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Degenerate principal series and nilpotent invariants

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Abstract

We study two nilpotent invariants, namely associated cycles and wave front cycles, attached to irreducible constituents of degenerate principal series representations of \(\mathrm {Sp}(2n,{\mathbb {R}})\). We compute the associated cycles of those constituents with the largest Gelfand–Kirillov dimension, as well as the dimensions of the space of generalized Whittaker models associated to nilpotent orbits occurring in the wave front cycles of these constituents. Furthermore, for these constituents, we prove that the coefficients of nilpotent orbits occurring in the wave front cycles equal the dimensions of the space of generalized Whittaker models associated to the corresponding nilpotent orbits. Our main approach is based on the result of Loke and Ma (Compos Math 151(1):179–206, 2015), and, Gomez and Zhu’s (Geom Funct Anal 24(3):796–853, 2014), on the behavior of these nilpotent invariants under the local theta correspondence.

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Acknowledgements

This paper contains part of the author’s doctorial thesis at the National University of Singapore. The author would like to express her deepest gratitude to Professor Chen-Bo Zhu for useful discussions and generous support. She would like to thank Professor Dmitry Gourevitch for kindly answering her questions. She also wants to thank Professor Hung Yean Loke and Professor Jing-Song Huang for helpful suggestions on this problem. She also would like to thank Professor Jiajun Ma and Dr. Xu Song for useful discussions. She is also grateful to the referees for their careful reading and useful comments, which improve the paper.

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  1. Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road, Beijing, 100871, China

    Ning Li

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Li, N. Degenerate principal series and nilpotent invariants. Math. Z. 300, 2117–2145 (2022). https://doi.org/10.1007/s00209-021-02854-z

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