On the P = W conjecture for $$SL_n$$ S L n
Abstract Let p be a prime number. We prove that the $$P=W$$ P = W...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer International Publishing
2022
|
| Online Access: | https://hdl.handle.net/1721.1/145826 |
| Summary: | 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. |
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