Derived Equivalences and Mixed Categories in Representation Theory

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A series of major advances in geometric representation theory in the past fifteen years have taken the form of "coherent-constructible equivalences": these theorems assert that the category of coherent sheaves on some variety associated to a reductive group is equivalent (or derived-equivalent) to the category of constructible (or perverse) sheaves on a different variety, associated to the Langlands dual group. Perhaps the best-known such result is the "geometric Satake equivalence," due to Lusztig, Ginzburg, and Mirkovic-Vilonen. It relates representations of a group G (equivalently: coherent sheaves on a point) to spherical perverse sheaves on the affine Grassmannian for the dual group. Another major result is the Arkhipov-Bezrukavnikov-Ginzburg (ABG) equivalence, which states that there is a derived equivalence between perverse sheaves on the affine Grassmannian and coherent sheaves on the cotangent bundle of the flag variety of G. Both these results may be regarded as part of the geometric Langlands program. The P.I. hopes to contribute to this picture with the following two projects: (I) In collaboration with S. Riche, the P.I. hopes to prove parabolic versions of several known coherent-constructible equivalences; these results would encompass the ABG and geometric Satake equivalences as special cases. (II) The P.I. hopes to develop a new axiomatic framework which is expected to lead to theorems on derived equivalences and higher Ext-vanishing in a very general setting, with a view to applications to coherent sheaves on the nilpotent cone and on the cotangent bundle of the flag variety.A "matrix group" is a set of invertible square matrices that contains all products and inverses of its members. A typical example is SU(2), the group of 2x2 unitary complex matrices. A "representation" of such a group is a rule that assigns to each member of the group a linear transformation of some vector space, in such a way that it transforms matrix multiplication into composition of linear transformations. SU(2) has a natural 2-dimensional representation--the rule assigning to each element of SU(2) itself--but there are many others as well: for instance, SU(2) has a representation on the space of polynomials in two variables, given by linear substitutions in the variables. SU(2) is also a topological space: in fact, it is topologically equivalent to a 3-sphere. A number of modern results in representation theory involve the geometry of matrix groups and related spaces. Two celebrated results are the geometric Satake equivalence and the Arkhipov-Bezrukavnikov-Ginzburg (ABG) equivalence, both of which relate representations of a matrix group to "D-modules" (a sophisticated way of working with spaces of local solutions of differential equations) on a certain infinite-dimensional space called the "affine Grassmannian." The proposed research includes two projects: (I) the P.I., in collaboration with S. Riche, hopes to prove a general "parabolic-parahoric equivalence theorem" that would include the geometric Satake and ABG equivalences as special cases; and (II) the P.I. hopes to study in an axiomatic way certain "positivity phenomena" that occur in the known equivalence theorems, with a view to generalizing those theorems to other settings.