Symplectic Representation Theory

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This proposal will apply powerful tools and techniques in geometry to solve certain problems in representation theory, a major branch of algebra interacting strongly with geometry and mathematical physics. Pure mathematics aims to abstract and distil the essence of familiar concepts: for instance, in the case of symmetries this leads to the definition of a group, the collection of symmetries of a given object. However, the mathematical definition is far more axiomatic, to the point that the underlying object that the group is describing all but disappears. In these cases it is important to try to recover this object, or more specifically to find all objects whose symmetries give rise to the group in question. This is the motivating idea behind representation theory. Despite this seemingly abstract problem, representation theory is crucially important in many areas of science such as physics (e.g. string theory / mirror symmetry), chemistry (study of molecular vibrations) and computer science, as well as being central for mathematics.Algebra and geometry have been kindred spirits from the very conception of modern mathematics, with ideas and motivating problems passing to and fro all the time. For instance, continuous groups, the bedrock of Lie theory and modern representation theory, came to prominence thanks to Sophus Lie's program applying algebra to the study and classification of geometries. Since the conception of Lie theory, geometry has played a key role, time and again, in moving the subject forward. Conversely, commutative and homological algebra has been pivotal in the modern development of algebraic geometry, enabling the giants, such as Grothendieck, to rebuild the subject on firm mathematical foundations. In this intradisciplinary proposal we aim once again to exploit powerful geometric results in the study of algebra, this time by developing the foundations of a theory of mixed Hodge structures on conic symplectic manifolds, thereby bringing the theory of mixed Hodge structures to bear on a host of (seemingly intractable) problems in representation theory. We also expect that the development of such a theory would also have myriad applications to the understanding of the geometry of conic symplectic manifolds. The first key step to develop this theory of mixed Hodge structures on Deformatio-Quantization (DQ)-modules, is to generalise the construction of nearby and vanishing functors for D-modules to this setting. Secondly, we will use these functors to reconstruct the categories of interest as categories glued out of simpler subquotients. We also propose to develop a geometric analogue of Soergel's V-functor in this setting, allowing us to apply Rouquier's theory of quasi-hereditary covers to DQ-modules.