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Transfer operators are conjectural “operators of functoriality,” which transfer test measures and (relative) characters from one homogeneous space to another. In the previous work (Yiannis Sakellaridis, Duke Math. J. 170(2), 279–364 (2021); Yiannis Sakellaridis, Spherical varieties, functoriality, and quantization, ICM—International Congress of Mathematicians, Vol. 4, Sections 5–8, pp. 2998–3037, EMS Press, Berlin (2023)), I computed transfer operators associated to spherical varieties of rank one and gave an interpretation of them in terms of geometric quantization. In this paper, I study how these operators vary in the horospherical limits of these varieties, where they have a conceptual interpretation related to scattering theory. I also revisit Jacquet’s Hankel transform for the Kuznetsov formula, which is related to the functional equation of the standard L-function of \(\operatorname {GL}_n\) , and provide an interpretation of it in terms of quantization.

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Transfer Operators and Hankel Transforms: Horospherical Limits and Quantization

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  • 发表日期：

    2025年

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  • Yiannis Sakellaridis

摘要

Transfer operators are conjectural “operators of functoriality,” which transfer test measures and (relative) characters from one homogeneous space to another. In the previous work (Yiannis Sakellaridis, Duke Math. J. 170(2), 279–364 (2021); Yiannis Sakellaridis, Spherical varieties, functoriality, and quantization, ICM—International Congress of Mathematicians, Vol. 4, Sections 5–8, pp. 2998–3037, EMS Press, Berlin (2023)), I computed transfer operators associated to spherical varieties of rank one and gave an interpretation of them in terms of geometric quantization. In this paper, I study how these operators vary in the horospherical limits of these varieties, where they have a conceptual interpretation related to scattering theory. I also revisit Jacquet’s Hankel transform for the Kuznetsov formula, which is related to the functional equation of the standard L-function of \(\operatorname {GL}_n\) , and provide an interpretation of it in terms of quantization.