DRINFELD-PLÜCKER RELATIONS FOR HOMOGENEOUS SPACES, (AFFINE) FLAG VARIETIES, AND RAPOPORT-ZINK MODELS

THOMAS J. HAINES

1. Introduction

Let F be a local field (or any function field k((ϖ))), with ring of integers O_{F}. The main object of this manuscript is to provide a first step in defining Rapoport-Zink “local models” Mμ̌ attached to an arbitrary split reductive O_{F} -group G and an arbitrary dominant coweight μ̌ for G. As it stands at this time, this paper proposes a definition for Mμ̌ when F = k((ϖ)) and char(k) = 0 (the “equi-char 0” case).

The main idea is to use every representation of G, rather than just one (the “standard representation” is used to define local models attached to classical groups – this does not seem to be adequate for orthogonal groups). Thus we are looking for a “Tannakian” description of Mμ̌.

Let G denote a linear algebraic group over an algebraically closed field k. Let Rep_{k}G denote the category of finite dimensional representations of G over the field k. By a Tannakian description of a variety X associated to G (e.g. a flag variety), we mean a bijection between the set of points of X and the set of collections of certain data (e.g. lines, lattices) indexed by the objects of Rep_{k}G, subject to certain compatibilities with respect to G-morphisms and tensor products. If G is a connected reductive group with a maximal torus T and a Borel subgroup B = T U, we let X^{+}(T) denote the set of B-dominant characters of T. By a Plücker description of X we mean a similar bijection where data is only specified for every Weyl module V (λ) (λ ∈ X^{+}(T)), rather than for every representation of G.

For H any closed subgroup of a linear algebraic group G, we give a Tannakian description for the affine closure \overline{G/H} of the quotient G/H. Using a theorem of Grosshans [6], the case H = U yields a Plücker description for G/U, where U is a maximal unipotent subgroup of a connected reductive group G. This description of G/U, apparently due to Drinfeld, is announced in work of Kuznetsov [11] (when char(k) = 0) and also Braverman-Gaitsgory [2]. We refer to the tensor compatibility conditions which arise in this and any similar context as Drinfeld-Plücker relations.

Letting B = T U be a Borel subgroup, the torus T acts freely on G/U and the quotient is the finite flag variety G/B. Translating this in terms of the Plücker description for G/U we arrive at a Plücker description for G/B, which is also due to Drinfeld (unpublished) and is used in the characteristic zero setting by Kuznetsov [11] and Finkelberg-Mirkovic [4]. The version presented here works equally well in any characteristic.

The first goal of this manuscript is simply to present complete proofs of the aforementioned results (to my knowledge, no proofs have appeared in the literature before now). Also, the description of \overline{G/H} for arbitrary subgroups H does not seem to have appeared before.

The second goal is to give similar Plücker descriptions for the affine Grassmanian and the affine flag variety, and some indications for a Plücker description for Rapoport-Zink local models (we describe “equi-characteristic 0” local models).

More precisely, following [4] we prove a Plücker description for the affine Grassmanian

G(k((t)))/G(k[[t]]),

when char(k) = 0. (This is apparently due to Beilinson-Drinfeld.) We give a similar description of the closure \overline{Q}μ̌ of the G(k[[t]])-orbit Qμ̌ indexed by the dominant coweight μ̌.

We also give a Plücker description of the affine flag variety G(k((t)))/I_{k}, where I_{k} is the Iwahori subgroup I_{k} = π^{−1}(B), where π : G(k[[t]]) → G(k) is the homomorphism given by t ↦ 0. We describe as well the finite-dimensional analogues Fl_{≤μ̌} of \overline{Q}μ̌.

Adding a parameter ϖ, we give the Plücker relations for Beilinson’s deformation of the affine Grassmanian to the affine flag variety, which played a crucial role in Gaitsgory’s construction of the center of the affine Hecke algebra via nearby cycles, see [5] and [7]. More precisely, we define a model over Spec(k[[ϖ]]) which on the generic fiber is the affine Grassmanian of G over k((ϖ)), and on the special fiber is the affine flag variety of G over k. The finite-dimensional scheme Mμ̌ corresponding to a minuscule μ̌ looks like a local model for a Shimura variety with Iwahori level structure: the generic fiber is the subvariety Q̄μ̌ of the affine Grassmanian, and the special fiber Fl_{≤μ̌} has a stratification indexed by a set which contains the μ̌-admissible orbits of the Iwahori subgroup I_{k} acting on the affine flag variety. This scheme Mμ̌ is our candidate for the definition of a Rapoport-Zink model attached to G and μ̌.

Speculative remarks:

Similar ideas should be useful in giving a definition of Mμ̌ over a p-adic number ring. (However, I expect that only a Tannakian description, not a Plücker description, will be possible for a general group.) At least when μ̌ is minuscule, it should be possible to define the correct object this way (i.e. a Witt-vector description of the affine Grassmanian is not needed in this case). Also, these ideas should give a Tannakian description of the Bruhat-Tits parahoric group schemes. It is conceivable that the Bruhat-Tits building itself has a Tannakian description (a point would be identified with a “Moy-Prasad filtration” on every finite-type rational representation of G over O_{F}, the filtrations being compatible in a certain sense with tensor products). These things will be studied as this project continues.

2. The affine closure of G/H

In this section G will be a linear algebraic group over k = k̄, and H ⊂ G will be a closed subgroup. By abuse we will often write G instead of the group of k-points G(k). We write k[G] for the ring of regular functions on G. The quotient G/H exists and its ring of regular functions satisfies

k[G/H] = k[G]^{H}

where H acts on k[G] via the right regular representation: h · f(g) = f(gh).

2.1. Affine closure. We define the affine closure of G/H to be

\overline{G/H} = Spec(k[G]^{H}).

The left action of G on G/H clearly extends to an action on \overline{G/H}. We denote the action morphism by a : G × \overline{G/H} → \overline{G/H} and its comorphism by a^{∗} : k[G]^{H} → k[G]^{H} ⊗ k[G]. We will often abbreviate a(g, x) = g · x, for g ∈ G and x ∈ \overline{G/H}. (Similarly with respect to other actions.)

There are canonical dominant G-morphisms

G → G/H → \overline{G/H}.

The composition is denoted ξ : G → \overline{G/H}.

2.2. Reformulation of Tannaka’s theorem. For any object (V, a_{v}) ∈ Rep_{k}G, the action a_{V} : G × V → V is determined by its comorphism a_{V}^{∗} : V → V ⊗ k[G].

We let 𝕀 denote the 1-dimensionial trivial representation of G.

Theorem 2.1 (Tannaka). The map g ↦ (a_{V} (g))_{V} defines a bijection between G and the set of collections {α_{V} ∈ Aut_{k}V }_{V ∈Rep_{k}G} satisfying the following properties:

(1) α_{𝕀} = id,

(2) For any G-morphism V → W, the following diagram commutes

\xymatrix{V \ar[d] \ar[r]^{α_{V}} & V \ar[d] \\ W \ar[r]^{α_{W}} & W}

(3) α_{V ⊗W} = α_{V} ⊗ α_{W}, for any V, W ∈ Rep_{k}G.

We want to interpret the rule g ↦ {a_{V} (g)} in a different way, which is more amenable to generalization to quotients. Let e ∈ G denote the identity element.

Given g ∈ G, there is a unique G-equivariant morphism

ϕ_{g} : G → G

such that ϕ_{g}(e) = g (here G acts on the left). Its comorphism ϕ_{g}^{∗} : k[G] → k[G] fits into the following commutative diagram:

\xymatrix{V \ar[r]^{a_{v}^{∗}} \ar[d]_{a_{V} (g)} & V ⊗ k[G] \ar[d]^{id⊗ϕ_{g}^{∗}} \\ V \ar[r]^{a_{v}^{∗}} & V ⊗ k[G].}

Since the map a_{V}^{∗} is injective, the vertical map a_{V} (g) : V → V is the unique map making the diagram commute, hence is determined by id ⊗ ϕ_{g}^{∗}.

Tannaka’s map can thus be described as g ↦ {(id ⊗ ϕ_{g}^{∗})|_{V} }_{V}, where restriction is with respect to the inclusions a_{V}^{∗}.

2.3. Tannakian description of \overline{G/H}. For V ∈ Rep_{k}G, let V^{H} ⊂ V denote the subspace of invariants under H.

Let ρ denote the right regular representation of G on k[G]. It is easy to see that the action comorphism a_{V}^{∗} : V → V ⊗ k[G] intertwines the actions a_{V} and id ⊗ ρ. Hence a_{V}^{∗} induces an injective map

a_{V}^{∗} : V^{H} → V ⊗ k[G]^{H}.

Suppose x ∈ \overline{G/H}. Then there is a unique G-equivariant morphism

ϕ_{x} : G → \overline{G/H}

such that ϕ_{x}(e) = x. Let ϕ_{x}^{∗} : k[G]^{H} → k[G] denote its comorphism.

Lemma 2.2. The map

(id ⊗ ϕ_{x}^{∗}) ∘ a_{V}^{∗} : V^{H} → V ⊗ k[G]^{H} → V ⊗ k[G]

factors (uniquely) through a_{V}^{∗} : V → V ⊗ k[G]. That is, there is a unique map α_{V} : V^{H} → V such that the following diagram commutes

\xymatrix{V^{H} \ar[d]_{α_{V}} \ar[r]^{a_{V}^{∗}} & V ⊗ k[G]^{H} \ar[d]^{id⊗ϕ_{x}^{∗}} \\ V \ar[r]^{a_{V}^{∗}} & V ⊗ k[G].}

Proof. Since a_{V}^{∗} is injective, it is clear that we need only prove the existence of the factoring α_{V}, i.e., that (id ⊗ ϕ_{x}^{∗})(a_{V}^{∗} (V^{H})) ⊂ a_{V}^{∗} (V). Since ξ : G → \overline{G/H} has dense image, the proof consists in verifying the following points:

(1) If x = ξ(g), for some g ∈ G, then the factoring exists.

(2) For a fixed v ∈ V^{H}, the set of elements x ∈ \overline{G/H} such that

(id ⊗ ϕ_{x}^{∗})(a_{V}^{∗} (v)) ∈ a_{V}^{∗} (V)

is a closed subvariety. The first point (1) follows from 2.2. For point (2), fix v ∈ V^{H} and write f = a_{V}^{∗} (v) ∈ V ⊗k[G]^{H};

we will think of f as a V -valued polynomial function on G, invariant under H.

Let f′_{i} (i ∈ I(V)) denote a k-basis for a_{V}^{∗} (V) ⊂ V ⊗ k[G], and let us extend it to a k-basis f′_{i} (i ∈ I) for V ⊗ k[G].

By abuse we denote by

a^{∗} : V ⊗ k[G]^{H} → V ⊗ k[G]^{H} ⊗ k[G]

the map id_{V} ⊗ a^{∗}, where a^{∗} : k[G]^{H} → k[G]^{H} ⊗ k[G] is the comorphism for the G-action on \overline{G/H}. We think of a^{∗}f as a V -valued polynomial function on G × \overline{G/H}.

By definition we have [(id ⊗ ϕ_{x}^{∗})a_{V}^{∗} (v)](g) = f(g · x). Moreover, writing

a^{∗}f = ∑_{i∈I} f′_{i} ⊗ f′′_{i}

for some uniquely determined f′′_{i} ∈ k[G]^{H}, we see

f(g · x) = a^{∗}f(g, x) = ∑_{i∈I} f′′_{i}(x)f′_{i}(g),

which is in a_{V}^{∗} (V) if and only if f′′_{i}(x) = 0 for every i ∈ I\I(V). Thus

(id ⊗ ϕ_{x}^{∗})a_{V}^{∗} (v) ∈ a_{V}^{∗} (V) ⇔ f′′_{i}(x) = 0, ∀i ∈ I\I(V),

which proves point (2). The lemma is proved. □

By Lemma 2.2, for each x ∈ \overline{G/H} we may define a k-linear map α_{V} : V^{H} → V by α_{V} = (id_{V} ⊗ ϕ_{x}^{∗})|_{V^{H}}, where restriction is taken with respect to the inclusions a_{V}^{∗}.

The following proposition gives the Tannakian description of \overline{G/H}.

Proposition 2.3. The map Φ : x ↦ {α_{V} = (id_{V} ⊗ ϕ_{x}^{∗})|_{V^{H}}}_{V} defines an isomorphism between \overline{G/H} and the set of collections of k-linear maps \{α_{V} ∈ Hom_{k}(V^{H}, V)\}_{V} satisfying the following properties:

(1) α_{𝕀} = id,

(2) For every G-morphism V → W, the following diagram commutes

\xymatrix{V^{H} \ar[d] \ar[r]^{α_{v}} & W \ar[d] \\ W^{H} \ar[r]^{α_{W}} & W}

(3) For any V, W ∈ Rep_{k}G, the following diagram commutes

[figure: V^{H} ⊗ W^{H}; α_{V} ⊗α_{W}; can; V ⊗ W; (V ⊗ W)^{H}; α_{V ⊗W}; V ⊗ W]

Proof. We will define the map Ψ which is inverse to Φ. But first we must verify that Φ takes values in the right hand side. Fix x ∈ \overline{G/H}, and let Φ(x) = \{α_{V} \}_{V}, where α_{V} = (id_{V} ⊗ ϕ_{x}^{∗})|_{V^{H}} as above. Since ϕ_{x}^{∗} is a k-algebra homomorphism, it is clear that property (1) is satisfied.

To check (3), let diag^{∗} : k[G]⊗ → k[G] denote the comorphism of the diagonal map diag : G → G × G. Then (3) follows easily using the equalities

a_{V ⊗W}^{∗} = (id_{V ⊗W} ⊗ diag^{∗}) ∘ (a_{V}^{∗} ⊗ a_{W}^{∗})
ϕ_{x}^{∗} ∘ diag^{∗} = diag^{∗} ∘ (ϕ_{x}^{∗} ⊗ ϕ_{x}^{∗})

and the fact that a_{V ⊗W}^{∗} is injective.

To check (2), let θ : V → W be a G-morphism. Then we have a_{V}^{∗} (v)(h) = h · v, for every h ∈ G and v ∈ V. Thus

(θ ⊗ id) ∘ a_{V}^{∗} = a_{W}^{∗} ∘ θ,

which, together with the injectivity of a_{W}^{∗}, implies (2). Definition of Ψ:

Let {α_{V} }_{V} be a collection satisfying properties (1-3) in the proposition. We note that we can easily extend the family of maps α_{V} : V^{H} → V to include all locally finite representation of G, such that properties (1-3) continue to hold; in particular for the right regular representation ρ on k[G] we may define

α : k[G]^{H} → k[G],

by setting α(v) = α_{V} (v) whenever v ∈ V, where V ⊂ k[G] is a finite dimensional G-stable subspace.

Claim 2.4. α : k[G]^{H} → k[G] is a k-algebra homomorphism.

Proof. First α preserves the identity element since α_{𝕀} = id. To show α preserves multiplication, use properties (2) and (3) applied to the multiplication morphism:

m : k[G] ⊗ k[G] → k[G]

(this intertwines the G-module structures ρ ⊗ ρ and ρ). □

Using the claim, we get a morphism ϕ = α^{∗} : G → \overline{G/H}.

Lemma 2.5. ϕ is G-equivariant.

Proof. Denote the comultiplication by Δ : k[G] → k[G] ⊗ k[G]. It is easily seen that

ϕ is equivariant ⇔ (id_{k[G]} ⊗ α) ∘ a^{∗} = Δ ∘ α.

On the other hand applying (2) to Δ (which intertwines the G-module structures ρ and id ⊗ ρ), we get the commutative diagram

\xymatrix{k[G]^{H} \ar[d]_{α} \ar[r]^{a^{∗}} & k[G] ⊗ k[G]^{H} \ar[d]^{id⊗α} \\ k[G] \ar[r]^{Δ} & k[G] ⊗ k[G].}

This proves the lemma. □

Finally, we define Ψ(\{α_{V} \}_{V}) = ϕ(e) ∈ \overline{G/H}. It is straightforward to check that Φ and Ψ are mutually inverse. This proves the proposition.

□

Remark. In case G is reductive and H is a reductive subgroup of G, then G/H is affine and so \overline{G/H} = G/H. Hence in this case the proposition gives a Tannakian description of the homogeneous space G/H.

2.4. \overline{G/H} over a k-scheme S. Let S be any k-scheme. Let O_{S} denote its structure sheaf. For any V ∈ Rep_{k}G, let V ⊗ O_{S} denote the trivial vector bundle over S corresponding to V.

Using the same method as above, we get the following Tannakian description of the S-points \overline{G/H}(S).

Proposition 2.6. There is a bijection (functorial in S) between the set \overline{G/H}(S) and the set of all collections

{α_{V} : V^{H} ⊗ O_{S} → V ⊗ O_{S}}_{V ∈Rep_{k}G}

where each α_{V} is O_{S}-linear, and the collection satisfies the following properties:

(1) α_{𝕀} = id_{O_{S}}, (2) For any G-morphism V → W, the following diagram commutes

\xymatrix{V^{H} ⊗ O_{S} \ar[r]^{α_{V}} \ar[d] & V ⊗ O_{S} \ar[d] \\ W^{H} ⊗ O_{S} \ar[r]^{α_{W}} & W ⊗ O_{S}.}

(3) For V, W ∈ Rep_{k}G, the following diagram commutes

[figure: V^{H} ⊗ W^{H} ⊗ O_{S}; α_{V} ⊗α_{W}; can; V ⊗ W ⊗ O_{S}; (V ⊗ W)^{H} ⊗ O_{S}; α_{V ⊗W}; V ⊗ W ⊗ O_{S}.]

Proof. We define the maps Φ and Ψ as in Proposition 2.3; the proof then proceeds exactly as before. Definition of Ψ:

Given {α_{V} }_{V} satisfying the properties (1-3), we first extend the collection to include all locally finite representations V of G, such that (1-3) continue to hold. Taking V to be the right regular representation k[G], we get

α : k[G]^{H} ⊗ O_{S} → k[G] ⊗ O_{S},

which by the same method as Proposition 2.3 is seen to be a morphism of quasi-coherent O_{S}- algebras. Using the anti-equivalence of categories

{q.c sheaves of O_{S}-algebras} ⟷ {schemes X, affine over S},

α determines a map of schemes

ϕ = α^{∗} : G × S → \overline{G/H} × S.

By the same argument as in Lemma 2.5, this is G × S-equivariant. Now we set Ψ({α_{V} }) = ϕ(e) ∈ \overline{G/H}(S), where e ∈ (G × S)(S) is the identity element of the S-points of the S-group scheme G × S. Definition of Φ:

A point x ∈ \overline{G/H}(S) determines for every S-scheme S′ → S an element x(S′) ∈ \overline{G/H}(S′) and hence a (G × S)(S′)-equivariant map

ϕ_{x(S′)} : (G × S)(S′) → (\overline{G/H} × S)(S′)

taking e(S′) to x(S′). The family {ϕ_{x(S′)}} determines G × S-equivariant morphism of S-schemes

ϕ_{x} : G × S → \overline{G/H} × S

taking ′′e′′ to ′′x′′. By the anti-equivalence of categories above this determines a morphism of quasi-coherent O_{S}-algebras

α_{x} = ϕ_{x} : k[G]^{H} ⊗ O_{S} → k[G] ⊗ O_{S}.

We would like to define Φ(x) = {α_{V} }_{V} by α_{V} = (id_{V} ⊗ α_{x})|_{V^{H}⊗O_{S}} where restriction is taken relative to the inclusion a_{V}^{∗} : V^{H} ⊗ O_{S} → V ⊗ k[G]^{H} ⊗ O_{S}. But first we must prove the analogue of Lemma 2.2.

Lemma 2.7. The map

(id_{V} ⊗ α_{x}) ∘ a_{V}^{∗} : V^{H} ⊗ O_{S} → V ⊗ k[G] ⊗ O_{S}

factors uniquely through a_{V}^{∗} : V ⊗ O_{S} → V ⊗ k[G] ⊗ O_{S}.

Proof. We first observe that the case H = e is straightforward: this case is simply the S-scheme analogue of §2.2. We will use this case below to prove the general case.

By passing to an open cover of S, it is enough to check this for S = Spec(R), an affine k-scheme.

Let us denote

X(R) = {collections {α_{V} : V^{H} ⊗ R → V ⊗ R}_{V} over R, satisfying (1-3)}

and X′(R) = {collections {α_{V} }_{V} ∈ X(R) such that the desired factoring exists}.

These make sense for any R-algebra R′ in place of R; we denote by X and X′ the resulting Rschemes. Thus X′ is a subscheme of X. By §2.2, there is an obvious map Φ′ : Spec(k[G]) → X′ defined by the composition of Tannaka’s map with restriction map which takes a collection {α_{V} : V ⊗ R → V ⊗ R} to {α_{V} : V^{H} ⊗ R → V ⊗ R}. Write A = k[G]^{H} ↪ k[G] = B. Consider the commutative diagram

\xymatrix{Spec(B) \ar[r]^{Φ′} \ar[d]_{ξ} & X′ \ar[d] \\ Spec(A) \ar[r]^{Φ} & X.}

By the same argument as in Lemma 2.2, we see

Φ^{−1}(X′) is a closed subscheme of Spec(A).

Therefore we can write

Φ^{−1}(X′) = Spec(A/I)

for some ideal I ⊂ A. Since the image of Spec(B) under ξ lies in Spec(A/I), the homomorphism ξ^{∗} : A → B factors through A/I. But since ξ^{∗} is inclusion, this means that I = 0, and hence Φ takes all of Spec(A) into X′. This proves that the desired factoring exists, whence the lemma. □

Now for x ∈ \overline{G/H}(S) we define Φ(x) = \{(id_{V} ⊗ α_{x})|_{V^{H}⊗O_{S}}\}_{V} as above.

It is straightforward to check that Ψ and Φ are mutually inverse. This proves the proposition.

□

3. Frobenius reciprocity

Let G be a connected reductive group over k. Suppose P is a parabolic subgroup and H ◁ P is a normal subgroup. For U ∈ Rep_{k}P/H, we define the associated induced module:

Definition 3.1.

ind_{P}^{G}U = {f : G → U | f(gp) = p^{−1}f(g), ∀g ∈ G, p ∈ P }.

This is an object of Rep_{k}G where G acts by (g · f)(g′) = f(g^{−1}g′). Note that we have dim_{k}(ind_{P}^{G}U) < ∞, because there is a finite-rank locally free O_{G/P} -module L(U) associated to U, for which we have

ind_{P}^{G}U = H^{0}(G/P, L(U)).

For V ∈ Rep_{k}G, U ∈ Rep_{k}P, we have the usual Frobenius reciprocity:

Hom_{P} (V, U) = Hom_{G}(V, ind_{P}^{G}U),

where V is regarded as an object of Rep_{k}P on the left hand side.

For V ∈ Rep_{k}G, let V^{∗} denote the contragredient representation.

Definition 3.2. For U ∈ Rep_{k}P/H, define the associated Weyl module by

V (U) = (ind_{P}^{G}U^{∗})^{∗}.

The usual Frobenius reciprocity implies (by dualizing) the following version which we shall use.

Lemma 3.3. For V ∈ Rep_{k}G and U ∈ Rep_{k}P/H, we have a natural isomorphism

Hom_{G}(V (U), V) = Hom_{P/H}(U, V^{H}).

Consequently, we have the adjuction maps

ϵ_{U} : U → V (U)^{H} (P/H − linear)
δ_{V} : V (V^{H}) → V (G − linear).
4. Describing \overline{G/H} with Rep_{k}P/H

Let G continue to denote a connected reductive group over k.

For U_{1}, U_{2} ∈ Rep_{k}P/H, there is a canonical G-morphism

V (U_{1} ⊗ U_{2}) → V (U_{1}) ⊗ V (U_{2}),

defined to be the dual of the natural morphism

ind_{P}^{G}U_{1}^{∗} ⊗ ind_{P}^{G}U_{2}^{∗} → ind_{P}^{G}(U_{1}^{∗} ⊗ U_{2}^{∗}).

Note that since G/P is connected and complete, we have a canonical isomorphism V (𝕀) = 𝕀 (we denote the trivial representations of both G and P/H by 𝕀).

In this section we will consider collections

{β_{U}}_{U∈Rep_{k}P/H}

consisting of k-linear maps β_{U} : U → V (U) satisfying the following conditions:

(1) β_{𝕀} = id,

(2) For a P/H-morphism U_{1} → U_{2}, the following diagram commutes

\xymatrix{U_{1} \ar[r]^{β_{U_{1}}} \ar[d] & V (U_{1}) \ar[d] \\ U_{2} \ar[r]^{β_{U_{2}}} & V (U_{2}).}

(3) For U_{1}, U_{2} ∈ Rep_{k}P/H, the following diagram commutes

[figure: U_{1} ⊗ U_{2}; β_{U_{1}⊗U_{2}}; β_{U_{1}}⊗β_{U_{2}}; V (U_{1} ⊗ U_{2}); can; V (U_{1}) ⊗ V (U_{2}); V (U_{1}) ⊗ V (U_{2}).]

We call these conditions (1-3) tensor conditions. (We will use the same terminology for the previous conditions (1-3) imposed on a collection {α_{V}}_{V}.)

We will say an arbitrary collection of k-linear maps {β_{U}}_{U∈Rep_{k}P/H} is compatible with P/Hmorphisms provided it satisfies condition (2) above. Similarly, we say an arbitrary collection {α_{V}}_{V∈Rep_{k}G} is compatible with G-morphisms if it satisfies the corresponding property (2) from Proposition 2.3.

Now suppose {α_{V}}_{V∈Rep_{k}G} is any collection of k-linear maps α_{V} : V^{H} → V. We construct a collection {β_{U}}_{U∈Rep_{k}P/H} of k-linear maps β_{U} : U → V (U) by setting

β_{U} = α_{V (U)} ∘ ϵ_{U}.

Conversely, given {β_{U}}_{U∈Rep_{k}P/H}, we construct {α_{V}}_{V∈Rep_{k}G} by setting

α_{V} = δ_{V} ∘ β_{V^{H}}.

We have the following lemma.

Lemma 4.1. (a) The association

{α_{V}}_{V∈Rep_{k}G} ⟷ {β_{U}}_{U∈Rep_{k}P/H}.

described above gives a bijection between the set of collections compatible with G-morphisms and the set of collections compatible with P/H-morphisms.

(b) Moreover, {α_{V}}_{V∈Rep_{k}G} satisfies the tensor conditions if and only if the corresponding collection {β_{U}}_{U∈Rep_{k}P/H} does.

Proof. Part (a) follows easily using the following identities, which follow by adjunction:

δ_{V}|_{V (V^{H})^{H}} ∘ ϵ_{V^{H}} = id_{V^{H}}
δ_{V (U)} ∘ V (ϵ_{U}) = id_{V (U)}.

Part (b) is a straightforward diagram-chase, which we omit.

□

It is clear that this discussion goes over without change over an arbitrary k-scheme S. Therefore we have the following analogue of Proposition 2.6.

Proposition 4.2. There is a bijection (functorial in S) between the set \overline{G/H}(S) and the set of all collections

{β_{U} : U ⊗ O_{S} → V (U) ⊗ O_{S}}_{U∈Rep_{k}P/H}

consisting of O_{S}-linear morphisms satisfying the following tensor conditions:

(1) β_{𝕀} = id_{O_{S}}, (2) For a P/H-morphism U_{1} → U_{2}, the following diagram commutes:

\xymatrix{U_{1} ⊗ O_{S} \ar[r]^{β_{U_{1}}} \ar[d] & V (U_{1}) ⊗ O_{S} \ar[d] \\ U_{2} ⊗ O_{S} \ar[r]^{β_{U_{2}}} & V (U_{2}) ⊗ O_{S}.}

(3) For U_{1}, U_{2} ∈ Rep_{k}P/H, the following diagram commutes

[figure: U_{1} ⊗ U_{2} ⊗ O_{S}; β_{U_{1}⊗U_{2}}; β_{U_{1}}⊗β_{U_{2}}; V (U_{1} ⊗ U_{2}) ⊗ O_{S}; can; V (U_{1}) ⊗ V (U_{2}) ⊗ O_{S}; V (U_{1}) ⊗ V (U_{2}) ⊗ O_{S}.]

5. Plücker description of G/U

5.1. Strongly quasi-affine quotients. We say the quotient G/H is strongly quasi-affine if the canonical map

G/H → \overline{G/H}

is an open immersion.

We will make use of the following theorem of F. Grosshans [6], Thm. 4.3.

Theorem 5.1 (Grosshans). If U(P) is the unipotent radical of a parabolic subgroup P ⊂ G, then G/U(P) is strongly quasi-affine.

5.2. Tannakian description of G/U(P). It is easy to see which collections {α_{V}} correspond to elements in the open subset G/U(P) of \overline{G/U(P)}.

Lemma 5.2. In the Tannakian description for \overline{G/U(P)}, a collection \{α_{V}\} corresponds to an element in G/U(P) if and only if α_{V} is injective for every V ∈ Rep_{k}G.

Proof. Suppose x ∈ \overline{G/U(P)} corresponds to the collection \{α_{V}\}. Taking into account that each comorphism a_{V}^{∗} : V → V ⊗ k[G] is injective, Theorem 5.1 yields the following equivalences:

α_{V} is injective ∀V ∈ Rep_{k}G ⇔ α : k[G]^{U(P)} → k[G] is injective
⇔ ϕ_{x} = α^{∗} : G → \overline{G/U(P)} is dominant
⇔ im(ϕ_{x}) = G/U(P)
⇔ x ∈ G/U(P).

□

5.3. The Weyl modules V (λ). Let B = T U denote a Borel subgroup of G, where T is a maximal torus, and U = U(B). We denote by X^{∗}(T) the lattice of characters of T, and by X^{+} the cone of B-dominant characters in X^{∗}(T). We will use the following notation: For each λ ∈ X^{∗}(T), we let k_{λ} denote the 1-dimensional representation of B/U = T where T acts via by the character λ. When confusion is not possible, we denote k_{λ} simply by λ. We write

V (λ) = (ind_{B}^{G}(−λ))^{∗};

this representation is called the Weyl module for λ. It is known that

V (λ) ≠ 0 ⇔ λ ∈ X^{+}.

We suppose now that λ ∈ X^{+}. Then V (λ) ∈ Rep_{k}G enjoys the following properties (cf. [9], I.2.13).

• V (λ) possesses a B-highest weight vector v_{λ}, which generates a B-stable line 𝕃(λ) ⊂ V (λ).

• The G-module V (λ) is generated by 𝕃(λ).

• Any G-module generated by a B-stable line is a quotient of V (λ). In general V (λ) is not irreducible (although this holds if char(k) = 0). Let V_{λ} denote the

irreducible G-module with highest weight λ; by the above remarks, this is a quotient of V (λ). We have an exact sequence

0 → rad_{G}V (λ) → V (λ) → V_{λ} → 0,

where rad_{G}V (λ) is the intersection of all maximal proper G-submodules in V (λ).

We note also the following fact.

Lemma 5.3. For any weights λ, μ ∈ X^{+}, the canonical map

V (λ + μ) → V (λ) ⊗ V (μ)

is injective.

Proof. The dual map

H^{0}(G/B, L(−λ)) ⊗ H^{0}(G/B, L(−μ)) → H^{0}(G/B, L(−λ − μ))

(induced by cup-product) is surjective (cf. [9], II.14.20). □

5.4. Plücker description for G/U. Using Frobenius reciprocity as in Lemma 4.1, it is not hard to translate a collection {α_{V}}_{V∈Rep_{k}G} giving a point in G/U into a corresponding collection {β_{U}}_{U∈Rep_{k}B/U}. When we do so, the condition

α_{V} is injective ∀V ∈ Rep_{k}G

is converted into the condition

β_{V^{U}}^{−1}(ker(δ_{V})) = 0 ∀V ∈ Rep_{k}G.

Next we want to translate the collection {β_{U}}_{U∈Rep_{k}B/U} into an even simpler collection {β_{λ}}_{λ∈X^{+}}. The category Rep_{k}B/U is semi-simple, with simple objects k_{λ} (λ ∈ X^{∗}(T)). Furthermore the

functor V (·) : Rep_{k}B/U → Rep_{k}G commutes with direct sums. This means that any collection of k-linear maps

β_{U} : U → V (U),

(U ∈ Rep_{k}B/U), is uniquely determined by the sub-collection

β_{λ} : k_{λ} → V (λ),

(for λ ∈ X^{+}) – here we have used that V (λ) = 0 unless λ ∈ X^{+}. In fact it is not hard to translate the tensor conditions on the collection {β_{U}} into similar tensor conditions on {β_{λ}}.

In the following statement, we note that if λ = 0, we have k_{λ} = 𝕀 and V (λ) = V (0) = 𝕀. Also, by abuse, we write λ + μ for the representation k_{λ+μ} = k_{λ} ⊗ k_{μ}. We get the following Plücker description for G/U.

Proposition 5.4. There is a bijection (functorial in S) between the set (G/U)(S) and the set of all collections

{β_{λ} : λ ⊗ O_{S} → V (λ) ⊗ O_{S}}_{λ∈X^{+}}

of O_{S}-linear maps β_{λ} satisfying the following tensor conditions:

(1) β_{𝕀} = id_{O_{S}},

(2) For each λ, μ ∈ X^{+} the following diagram commutes:

[figure: (λ + μ) ⊗ O_{S}; β_{λ+μ}; β_{λ}⊗β_{μ}; V (λ + μ) ⊗ O_{S}; can; V (λ) ⊗ V (μ) ⊗ O_{S}; V (λ) ⊗ V (μ) ⊗ O_{S}.]

(3) For each λ ∈ X^{+}, we have β_{λ}(k_{λ} ⊗ O_{S}) ⊄ rad_{G}V (λ) ⊗ O_{S}.

Proof. Given a collection {β_{U}} we let {β_{λ}} denote the subcollection indexed by the B/U-modules k_{λ}, for λ ∈ X^{+}. Conversely, suppose we are given {β_{λ}}_{λ∈X^{+}}. We have a canonical decomposition

U = ⊕_{λ∈X^{∗}(T)}Hom_{B/U}(λ, U) ⊗ λ,

and thus

V (U) = ⊕_{λ∈X^{∗}(T)}Hom_{B/U}(λ, U) ⊗ V (λ).

We recall that V (λ) = 0 unless λ ∈ X^{+}, so we may define

β_{U} = ⊕_{λ∈X^{∗}(T)}id ⊗ β_{λ},

where we understand β_{λ} = 0 unless λ ∈ X^{+}.

It is straightforward to verify that this determines a bijection {β_{U}}_{U∈Rep_{k}B/U} ↔ {β_{λ}}_{λ∈X^{+}} which preserves the tensor conditions. Finally, one can check that

β_{V^{U}}^{−1}(ker(δ_{V})) = 0 ∀V ∈ Rep_{k}G ⇔ β_{λ}(k_{λ}) ⊄ rad_{G}V (λ) ∀λ ∈ X^{+}.
□

5.5. Plücker description for G/B. To simplify notation, in this section we will consider only the case S = Spec(k) (this is all we need for our intended applications).

The torus T acts freely on G/U by (t, gU) ↦ gtU, and the quotient of this action is the finite flag variety G/B. This transports to the following action of T on a collection {β_{λ}}_{λ∈X^{+}}:

(t, {β_{λ}}_{λ}) ↦ {λ(t)β_{λ}}_{λ}.

Given a collection {β_{λ}}_{λ∈X^{+}}, we define a collection of lines {L(λ) ⊂ V (λ)}_{λ∈X^{+}} by setting

L(λ) = β_{λ}(k_{λ})

for every λ ∈ X^{+}. Recall the “standard” line 𝕃(λ) ⊂ V (λ) as described in §5.3. We have the following Plücker description for G/B:

Proposition 5.5. The map g ↦ {g𝕃(λ)}_{λ∈x^{+}} determines a natural bijection between the finite flag variety G/B and the set of all collections

{L(λ)}_{λ∈X^{+}}

consisting of lines L(λ) ⊂ V (λ) satisfying the following tensor conditions:

(1) For λ, μ ∈ X^{+}, let i_{λ,μ} : V (λ + μ) ↪ V (λ) ⊗ V (μ) denote the canonical inclusion. Then i_{λ,μ}(L(λ + μ)) = L(λ) ⊗ L(μ).

(2) For λ ∈ X^{+}, we have L(λ) ⊄ rad_{G}V (λ).

Proof. The map F : {β_{λ}} ↦ {L(λ)} defined by L(λ) = β_{λ}(k_{λ}) clearly takes two collections in the same T -orbit to the same collection of lines. To construct an inverse map, note first that a collection of lines {L(λ)} satisfying the given tensor conditions is uniquely determined by the finite collection {L(ω_{i})}_{i}, where ω_{i} ranges over the fundamental weights of X^{+}. For each i, choose a generator v_{i} ∈ L(ω_{i}). We can define β_{ω_{i}} : k_{ω_{i}} → V (ω_{i}) by letting β_{ω_{i}}(γ) = γv_{i}, ∀γ ∈ k_{ω_{i}}. (Here we identify each k_{λ} with the underlying space k.)

Claim 5.6. If we have defined β_{λ} and β_{μ} such that β_{λ}(k) = L(λ) and β_{μ}(k) = L(μ), then there is a unique way to define β_{λ+μ} such that i_{λ,μ} ∘ β_{λ+μ} = β_{λ} ⊗ β_{μ} and β_{λ+μ}(k) = L(λ + μ).

Proof. This follows easily from the injectivity of i_{λ,μ} and the relation

i_{λ,μ}(L(λ + μ)) = L(λ) ⊗ L(μ).
□

By the claim, the choice of {β_{ω_{i}}}_{i} uniquely determines a collection {β_{λ}}_{λ∈X^{+}} which satisfies the tensor conditions and satisfies as well β_{λ}(k) = L(λ) for every λ. The T -orbit of {β_{λ}} depends only on {L(λ)} (i.e., it is independent of the choice of v_{i}), and this defines an inverse map to F.

□

Remark. If char(k) = 0, then the second condition in Proposition 5.5 can be omitted, since V (λ) = V_{λ} is irreducible. The Drinfeld-Plücker relations for G/B announced by Kuznetsov [11] differ from those appearing in Proposition 5.5, in that the tensor conditions here are expressed using the canonical inclusion V_{λ+μ} ↪ V_{λ}⊗V_{μ} and those in [11] use all G-projections V_{λ}⊗V_{μ} → V_{ν}, for any weight ν ⪯ λ + μ. The tensor conditions are in fact equivalent, so the above arguments provide a proof for the statement in [11], and also a generalization to characteristic p > 0. For the sequel we will find it useful to slightly reformulate the tensor conditions in Proposition 5.5 in a way that looks more like [11]. This yields the proposition below.

Proposition 5.7. Assume char(k) = 0. Then g ↦ {g𝕃(λ)}_{λ∈X^{+}} gives a bijection between the flag variety G/B and the set of collections

{L(λ)}_{λ∈X^{+}}

of lines L(λ) ⊂ V (λ) which satisfy the following tensor condition:

Let λ, μ, ν ∈ X^{+}. Then for every G-inclusion ϕ : V (ν) ↪ V (λ) ⊗ V (μ) we have

ϕ^{−1}(L(λ) ⊗ L(μ)) = \begin{cases}0, & if ν ≺ λ + μ, \\ L(λ + μ) & if ν = λ + μ. \end{cases}

6. Affine Grassmannians

6.1. Notation. For this section and the rest of this paper, we assume char(k) = 0. (The case of characteristic p > 0 will be considered in a subsequent version of this work.)

We assume G is a connected reductive group over k. Suppose its based root system is (X^{∗}, X_{∗}R, Ř, Π). Choose a maximal torus T and a Borel subgroup B = T U and identify X^{∗} with X^{∗}(T) and Π with the B-positive simple roots in R. Let Q̌ denote the subgroup of X_{∗} generated by the coroots Ř. Let

⟨ , ⟩ : X^{∗} × X_{∗} → ℤ

denote the canonical pairing.

At some points below, we shall assume the derived group G^{der} is simply-connected.

We will use the following notation for the remainder of this paper:

- μ̌ will always denote a dominant coweight for T, and λ will always denote a dominant weight for T.

- Given μ̌ (resp. λ), we will denote by Ω(μ̌) (resp. Ω(λ)) the set of coweights (resp. weights) in the Weyl module associated to μ̌ (resp. λ).

- λ̌ (resp. λ′) will always denote an element of Ω(μ̌) (resp. Ω(λ)).

- for x, y ∈ X_{∗} (resp. X^{∗}) the relation x ⪯ y will mean that y − x is a sum of positive coroots (resp. roots).

- We will denote the extended affine Weyl group by W̃ = X_{∗} ⋊ W, where W denotes the finite Weyl group. We let w_{0} denote the longest element in W.

- We fix a parameter t and let O_{t} = k[[t]] and K_{t} = k((t)).

- We denote translation elements in W̃ by t_{x} (x ∈ X_{∗}). We identify t_{x} with an element of T (K_{t}) ⊂ G(K_{t}) by the rule:

t_{x} = x(t^{−1})

where x is viewed as a cocharacter x ∈ X_{∗}(T). Thus if v_{λ′} is vector with weight λ′ in some G-module, then

t_{x} · v_{λ′} = t^{−⟨λ′,x⟩}v_{λ′}.

- For any V ∈ Rep_{k}G denote V^{0} = V ⊗ O_{t}. Thus V^{0} is the “standard” O_{t}-lattice in V ⊗ K_{t}.

6.2. Plücker description for affine Grassmannians. We denote the affine Grassmannian over Spec(O_{t}) by Q. In the forgoing it is possible to describe Q as an ind-scheme, giving the points Q(R) for every k-algebra R. To illustrate just the main ideas, we will only describe the k-points

Q(k) = G(K_{t})/G(O_{t}).

Proposition 6.1. The map g ↦ {gV (λ)^{0}}_{λ∈X^{+}} defines a bijection between G(K_{t})/G(O_{t}) and the set of collections

{L(λ)}_{λ∈X^{+}}

consisting of O_{t}-lattices L(λ) ⊂ V (λ) ⊗ K_{t} satsifying the following tensor condition: Let λ, μ, ν ∈ X^{+}. Then for every G-inclusion ϕ : V (ν) ↪ V (λ) ⊗ V (μ) we have

ϕ^{−1}(L(λ) ⊗ L(μ)) = L(ν),

where ϕ is also used to denote the induced map

ϕ ⊗ id : V (ν) ⊗ K_{t} → V (λ) ⊗ V (μ) ⊗ K_{t}.

Proof. The injectivity follows from

∩_{λ}Stab_{G(K_{t})}V (λ)^{0} = G(O_{t}),

which is a consequence of the Cartan decomposition

G(K_{t}) = ∐_{μ̌} G(O_{t})tμ̌G(O_{t}).

It remains to prove the surjectivity. Since the category Rep_{k}G is semi-simple, every G-module V has a canonical decomposition

V = ⊕_{λ}Hom_{G}(V (λ), V) ⊗ V (λ),

where G acts trivially on the factor Hom_{G}(V (λ), V). Given the data {L(λ)}_{λ∈X^{+}}, we define for each V ∈ Rep_{k}G an O_{t}-lattice in V ⊗ K_{t} by

L(V) = ⊕_{λ}Hom_{G}(V (λ), V) ⊗ L(λ).

Let ProjMod(O_{t}) denote the category of finite-type projective O_{t}-modules. It is straightforward to check that

L : Rep_{k}G → ProjMod(O_{t})

is a faithful, exact, ⊗-functor (i.e., it is a fiber functor). Therefore L comes from a unique Gtorsor 𝕃 over Spec(O_{t}), see [3]. Moreover, the isomorphisms L(λ) ⊗ K_{t} ≅ V (λ) ⊗ K_{t} induced by the inclusions L(λ) ↪ V (λ) ⊗ K_{t} determine a trivialization of 𝕃 over the generic fiber Spec(K_{t}). Now by the theorem of Beauville-Laszlo [1] there is a natural correspondence

G(K_{t})/G(O_{t}) ⟷ {G-torsors over Spec(O_{t}) with trivialization on Spec(K_{t})}.

Thus we see 𝕃 and hence the collection {L(λ)}_{λ} comes from a well-defined coset gG(O_{t}) ∈ G(K_{t})/G(O_{t}). This completes the proof.

□

6.3. Plücker description for \overline{Q}μ̌. By the Cartan decomposition, the G(O_{t})-orbits on Q are indexed by the dominant coweights μ̌. Let \overline{Q}μ̌ denote the closure of the orbit

Qμ̌ = G(O_{t})tμ̌G(O_{t})/G(O_{t})

corresponding to μ̌. Our description of \overline{Q}μ̌ is based on the following lemmas, whose proofs are straightforward.

Lemma 6.2. Suppose λ̌ and μ̌ are two dominant coweights, and μ̌ − λ̌ ∈ Q̌. Then the following conditions are equivalent:

(a) λ̌ ⪯ μ̌.

(b) ⟨λ, λ̌⟩ ≤ ⟨λ, μ̌⟩, for every λ ∈ X^{+}.

(c) tλ̌V (λ)^{0} ⊂ t^{−⟨λ,μ̌⟩}V (λ)^{0}, for every λ ∈ X^{+}.

(d) t^{−⟨w_{0}λ,μ̌⟩}V (λ)^{0} ⊂ tλ̌V (λ)^{0} ⊂ t^{−⟨λ,μ̌⟩}V (λ)^{0}, for every λ ∈ X^{+}.

For the next lemma, let us denote d_{λ} = dim_{k}V (λ), for λ ∈ X^{+}.

Lemma 6.3. Suppose G^{der} is simply connected (i.e. X_{∗}/Q̌ is torsion-free). Then the following statements are equivalent for any dominant coweights λ̌, μ̌ ∈ X_{∗}:

(a) μ̌ − λ̌ ∈ Q̌.

(b) ∧^{d_{λ}}tλ̌V (λ)^{0} = ∧^{d_{λ}}tμ̌V (λ)^{0}, for every λ ∈ X^{+}.

The closure relations in the affine Grassmannian are given by the partial order on X_{∗}:

Qλ̌ ⊂ \overline{Q}μ̌ ⟺ λ̌ ⪯ μ̌.

Hence from Lemmas 6.2, 6.3 and Proposition 6.1 we derive the following Plücker description for \overline{Q}μ̌:

Proposition 6.4. Suppose G^{der} = G^{sc}. Then there is a natural bijection between \overline{Q}μ̌ and the set of collections

{L(λ)}_{λ∈X^{+}}

consisting of O_{t}-lattices L(λ) ⊂ V (λ) ⊗ K_{t} which satisfy the following properties:

(1) Let ν, λ, μ ∈ X^{+}. For every G-injection ϕ : V (ν) → V (λ) ⊗ V (μ), we have ϕ^{−1}(L(λ) ⊗ L(μ)) = L(ν).

(2) ∧^{d_{λ}}L(λ) = ∧^{d_{λ}}tμ̌V (λ)^{0}, for every λ ∈ X^{+}.

(3) t^{−⟨w_{0}λ,μ̌⟩}V (λ)^{0} ⊂ L(λ) ⊂ t^{−⟨λ,μ̌⟩}V (λ)^{0} , for every λ ∈ X^{+}.

6.4. Ind-scheme structure on Q.

Assume G^{der} = G^{sc}.

As mentioned above, we can easily define \overline{Q}μ̌ as a k-scheme, using a similar definition as that given above. Let ω_{i} denote the fundamental weights for G. It is easy to see that a collection {L(λ)}_{λ∈X^{+}} is uniquely determined by the finitely many lattices in the subcollection {L(ω_{i})}_{i}. In fact, the forgetful functor

{L(λ)}_{λ} ↦ {L(ω_{i})}_{i}

defines a closed immersion of \overline{Q}μ̌ into a finite-dimensional projective k-scheme.

By bounding the relative positions of the finitely-many lattices L(ω_{i}), we see as well that every element in Q is contained in some \overline{Q}μ̌. Hence we have the following ind-scheme structure on Q:

Proposition 6.5. The equality

Q = ∪μ̌\overline{Q}μ̌

realizes the affine Grassmannian as an inductive limit of finite-dimensional projective schemes over k.

7. Affine flag varieties

7.1. Notation. Let π : G(O_{t}) → G(k) be the homomorphism determined by t ↦ 0. Let I = I_{k} ⊂ G(O_{t}) be the “standard” Iwahori subgroup:

I = π^{−1}(B).

Recall that each Weyl module V (λ) is generated by its “standard” highest-weight line 𝕃(λ) ⊂ V (λ). By definition, this line is the image of the adjunction map ϵ_{λ} : λ → V (λ)^{U} (cf. Lemma 3.3).

Choose a generator v_{λ} ∈ 𝕃(λ). We define the standard two-step lattice chain

V (λ)^{•} = (V (λ)^{0} ⊂ V (λ)^{1} ⊂ t^{−1}V (λ)^{0})

by defining V (λ)^{0} ⊂ V (λ) ⊗ K_{t} as before, and by defining

V (λ)^{1} = t^{−1}O_{t}v_{λ} ⊕ (⊕_{λ′≺λ}O_{t}v_{λ′}).

Here the summand on the right denotes the O_{t}-span of all weight vectors in V (λ) with weight less than λ.

There is a natural isomorphism

t^{−1}V (λ)^{0}/V (λ)^{0} = V (λ)

under which we have the identification

V (λ)^{1}/V (λ)^{0} = 𝕃(λ).

Furthermore, if λ, μ ∈ X^{+}, then there is a natural isomorphism

t^{−1}(V (λ)^{0} ⊗ V (μ)^{0})/V (λ)^{0} ⊗ V (μ)^{0} = V (λ) ⊗ V (μ)

under which we have the identification

[(V (λ)^{0} ⊗ V (μ)^{0}) + t(V (λ)^{1} ⊗ V (μ)^{1})]/V (λ)^{0} ⊗ V (μ)^{0} = 𝕃(λ) ⊗ 𝕃(μ).

7.2. Plücker description for affine flag varieties. We denote the affine flag variety by Fl. As for affine Grassmannians, we can describe it as an ind-scheme, but for simplicity we describe here only its k-points

Fl(k) = G(k((t)))/I.

Proposition 7.1. The map g ↦ {gV (λ)^{•}}_{λ} determines a natural bijection between G(k((t)))/I and the set of collections {L(λ)^{•}}_{λ∈X^{+}} consisting of two-step lattice chains

L(λ)^{•} = (L(λ)^{0} ⊂ L(λ)^{1} ⊂ t^{−1}L(λ)^{0})

in V (λ) ⊗ K_{t} which satisfy the following properties:

(1) dim_{k}L(λ)^{1}/L(λ)^{0} = 1.

(2) Let ν, λ, μ ∈ X^{+}. For every G-inclusion ϕ : V (ν) → V (λ) ⊗ V (μ), we have:

ϕ^{−1}(L(λ)^{0} ⊗ L(μ)^{0}) = L(ν)^{0}.

(3) For ϕ as above, we have

ϕ^{−1}[L(λ)^{0} ⊗ L(μ)^{0} + t(L(λ)^{1} ⊗ L(μ)^{1})] = \begin{cases}L(ν)^{0}, & if ν ≺ λ + μ, \\ L(ν)^{1}, & if ν = λ + μ. \end{cases}

Proof. The map is injective, because

I = ∩_{λ}Stab_{G(K_{t})}(V (λ)^{•}).

To see the surjectivity, it suffices to prove that G(K_{t}) acts transitively on the set of collections of two-step lattice chains. This follows from Proposition 6.1 and the fact that the simultaneous stabilizer G(O_{t}) of the standard lattices V (λ)^{0} acts transitively on the collections of lines in V (λ) which satisfy the conditions in Proposition 5.7. □

7.3. Plücker definition of the subvariety Fl_{≤μ̌}.

For the rest of this section, we assume G^{der} = G^{sc}.

Given a dominant coweight μ̌, we will define here a finite-dimensional projective subscheme Fl_{≤μ̌} ⊂ Fl. This will be a natural analogue of the subscheme \overline{Q}μ̌ ⊂ Q.

Definition 7.2. We denote by Fl_{≤μ̌} the set of collections {L(λ)^{•}}_{λ} ∈ Fl which satisfy the following properties:

(1) ∧^{d_{λ}}L(λ)^{0} = ∧^{d_{λ}}tμ̌V (λ)^{0}, for every λ ∈ X^{+}.

(2) t^{−⟨w_{0}λ,μ̌⟩}V (λ)^{i} ⊂ L(λ)^{i} ⊂ t^{−⟨λ,μ̌⟩}V (λ)^{i}, (i ∈ {0, 1}), for every λ ∈ X^{+}.

7.4. Ind-scheme structure on Fl.

By the same argument as in §6.4, the forgetful functor

{L(λ)^{•}}_{λ∈X^{+}} ↦ {L(ω_{i})^{•}}_{i}

defines a closed immersion of Fl_{≤μ̌} into a finite-dimensional projective k-scheme.

Moreoever, by bounding the relative positions of the finitely-many two-step lattice chains L(ω_{i})^{•}, we see that any element of Fl is contained in some Fl_{≤μ̌}. Hence we have the following ind-scheme structure on Fl:

Proposition 7.3. The equality

Fl = ∪μ̌Fl_{≤μ̌}

realizes the affine flag variety as an inductive limit of finite-dimensional projective schemes over k.

7.5. The stratification of Fl_{≤μ̌}. For w ∈ W̃, let Y (w) = IwI/I ⊂ Fl denote the corresponding Bruhat cell, and let X(w) = \overline{Y (w)} denote the corresponding affine Schubert variety. Recall the the closure relation between Schubert varieties is given by the Bruhat order on W̃:

X(w) ⊂ X(w′) ⟺ w ≤ w′.

The ind-scheme Fl carries an obvious action of the Iwahori subgroup I, and this action preserves the subscheme Fl_{≤μ̌}. Since the latter is finite-dimensional, the I-orbits in Fl_{≤μ̌} are indexed by a finite set

Strat(μ̌) ⊂ W̃.

The following result gives a combinatorial description of this set.

Proposition 7.4. Let x ∈ W̃, and write x = tλ̌w, with λ̌ ∈ X_{∗} and w ∈ W. Then x ∈ Strat(μ̌) if and only if the following properties are satisfied:

(1) λ̌ ∈ Ω(μ̌); equivalently, ⟨w_{0}λ, μ̌⟩ ≤ ⟨λ′, λ̌⟩ ≤ ⟨λ, μ̌⟩, for every λ ∈ X^{+} and every λ′ ∈ Ω(λ).

(2) For every λ ∈ X^{+}, the following condition holds: if wλ ≠ λ then • ⟨λ, w^{−1}λ̌⟩ < ⟨λ, μ̌⟩; and • ⟨w_{0}λ, μ̌⟩ < ⟨λ, λ̌⟩.

The set Strat(μ̌) shares certain properties with the sets Perm(μ̌) and Adm(μ̌) defined by Kottwitz-Rapoport [10]. Some of these are listed below. It is to be hoped that Strat(μ̌) is a better approximation to Adm(μ̌) than was the case for Perm(μ̌) (cf. [8]). In some sense, this hope and the corollary below were the motivation for this approach to local models.

Corollary 7.5. For every dominant coweight μ̌, we have the following properties of Strat(μ̌): (1) Adm(μ̌) ⊂ Strat(μ̌).

(2) Adm(μ̌) ∩ X_{∗} = Strat(μ̌) ∩ X_{∗} = Perm(μ̌) ∩ X_{∗} = Ω(μ̌).

Proof. It is clear from Proposition 7.4 that

Strat(μ̌) ∩ X_{∗} = Ω(μ̌).

It is already known that this set coincides with both Perm(μ̌) ∩ X_{∗} and Adm(μ̌) ∩ X_{∗}. Thus (2) is proved.

Since Fl_{≤μ̌} is a projective scheme, it is a closed subscheme in Fl. Therefore the set Strat(μ̌) is closed with respect to the Bruhat order. But by (2), the translation elements t_{wμ̌} (w ∈ W) lie in Strat(μ̌). Hence all of Adm(μ̌) lies in Strat(μ̌), and (1) is proved. □

Unfortunately, as was the case for Perm(μ̌) [8], the set Strat(μ̌) does not coincide with Adm(μ̌) in general:

Proposition 7.6. Suppose G is not of type A_{n} and is of rank ≥ 4. Then for every regular dominant coweight μ̌, we have

Adm(μ̌) ≠ Strat(μ̌).

Proof. We use Deodhar’s examples, as in [8]. For groups of the given type, there exist w, w′ ∈ W, such that

- w ≠ w′,

- l(w) = l(w′),

- wλ − w′λ is a sum of positive roots, for every λ ∈ X^{+}. Now one can see by Proposition 7.4 and by similar reasoning as in [8], that the element

x = t_{w^{−1}μ̌}w^{−1}w′

belongs to Strat(μ̌) but not Adm(μ̌). □

It is still possible that Adm(μ̌) = Strat(μ̌) whenever μ̌ is a minuscule coweight, or more generally, a sum of minuscule coweights.

8. Rapoport-Zink models

8.1. Notation. Let ϖ be a parameter. Write O_{F} = k[[ϖ]] and F = k((ϖ)). Let R be any O_{F} -algebra. We are going to consider two-step chains of R[t]-modules contained in the module V (λ) ⊗ R[t, t^{−1}, (t + ϖ)^{−1}].

We define the “standard” two-step chain of R[t]-modules in V (λ) ⊗ R[t, t^{−1}, (t + ϖ)^{−1}]

V (λ)_{R}^{•} = (V (λ)_{R}^{0} ⊂ V (λ)_{R}^{1} ⊂ (t + ϖ)^{−1}V (λ)_{R}^{0})

as follows: we define V (λ)_{R}^{0} = V (λ) ⊗ R[t] and

V (λ)_{R}^{1} = (t + ϖ)^{−1}R[t]v_{λ} ⊕ (⊕_{λ′≺λ}R[t]v_{λ′}).

Here the summand on the right denotes the R[t]-span of all weight vectors in V (λ) having weight less than λ.

If ϕ : V → W is any G-morphism in Rep_{k}G, then we denote by

ϕ̃: V ⊗ R[t, t^{−1}, (t + ϖ)^{−1}] → W ⊗ R[t, t^{−1}, (t + ϖ)^{−1}]

the induced map on the tensor product with R[t, t^{−1}, (t + ϖ)^{−1}].

8.2. Plücker definition for Rapoport-Zink models. In this section we propose a general Plücker definition of “Rapoport-Zink” models Mμ̌. The models defined here are the “Plücker analogues” of the ones for GL_{n} and GSp_{2n}, used in [7], but here are defined only in the case of equi-characteristic 0 function fields.

Definition 8.1. For any O_{F} -algebra R, we define the R-points Mμ̌(R) to be the set of all collections {L(λ)^{•}}_{λ∈X^{+}} of two-step chains of R[t]-submodules in V (λ) ⊗ R[t, t^{−1}, (t + ϖ)^{−1}]

L(λ)^{•} = (L(λ)^{0} ⊂ L(λ)^{1} ⊂ (t + ϖ)^{−1}L(λ)^{0})

which satisfy the following properties:

(1) ∧_{R[t]}^{d_{λ}}L(λ)^{i} = ∧_{R[t]}^{d_{λ}}tμ̌V (λ)_{R}^{i}, (i ∈ {0, 1}), for every λ ∈ X^{+}.

(2) t^{−⟨w_{0}λ,μ̌⟩}V (λ)_{R}^{i} ⊂ L(λ)^{i} ⊂ t^{−⟨λ,μ̌⟩}V (λ)_{R}^{i} (i ∈ {0, 1}), for every λ ∈ X^{+}.

(3) t^{−⟨λ,μ̌⟩}V (λ)_{R}^{i}/L(λ)^{i} (i ∈ {0, 1}) is R-locally free, for every λ ∈ X^{+}.

(4) Let ν, λ, μ ∈ X^{+}. Then for every G-inclusion ϕ : V (ν) → V (λ) ⊗ V (μ), we have ϕ̃^{−1}(L(λ)^{0} ⊗ L(μ)^{0}) = L(ν)^{0}.

(5) For ϕ as above, we have

ϕ̃^{−1}[L(λ)^{0} ⊗ L(μ)^{0} + (t + ϖ)(L(λ)^{1} ⊗ L(μ)^{1})] = \begin{cases}L(ν)^{0}, & if ν ≺ λ + μ, \\ L(ν)^{1}, & if ν = λ + μ. \end{cases}

By arguing as in [7], we can easily describe the generic and special fibers for these models. One finds that

Mμ̌(k((ϖ))) = \overline{Q}μ̌(k((ϖ)))
Mμ̌(k) = Fl_{≤μ̌}(k).

The functor R ↦ Mμ̌(R) is represented by a finite-dimensional projective O_{F} -scheme; this is seen as in §6.4,7.4, by considering the forgetful functor

{L(λ)^{•}}_{λ∈X^{+}} ↦ {L(ω_{i})^{•}}_{i}.

We remark that the union ∪μ̌Mμ̌ gives an ind-scheme structure to Beilinson’s deformation of the affine Grassmannian to the affine flag variety, as mentioned in the introduction.

Finally, we point out one of the main reasons that Weyl modules look promising for use in giving the analogue of Definition 8.1 over ℤ_{p} instead of k[[ϖ]]: all that is needed to transfer the definition, nearly word-for-word, to one over ℤ_{p} are a suitable integral structure V (λ)_{ℤ_{p}}^{0} for V (λ) over ℚ_{p}, and also a good notion for V (λ)_{ℤ_{p}}^{1}. But if G is a split reductive group over ℤ_{p}, and if λ ∈ X^{+}, then the choice of a highest weight vector v_{λ} ∈ V (λ)_{ℚ_{p}} (the irreducible Weyl module over ℚ_{p}) determines an integral structure

V (λ)_{ℤ_{p}}^{0} = Dist(G_{ℤ_{p}}) · v_{λ}

for V (λ)_{ℚ_{p}}, where Dist(G_{ℤ_{p}}) denotes the algebra of distributions (cf. [9], II.8.2).

Moreover, this integral structure has the property

V (λ)_{k} = V (λ)_{ℤ_{p}}^{0} ⊗ k,

where the left hand side is the Weyl module attached to λ over k, for every field k which is a ℤ_{p}-algebra. We can therefore set

V (λ)_{ℤ_{p}}^{1} = p^{−1}ℤ_{p}v_{λ} ⊕ (⊕_{λ′≺λ}ℤ_{p}v_{λ′}),

and get the standard two-step chain as above. Then one can use this to define Mμ̌ over Spec(ℤ_{p}). The problem is that the special fiber could be too large: one cannot necessarily describe the affine flag variety or the affine Grassmannian over a characteristic p > 0 field using only Weyl modules (at least not for general groups: it can be done for GL_{n} however).

Thus, to define the models so that the special fiber is correct, it should then be necessary to use more representations than just the Weyl modules in the definition of Mμ̌.

Questions that will be considered in the continuation of this work:

- Is Adm(μ̌) = Strat(μ̌) for every minuscule coweight μ̌? ^{1}

- Let M RZ ; μ̌ denote a Rapoport-Zink local model. There is (at least in examples I have considered) a natural map

M RZ ; μ̌ → Mμ̌.

Is this an isomorphism?

- Define the “Tannakian” models Mμ̌ over a p-adic number ring.

- Are these models flat?

- Use the models Mμ̌ to prove Kottwitz’ conjecture for semi-simple trace of Frobenius on nearby cycles for these models, following [7].

References

[1] A. Beauville, Y. Laszlo, Un lemme de descente, C.R. Acad. Sci. Paris Sér. 1 Math. 320 no. 3, 335-340 (1995).

[2] A. Braverman, D. Gaitsgory, Geometric Eisenstein series, preprint, arXiv:math.AG/9912097 (1 Dec 2000).

[3] P. Deligne, J.S. Milne, Tannakian categories, in Hodge Cycles, Motives, and Shimura varieties, LNM 900, Springer Verlag, 101-228 (1982).

[4] M. Finkelberg, I. Mirkovic, Semiinfinite flags. I. Case of global curve ℙ^{1}., preprint, arXiv:alg-geom/9707010 (25 Mar 1998).

[5] D. Gaitsgory, Construction of central elements in the affine Hecke algebra via nearby cycles, Inv. Math. 144, 253-280 (2001).

[6] F. Grosshans, Algebraic homogeneous spaces and invariant theory, LNM 1673, Springer (1997).

[7] T. Haines, B.C. Ngô, Nearby cycles for local models of some Shimura varieties, preprint (1999), to appear Comp. Math.

[8] T. Haines, B.C. Ngô, Alcoves associated to special fibers of local models, preprint (2001), to appear Amer. J. Math.

[9] J. C. Jantzen, Representations of algebraic groups, Pure and applied mathematics 131, Academic Press (1987).

[10] R. Kottwitz, M. Rapoport, Minuscule alcoves for Gl_{n} and GSp_{2n}, manuscripta math. 102, 403-428 (2000).

[11] A. Kuznetsov, Laumon’s resolution of Drinfeld’s compactification is small, Math. Res. Letters 4, 349-364 (1997).

1Added in August 2003: I have found that in fact this equality is almost never true. Therefore the set Strat(μ) and the model giving rise to it are not the correct ones to consider...

University of Maryland Department of Mathematics College Park, MD 20742-4015 tjh@math.umd.edu