Oyadare, O. O. (2024). A note on the 𝐿2−harmonic analysis of the Joint-Eigenspace Fourier transform. arXiv:2410.09075v1 [math.FA].
This paper investigates the irreducibility of the regular representation of a noncompact semisimple Lie group G on the Hilbert space formed by the image of the Joint-Eigenspace Fourier transform on its corresponding symmetric space G/K.
The paper utilizes the Joint-Eigenspace Fourier transform on the symmetric space X = G/K to decompose the space of square-integrable functions L2(X) into a direct integral. This decomposition is then used to analyze the irreducibility of the regular representation.
The paper concludes that the irreducibility of the regular representation Tλ on the image of the Joint-Eigenspace Fourier transform can be completely characterized by the simplicity of λ and -λ in a∗C.
This research contributes to the understanding of harmonic analysis on noncompact symmetric spaces, particularly by highlighting the utility of the Joint-Eigenspace Fourier transform in decomposing function spaces and representations. The characterization of irreducibility has potential applications in classifying specific types of symmetric spaces.
The paper focuses on a specific type of Fourier transform and its properties on symmetric spaces. Further research could explore the applicability of these findings to other types of Fourier transforms or more general spaces. Additionally, exploring the practical implications of the irreducibility characterization in specific geometric contexts could be a fruitful avenue for future work.
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by O. O. Oyadar... at arxiv.org 10-15-2024
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