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arXiv · 2610.17888

Multiplicity One for Five Strongly Tempered Spherical Varieties

Yuxuan Ge
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Abstract

Let $F$ be a non-Archimedean local field of characteristic zero. We prove multiplicity one for every irreducible smooth admissible representation in the split forms of Wan-Zhang Models 6-10. Combining the Gelfand-Kazhdan method with nilpotent-wavefront restrictions on Bernstein-finite equivariant distributions, we establish the required symmetries through rational orbit analysis, parabolic descent, stabilizer characters, and conormal arguments. The type-$D_6$ analysis treats both the vector-similitude group of Model 9 and the semispin form arising in the exceptional $E_7$ model. For Model 10, the distribution theorem gives a product bound for periods; the individual multiplicity bound follows from Langlands classification and a vanishing theorem for standard modules induced from nongeneric tempered data on the maximal $E_6$ Levi.

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Yuxuan Ge. 2026-10-13. Multiplicity One for Five Strongly Tempered Spherical Varieties. https://arxiv.org/abs/2610.17888

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