## Exotic t-structures for two-block Springer fibres

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Rina Anno and Vinoth Nandakumar
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## 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.## References

- R. Anno,
*Multiplication in Khovanov cohomology via triangulated categorifications*, in preparation. - Rina Anno and Timothy Logvinenko,
*On adjunctions for Fourier-Mukai transforms*, Adv. Math.**231**(2012), no. 3-4, 2069–2115. MR**2964634**, DOI 10.1016/j.aim.2012.06.007 - Rina Anno and Timothy Logvinenko,
*Orthogonally spherical objects and spherical fibrations*, Adv. Math.**286**(2016), 338–386. MR**3415688**, DOI 10.1016/j.aim.2015.08.027 - Rina Anno and Timothy Logvinenko,
*Spherical DG-functors*, J. Eur. Math. Soc. (JEMS)**19**(2017), no. 9, 2577–2656. MR**3692883**, DOI 10.4171/JEMS/724 - Joseph Bernstein, Igor Frenkel, and Mikhail Khovanov,
*A categorification of the Temperley-Lieb algebra and Schur quotients of $U(\mathfrak {sl}_2)$ via projective and Zuckerman functors*, Selecta Math. (N.S.)**5**(1999), no. 2, 199–241. MR**1714141**, DOI 10.1007/s000290050047 - J. Elisenda Grigsby, Anthony M. Licata, and Stephan M. Wehrli,
*Annular Khovanov homology and knotted Schur-Weyl representations*, Compos. Math.**154**(2018), no. 3, 459–502. MR**3731256**, DOI 10.1112/S0010437X17007540 - I. Mirkovic and M. Vybornov,
*Quiver varieties and Beilinson-Drinfeld Grassmannians of type A*, Preprint, arXiv:07124160v2. - Roman Bezrukavnikov and Ivan Mirković,
*Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution*, Ann. of Math. (2)**178**(2013), no. 3, 835–919. With an appendix by Eric Sommers. MR**3092472**, DOI 10.4007/annals.2013.178.3.2 - Roman Bezrukavnikov, Ivan Mirković, and Dmitriy Rumynin,
*Localization of modules for a semisimple Lie algebra in prime characteristic*, Ann. of Math. (2)**167**(2008), no. 3, 945–991. With an appendix by Bezrukavnikov and Simon Riche. MR**2415389**, DOI 10.4007/annals.2008.167.945 - Roman Bezrukavnikov and Simon Riche,
*Affine braid group actions on derived categories of Springer resolutions*, Ann. Sci. Éc. Norm. Supér. (4)**45**(2012), no. 4, 535–599 (2013) (English, with English and French summaries). MR**3059241**, DOI 10.24033/asens.2173 - Jonathan Brundan and Catharina Stroppel,
*Highest weight categories arising from Khovanov’s diagram algebra III: category $\scr O$*, Represent. Theory**15**(2011), 170–243. MR**2781018**, DOI 10.1090/S1088-4165-2011-00389-7 - Sabin Cautis and Joel Kamnitzer,
*Knot homology via derived categories of coherent sheaves. I. The ${\mathfrak {sl}}(2)$-case*, Duke Math. J.**142**(2008), no. 3, 511–588. MR**2411561**, DOI 10.1215/00127094-2008-012 - Yanfeng Chen and Mikhail Khovanov,
*An invariant of tangle cobordisms via subquotients of arc rings*, Fund. Math.**225**(2014), no. 1, 23–44. MR**3205563**, DOI 10.4064/fm225-1-2 - Neil Chriss and Victor Ginzburg,
*Representation theory and complex geometry*, Birkhäuser Boston, Inc., Boston, MA, 1997. MR**1433132** - Eric M. Friedlander and Brian J. Parshall,
*Deformations of Lie algebra representations*, Amer. J. Math.**112**(1990), no. 3, 375–395. MR**1055649**, DOI 10.2307/2374747 - Jens Carsten Jantzen,
*Representations of Lie algebras in positive characteristic*, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, pp. 175–218. MR**2074594**, DOI 10.2969/aspm/04010175 - D. Huybrechts,
*Fourier-Mukai transforms in algebraic geometry*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR**2244106**, DOI 10.1093/acprof:oso/9780199296866.001.0001 - Mikhail Khovanov,
*Link homology and categorification*, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 989–999. MR**2275632** - Mikhail Khovanov,
*A functor-valued invariant of tangles*, Algebr. Geom. Topol.**2**(2002), 665–741. MR**1928174**, DOI 10.2140/agt.2002.2.665 - V. Nandakumar and D. Yang,
*Modular representations of $\mathfrak {sl}_n$ with a two-row nilpotent $p$-character*, in progress - Heather M. Russell,
*A topological construction for all two-row Springer varieties*, Pacific J. Math.**253**(2011), no. 1, 221–255. MR**2869443**, DOI 10.2140/pjm.2011.253.221 - Alain Lascoux and Marcel-Paul Schützenberger,
*Polynômes de Kazhdan & Lusztig pour les grassmanniennes*, Young tableaux and Schur functors in algebra and geometry (Toruń, 1980), Astérisque, vol. 87, Soc. Math. France, Paris, 1981, pp. 249–266 (French). MR**646823** - Catharina Stroppel,
*Categorification of the Temperley-Lieb category, tangles, and cobordisms via projective functors*, Duke Math. J.**126**(2005), no. 3, 547–596. MR**2120117**, DOI 10.1215/S0012-7094-04-12634-X - Catharina Stroppel and Ben Webster,
*2-block Springer fibers: convolution algebras and coherent sheaves*, Comment. Math. Helv.**87**(2012), no. 2, 477–520. MR**2914857**, DOI 10.4171/CMH/261 - Ben Webster,
*Tensor product algebras, Grassmannians and Khovanov homology*, Physics and mathematics of link homology, Contemp. Math., vol. 680, Amer. Math. Soc., Providence, RI, 2016, pp. 23–58. MR**3591642**, DOI 10.1090/conm/680

## 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: ranno@ksu.edu
**Vinoth Nandakumar**- Affiliation: Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, Utah, 84102
- MR Author ID: 1047604
- Email: vinoth@math.utah.edu
- Received by editor(s): December 3, 2019
- Received by editor(s) in revised form: January 7, 2022
- Published electronically: December 8, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**376**(2023), 1523-1552 - MSC (2020): Primary 14F08; Secondary 18N25
- DOI: https://doi.org/10.1090/tran/8765
- MathSciNet review: 4549684

Dedicated: Dedicated to our teacher, Roman Bezrukavnikov