On the P = W conjecture for $$SL_n$$ S L n
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29_2022_803_ReferencePDF.pdf
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Author(s)
de Cataldo, Mark A.
Maulik, Davesh
Shen, Junliang
Date Issued
October 13, 2022
Publisher
Springer International Publishing
Citation
Selecta Mathematica. 2022 Oct 13;28(5):90
Version
Author's final manuscript
Abstract
Abstract
Let p be a prime number. We prove that the
$$P=W$$
P
=
W
conjecture for
$$\mathrm {SL}_p$$
SL
p
is equivalent to the
$$P=W$$
P
=
W
conjecture for
$$\mathrm {GL}_p$$
GL
p
. As a consequence, we verify the
$$P=W$$
P
=
W
conjecture for genus 2 and
$$\mathrm {SL}_p$$
SL
p
. For the proof, we compute the perverse filtration and the weight filtration for the variant cohomology associated with the
$$\mathrm {SL}_p$$
SL
p
-Hitchin moduli space and the
$$\mathrm {SL}_p$$
SL
p
-twisted character variety, relying on Gröchenig–Wyss–Ziegler’s recent proof of the topological mirror conjecture by Hausel–Thaddeus. Finally we discuss obstructions of studying the cohomology of the
$$\mathrm {SL}_n$$
SL
n
-Hitchin moduli space via compact hyper-Kähler manifolds.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1007/s00029-022-00803-0
