Morningside Center of Mathematics

Chinese Academy of Sciences

中国科学院晨兴数学中心

Morningside Center of Mathematics

Chinese Academy of Sciences

Title: | Generalized special cycles and theta series |

Speaker: | Yousheng Shi (University of Wisconsin-Madison) |

Time: | 2020-10-22, 9:00am-10:00am |

Place: | http://www.mcm.ac.cn/activities/programs/znts |

Abstract: | We study generalized special cycles on Hermitian locally symmetric spaces $\Gamma \backslash D$ associated to the groups $G = U(p, q), \ \mathrm{Sp}(2n, \mathbb R)$ and $\mathrm{O}(2n)$. These cycles are algebraic and covered by symmetric spaces associated to subgroups of $G$ which are of the same type. Using differential forms with values in the Weil representation, we show that Poincare duals of these generalized special cycles can be viewed as Fourier coefficients of a theta series. This gives new cases of theta lifts from the cohomology of Hermitian locally symmetric manifolds associated to $G$ to vector-valued automorphic forms associated to the groups $G' = \mathrm{U}(m, m), \ \mathrm{O}(m, m)$ or $\mathrm{Sp}(m, m)$ which are members of a dual pair with $G$ in the sense of Howe. This partially generalizes the work of Kudla and Millson on the special cycles on Hermitian locally symmetric spaces associated to the unitary groups. |

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