FRG: Collaborative Research: Atlas of Lie Groups and Representations

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AbstractTrapaThe proposed research is most naturally understood in terms of the geometric framework of the representation theoretic part of the local Langlands conjecture, a framework which originated in the work of Deligne and Kazhdan-Lusztig. Roughly speaking, the conjecture asserts that the representation theory of a reductive algebraic group over a local field is dual to a geometric category of equivariant sheaves on an associated dual space. In this formulation, the Arthur conjectures may be interpreted as suggesting a very precise relationship between local components of automorphic forms and the microlocal geometry of the dual space (codified in the characteristic variety of certain equivariant perverse sheaves). When the local field is archimedian, the local Langlands conjecture is a theorem, but the Arthur conjectures are ineffective in the sense that they fall well short of providing a conjectural enumeration of the automorphic spectrum. We propose a project to extend the framework of the local Langlands conjecture to real groups that arise as nonalgebraic metaplectic double covers of the real points of algebraic groups. Our approach is predicated on an intricate relationship between the algebraic and nonalgebraic theories which may be interpreted as a kind of ramified local Shimura correspondence. This allows us to pursue some of Arthurs' ideas in the nonalgebraic setting. Finally in a separate project we propose a series of nontrivial and computationally powerful restrictions on the form that the characteristic variety computations can take. (These computations are relevant for the Arthur theory for both algebraic and nonalgebraic groups.) The restrictions we pursue give insight into making the Arthur conjectures effective for real groups.The proposed research deals with understanding the theory of semisimple (real) Lie groups. These objects arise most naturally as symmetries of finite-dimensional structures; the modifier semisimple means, roughly speaking, that the Lie group cannot be broken into smaller Lie groups. An example of historical import is the Lie group of rotations in three dimensions (which codifies the symmetry of the spherically symmetric hydrogen atom). One is often interested in how Lie groups can act on infinite-dimensional linear spaces (like the solution space of the Schroedinger equation for the hydrogen atom). Such actions are called representations. In the 1960s, Langlands suggested an astonishing relationship between the representation theory of semisimple Lie groups and a set of data of essentially arithmetic origin. This is remarkable because it connects two subjects representation theory and number theory which are ostensible unrelated. Subsequently the arithmetic parameters that Langlands suggested (and much more of Langlands ideas) became understood as a shadow of a deep geometric theory of singular algebraic varieties. The proposer's research seeks to extend some of the ideas of Langlands, and consequent refinements due to Arthur, to a larger class of nonalgebraic Lie groups than originally considered in the original algebraic setting. (The distinction is a little too delicate to make precise here, but suffice it to say that the nonalgebraic setting has been demonstrated to be very interesting from many perspectives.) Our research has a representation theoretic part and a geometric part. Qualitatively the representation theoretic part is predicated on a precise relationship between the nonalgebraic and algebraic theories (revealing new information about each) and the geometric part turns on a latent structure of the parameter set mentioned above. It is worth mentioning that for particular examples, these latent structures amount to a lovely (and very concrete) combinatoric which is accessible to bright undergraduates; I plan to direct undergraduate research projects based on this combinatorial theory.