Global topology of the Hitchin system - GEOM

Hard Lefschetz for Weight and Perverse Filtrations
E( ˆˇMB; 1/q) = q d E( ˆˇMB; q) ⇐ Alvis-Curtis in Irr(G(Fq))
Weight filtration: W0 ⊂ · · · ⊂ Wi ⊂ · · · ⊂ W2k = H k (X )
palindromicity ❀ Curious Hard Lefschetz Conjecture:
L l : Gr W d−2l (Hi−l (MB)) ∼ = → Gr W d+2l H i+l (MB)
x ↦→ x ∪ α l ,
where α ∈ W4H 2 (MB)
Perverse filtration: P0 ⊂ · · · ⊂ Pi ⊂ . . . Pk(X ) ∼ = H k (X )
for f : X → Y proper X smooth Y affine
(de Cataldo-Migliorini, 2008):
take Y0 ⊂ · · · ⊂ Yi ⊂ . . . Yd = Y
s.t. Yi generic with dim(Yi) = i then
Pk−i−1H k (X ) = ker(H k (X) → H k (f −1 (Yi)))
the Relative Hard Lefschetz Theorem holds:
L l : Gr P d−l (H∗ (X )) ∼ = → Gr P d+l H ∗+2l (X )
x ↦→ x ∪ α l
where α ∈ W2H 2 (X ) is a relative ample class 36 / 39