Study of local Langlands conjecture and harmonic analysis

朗兰兹猜想是数论领域中的一个重要猜想。本文证明了SL（N）的表示的不变量与相应的Galois表示的不变量之间的关系，并证明了Kottwitz-Shelstad猜想，另一方面，Langlands猜想的一个重要组成部分是内窥镜.本文研究了SL（2）的覆盖群的内视，并将Kohnen+空间推广到全真实的数域的情形。

The Langlands conjecture is important in the field of number theory. It predicts a good relation between the representations of the reductive groups and the Galois representations.In this research, the Kottwitz-Shelstad conjecture, which can be regarded as a relation between an invariant of a representation of SL(N) and an invariant of the corresponding Galois representation, is proved.On the other hand, one of the important parts of the Langlands conjecture is the endoscopy. In this research, the endoscopy for the covering groups of SL(2) is studied and the Kohnen plus space is generalized to the totally real number field case.