假設F是一個非阿基米德局部體，而π1 、π2 為GLn(F)的不可約表現。對GLn 而言，伽瑪因子的穩定性指的是說，如果π1 、π2 這兩個表現有一樣的中心特徵，而χ是一個高度分歧特徵，那麼π1 × χ和π2 × χ所對應的伽瑪因子就會一樣。這個漂亮的定理出自Jacquet-Shalika，在應用逆定理證明Langlands猜想時扮演了重要的角色。近年來，有越來越多的研究試著把伽瑪因子的穩定性從GLn 推廣到其它約化群，以便能證明更多Langlands猜想。在這個計畫中，我們提議在GLn 的n摺覆蓋上，考慮相同的問題。處理這樣的亞辛群一個有趣的地方是他們的伽瑪因子不再只是個係數，而是一個矩陣，所以穩定性的描繪也就更為巧妙；我們希望在亞辛群上也可以建立類似Jacquet-Shalika的穩定性結果，那麼這個穩定性就可以用來幫忙造出GLn 的n摺覆蓋上的傑異表現；另外我們也想試著算出這個伽瑪因子的穩定值，並且預期結果應該經由廣義的Shimura對應和來自GLn 的伽瑪因子相關。 Let F be a nonarchimedean local field, and π1 ,π2 irreducible representations of GLn(F). The stability of Gamma factors for GLn states that the Gamma factors associated to π1 × χ and π2 × χ are the same, as long as χ is a highly ramified character andπ1 ,π2 share the same central character. This remarkable theorem of Jacquet-Shalika plays an important role in the proof of Langlands functorial liftings via the converse theorem. In recent years, it has become an active research topic trying to generalize this property of stability from GLn to other reductive groups to achieve more cases of functoriality. In this project we propose to study the same problem in the setting of n-fold metaplectic covers of GLn. What is interesting with the metaplectic case is that the Gamma factors are now realized as matrices instead of scalars, so the format of stability is more delicate. We hope to establish similar consequence in the metaplectic case, and then the stability will have applications in the construction of certain distinguished representations of n-fold covers of GLn . It will also be interesting to figure out the precise values of these stabilized Gamma factors, and we expect that they show some evidence of the connection between the representations of metaplectic covers of GLn and those of GLn through the generalized Shimura correspondence. 研究期間：10008 ~ 10107
