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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Exotic t-structures for two-block Springer fibres
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by Rina Anno and Vinoth Nandakumar HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 1523-1552 Request permission


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.
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Additional Information
  • Rina Anno
  • Affiliation: Department of Mathematics, Kansas State University, 138 Cardwell Hall, 1228 N. 17th Street, Manhattan, Kansas 66506-2602
  • MR Author ID: 723023
  • Email:
  • Vinoth Nandakumar
  • Affiliation: Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, Utah, 84102
  • MR Author ID: 1047604
  • Email:
  • Received by editor(s): December 3, 2019
  • Received by editor(s) in revised form: January 7, 2022
  • Published electronically: December 8, 2022

  • Dedicated: Dedicated to our teacher, Roman Bezrukavnikov
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 1523-1552
  • MSC (2020): Primary 14F08; Secondary 18N25
  • DOI:
  • MathSciNet review: 4549684