Introduction The Siegel case Affine Deligne-Lusztig varieties

Affine Deligne-Lusztig varieties

Ulrich Gortz, University of Duisburg-Essen

NCTS, June 2011

Introduction The Siegel case Affine Deligne-Lusztig varieties

Study the reduction at p of Shimura varieties of PEL type withIwahori level structure at p:

Find “good” integral model over ring of integers of Ep (E the reflexfield, p|p). What is the geometric structure of the special fiber?

Motivation: Study arithmetic properties of these Shimura varieties.Can hope for applications in the Langlands program.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Example: The Siegel case

g ≥ 1, p a prime number

“Hyperspecial level structure” (good reduction)

Ag , the moduli space of principally polarized abelian varieties ofdimension g

Iwahori level structure

AI , the moduli space of chains

A0α // A1

α // · · · α // Ag

of chains of isogenies of order p between g -dimensional abelianvarieties, together with principal polarizations λ0, λg on A0, Ag ,resp., such that (αg )∗λg = pλ0.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Example: The Siegel case

g ≥ 1, p a prime number

“Hyperspecial level structure” (good reduction)

Ag , the moduli space of principally polarized abelian varieties ofdimension g

Iwahori level structure

AI , the moduli space of chains

A0α // A1

α // · · · α // Ag

of chains of isogenies of order p between g -dimensional abelianvarieties, together with principal polarizations λ0, λg on A0, Ag ,resp., such that (αg )∗λg = pλ0.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Goal: “Determine” geometric structure of special fiber (aim atdescription in terms of the combinatorial structure of theunderlying algebraic group).

e.g. Newton stratification

Specifically, interested in supersingular locus Sg ⊂ Ag andSI ⊂ AI , resp.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Goal: “Determine” geometric structure of special fiber (aim atdescription in terms of the combinatorial structure of theunderlying algebraic group).

e.g. Newton stratification

Specifically, interested in supersingular locus Sg ⊂ Ag andSI ⊂ AI , resp.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Supersingular locus

Hyperspecial case:

[Li, Oort] Sg is equi-dimensional of dimension[g2

4

], and

connected if g > 1.

Iwahori case:SI is not equidimensional (if g ≥ 2), its dimension is not known ingeneral.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Supersingular locus

Hyperspecial case:

[Li, Oort] Sg is equi-dimensional of dimension[g2

4

], and

connected if g > 1.

Iwahori case:SI is not equidimensional (if g ≥ 2), its dimension is not known ingeneral.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Supersingular locus

Hyperspecial case:

[Li, Oort] Sg is equi-dimensional of dimension[g2

4

], and

connected if g > 1.

Iwahori case:SI is not equidimensional (if g ≥ 2), its dimension is not known ingeneral.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Theorem (G., Yu)

1 Let g be even. Then dim SI = g2

2 .

2 Let g be odd. Then

g(g − 1)

2≤ dim SI ≤

(g + 1)(g − 1)

2.

Conjecture

Let g be odd. Then

g(g − 1)

2= dim SI .

Introduction The Siegel case Affine Deligne-Lusztig varieties

Theorem (G., Yu)

1 Let g be even. Then dim SI = g2

2 .

2 Let g be odd. Then

g(g − 1)

2≤ dim SI ≤

(g + 1)(g − 1)

2.

Conjecture

Let g be odd. Then

g(g − 1)

2= dim SI .

Introduction The Siegel case Affine Deligne-Lusztig varieties

Affine Deligne-Lusztig varieties

k = Fq, k algebraic closure,

L = k((t)).

σ Frobenius on k , L, . . . .

G a split connected reductive group over k ,

G ⊃ B ⊃ A Borel, split max. torus.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Affine Deligne-Lusztig varieties

k = Fq, k algebraic closure,

L = k((t)).

σ Frobenius on k , L, . . . .

G a split connected reductive group over k ,

G ⊃ B ⊃ A Borel, split max. torus.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Hyperspecial case

Let µ be a coweight, b ∈ G (L). K = G (k[[t]]).

Definition (Affine DL variety for µ, b ∈ G (L))

Xµ(b) = {g ∈ G (L)/K ; g−1bσ(g) ∈ Kµ(t)K},

a locally closed subscheme of the affine Grassmannian Grass,locally of finite type /k.

For µ the minuscule coweight of the Shimura variety, b“supersingular”:

dim Sg = dimXµ(b)

(is expected).

Introduction The Siegel case Affine Deligne-Lusztig varieties

Hyperspecial case

Let µ be a coweight, b ∈ G (L). K = G (k[[t]]).

Definition (Affine DL variety for µ, b ∈ G (L))

Xµ(b) = {g ∈ G (L)/K ; g−1bσ(g) ∈ Kµ(t)K},

a locally closed subscheme of the affine Grassmannian Grass,locally of finite type /k.

For µ the minuscule coweight of the Shimura variety, b“supersingular”:

dim Sg = dimXµ(b)

(is expected).

Introduction The Siegel case Affine Deligne-Lusztig varieties

Sketch of connection to Shimura varieties

Replace L by Qnrp . Denote by Λ ⊂ Ln the standard lattice.

For g ∈ Xµ(b), gΛ is a lattice with

pgΛ ⊂ (bσ)gΛ ⊂ gΛ.

Hence, using Dieudonne theory, we can identify the set Xµ(b) with

the set of k-valued points of the corresponding moduli space ofp-divisible groups (Rapoport-Zink space).

Problem: In this case, we have no ind-scheme structure onG (L)/K .

Introduction The Siegel case Affine Deligne-Lusztig varieties

Sketch of connection to Shimura varieties

Replace L by Qnrp . Denote by Λ ⊂ Ln the standard lattice.

For g ∈ Xµ(b), gΛ is a lattice with

pgΛ ⊂ (bσ)gΛ ⊂ gΛ.

Hence, using Dieudonne theory, we can identify the set Xµ(b) with

the set of k-valued points of the corresponding moduli space ofp-divisible groups (Rapoport-Zink space).

Problem: In this case, we have no ind-scheme structure onG (L)/K .

Introduction The Siegel case Affine Deligne-Lusztig varieties

Sketch of connection to Shimura varieties

Replace L by Qnrp . Denote by Λ ⊂ Ln the standard lattice.

For g ∈ Xµ(b), gΛ is a lattice with

pgΛ ⊂ (bσ)gΛ ⊂ gΛ.

Hence, using Dieudonne theory, we can identify the set Xµ(b) with

the set of k-valued points of the corresponding moduli space ofp-divisible groups (Rapoport-Zink space).

Problem: In this case, we have no ind-scheme structure onG (L)/K .

Introduction The Siegel case Affine Deligne-Lusztig varieties

Sketch of connection to Shimura varieties

Replace L by Qnrp . Denote by Λ ⊂ Ln the standard lattice.

For g ∈ Xµ(b), gΛ is a lattice with

pgΛ ⊂ (bσ)gΛ ⊂ gΛ.

Hence, using Dieudonne theory, we can identify the set Xµ(b) with

the set of k-valued points of the corresponding moduli space ofp-divisible groups (Rapoport-Zink space).

Problem: In this case, we have no ind-scheme structure onG (L)/K .

Introduction The Siegel case Affine Deligne-Lusztig varieties

Dimension formula in the hyperspecial case

Theorem (G., Haines, Kottwitz, Reuman; Viehmann)

Assume that Xµ(b) 6= ∅, let νdom be the Newton vector of b. Then

dimXµ(b) = 〈ρ, µ− νdom〉 −1

2defG b.

“Generalization of Theorem of Li and Oort”

Hartl, Viehmann: All ADLV Xµ(b) ⊂ Grass are equi-dimensional.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Dimension formula in the hyperspecial case

Theorem (G., Haines, Kottwitz, Reuman; Viehmann)

Assume that Xµ(b) 6= ∅, let νdom be the Newton vector of b. Then

dimXµ(b) = 〈ρ, µ− νdom〉 −1

2defG b.

“Generalization of Theorem of Li and Oort”

Hartl, Viehmann: All ADLV Xµ(b) ⊂ Grass are equi-dimensional.

Introduction The Siegel case Affine Deligne-Lusztig varieties

ADLV in the Iwahori case

I ⊂ K Iwahori subgroup

Definition (Affine DL variety for w ∈ W , b ∈ G (L))

Xw (b) = {g ∈ G (L)/I ; g−1bσ(g) ∈ IwI},

a locally closed subscheme of the affine flag variety Flag , locallyof finite type /k .

Similarly as in the hyperspecial case, can expect that

dim SI = maxw∈Adm(µ)

dimXw (b).

Introduction The Siegel case Affine Deligne-Lusztig varieties

ADLV in the Iwahori case

I ⊂ K Iwahori subgroup

Definition (Affine DL variety for w ∈ W , b ∈ G (L))

Xw (b) = {g ∈ G (L)/I ; g−1bσ(g) ∈ IwI},

a locally closed subscheme of the affine flag variety Flag , locallyof finite type /k .

Similarly as in the hyperspecial case, can expect that

dim SI = maxw∈Adm(µ)

dimXw (b).

Introduction The Siegel case Affine Deligne-Lusztig varieties

ADLV in the Iwahori case

I ⊂ K Iwahori subgroup

Definition (Affine DL variety for w ∈ W , b ∈ G (L))

Xw (b) = {g ∈ G (L)/I ; g−1bσ(g) ∈ IwI},

a locally closed subscheme of the affine flag variety Flag , locallyof finite type /k .

Similarly as in the hyperspecial case, can expect that

dim SI = maxw∈Adm(µ)

dimXw (b).

Introduction The Siegel case Affine Deligne-Lusztig varieties

ADLV for G = GSp4, Iwahori case, b supersingular.

Introduction The Siegel case Affine Deligne-Lusztig varieties

ADLV for G = GSp4, Iwahori case, b supersingular.

Introduction The Siegel case Affine Deligne-Lusztig varieties

ADLV for G = GSp4, Iwahori case, b supersingular.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Questions

Fur which w , b is Xw (b) 6= ∅?What is the dimension of Xw (b)?

Introduction The Siegel case Affine Deligne-Lusztig varieties

Further notation

W the finite Weyl groups, S the set of simple reflections

For w ∈W let supp(w) ⊆ S denote the set of simple reflectionsneeded to express w .

W the extended affine Weyl group (∼= W n X∗(A))

η : W →W , η(wtµv) = vw

for w , v ∈W such that tµv lies in the dominant chamber.

W ′ the “union of the shrunken Weyl chambers”

Introduction The Siegel case Affine Deligne-Lusztig varieties

Further notation

W the finite Weyl groups, S the set of simple reflections

For w ∈W let supp(w) ⊆ S denote the set of simple reflectionsneeded to express w .

W the extended affine Weyl group (∼= W n X∗(A))

η : W →W , η(wtµv) = vw

for w , v ∈W such that tµv lies in the dominant chamber.

W ′ the “union of the shrunken Weyl chambers”

Introduction The Siegel case Affine Deligne-Lusztig varieties

ADLV for G = GSp4, Iwahori case, b supersingular.

Introduction The Siegel case Affine Deligne-Lusztig varieties

Theorem (G., Haines, Kottwitz, Reuman; G., He)

Let b be basic (“supersingular”). Let w = tµv ∈ W ′, v ∈W, andsuppose that w “lies in the same connected component as b”.Then Xw (b) 6= ∅ if and only if

supp(η(w)) = S

and in this case, at least if µ is regular or G is of type A,

dimXw (b) =1

2(`(w) + `(η(w))− def(b)).

Introduction The Siegel case Affine Deligne-Lusztig varieties

Method of proof I

To prove emptiness, analyze structure of the t-adic group G (L).All w with Xw (b) = ∅ arise as follows:

There exists a parabolic subgroup P = MN, A ⊂ P ( G , such thatw is a P-alcove.

Theorem (G., Haines, Kottwitz, Reuman)

If w is a P-alcove, then every element of IwI is σ-conjugate underI to an element of (I ∩M)w(I ∩M).

Introduction The Siegel case Affine Deligne-Lusztig varieties

Method of proof II

To prove non-emptiness and the dimension formula, usecombinatorics of the extended affine Weyl group together with

Deligne-Lusztig reduction

1 If `(sxs) = `(x), then there exists a universal homeomorphismXx(b)→ Xsxs(b).

2 If `(sxs) = `(x)− 2, then Xx(b) can be written as a disjointunion Xx(b) = X1 t X2 where X1 is closed and X2 is open,and such that there exist morphisms X1 → Xsxs(b) andX2 → Xsx(b) which are compositions of a Zariski- locallytrivial fiber bundle with one-dimensional fibers and a universalhomeomorphism.