亞辛群的傑異表現可視為θ函數在表現理論架構下的推廣，這些傑異表現的特質是它們的Whittaker 模型具有唯一性。正如同θ函數在數論中扮演了極具重要的角色，傑異表現豐富的算術性質也成為研究的焦點。本計畫的目的是了解GL(4)的二摺覆蓋群上的傑異表現，我們希望能證明這些表現都是來自於GL(2)的尖點自守表現，並對尖點傑異表現做進一步的刻劃。這個問題有趣的地方在於之前大家所熟悉的尖點傑異表現都是源自於GL(1)的自守表現, 而非GL(2)，所以計畫的結果應該可以對傑異表現有更深一層的認識。我們打算應用廣義的Shimura 對應及逆定理來得到想要的自守表現。 Distinguished representations of metaplectic groups are the analogue of theta functions in the representation-theoretic setting. They are characterized by the uniqueness of their Whittaker models. Just as classical theta functions play an important role in number theory, distinguished representations carry rich arithmetic information and they become a focus of active research. In this project we propose to classify distinguished representations on the two-fold covers of GL(4) by showing that they are all coming from cuspidal automorphic representations of GL(2). We would also like to give criterions for a distinguished representation to be cuspidal. This project is of particular interest since the earlier known examples of cuspidal distinguished representations are all coming from automorphic representations of GL(1) instead of GL(2). Our approach uses generalized Shimura correspondence and the converse theorem to construct desired automorphic representations. 研究期間：9904 ~ 10007
