoplus_{\lambda\i

Theorems & Definitions (49)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 39 more