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Polynomial growth of holomorphic extensions of orbit maps of $K$-finite vectors at the boundary of the crown

Tobias Simon

TL;DR

The paper extends the Krötz--Stanton Extension Theorem to arbitrary connected semisimple Lie groups and proves polynomial growth bounds for the holomorphic extensions of orbit maps of $K$-finite vectors at the boundary of the crown domain. Central to the approach is a detailed analysis of the complexified Iwasawa component maps, the crowns’ complex geometry, and maximal scale functions, which yield bounds of the form $Cig( rac{oldsymbol{ extpi}}{2}-r_{ m spec}({ m ad}(x))ig)^{-N}$ as $x$ approaches the boundary. These growth estimates enable boundary values to exist in distribution spaces and underpin applications to Poisson transforms of distributional sections, with explicit constants depending on the inducing data and $K$-types. The results bridge holomorphic extension theory, Harish-Chandra module globalization, and distributional boundary phenomena, offering tools for analysis of boundary behavior in harmonic analysis on symmetric spaces. In particular, the work provides a framework for extending holomorphic orbit maps to crown domains for principal series and general Hilbert globalizations, with quantitative control essential for applications in representation theory and PDE on Lie groups.

Abstract

The Krötz-Stanton Extension Theorem states that the orbit map of a K-finite vector in a Hilbert representation of a linear Lie group extends to a holomorphic map to a principal fibre bundle over the complex crown domain associated to the Riemannian symmetric space $G/K$. We extend this theorem to arbitrary connected semisimple Lie groups and prove polynomial growth estimates at the boundary. Using this, we show that the boundary values of these holomorphic extensions exist in the space of distribution vectors.

Polynomial growth of holomorphic extensions of orbit maps of $K$-finite vectors at the boundary of the crown

TL;DR

The paper extends the Krötz--Stanton Extension Theorem to arbitrary connected semisimple Lie groups and proves polynomial growth bounds for the holomorphic extensions of orbit maps of -finite vectors at the boundary of the crown domain. Central to the approach is a detailed analysis of the complexified Iwasawa component maps, the crowns’ complex geometry, and maximal scale functions, which yield bounds of the form as approaches the boundary. These growth estimates enable boundary values to exist in distribution spaces and underpin applications to Poisson transforms of distributional sections, with explicit constants depending on the inducing data and -types. The results bridge holomorphic extension theory, Harish-Chandra module globalization, and distributional boundary phenomena, offering tools for analysis of boundary behavior in harmonic analysis on symmetric spaces. In particular, the work provides a framework for extending holomorphic orbit maps to crown domains for principal series and general Hilbert globalizations, with quantitative control essential for applications in representation theory and PDE on Lie groups.

Abstract

The Krötz-Stanton Extension Theorem states that the orbit map of a K-finite vector in a Hilbert representation of a linear Lie group extends to a holomorphic map to a principal fibre bundle over the complex crown domain associated to the Riemannian symmetric space . We extend this theorem to arbitrary connected semisimple Lie groups and prove polynomial growth estimates at the boundary. Using this, we show that the boundary values of these holomorphic extensions exist in the space of distribution vectors.
Paper Structure (14 sections, 47 theorems, 298 equations)

This paper contains 14 sections, 47 theorems, 298 equations.

Table of Contents

  1. Hilbert representations of semisimple Lie groups
  2. Harish Chandra modules and Fréchet globalizations
  3. The Krötz--Stanton Extension Theorem
  4. Growth of the complexified Iwasawa component maps
  5. The complexified Iwasawa domain
  6. Scaling properties of the Iwasawa component maps
  7. Asymptotic behaviour at the boundary of the crown
  8. Applications of the growth estimates
  9. Growth of Poisson transforms of distributional sections
  10. Dependence of the constants on the representation
  11. Distributional limits at the boundary of the crown
  12. Analytic vectors of slow growth
  13. Tempered vectors in unitary representations
  14. Holomorphic extension of $(Z(\mathfrak{g}),K)$-finite functions

Key Result

Theorem 1

Let $G$ be a connected semisimple Lie group and let $(\pi,\mathcal{H})$ be For $v\in \mathcal{H}^{[K]}$, there exist $C,N>0$ such that In particular, $\lim_{t\nearrow 1}e^{it\partial\pi(x)}v\in \mathcal{H}^{-\infty}$ exists in the weak-$\ast$-topology, for all $x_0\in \partial\Omega_\mathfrak{p}$.

Theorems & Definitions (123)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1.1.1
  • Remark 1.1.2
  • Definition 1.1.3
  • Definition 1.1.4
  • Definition 1.1.5
  • Theorem 1.1.6
  • proof
  • ...and 113 more