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We describe the loci of non-rationally smooth (nrs) points and of singular points for any non-spiral Schubert variety of $\tilde{A}_2$ in terms of the geometry of the (affine) Weyl group action on the plane $\mathbb{R}^2$. Together with the results of Graham and Li for spiral elements, this allows us to explicitly identify the maximal singular and nrs points in any Schubert variety of type $\tilde{A}_2$. Comparable results are not known for any other infinite-dimensional Kac-Moody flag variety (except for type $\tilde{A}_1$, where every Schubert variety is rationally smooth). As a consequence, we deduce that if $x$ is a point in a non-spiral Schubert variety $X_w$, then $x$ is nrs in $X_w$ if and only if there are more than $\dim X_w$ curves in $X_w$ through $x$ which are stable under the action of a maximal torus, as is true for Schubert varieties in (finite) type $A$. Combined with the work of Graham and Li for spiral Schubert varieties, this implies the Lookup Conjecture for $\tilde{A}_2$., Comment: v1: 24 figures, 1 table; v2: added references

We consider a generalization of the Springer resolution studied in earlier work of the authors, called the extended Springer resolution. In type $A$, this map plays a role in Lusztig's generalized Springer correspondence comparable to that of the Springer resolution in the Springer correspondence. The fibers of the Springer resolution play a key part in the latter story, and connect the combinatorics of tableaux to geometry. Our main results prove the same is true for fibers of the extended Springer resolution -- their geometry is governed by the combinatorics of tableaux. In particular, we prove that these fibers are paved by affines, up to the action of a finite group, and give combinatorial formulas for their Betti numbers. This yields, among other things, a simple formula for dimensions of stalks of the Lusztig sheaves arising in the study of the generalized Springer correspondence, and shows that there is a close resemblance between each Lusztig sheaf and the Springer sheaf for a smaller group., Comment: 37 pages, 2 figures

The set of T-invariant curves in a Schubert variety through a T-fixed point is relatively easy to characterize in terms of its weights, but the tangent space is more difficult. We prove that the weights of the tangent space are contained in the rational cone generated by the weights of the T-invariant curves. In simply laced types, this remains true if "rational" is replaced by "integral". We also obtain conditions under which every weight of the tangent space is the weight of a T-invariant curve, as well as a smoothness criterion. The results rely on equivariant K-theory, as well as the study of different notions of decomposability of roots., Comment: 37 pages

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 Lusztig's generalized Springer correspondence., Comment: 28 pages, 1 figure

Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri. This result was extended to the other classical types by Lakshmibai. We give a uniform characterization of tangent spaces to Schubert varieties in cominuscule G/P. Our results extend beyond cominuscule G/P; they describe the tangent space to any Schubert variety in G/B at a point xB, where x is a cominuscule Weyl group element in the sense of Peterson. Our results also give partial information about the tangent space to any Schubert variety at any point. Our method is to describe the tangent spaces of Kazhdan-Lusztig varieties, and then recover results for Schubert varieties. Our proof uses a relationship between weights of the tangent space of a variety with torus action, and factors of the class of the variety in torus equivariant K-theory. The proof relies on a formula for Schubert classes in equivariant K-theory due to Graham and Willems, as well as a theorem on subword complexes due to Knutson and Miller., Comment: 20 pages, final version, to appear in Michigan Mathematical Journal

The Springer resolution of the nilpotent cone of a semisimple Lie algebra has played an important role in representation theory. The nilpotent cone is equal to Spec R, where R is the ring of regular functions on the nilpotent cone. This paper constructs and studies an analogue of the Springer resolution for the variety Spec S, where S is the ring of regular functions on the universal cover of the regular nilpotent orbit. The construction makes use of the theory of toric varieties. Using this construction, we provide new proofs of results of Broer and of Graham about Spec S. Finally, we show that the construction can be adapted to covers of an arbitrary nilpotent orbit.

We analyse waves that propagate along the interface between a dielectric half-space and a half-space filled with a Lorentz material. We show that the corresponding interface condition leads to a generalisation of the classical Leontovich condition on the boundary of a dielectric half-space. We study when this condition supports propagation of (dispersive) surface waves. We derive the related dispersion relation for waves propagating along the boundary of a stratified half-space and determine the relationship between the loss parameter, frequency and wavenumber for which interfacial waves exist., Comment: 16 pages, 6 figures

We introduce the notion of a cominuscule point in a Schubert variety in a generalized flag variety for a semisimple group. We derive formulas expressing the Hilbert series and multiplicity of a Schubert variety at a cominuscule point in terms of the restrictions of classes in torus-equivariant K-theory and cohomology to that point, generalizing previously known formulas for flag varieties of cominuscule type. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. The formulas for Schubert varieties are special cases of more general formulas valid at generalized cominuscule points of schemes with torus actions., Comment: 25 pages; v2: shortened

We give combinatorial descriptions of the restrictions to T-fixed points of the classes of structure sheaves of Schubert varieties in the T-equivariant K-theory of Grassmannians and of maximal isotropic Grassmannians of orthogonal and symplectic types. We also give formulas, based on these descriptions, for the Hilbert series and Hilbert polynomials at T-fixed points of the corresponding Schubert varieties. These descriptions and formulas are given in terms of two equivalent combinatorial models: excited Young diagrams and set-valued tableaux. The restriction fomulas are positive, in that for a Schubert variety of codimension d, the formula equals (-1)^d times a sum, with nonnegative coefficients, of monomials in the expressions (e^{-\alpha}-1), as \alpha runs over the positive roots. In types A_n and C_n the restriction formulas had been proved earlier by [Kreiman 05], [Kreiman 06] by a different method. In type A_n, the formula for the Hilbert series had been proved earlier by [Li-Yong 12]. The method of this paper, which relies on a restriction formula of [Graham 02] and [Willems 06], is based on the method used by [Ikeda-Naruse 09] to obtain the analogous formulas in equivariant cohomology. The formulas we give differ from the K-theoretic restriction formulas given by [Ikeda-Naruse 11], which use different versions of excited Young diagrams and set-valued tableaux. We also give Hilbert series and Hilbert polynomial formulas which are valid for Schubert varieties in any cominuscule flag variety, in terms of the 0-Hecke algebra., Comment: 54 pages, added reference to and comparison with results of [Ikeda-Naruse 11]

We consider the Belkale-Kumar cup product $\odot_t$ on $H^*(G/P)$ for a generalized flag variety $G/P$ with parameter $t \in \C^m$, where $m=\dim(H^2(G/P))$. For each $t\in \C^m$, we define an associated parabolic subgroup $P_K \supset P$. We show that the ring $(H^*(G/P), \odot_t)$ contains a graded subalgebra $A$ isomorphic to $H^*(P_K/P)$ with the usual cup product, where $P_K$ is a parabolic subgroup associated to the parameter $t$. Further, we prove that $(H^*(G/P_K), \odot_0)$ is the quotient of the ring $(H^*(G/P), \odot_t)$ with respect to the ideal generated by elements of positive degree of $A$. We prove the above results by using basic facts about the Hochschild-Serre spectral sequence for relative Lie algebra cohomology, and most of the paper consists of proving these facts using the original approach of Hochschild and Serre.

We study the Belkale-Kumar family of cup products on the cohomology of a generalized flag variety. We give an alternative construction of the family using relative Lie algebra cohomology, and in particular, identify the Belkale-Kumar cup product with a relative Lie algebra cohomology ring for every value of the parameter. As a consequence, we extend a fundamental disjointness result of Kostant to a family of Lie algebras. In an appendix, written jointly with Edward Richmond, we extend a Levi movability result of Belkale and Kumar to arbitrary parameters., Comment: 26 pages; some small revisions from previous version

We prove some general results on the T-equivariant K-theory K_T(G/P) of the flag variety G/P, where G is a semisimple complex algebraic group, P is a parabolic subgroup and T$ is a maximal torus contained in P. In particular, we make a conjecture about a positivity phenomenon in K_T(G/P) for the product of two basis elements written in terms of the basis of K_T(G/P) given by the dual of the structure sheaf (of Schubert varieties) basis. (For the full flag variety G/B, this dual basis is closely related to the basis given by Kostant-Kumar.) This conjecture is parallel to (but different from) the conjecture of Griffeth-Ram for the structure constants of the product in the structure sheaf basis. We give explicit expressions for the product in the T-equivariant K-theory of projective spaces in terms of these bases. In particular, we establish our conjecture and the conjecture of Griffeth-Ram in this case.

Let G be a connected reductive algebraic group acting on a scheme X. Let R(G) denote the representation ring of G, and let I be the ideal in R(G) of virtual representations of rank 0. Let G(X) (resp. G(G,X)) denote the Grothendieck group of coherent sheaves (resp. G-equivariant coherent sheaves) on X. Merkurjev proved that if the fundamental group of G is torsion-free, then the map of G(G,X)/IG(G,X) to G(X) is an isomorphism. Although this map need not be an isomorphism if the fundamental group of G has torsion, we prove that without the assumption on the fundamental group of G, this map is an isomorphism after tensoring with the rational numbers.

Let $G$ be a complex, linear algebraic group acting on an algebraic space $X$. The purpose of this paper is to prove a Riemann-Roch theorem (Theorem 5.3) which gives a description of the completion of the equivariant Grothendieck group $G_0(G,X)$ at any maximal ideal of the representation ring $R(G) \otimes \C$ in terms of equivariant cycles. The main new technique for proving this theorem is our non-abelian completion theorem (Theorem 4.3) for equivariant $K$-theory. Theorem 4.3 generalizes the classical localization theorems for diagonalizable group actions to arbitrary groups., Comment: 35 pages, Latex2e, accepted Duke Math Journal