Exotic t-structures for two-block Springer fibres. (English) Zbl 1524.14037
Summary: We study the category of representations of \(\mathfrak{sl}_{m+2n}\) in positive characteristic, with \(p\)-character given by a nilpotent with Jordan type \((m+n,n)\). Recent work of R. Bezrukavnikov and I. Mirković [Ann. Math. (2) 178, No. 3, 835–919 (2013; Zbl 1293.17021)] implies that this representation category is equivalent to \(\mathcal{D}_{m,n}^0\), the heart of the exotic t-structure on the derived category of coherent sheaves on a Springer fibre for that nilpotent. Using work of S. Cautis and J. Kamnitzer [Duke Math. J. 142, No. 3, 511–588 (2008; Zbl 1145.14016)], we construct functors indexed by affine tangles, between these categories \(\mathcal{D}_{m,n} \) (i.e. for different values of \(n)\). This allows us to describe the irreducible objects in \(\mathcal{D}_{m,n}^0\) and enumerate them by crossingless \((m,m+2n)\) matchings. We compute the Ext spaces between the irreducible objects, and conjecture that the resulting Ext algebra is an annular variant of Khovanov’s arc algebra. In subsequent work, we use these results to give combinatorial dimension formulae for the irreducible representations. These results may be viewed as a positive characteristic analogue of results about two-block parabolic category \(\mathcal{O}\) due to A. Lascoux and M.-P. Schuetzenberger [Astérisque 87–88, 249–266 (1981; Zbl 0504.20007)], J. Bernstein et al. [Sel. Math., New Ser. 5, No. 2, 199–241 (1999; Zbl 0981.17001)], J. Brundan and C. Stroppel [Represent. Theory 15, 170–243 (2011; Zbl 1261.17006)], et al.
MSC:
| 14F08 | Derived categories of sheaves, dg categories, and related constructions in algebraic geometry |
| 18N25 | Categorification |
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