Stratifications of affine Deligne-Lusztig varieties Defjnition - PowerPoint PPT Presentation

Ulrich Görtz Sydney, August 9, 2019 Stratifications of affine Deligne-Lusztig varieties

Defjnition (Deligne–Lusztig variety) Fix . We set . Properties locally closed in , smooth of dimension , acts on , hence on . Classical Deligne-Lusztig varieties G 0 connected reductive group / fjnite fjeld F q , T 0 ⊂ B 0 ⊂ G 0 . G base change to F q , B , W , Frobenius σ acts on G , W , …

Defjnition (Deligne–Lusztig variety) Properties locally closed in , smooth of dimension , acts on , hence on . Classical Deligne-Lusztig varieties G 0 connected reductive group / fjnite fjeld F q , T 0 ⊂ B 0 ⊂ G 0 . G base change to F q , B , W , Frobenius σ acts on G , W , … Fix w ∈ W . We set X w = { g ∈ G / B ; g − 1 σ ( g ) ∈ BwB } .

Defjnition (Deligne–Lusztig variety) Properties Classical Deligne-Lusztig varieties G 0 connected reductive group / fjnite fjeld F q , T 0 ⊂ B 0 ⊂ G 0 . G base change to F q , B , W , Frobenius σ acts on G , W , … Fix w ∈ W . We set X w = { g ∈ G / B ; g − 1 σ ( g ) ∈ BwB } . locally closed in G / B , smooth of dimension ℓ ( w ) , G 0 ( F q ) acts on X w , hence on H ∗ ( X w , Q ℓ ) .

(‘positive’) affjne fmag variety fjxed rational Iwahori subgroup, . , Let Defjnition (Affjne Deligne–Lusztig variety – Rapoport) simple affjne refmections Iwahori–Weyl group, G , G quasi-simple connected reductive group over Frobenius , Setup, affine DL varieties (equal characteristic: F = F q (( t )) , F local fjeld mixed characteristic: F / Q p fjnite).

(‘positive’) affjne fmag variety G . , Let Defjnition (Affjne Deligne–Lusztig variety – Rapoport) simple affjne refmections Iwahori–Weyl group, fjxed rational Iwahori subgroup, , G quasi-simple connected reductive group over Setup, affine DL varieties (equal characteristic: F = F q (( t )) , F local fjeld mixed characteristic: F / Q p fjnite). σ : � a i t i �→ � a q ˘ i t i Frobenius F = F q (( t )) ,

(‘positive’) affjne fmag variety fjxed rational Iwahori subgroup, . , Let Defjnition (Affjne Deligne–Lusztig variety – Rapoport) simple affjne refmections Iwahori–Weyl group, Setup, affine DL varieties (equal characteristic: F = F q (( t )) , F local fjeld mixed characteristic: F / Q p fjnite). σ : � a i t i �→ � a q ˘ i t i Frobenius F = F q (( t )) , G = G ( ˘ ˘ G quasi-simple connected reductive group over F , F )

(‘positive’) affjne fmag variety Defjnition (Affjne Deligne–Lusztig variety – Rapoport) . , Let Setup, affine DL varieties (equal characteristic: F = F q (( t )) , F local fjeld mixed characteristic: F / Q p fjnite). σ : � a i t i �→ � a q ˘ i t i Frobenius F = F q (( t )) , G = G ( ˘ ˘ G quasi-simple connected reductive group over F , F ) I ⊂ ˘ ˘ G fjxed rational Iwahori subgroup, ˜ ˜ S simple affjne refmections W Iwahori–Weyl group,

(‘positive’) affjne fmag variety Defjnition (Affjne Deligne–Lusztig variety – Rapoport) Setup, affine DL varieties (equal characteristic: F = F q (( t )) , F local fjeld mixed characteristic: F / Q p fjnite). σ : � a i t i �→ � a q ˘ i t i Frobenius F = F q (( t )) , G = G ( ˘ ˘ G quasi-simple connected reductive group over F , F ) I ⊂ ˘ ˘ G fjxed rational Iwahori subgroup, ˜ ˜ S simple affjne refmections W Iwahori–Weyl group, Let w ∈ ˜ W , b ∈ ˘ G . X w ( b ) = { g ∈ ˘ G /˘ I ; g − 1 bσ ( g ) ∈ ˘ I w ˘ I } .

(‘positive’) affjne fmag variety Defjnition (Affjne Deligne–Lusztig variety – Rapoport) Setup, affine DL varieties (equal characteristic: F = F q (( t )) , F local fjeld mixed characteristic: F / Q p fjnite). σ : � a i t i �→ � a q ˘ i t i Frobenius F = F q (( t )) , G = G ( ˘ ˘ G quasi-simple connected reductive group over F , F ) I ⊂ ˘ ˘ G fjxed rational Iwahori subgroup, ˜ ˜ S simple affjne refmections W Iwahori–Weyl group, Let w ∈ ˜ W , b ∈ ˘ G . X w ( b ) = { g ∈ ˘ G /˘ I ; g − 1 bσ ( g ) ∈ ˘ I w ˘ I } .

Relative position map: what are the possible relative positions of id or ) , Example ( ? and odd I ∼ inv : ˘ G /˘ I × ˘ G /˘ → ˘ I \ ˘ G /˘ ˜ I − = W → g − 1 h ( g, h ) �−

Relative position map: Example ( id or ) , odd I ∼ inv : ˘ G /˘ I × ˘ G /˘ → ˘ I \ ˘ G /˘ ˜ I − = W → g − 1 h ( g, h ) �− � what are the possible relative positions of g and σ ( g ) ?

Relative position map: I ∼ inv : ˘ G /˘ I × ˘ G /˘ → ˘ I \ ˘ G /˘ ˜ I − = W → g − 1 h ( g, h ) �− � what are the possible relative positions of g and σ ( g ) ? Example ( SL 2 , b = 1 ) X w (1) � = ∅ ⇐ ⇒ w = id or ℓ ( w ) odd

Example: GSp 4 , b = τ � = id , ℓ ( τ ) = 0 10 10 7 7 7 6 5 6 7 7 8 8 8 6 5 5 6 6 7 7 9 9 8 8 7 6 6 5 5 4 5 6 8 9 8 9 7 4 4 3 4 5 5 8 7 5 3 3 4 4 6 6 8 8 7 7 5 5 4 3 3 2 3 5 7 8 6 5 7 7 5 3 1 2 4 6 6 4 2 0 1 3 5 7 7 5 3 1 2 3 4 5 6 7 8 8 6 4 5 7 9 7 5 2 3 6 8 8 7 6 5 4 3 3 4 6 8 8 6 4 3 4 4 5 6 7 8 9 9 7 8 10 8 5 5 4 5 6 6 9 9 8 7 6 6 5 5 6 7 9 9 7 6 5 6 6 7 7 8 9 10 10

The admissible set Fix t µ ∈ ˜ W translation element. Adm ( µ ) = { w ∈ ˜ W ; ∃ v ∈ W 0 : w ≤ t v ( µ ) } .

The admissible set Fix t µ ∈ ˜ W translation element. Adm ( µ ) = { w ∈ ˜ W ; ∃ v ∈ W 0 : w ≤ t v ( µ ) } .

The admissible set Fix t µ ∈ ˜ W translation element. Adm ( µ ) = { w ∈ ˜ W ; ∃ v ∈ W 0 : w ≤ t v ( µ ) } .

The admissible set Fix t µ ∈ ˜ W translation element. Adm ( µ ) = { w ∈ ˜ W ; ∃ v ∈ W 0 : w ≤ t v ( µ ) } .

The admissible set Fix t µ ∈ ˜ W translation element. Adm ( µ ) = { w ∈ ˜ W ; ∃ v ∈ W 0 : w ≤ t v ( µ ) } .

The admissible set Fix t µ ∈ ˜ W translation element. Adm ( µ ) = { w ∈ ˜ W ; ∃ v ∈ W 0 : w ≤ t v ( µ ) } .

The admissible set Fix t µ ∈ ˜ W translation element. Adm ( µ ) = { w ∈ ˜ W ; ∃ v ∈ W 0 : w ≤ t v ( µ ) } .

. -conjugacy class element for a unique length , Given . in Can choose of . depend only on , be the projection. Let . , Parahoric variant: Main object of study: X ( µ, b ) K � X ( µ, b ) := X w ( b ) . w ∈ Adm ( µ )

. , element for a unique length , Given . in Can choose of . -conjugacy class depend only on Main object of study: X ( µ, b ) K � X ( µ, b ) := X w ( b ) . w ∈ Adm ( µ ) Parahoric variant: K ⊂ ˜ � ˘ K ⊂ ˘ S , σ ( K ) = K G . Let π K : ˘ G /˘ I → ˘ G / ˘ K be the projection.

. , element for a unique length , Given . in Can choose of . -conjugacy class depend only on Main object of study: X ( µ, b ) K � X ( µ, b ) := X w ( b ) . w ∈ Adm ( µ ) Parahoric variant: K ⊂ ˜ � ˘ K ⊂ ˘ S , σ ( K ) = K G . Let π K : ˘ G /˘ I → ˘ G / ˘ K be the projection. G / ˘ X ( µ, b ) K = π K ( X ( µ, b )) ⊂ ˘ K

Main object of study: X ( µ, b ) K � X ( µ, b ) := X w ( b ) . w ∈ Adm ( µ ) Parahoric variant: K ⊂ ˜ � ˘ K ⊂ ˘ S , σ ( K ) = K G . Let π K : ˘ G /˘ I → ˘ G / ˘ K be the projection. G / ˘ X ( µ, b ) K = π K ( X ( µ, b )) ⊂ ˘ K X w ( b ) , X ( µ, b ) depend only on σ -conjugacy class [ b ] of b . Can choose b in ˜ W . Given µ , X ( µ, τ ) � = ∅ for a unique length 0 element τ ∈ ˜ W .

NB: Usually not equi-dimensional. . . dim odd, Bonan: For . dim odd, For dim even, For Theorem (G–Yu) principally polarized abelian varieties with Iwahori level structure at dim X ( µ, τ ) = ? Say G = GSp 2 g , µ = ω ∨ g . Then dim X ( µ, τ ) equals the dimension of the supersingular locus of the moduli space of g -dimensional p , over F p . (For g = 1 : supersingular points in modular curve X 0 ( p ) over F p .)

NB: Usually not equi-dimensional. Theorem (G–Yu) . dim principally polarized abelian varieties with Iwahori level structure at odd, Bonan: For dim X ( µ, τ ) = ? Say G = GSp 2 g , µ = ω ∨ g . Then dim X ( µ, τ ) equals the dimension of the supersingular locus of the moduli space of g -dimensional p , over F p . (For g = 1 : supersingular points in modular curve X 0 ( p ) over F p .) dim X ( µ, τ ) = g 2 /2 . For g even, g ( g − 1)/2 ≤ dim X ( µ, τ ) ≤ ( g + 1)( g − 1)/2 . For g odd,

principally polarized abelian varieties with Iwahori level structure at Theorem (G–Yu) NB: Usually not equi-dimensional. dim X ( µ, τ ) = ? Say G = GSp 2 g , µ = ω ∨ g . Then dim X ( µ, τ ) equals the dimension of the supersingular locus of the moduli space of g -dimensional p , over F p . (For g = 1 : supersingular points in modular curve X 0 ( p ) over F p .) dim X ( µ, τ ) = g 2 /2 . For g even, g ( g − 1)/2 ≤ dim X ( µ, τ ) ≤ ( g + 1)( g − 1)/2 . For g odd, Bonan: For g ≤ 5 odd, g ( g − 1)/2 = dim X ( µ, τ ) .

. Let -stratifjcation on , get Intersecting with . inv inv : for all lie in the same stratum Defjnition (Chen–Viehmann) The J -stratification Relative position (for K ⊂ ˜ S � ˘ K ⊂ ˘ G ) inv K : ˘ G / ˘ K × ˘ G / ˘ K → ˘ K \ ˘ G / ˘ K ∼ = W K \ ˜ W / W K ∼ = K W K .