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Register a new userOn the cohomology of Fano varieties and the Springer correspondence
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In this paper we compute the cohomology of the Fano varieties of $k$-planes in the smooth complete intersection of two quadrics in $mathbb{P}^{2g+1}$, using Springer theory for symmetric spaces.
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In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [CVX2].
The Springer resolution of the nilpotent cone is used to give a geometric construction of the irreducible representations of Weyl groups. Borho and MacPherson obtain the Springer correspondence by applying the decomposition theorem to the Springer resolution, establishing an injective map from the set of irreducible Weyl group representations to simple equivariant perverse sheaves on the nilpotent cone. In this manuscript, we consider a generalization of the Springer resolution using a variety defined by the first author. Our main result shows that in the type A case, applying the decomposition theorem to this map yields all simple perverse sheaves on the nilpotent cone with multiplicity as predicted by Lusztigs generalized Springer correspondence.
One can associate an invariant to a large class of regular codimension two defects of the six dimensional $(0,2)$ SCFT $mathscr{X}[j]$ using the classical Springer correspondence. Such an association allows a simple description of S-duality of associated Gaiotto-Witten boundary conditions in $mathcal{N}=4$ SYM for arbitrary gauge group and by extension, a determination of certain local aspects of class $mathcal{S}$ constructions. I point out that the problem of textit{classifying} the corresponding boundary conditions in $mathcal{N}=4$ SYM is intimately tied to possible symmetry breaking patterns in the bulk theory. Using the Springer correspondence and the representation theory of Weyl groups, I construct a pair of functors between the class of boundary conditions in the theory in the phase with broken gauge symmetry and those in the phase with unbroken gauge symmetry.
We show the rationality of a generating series from the affine Springer fibers. The main ingredient is the homogeneity of the Arthur-Shalika germ expansion for the weighted orbital integrals.
This is an expository lecture, for the Abel bicentennial (Oslo, 2002), describing some recent work on the (small) quantum cohomology ring of Grassmannians and other homogeneous varieties.
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