[Paper Review] On harmonic analysis of spherical convolutions on semisimple Lie groups

Olufemi O. Oyadare|arXiv (Cornell University)|Jun 21, 2017

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper introduces spherical convolutions as a generalization of Harish-Chandra transforms on connected semisimple Lie groups with finite center, establishing their role in harmonic analysis through a new Plancherel formula. The key contribution is a unitary isomorphism between $ L^2(G//K) $ and $ L^2(rak{F}^1, d ilde{ u}_{x, u})^\mathfrak{w} $, with an explicitly computed Plancherel measure involving the Harish-Chandra $ c $-function and spherical Fourier transforms.

ABSTRACT

This paper contains a non-trivial generalization of the Harish-Chandra transforms on a connected semisimple Lie group $G,$ with finite center, into what we term spherical convolutions. Among other results we show that its integral over the collection of bounded spherical functions at the identity element $e \in G$ is a weighted Fourier transforms of the Abel transform at $0.$ Being a function on $G,$ the restriction of this integral of its spherical Fourier transforms to the positive-definite spherical functions is then shown to be (the non-zero constant multiple of) a positive-definite distribution on $G,$ which is tempered and invariant on $G=SL(2,\mathbb{R}).$ These results suggest the consideration of a calculus on the Schwartz algebras of spherical functions. The Plancherel measure of the spherical convolutions is also explicitly computed.

Motivation & Objective

• To generalize Harish-Chandra transforms via a new class of functions called spherical convolutions on semisimple Lie groups.
• To establish a unitary isomorphism between $ L^2(G//K) $ and a weighted $ L^2 $-space on the space of elementary spherical functions.
• To compute the Plancherel measure for spherical convolutions explicitly in terms of the Harish-Chandra $ c $-function and spherical Fourier transforms.
• To explore the relationship between spherical convolutions and tempered, invariant distributions on $ G $, particularly in the case $ G = SL(2,\mathbb{R}) $.
• To lay the foundation for a calculus on Schwartz algebras of spherical functions using the spherical Bochner theorem and Plancherel theory.

Proposed method

• Define spherical convolutions as $ \mathcal{H}_{x,\lambda}f = (f * \varphi_\lambda)(x) $, where $ f \in \mathcal{C}^p(G//K) $ and $ \varphi_\lambda $ is a spherical function.
• Use the spherical Fourier transform $ \widehat{f}(\lambda) = \int_A f(a) e^{\lambda(\log a)} da $ to relate $ f $ to its transform on $ \frak{F}^1 $, the space of elementary spherical functions.
• Establish the Plancherel formula via the measure $ d\zeta_{x,\lambda}(\nu) = |\widehat{\varphi_\lambda}(\nu)|^{-2} d\mu(\lambda) $, normalizing the spherical Bochner measure.
• Prove that the map $ f \mapsto \widehat{\mathcal{H}_{x,\lambda}f} $ is a unitary isomorphism from $ L^2(G//K) $ to $ L^2(\frak{F}^1, d\zeta_{x,\lambda})^\mathfrak{w} $, using known Plancherel theory and properties of $ \widehat{\varphi_\lambda} $.
• Explicitly compute the Plancherel measure as $ d\zeta_{x,\lambda}(\nu) = |\mathfrak{w}|^{-1} |\widehat{\varphi_\lambda}(\nu)|^{-2} |c(\lambda)|^{-2} d\mu(\lambda) $, where $ c(\lambda) $ is the Harish-Chandra $ c $-function.
• Use the Weyl group invariance of $ \mathcal{H}_{x,\lambda}f $ and the structure of $ G = KAN $ and $ G = K \cdot \overline{A^+} \cdot K $ to analyze the behavior of convolutions at $ x = e $ and $ \lambda = 0 $.

Experimental results

Research questions

• RQ1 How can the Harish-Chandra transform be generalized to a broader class of functions on semisimple Lie groups?
• RQ2 What is the structure of the spherical convolution $ \mathcal{H}_{x,\lambda}f $ as a function on $ G $, and how does it relate to the spherical Fourier transform?
• RQ3 Under what conditions does the Plancherel measure for spherical convolutions reduce to the classical Plancherel measure on $ G $?
• RQ4 What is the precise form of the Plancherel measure $ d\zeta_{x,\lambda}(\nu) $ for spherical convolutions, and how does it depend on the $ c $-function and spherical Fourier transforms?
• RQ5 How do spherical convolutions behave at the identity $ x = e $, and what does this imply about their relationship with the Abel transform and the $ \Xi $-function?

Key findings

• The spherical convolution $ \mathcal{H}_{x,\lambda}f $ is well-defined on $ \mathcal{P} $, Weyl group invariant, and its spherical Fourier transform integrates to a non-zero constant multiple of $ T[\varphi_\lambda] $, where $ T $ is a positive-definite distribution.
• The integral of the spherical Fourier transform of $ \mathcal{H}_{x,\lambda}f $ over $ \mathcal{P} $ is shown to be a positive-definite, tempered, and $ G $-invariant distribution on $ SL(2,\mathbb{R}) $.
• The Plancherel formula for spherical convolutions is established as $ \int_G |f(y)|^2 dy = \int_{\frak{F}^1} |\widehat{\mathcal{H}_{x,\lambda}f}(\nu)|^2 d\zeta_{x,\lambda}(\nu) $, with $ d\zeta_{x,\lambda}(\nu) = |\widehat{\varphi_\lambda}(\nu)|^{-2} d\mu(\lambda) $.
• The Plancherel measure for spherical convolutions is explicitly computed as $ d\zeta_{x,\lambda}(\nu) = |\mathfrak{w}|^{-1} |\widehat{\varphi_\lambda}(\nu)|^{-2} |c(\lambda)|^{-2} d\mu(\lambda) $, linking it to the Harish-Chandra $ c $-function.
• At $ x = e $, the spherical convolution reduces to $ \mathcal{H}_{e,\lambda}f = \widehat{f}(\lambda) $, recovering the classical spherical Fourier transform and linking the new framework to known harmonic analysis.
• The inverse of the spherical convolution transform is given by the wave packet formula $ (\mathcal{H}_{x,\lambda}f)^{-1}(b)(y) = \int_{\frak{F}^1} b(\lambda) \varphi_\lambda(y) d\zeta_{x,\lambda}(\nu) $, which is a normalized exact wave packet in $ L^2 $.