Lie Groups and Their Discrete Subgroups

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The Principal Investigator (John Millson) proposes three main lines of research all within the general framework of reductive algebraic groups and geometry. In the first and most important main line, Nicolas Bergeron and Colette Moeglin (both of the University of Paris) and the PI will try to generalize to the unitary groups SU(p,q) their result that the Poincare duals of classes carried by totally geodesic cycles in the locally symmetric spaces of simple arithmetic type associated to the orthogonal groups SO(p,q) span the cohomology groups of low degree and special refined Hodge type, see "Hodge type theorems for arithmetic manifolds associated to orthogonal groups." Also, the PI proposes generalizing to the unitary case his earlier joint work (comprising five papers) with Jens Funke (from Durham University) on the boundary values of special cohomology classes associated to orthogonal groups. The second main line deals with the ring R of projective invariants of n ordered points on complex projective m space and the equivariant symplectic geometry of the corresponding moduli spaces and their toric degenerations. All of the completed work referred to above was supported by the NSF grant DMS-0907446. The third main line is an attempt to generalize to Kac Moody Lie algebras the PI's previous work with Michael Kapovich (and in parts with Thomas Haines, Shrawan Kumar, and Bernhard Leeb) dealing with the generalized triangle inequalities and related (saturation) problems from reductive Lie algebras. This project was the subject of the PI's earlier FRG grant DMS-0554254 with Prakash Belkale, Thomas Haines, Michael Kapovich and Shrawan Kumar. The problem looks difficult but there is a test example that will indicate whether the earlier theory will generalize. That example is affine SL(2). The PI proposes three main lines of research all within the general framework of reductive algebraic groups and geometry. The first part of the proposal deals with a remarkable and unexpected interaction between geometry, analysis and two different areas of representation theory (the oscillator/Weil representation and the work of James Arthur on the Selberg trace formula based in part on ideas of Robert Langlands). This work should have applications to number theory along the lines described by Steven Kudla in his 2002 International Congress of Mathematicians talk. The second and third parts of the proposal are motivated in part because they are related to much studied problems going back to the beginning of invariant theory in the late nineteenth century. The earlier work of the PI on the third part deals with basic problems in representation theory, e.g. decomposing tensor products and branching formulas which are much used in a number of disciplines. The current proposal outlines an extension of this work to other settings. All the above projects are in collaboration with other mathematicians from within the USA or abroad. In the last four years, the PI has had collaborations with ten mathematicians continuing a history of extensive collaboration (over fifty joint papers).