We study the exotic t-structure on Dn, the derived category of coherent sheaves on two-block Springer fibre (i.e. for a nilpotent matrix of type (m + n, n) in type A). The exotic t-structure has been defined by Bezrukavnikov and Mirkovic for Springer theoretic varieties in order to study representations of Lie algebras in positive characteristic. Using work of Cautis and Kamnitzer, we construct functors indexed by affine tangles, between categories of coherent sheaves on different two-block Springer fibres (i.e. for different values of n). After checking some exactness properties of these functors, we describe the irreducible objects in the heart of the exotic t-structure on Dn and enumerate them by crossingless (m, m + 2n) matchings. We compute the Ext's between the irreducible objects, and show that the resulting algebras are an annular variant of Khovanov's arc algebras. In subsequent work we will make a link with annular Khovanov homology, and use these results to give a characteristic p analogue of some categorification results using two-block parabolic category O (by Bernstein-Frenkel-Khovanov, Brundan, Stroppel, et al).

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson-Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of the exotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces. Contents

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Abstract. Let G = Sp 2n (C), and N be Kato's exotic nilpotent cone. Following techniques used by Bezrukavnikov in 2003 to establish a bijection between Λ + , the dominant weights for an arbitrary simple algebraic group H, and O, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution N have vanishing higher cohomology, and compute their global sections using techniques of Broer. This allows us to show that the direct images of these dominant line bundles constitute a quasi-exceptional set generating the category D b (Coh G (N)), and deduce that the resulting t-structure on D b (Coh G (N)) coincides with the perverse coherent t-structure. The desired result now follows from the bijection between costandard objects and simple objects in the heart of the t-structure on D b (Coh G (N)). Contents

Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztig's quiver varieties, which are varieties of nilpotent representations of a pre-projective algebra. We generalize this to symmetrizable Kac-Moody algebras by replacing Lusztig's preprojective algebra with a more general one due to Dlab and Ringel. In non-symmetric types we are forced to work over non-algebraically-closed fields.2010 Mathematics Subject Classification. 17B37.

Recently, Anno, Bezrukavnikov and Mirkovic have introduced the notion of a real variation of stability conditions (which is related to Bridgeland's stability conditions), and construct an example using categories of coherent sheaves on Springer fibers. Here we construct another example, by studying certain sub-quotients of category O with a fixed Gelfand-Kirillov dimension. We use the braid group action on the derived category of category O, and certain leading coefficient polynomials coming from translation functors. Consequently, we use this to explicitly describe a sub-manifold in the space of Bridgeland stability conditions on these sub-quotient categories, which is a covering space of a hyperplane complement in the dual Cartan.

Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak {sl}_2$ , where they use singular blocks of category $\mathcal {O}$ for $\mathfrak {sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak {s}\mathfrak {l}_n$ over a field $\mathbf {k}$ of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.

This paper presents a review of mono and hybrid nanofluid using heat sink technologies, which is the foremost task of new generation technology in cooling electronic devices. Heat generation in a tiny electronic device is the main factor to be prevented to enhance the heat transfer. One prominent remedy for this problem is to adopt mono and hybrid nanofluid based microchannel heat sinks are considered to be the recent trends.In this article, a state-of-the-art review of heat sinks, nanofluids preparation and characterization techniques have been carried out. The study begins with an overview of the heat sink, designing parameters, research work carried out in the last decade using mono nanofluids and hybrid nanofluids followed by the analysis of the research work carried out in the last decade in terms of different geometries of MCHS to examine the diverse factors like pressure drop, heat transfer coefficient, and critical heat flux. Current challenges and opportunities for future research are presented as well.