Exotic t-structures for two-block Springer fibres

Abstract:

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 Bezrukavnikov-Mirkovic [Ann. of Math. (2) 178 (2013), pp. 835–919] 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 Cautis and Kamnitzer [Duke Math. J. 142 (2008), pp. 511–588], 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 $\mathrm {Ext}$ 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 Lascoux-Schutzenberger [Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266], Bernstein-Frenkel-Khovanov [Selecta Math. (N.S.) 5 (1999), pp. 199–241], Brundan-Stroppel [Represent. Theory 15 (2011), pp. 170–243], et al.