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Events for May – September 2012

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May. 01, 2012 2pm (MATH 350)	Lie Theory Rob Maier (U. Arizona) X Coxeter groups occur as both automorphism groups and monodromy groups of certain Fuchsian differential equations. This includes the Picard-Fuchs equations associated to families of elliptic curves (i.e., associated to elliptic surfaces). Such equations are second-order differential equations that are extensions of the Gauss hypergeometric equation (e.g., they may be of the Heun type). The appearance of Coxeter groups as their automorphism groups will be explained, and it will also be briefly explained how Coxeter groups appear as the monodromy groups of certain generalized hypergeometric equations (those with algebraic solutions), as has been known since work of Frits Beukers. Coxeter Groups and Certain Differential Equations
Sep. 04, 2012 2pm (MATH 350)	Lie Theory Matt Douglass (U. North Texas) X In their proof of the Deligne-Langlands conjecture, Kazhdan and Lusztig construct an explicit isomorphism between the equivariant K-group of the Steinberg variety of a complex, reductive, algebraic group G and the extended affine Hecke algebra of the Weyl group of G. Later, Ostrik used their construction to define a "Kazhdan-Lusztig" type basis for the equivariant K-group of the nilpotent cone in the Lie algebra of G. In these talks, I'll explain known and conjectural connections between (1) Ostrik's construction, (2) the lowest two-sided cell ideal and some special Kazhdan-Lusztig basis elements in the extended, affine Hecke algebra, and (3) G-equivariant vector bundles supported on {0} in the nilpotent cone. In order to keep the talks as accessible as possible, I'll talk only about the group SL_2(C). Equivariant vector bundles supported on a point and Kazhdan-Lusztig theory I
Sep. 11, 2012 2pm (MATH 350)	Lie Theory Matt Douglass (U. North Texas) X In their proof of the Deligne-Langlands conjecture, Kazhdan and Lusztig construct an explicit isomorphism between the equivariant K-group of the Steinberg variety of a complex, reductive, algebraic group G and the extended affine Hecke algebra of the Weyl group of G. Later, Ostrik used their construction to define a "Kazhdan-Lusztig" type basis for the equivariant K-group of the nilpotent cone in the Lie algebra of G. In these talks, I'll explain known and conjectural connections between (1) Ostrik's construction, (2) the lowest two-sided cell ideal and some special Kazhdan-Lusztig basis elements in the extended, affine Hecke algebra, and (3) G-equivariant vector bundles supported on {0} in the nilpotent cone. In order to keep the talks as accessible as possible, I'll talk only about the group SL_2(C). Equivariant vector bundles supported on a point and Kazhdan-Lusztig theory II
Sep. 18, 2012 2pm (MATH 350)	Lie Theory Nicolas Proudfoot (Oregon) Toric and hypertoric varieties
Sep. 25, 2012 2pm (MATH 350)	Lie Theory Sarah Kitchen (Freiburg) X This is a report on work in progress with Achar, McGerty, and Nevins. Koszul duality is a type of derived equivalence that appears in representation theory. It is expected to relate various categories of representations for Langlands dual pairs of Lie algebras, and underlie observed examples of duality in symplectic geometry. For a given abelian category, the derived equivalence does not occur at the level of the derived category of our abelian category, but for some graded version of the category. Obtaining a graded version and verifying that we have given one suitable for Koszul duality are not usually easy processes. In earlier work with Achar, we produced a grading on perverse sheaves with a fixed stratification from a candidate subcategory of mixed Hodge modules. In this talk, I will describe a more general context in which we can check if a mixed category is a grading suitable for Koszul duality. Obtaining Koszul Gradings Sponsored by the Meyer Fund

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