Poisson transform and unipotent complex geometry

Gimperlein, H and Krötz, B and Roncal, L and Thangavelu, S (2025) Poisson transform and unipotent complex geometry. In: Journal of Functional Analysis, 288 (3).

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Abstract

Our concern is with Riemannian symmetric spaces Z=G/K of the non-compact type and more precisely with the Poisson transform Pλ which maps generalized functions on the boundary �Z to λ-eigenfunctions on Z. Special emphasis is given to a maximal unipotent group N<G which naturally acts on both Z and �Z. The N-orbits on Z are parametrized by a torus A=(R>0)r<G (Iwasawa) and letting the level a�A tend to 0 on a ray we retrieve N via lima�0�Na as an open dense orbit in �Z (Bruhat). For positive parameters λ the Poisson transform Pλ is defined and injective for functions f�L2(N) and we give a novel characterization of Pλ(L2(N)) in terms of complex analysis. For that we view eigenfunctions �=Pλ(f) as families (�a)a�A of functions on the N-orbits, i.e. �a(n)=�(na) for n�N. The general theory then tells us that there is a tube domain T=Nexp�(i�)�NC such that each �a extends to a holomorphic function on the scaled tube Ta=Nexp�(iAd(a)�). We define a class of N-invariant weight functions wλ on the tube T, rescale them for every a�A to a weight wλ,a on Ta, and show that each �a lies in the L2-weighted Bergman space B(Ta,wλ,a):=O(Ta)�L2(Ta,wλ,a). The main result of the article then describes Pλ(L2(N)) as those eigenfunctions � for which �a�B(Ta,wλ,a) and ���:=supa�A�aReλ�2���a�Ba,λ<� holds. © 2024 The Author(s)

Item Type:	Journal Article
Publication:	Journal of Functional Analysis
Publisher:	Academic Press Inc.
Additional Information:	The copyright for this article belongs to publishers.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	29 Nov 2024 09:22
Last Modified:	29 Nov 2024 09:22
URI:	http://eprints.iisc.ac.in/id/eprint/86870

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