Studies in Representation Theory

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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.