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Explore the construction of dual groups of spherical varieties and weak spherical data in this advanced mathematical lecture focusing on exceptional theta correspondence. Delve into the calculation of B-eigenfunctions, restricted root systems, and the structure of dual groups specifically for the case F4/Spin(9). Examine the intricate relationships between spherical varieties and their dual counterparts through detailed mathematical analysis and theoretical frameworks. Learn about the computational aspects of determining eigenfunction properties and understand how restricted root systems contribute to the overall structure of these mathematical objects. Investigate the specific characteristics of the F4/Spin(9) case as a concrete example of these abstract concepts, gaining insight into how exceptional Lie groups interact with spherical geometry. Develop understanding of the theoretical foundations underlying theta correspondence and its applications in modern algebraic geometry and representation theory.
