Studies in Representation Theory

摘要：在早期的工作中，我注意到可以通过沿实轴对l -函数的（分布）边界值进行Mellin变换来推导附属于mass模形式的函数方程，而不是传统的沿虚轴对模形式积分的方法。我的方法似乎也适用于更高级别的群体，到目前为止，这些群体一直抵制通常的论点。我打算把这个想法扩展到更高阶的情况。这涉及到许多表示理论的子问题，其中几个子问题本身就很有趣。我还计划继续与Vilonen合作，研究实约化群表示的特征循环。通过贝林森-伯恩斯坦构造和黎曼-希尔伯特对应关系，这些表征被附加到旗变体上的可构造轴上。这些帧依次具有特征循环，这些特征循环编码有关表示的深层信息。通过对特征环的研究，我们得到了表示的几何不变量，阐明了单幂的意义，并可能指出了从几何上构造单幂表示的方法。提案的总体主题是表征理论。费利克斯·克莱因（Felix Klein）在18世纪晚期阐明了一个原则，即群体和群体作用的概念形式化了对称的概念。例如，经典力学的定律在旋转和平移下是不变的，它们共同构成了牛顿力学的对称群。狭义相对论——在克莱因之后被发现——有一个不同的对称群，这种差异解释了相对论力学的根本不同行为；事实上，对称群的差异甚至在相对论的初期就被很好地理解了。力学的对称性主要体现在表面上。不太明显的——有时是隐藏得很深的——对称性出现在很多情况下，不仅是物理，还有几何、数论和微分方程。表征是“原子”，即群的最基本成分。表征理论既研究表征本身，也研究对称思想的应用，在这种应用中，某些表征的不太明显的特性会带来新的见解。我的建议的第二部分是前者的类型，第一部分是后者的类型。

Abstract:DMS-0070714In earlier work, I observed that the functional equation for L-functionsattached to Maass modular forms can be derived by taking the Mellin transform of their (distribution) boundary values along the real axis, rather than the traditional way of integrating the modular forms alongthe imaginary axis. My method appears likely to apply also to higherrank groups which, until now, have resisted the usual arguments. I intendto extend the idea to higher rank cases. This involves a number of representation theoretic sub-problems, several of which are interestingin their own right. I also plan to continue my joint work with Vilonen,on characteristic cycles of representations of real reductive groups. Such representations are attached to constructible sheaves on the flag variety, via the Beilinson-Bernstein construction and the Riemann-Hilbert correspondence. These sheaves, in turn, have characteristic cycleswhich encode deep information about the representations. By studying thecharacteristic cycles, we intend to obtain geometric invariants of representations which clarify the meaning of unipotence, and possiblypoint to ways of constructing unipotent representations geometrically.The overall theme of the proposal is representation theory. Felix Klein,late in the eighteenth century, enunciated the principle that the notionsof a group and group action formalize the idea of symmetry. The laws ofclassical mechanics, for example, are invariant under rotations and translations, which together form the symmetry group of Newtonianmechanics. Special relativity -- discovered after Klein -- has a different symmetry group, and this difference explains the fundamentally different behavior relativistic mechanics; indeed, the difference in symmetry groupswas well understood even in the infancy of the theory of relativity. The symmetries of mechanics lie very much on the surface. Less obvious -- sometimes deeply hidden -- symmetries arise in many contexts, not only physics, but also geometry, number theory, and differential equations. Representations are the "atoms", i.e., the most basic ingredients, of groupactions. Representation theory studies both the representations themselves, and applications of the idea of symmetry where the less obvious propertiesof certain representations lead to new insights. The second part of myproposal is of the former type, and the first part, of the latter.