Normality and Cohen-Macaulayness of parahoric local models

We study the singularities of parahoric local models and hence of the corresponding integral models of Shimura varieties and moduli stacks of shtukas. We show the normality and, if p>2, also the Cohen-Macaulayness of all Pappas-Zhu and Levin mixed characteristic local models, and of all equal characteristic local models attached to groups whose absolutely simple factors are tamely ramified. This proves a conjecture of Pappas and Zhu. Along the way, we prove that all Schubert varieties for such groups are normal and all local models have reduced special fiber, and that both Schubert varieties and local models are invariant under passing to the adjoint group.


Introduction
Parahoric local models are flat projective schemes over a discrete valuation ring which are designed to model theétale local structure in the bad reduction of Shimura varieties [KP18] or moduli stacks of shtukas [AH19] with parahoric level structure. Thus, local models provide a tool to study the singularities appearing in the reduction of these spaces. The subject dates back to early 1990's, starting with the work Deligne-Pappas [DP94], Chai-Norman [CN90,CN92] and de Jong [dJ93], and was formalized in the book of Rapoport-Zink [RZ96]. For further details and references, we refer to the survey of Pappas, Rapoport and Smithling [PRS13].
The simplest example is the case of the classical modular curve X 0 (p) with Γ 0 (p)-level structure. In this case, the local model is a P 1 Zp blown up in the origin of the special fiber P 1 Fp . This models the reduction modulo p of X 0 (p), which is visualized as the famous picture of two irreducible components crossing transversally at the supersingular points.
In a breakthrough, a group theoretic construction for parahoric local models in mixed characteristic was given by Pappas-Zhu [PZ13]: fix a local model triple (G, {µ}, K) where G is a reductive group over Q p , K ⊂ G(Q p ) is a parahoric subgroup and {µ} is a (not necessarily minuscule) geometric conjugacy class of one-parameter subgroups in G. Under a tamely ramified hypothesis on G which was further relaxed by Levin [Lev16], Pappas and Zhu construct a flat projective Z p -scheme M {µ} = M (G,{µ},K) called the (parahoric) local model. Assume now that p does not divide the order of π 1 (G der ), e.g., the case G = PGL n and p | n is excluded. Then it is shown in [PZ13] (cf. also [Lev16]) that M {µ} is normal with reduced special fiber. Further, they conjecture that M {µ} is Cohen-Macaulay, cf. [PZ13,Rmk. 9.5 (b)]. This conjecture was proven in [PZ13,Cor. 9.4] when K where A(G, {µ}) ⊂ F G ,f ⊗ kk denotes the union of the (f , f )-Schubert varieties indexed by the {µ}-admissible set of Kottwitz-Rapoport.
The following theorem proves results on the geometry of local models under weaker hypotheses than the hypothesis p |π 1 (G der )| in [PZ13,Thm. 9.3] and [Lev16,Thm. 4.3.2]. We recover as very special cases the results of [PZ13] in this direction and those of [He13], which treats unramified groups, Iwahori level, and minuscule {µ} 1 .
Theorem 2.1. Let (G, {µ}, G f ) be as above, and assume that Gr ≤{µ} G,F is normal and that all maximal (f , f )-Schubert varieties inside A(G, {µ}) are normal (see Remark 2.2 ii ) below for situations when this is satisfied ).
i) The special fiber M {µ} is geometrically reduced. More precisely, as closed subschemes of F G ,f ⊗ kk , one has M {µ} ⊗ k Ek = A(G, {µ}). Remark 2.2. i) In view of the results in [KP18] (resp. [AH19,§3]) the corresponding integral models of Shimura varieties (resp. moduli stacks of shtukas) with parahoric level structure are normal and Cohen-Macaulay as well.
Thus, Theorem 2.1 i) gives new cases of normal local models with reduced special fiber. The proof follows the original argument of Pappas-Zhu, using as a key input the Coherence Conjecture proved by Zhu [Zhu14] (cf. also [PZ13,§9.2.2, (9.19)]). The application of the Coherence Conjecture is justified by our assumption on the normality of Schubert varieties inside A(G, {µ}).
For ii), the normality of M {µ} is an immediate consequence of i) by [PZ13, Prop. 9.2]. As mentioned above, the Cohen-Macaulayness of M {µ} can be deduced from Proposition 5.5 below combined with the well-known theorem of Zhu [Zhu14,Thm. 6.5] which is also the key to the Coherence Conjecture. In particular, our method avoids using any finer geometric structure of the admissible locus A(G, {µ}) as for example in [Go01, §4.5.1] or [He13].
map of local model triples as above which induces G ad G ad on adjoint groups. Then the induced map on local models is a finite, birational universal homeomorphism where E /F denotes the reflex field of {µ }, naturally an overfield of E. In particular, if M (G,{µ},G f ) is normal (see Theorem 2.1), this map is an isomorphism.

Preliminaries on Schubert varieties
We introduce temporary notation for use within §3. Let k be an algebraically closed field, and let F = k((t)) denote the Laurent series field. Let G be a connected reductive F -group. Let f , f ⊂ B(G, F ) be facets of the Bruhat-Tits building. Let G f (resp. G f ) be the associated parahoric O F -group scheme. The loop group LG (resp. L + G f ) is the functor on the category of k-algebras R defined by LG(R) = G(R((t))) (resp. L + G f (R) = G f (R[[t]])). Then L + G f ⊂ LG is a subgroup functor, and the twisted affine flag variety is theétale quotient Let G sc → G der ⊂ G denote the simply connected cover of the derived group. Then G sc j∈J Res Fj /F (G j ) where J is a finite index set, each F j /F is a finite separable field extension, and each G j is an absolutely almost simple, simply connected, reductive F j -group. Under the induced map on buildings B(G sc , F ) → B(G, F ) the facets correspond bijectively to each other, and we denote by f j ⊂ B(G j , F j ) = B(Res Fj /F (G j ), F ) (cf. [HRb,Prop. 4.6]) the factor corresponding to j ∈ J of the facet f ⊂ B(G, F ).
Proposition 3.1. Let S w = S w (f , f ) be any Schubert variety.
i) The normalizationS w → S w is a finite birational universal homeomorphism. Further,S w is also Cohen-Macaulay, with only rational singularities, and Frobenius split if char(k) > 0.
ii) For each j ∈ J, there exists an alcove a j ⊂ B(G j , F j ) containing f j in its closure and an element w j ∈ LG j (k) together with an isomorphism of k-schemes where eachS wj is the normalization of a (a j , f j )-Schubert variety S wj = S wj (a j , f j ) ⊂ F Gj ,fj . Further, ifS w = S w is normal, then eachS wj = S wj is normal as well.
We first reduce the proof of Proposition 3.1 to the case where f = a is an alcove, and where The proof of part i) is given in §3.3, and of part ii) in §3.6 below.
3.1. Alcove reduction. By [BT72, Thm 7.4.18 (i)], there exists a maximal F -split torus S ⊂ G such that f , f are both contained in the apartment A (G, S, F ). The centralizer T = C G (S) is a maximal torus (because G is quasi-split over F by Steinberg's theorem), and we denote by N = N G (S) the normalizer. LetW = N (F )/T (O F ) be the Iwahori-Weyl group (or extended affine Weyl group) where T denotes the neutral component of the lft Néron model of T . We also denote Step 1. AsW acts transitively on the set of alcoves in A (G, S, F ), there exists an n ∈ N (F ) such that n · f and f are contained in the closure of the same alcove. Further, left multiplication by n on F G,f induces an isomorphism where we use that S n·w (n · f , f ) denotes the reduced orbit closure of n · w ∈ LG(k) under the group L + G n·f = nL + G f n −1 . Hence, we may assume the facets f , f are in the closure of a single alcove a ⊂ A (G, S, F ).

3.1.3.
Step 3. Corresponding to the choice of a, we have the decompositionW = W af Ω where W af = W af (Σ) is the affine Weyl group for theéchelonnage root systemΣ, and Ω = Ω a denotes the stabilizer of a inW , cf. [HR08, Prop 12 ff.]. ThusW has the structure of a quasi-Coxeter group.

3.1.4.
Step 4. The class of w inW can be written uniquely in the form τ · w af with w af ∈ W af , τ ∈ Ω. Left multiplication by any liftτ ∈ N (F ) induces an isomorphism of Schubert varieties Sẇ af (a, f ) = S w (a, f ). Hence, we reduce to the case where τ = 1, i.e., w = w af ∈ W af . This means that S w (a, f ) is contained in the neutral component F o G,f ⊂ F G,f (see [PR08,§5]). Corollary 3.2. If the statement of Proposition 3.1 holds for the Schubert varieties S w (a, f ) for all maximal split F -tori S ⊂ G, one choice of alcove a ⊂ A (G, S, F ) containing f in its closure and all w ∈ W af , then Proposition 3.1 holds for the Schubert varieties S w (f , f ) for all facets f , f ⊂ B(G, F ) and all w ∈ LG(k).
Proof. The corollary is immediate from Steps 1-4 above. Note that it is enough to show Proposition 3.1 ii) for a single alcove f = a ⊂ A (G, S, F ) because W af acts (simply) transitively on these alcoves.

3.2.
A lemma on orbits. The following lemma helps to control orbits in partial affine flag varieties.
Lemma 3.3. Let w ∈W be right f -minimal. Then the natural map Proof. In the split case, this was proved in [HRa,Lem. 4.2] using negative loop groups. In the general case the required properties of negative loop groups are not yet available, so we give a different approach. Denote B = L + G a and P = L + G f . Recall that (LG/P )(k) = LG(k)/P (k), cf. e.g. [PR08, Thm. 1.4]. The map ι : BwB/B → BwP/P is clearly surjective on k-points. To see that it is injective on k-points one uses the decomposition P/B = w ∈W f Bw B/B (cf. [Ri13, Rem 2.9]) together with the fact l(w · w ) = l(w) + l(w ) for all w ∈ W f which holds by the right f -minimality. Hence, the map is bijective on k-points, and, using Zariski's Main Theorem, one can show it is enough to prove that ι is smooth. Let w 0 ∈ W f be the longest element. Consider the commutative diagram of k-schemes The inclusion Bw 0 B/B ⊂ P/B is an open immersion. As w is assumed to be right f -minimal, we have l(w · w 0 ) = l(w) + l(w 0 ), and standard properties of partial Demazure resolutions show that π is an open immersion. We claim that both projections pr 1 and pr are smooth and surjective; in that case [StaPro,02K5] shows that the morphism BwB/B → BwP/P is smooth as well. The assertion for both pr and pr 1 results from the following general lemma applied with P = "smooth" (usingétale descent [StaPro,02VL] Lemma 3.4. Let S be a scheme. Let X, Y beétale sheaves on AffSch/S (affine schemes equipped with a map to S), and let G be anétale sheaf of groups over S. Suppose X (resp. Y ) carries a right (resp. left) G-action such that G acts freely on X. Let P be a property of maps of algebraic spaces which is stable under base change, and which isétale local on the target. If Y → S is an algebraic space (resp. and has property P), then the canonical projection pr 1 : X × G Y → X/G is representable (resp. and has property P).
Proof. We claim that there is a Cartesian diagram ofétale sheaves where X → X/G and X × Y → X are the canonical projections. The claim implies the representability of pr 1 by [LMB00, Cor. (1.6.3)] applied to the epimorphism X → X/G using that it is obvious for X × Y → X (cf. [LMB00, Rem. (1.5.1)]). Also this shows that if Y → S has property P, then pr 1 has property P. To prove the claim we note that the map is an isomorphism ofétale sheaves. This is elementary to check using the freeness of the G-action on X.

Proof of Proposition 3.1 i). Consider a general Schubert variety
LG(k). By Corollary 3.2, there exist a maximal F -split torus S ⊂ G, an alcove f ⊂ā ⊂ A (G, S, F ), and an element w af ∈ W af such that S w (f , f ) S w af (a, f ) as k-schemes. Thus, we may and do assume that f = a and that w = w af ∈ W af is right f -minimal by Lemma 3.3.
Further, if f = a, then Proposition 3.1 i) is proven in [PR08,Prop. 9.7], and we explain how to extend the arguments to the general case. Let π : F G,a → F G,f be the canonical projection. Denote by w 0 ∈ w · W f the maximal length representative inside the coset. Then π −1 (S w (a, f )) = S w0 (a, a).
The right vertical map is anétale locally trivial fibration with typical fiber the homogenous space , and hence so isS w by smoothness of p. For the property "Frobenius split if char(k) > 0" (resp. "has rational singularities"), we note first that the counit of the adjunction is an isomorphism. Indeed, byétale descent for coherent sheaves we may argue locally in thé etale topology. Using flat base change [StaPro,02KH], it remains to prove (3.2) for maps of type pr : Y × X → X for some k-variety X. As the push forward of O Y along the structure map , the assertion is immediate by flat base change. Further, it is clear that R q p * (O Y ) = 0 when q > 0, for example by a very special case of the Borel-Bott-Weil theorem (Y is isomorphic to a classical flag variety over k, cf. [HRa, §4.2.2]). If k is a field of characteristic p > 0, thenS w0 is Frobenius split by [PR08, Prop. 9.7 c)], and the push forward of a splitting defines a splitting ofS w by using (3.2). It remains to construct a rational resolution ontoS w . Recall that by [PR08, Prop. 9.7 b)] the natural closed immersion S w (a, a) ⊂ S w0 (a, a) = S w0 lifts to a closed immersionS a w :=S w (a, a) ⊂ S w0 . Let p w := p|S a w :S a w →S w (a, f ) =S w be the restriction. Fix a reduced decomposition w = s 1 · . . . · s n into simple reflections. The Demazure resolution D(w) → S a w factors through the normalization, and we claim that the composition It is enough to show that f is trivial. Indeed, if char(k) = 0, R q ω D(w) = 0 for q > 0 would follow from the Grauert-Riemenschneider vanishing theorem; if char(k) = p > 0, as D(w) is Frobenius split (cf. [Go01,Prop 3.20]) the vanishing would follow from the Grauert-Riemenschneider vanishing for Frobenius split varieties, cf. [MvdK92, Thm. 1.2]. Further note that both morphisms f = p w • π w are surjective and birational; for p w birational, use Lemma 3.3 and recall that w is chosen to be right f -minimal. Therefore f is a resolution of singularities. SinceS w is normal and integral, the Extend the reduced decomposition of w to a reduced decomposition of w 0 , and consider the following diagram Here the square is Cartesian, pr w is the natural projection (onto the first l(w) factors), and the dotted arrow h exists because f • pr w = p • π w0 . Since π w0 and g are both birational, so is h. We claim that g is trivial. By the Leray spectral sequence it suffices to prove that π w0 and h are trivial. The triviality of π w0 is proven in [PR08,Prop. 9 by the Stein factorization of h, as D (w) is smooth and integral and h is birational. This shows that h, hence g, is trivial. Now the required vanishing of R q f * (O D(w) ) for q > 0 follows from flat base change applied to the Cartesian square. This finishes the proof.
3.4. Central extensions. Let φ : G → G be a map of (connected) reductive F -groups which induces an isomorphism on adjoint groups G ad G ad (or equivalently on simply connected groups T is a maximal F -split torus contained in a maximal torus. This induces a map on apartments A (G , S , F ) → A (G, S, F ) under which the facets correspond bijectively to each other. We denote the image of f by the same letter. The map G → G extends to a map on parahoric group schemes G := G f → G f =: G, and hence to a map on twisted partial affine flag varieties F G ,f → F G,f . We are interested in comparing their Schubert varieties.
There is a natural map on Iwahori-Weyl groups which is compatible with the action on the apartments A (G , S , F ) → A (G, S, F ). For w ∈W denote by w ∈W its image. As the map F G ,f → F G,f is equivariant compatibly with the map L + G → L + G, we get a map of projective k-varieties Proposition 3.5. For each w ∈W , the map (3.3) is a finite birational universal homeomorphism, and induces an isomorphism on the normalizations.
We need some preparation. The Iwahori-Weyl groups are equipped with a Bruhat order ≤ and a length function l according to the choice of a.
Lemma 3.6. The mapW →W induces an isomorphism of affine Weyl groups compatible with the simple reflections, and thus compatible with ≤ and l.
Proof. LetW sc be the Iwahori-Weyl group associated to the simply connected cover ψ : G sc → G der and the torus S sc := ψ −1 (S ∩ G der ) o . By [BT84,5.2.10],W sc can be identified with the affine Weyl group for (G, S, a) as well as (G , S , a), cf. also [HR08,Prop. 13].
We denote the affine Weyl W af ofW , resp.W by the same symbol.
Corollary 3.7. For each w ∈W , the mapW →W induces a bijection under which the right f -minimal elements correspond to each other.
Lemma 3.8. For each w ∈W , the map Proof. By the proof of Corollary 3.7, we may assume that w ∈ W af , and that w is right f -minimal. By Lemma 3.3, we may further assume that f = a is the alcove. Let w = s 1 · . . . · s n be a reduced decomposition into simple reflections. By Lemma 3.6, the decomposition w = s 1 · . . . · s n is a reduced decomposition of w. Let π w : D(w ) → S w (resp. π w : D(w) → S w ) be the Demazure resolution associated with the reduced decomposition, cf. [PR08,Prop 8.8]. There is a commutative diagram of k-schemes where the vertical maps are birational and isomorphisms onto the open cells. Hence, it is enough to show that D(w ) → D(w) is an isomorphism. By induction on l(w ) = n, we reduce to the case w = s (and hence w = s) is a simple reflection. In this case, π w and π w are isomorphisms, and we have to show that P 1 k S w → S w P 1 k is an isomorphism. The crucial observation is now that the map G → G on Bruhat-Tits groups schemes is the identity on the O F -extension of the root subgroups (cf. [BT84, 4.6.3, 4.6.7]). Hence, the map S w → S w restricted to the open cells A 1 k ⊂ P 1 k is the identity. The lemma follows.
Proof of Proposition 3.5. The L + G a -orbits (resp. L + G a -orbits) in S w (resp. S w ) correspond under the map S w → S w bijectively to each other, cf. Corollary 3.7. Hence, Lemma 3.8 implies that the map S w → S w is birational and bijective on k-points. As being quasi-finite and proper implies finite, the map in question must be finite. To see that the map is a universal homeomorphism consider the commutative diagram of k-schemes where the vertical maps are the normalization morphisms. By Proposition 3.1 i), the vertical maps are finite birational universal homeomorphisms. In particular, the mapS w →S w is a birational bijective proper, hence finite, morphism of normal varieties, and therefore it is an isomorphism. This shows that the map S w → S w is a universal homeomorphism, and the proposition follows.
3.5. Simple reduction. There is a finite index set J, and an isomorphism of F -groups where each F j /F is a finite separable extension, and each G j is an absolutely almost simple, simply connected F j -group. Under the identification of buildings B(G sc , F ) = j∈J B(G j , F j ) the facet f corresponds to facets f j ⊂ B(G j , F j ) for each j ∈ J.
Lemma 3.9. There is an isomorphism of k-ind-schemes under which the Schubert varieties correspond isomorphically to each other.
Proof. It is enough to treat the following two cases separatedly.
Products. If G = G 1 × G 2 is a direct product of two F -groups, then we have a direct product of affine Weyl groups W af = W af,1 × W af,2 . Now, for each w = (w 1 , w 2 ) ∈ W af , there is an equality on Schubert varieties S w = S w1 × S w2 which is easy to prove. In particular, if both S w1 and S w2 are normal, then S w is normal by [StaPro,06DG].
Restriction of scalars. Let G = Res F /F (G ) where F /F is a finite separable extension, and G is an F -group. By [HRb,Prop 4.7], we have . This gives an equality on loop groups L + G f = L + G f resp. LG = LG . Hence, there is an equality on twisted affine flag varieties F G,f = F G ,f under which the Schubert varieties correspond to each other.
3.6. Proof of Proposition 3.1 ii). Consider a general Schubert variety S w (f , f ) for some f , f ⊂ B(G, F ), w ∈ LG(k). By Corollary 3.2, there exist a maximal F -split torus S ⊂ G, an alcove f ⊂ā ⊂ A (G, S, F ), and an element w af ∈ W af such that S w (f , f ) S w af (a, f ) as k-schemes. Proposition 3.5 applied to G sc → G shows that the normalizationS w af (a, f ) is isomorphic to the normalization of a Schubert variety inside cf. Lemma 3.9. Thus,S w af (a, f ) j∈JS wj (a j , f j ) for some w j ∈ LG j (k). Now assume that S w S w af (a, f ) =: S w af is normal. It remains to prove that each variety S wj := S wj (a j , f j ) is normal as well. First note that the canonical map j∈J S wj → S w af must be an isomorphism by Proposition 3.5, so that the product of the varieties S wj is normal. Fix some j 0 ∈ J, and consider whereS wj 0 → S wj 0 denotes the normalization. By Proposition 3.1 i), this map is finite and birational. As the target is normal, it must be an isomorphism, so thatS wj 0 → S wj 0 must be an isomorphism (because being an isomorphism is fpqc local on the target). This proves Proposition 3.1 ii).

Reducedness of special fibers of local models
4.1. Weil restricted local models in mixed characteristic. We now switch back to the notation of §2. We first treat the case where F/Q p is a mixed characteristic local field. Recall that in this case we are assuming G = Res K/F (G 1 ) where K/F is a finite extension with residue field k of characteristic p > 0, and G 1 is a tamely ramified connected reductive K-group. To simplify the discussion and notation we first assume that K/F is totally ramified. The extension to the more general case is easy and is explained in Remark 4.2 below. We fix a uniformizer ∈ K, and let Q ∈ O F [u] be its minimal polynomial (an Eisenstein polynomial). The reader who is only concerned with Pappus-Zhu local models may take K = F and G 1 = G throughout this discussion.
Under the identification B(G, F ) = B(G 1 , K) (cf. [HRb,Prop. 4.6]), the facet f corresponds to a facet denoted f 1 . We denote by G 1 = G f1 the parahoric O K -group scheme of G 1 associated with [HRb,Cor. 4.8]. We let A 1 ⊂ G 1 be a maximal K-split torus whose apartment A (G 1 , A 1 , K) contains f 1 .  [HRb,Prop. 4.10 i)]) together with an isomorphism  Informally, we think about the k((u))-group G as being a connected reductive group of the "same type" as the K-group G 1 . By the discussion above, it is equipped with an identification of apartments A (G , A , k((u))) = A (G 1 , A 1 , K) where A := A 1 ⊗ k((u)).
Recall we fixed a conjugacy class {µ} of geometric cocharacters in G with reflex field E/F . This defines a closed subscheme Gr By definition, the local model is a reduced flat projective O E -scheme, and equipped with an embedding of its special fiber We now define the admissible locus A(G, {µ}) ⊂ F G ,f ⊗ kk . Recall from [HRb,§5.4] that there are identifications of Iwahori-Weyl groups W = W (G, A,F ) = W (G 1 , A 1 ,K) = W (G , A ,k((u))). Note that the inclusion A(G, {µ}) ⊆ M {µ} ⊗ k Ek is proven as part of [HRb, Thm. 5.14]. Let V 1 = Lie G 1 denote the Lie algebra, which is a free O F [u]-module of rank dim K (G 1 ). The adjoint representation G 1 → GL(V 1 ) induces by functoriality a morphism of ind-projective O F -ind-schemes

Then the admissible locus A(G, {µ}) is defined as the union of the (f , f )-Schubert varieties
where the target is defined as in (4.3) using the O F [u]-group scheme GL(V 1 ).
Let L det be the determinant line bundle on the target, and denote by L := ad * (L det ) its pullback. Let LF (resp. Lk) denote the restriction of L to the geometric generic (resp. geometric special) fiber Gr G,F = Gr G,F (resp. Gr G,k = F G ,f ⊗k). Let G ad = j∈J Res Fj /F (G j,ad ) where J is a finite index set, F j /F are finite field extensions containing K and G j,ad are absolutely simple, tamely ramified F j -groups. The geometric generic fiber becomes i,k }) k=1,...,mj i=1,...,lj according to Res Fj /F (G j,ad ) = Res Fj /F nr j (Res F nr j /Fj (G j,ad )). As F j /F ur j is totally ramified, we obtain in the geometric special fiber 3 Again (4.8) comes from the analogous decomposition for F G sc ,f sc where f sc ⊂ B(G sc , k((u))) denotes the facet corresponding to f . Here one has to remember G = Res K/F (G 1 ) and that the composite field K · F ur j is a subfield of F j /F ur j . Also LF (resp. Lk) decomposes according to (4.7) (resp. (4.8)), so that its ampleness on M o G,{µ},F (resp. on A(G, {µ})) now follows from the explicit formula given in [Zhu,Lem. 4.2] (see also [Lev16,Prop. 4.3.6]). Note that we are using here that a line bundle is ample if and only if its restriction to the reduced locus is ample, cf. [EGA II, Prop 4.5.13]. For each j ∈ J, the remaining claim (4.5) now reads By [Lev16,Prop. 4.3.8] (and the references cited there), we have the product formula Thus (4.9) follows from the equality Again its generic fiber is isomorphic to the usual affine Grassmannian Gr G , equivariantly for the action of (L + GL +,Q(u) G 1 ) η = L + z G, where z = u − . Its special fiber is isomorphic to F G ,f where now and f ⊂ B(G , k((u))) corresponds to f 1 under B(G , k((u))) = B(G 1 , k 0 ((u))) = B (G 1 , K).

4.2.
Equal characteristic local models. We now treat the case where F k((t)) is of equal characteristic. In this situation, we assume that in the simply connected group G sc j∈J Res Fj /F (G j ) each absolutely almost simple factor G j splits over a tamely ramified extension of F j . We also fix a , and thus is equivariantly (for the left action of the positive loop group) isomorphic to the usual affine Grassmannian Gr G over F formed using the parameter Its special fiber is canonically the twisted affine flag variety in the sense of [PR08].
Recall we fixed a conjugacy class {µ} of geometric cocharacters in G with reflex field E/F . This defines a closed subscheme Gr ≤{µ} G ⊂ Gr G ⊗ F E inside the affine Grassmannian which is a (geometrically irreducible) projective E-variety. As in mixed characteristic above (see [Zhu14,Ri16]), the local model (or global Schubert variety) M {µ} = M (G, G f , {µ}, t) is the scheme theoretic closure of the locally closed subscheme By definition, the local model is a reduced flat projective O E -scheme, and equipped with an embedding of its special fiber where the target is defined as in (4.11) using the O F -group scheme GL(V). Also we let L det be the determinant line bundle on the target, and denote by L := ad * (L det ) its pullback. Let LF (resp. Lk) denote the restriction of L to the geometric generic (resp. geometric special) fiber Gr G,F = Gr G,F (resp. Gr G,k = F G,f ⊗ kk ). The rest of the argument is the same as in mixed characteristic above using the following lemma. Proof. The proof relies on Proposition 3.1 and the Coherence Conjecture [Zhu14], and proceeds in the same steps as in Lemma 4.1. The restriction on the group G is a little milder in equal characteristic due the existence of local models for any, possibly wildly ramified, reductive group. 4.3. Proof of Corollary 2.3. We treat mixed and equal characteristic by the same argument, so that now the local field F is either a finite extension of Q p or isomorphic to k F ((t)). Let where E /F (resp. E/F ) denotes the reflex field of {µ } (resp. {µ}). Note that E ⊂ E is naturally a subfield. Now on geometric generic fibers (4.12) is the canonical map Gr on Schubert varieties which is finite and birational by Proposition 3.5. In particular, (4.12) is birational. To show that (4.12) is finite, we observe that this map is proper (because source and target are proper), and hence it is enough, by [StaPro,0A4X], to show that (4.12) is quasi-finite. As we already know that (4.12) is (quasi-)finite in generic fibers, it remains to show that it is quasi-finite on (reduced geometric) special fibers. By [HRb,Thm. 5.14], the map (4.12) identifies on reduced geometric special fibers with the canonical map A(G , {µ }) → A(G, {µ}). Applying Proposition 3.5 again, we see that the latter map is finite. This shows that (4.12) is birational and finite.
For universal homeomorphism, we have to show that map (4.12) is integral, universally injective and surjective, cf. [StaPro,04DC]. Being finite this map is integral. Universally injective and surjective can be checked on geometric points over the fibers of Spec(O E ) where it again follows from Proposition 3.5. Now assume the normality of Gr ≤{µ} G,F and all maximal (f , f )-Schubert varieties inside A(G, {µ}). Then Theorem 2.1 i) applies to show that the special M (G,{µ},G f ),k E is reduced (because it is geometrically reduced). Since its generic fiber is normal, we can apply [PZ13,Prop. 9.2] to prove the normality of M (G,{µ},G f ) ⊗ O E O E . As the map (4.12) is finite and birational, it must be an isomorphism.

Cohen-Macaulayness of local models
5.1. Recollections on Frobenius splittings. Let X be a scheme in characteristic p > 0, and denote by F = F X : X → X its absolute p-th power Frobenius. The scheme X is called Frobenius split if the map O X → F * O X splits as a map of O X -modules. In this case a splitting map ϕ : F * O X → O X is called a Frobenius splitting. Also recall the following notions.
Definition 5.1. Let X be Frobenius split.
i) We say X splits compatibly with some closed subscheme Z = V (I) ⊂ X if there exists a splitting ϕ : splits as a map of O X -modules.
Lemma 5.2. Let X be Frobenius split, and let D ⊂ X be an effective Cartier divisor. If X splits compatibly with D, then X splits relative to (p − 1) · D. In this case, X splits relative to D.
Proof. Let ϕ : i.e., X splits relative to (p − 1) · D. The last assertion is immediate from the factorization Remark 5.3. If X is a smooth variety over an algebraically closed field, then the converse to Lemma 5.2 holds. Namely, X splits compatibly with D if and only if X splits relative to (p − 1) · D. This is stated in [BK05, Thm. 1.4.10], but we do not need this sharper result.
Proof. For i = 1, 2, let Z i = V (I i ) and let ϕ be a splitting with ϕ(F * I i ) ⊂ I i . This automatically induces splittings on each Z i , and it is elementary to see that ϕ(F * (I 1 + I 2 )) ⊂ I 1 + I 2 . Since V (I 1 + I 2 ) = Z 1 ∩ Z 2 , the lemma follows.
Proposition 5.5. Let X be Frobenius split and locally of finite type over a field (or a Dedekind domain), and let D ⊂ X be an effective Cartier divisor. If X splits relative to D and X\D is Cohen-Macaulay, then X is Cohen-Macaulay.
Proof. This is [BS13, Ex. 5.4], and the following proof was communicated to us by K. Schwede 4 . We have to show that all local rings O X,x for x ∈ D are Cohen-Macaulay. Without loss of generality we may assume that X = Spec(R) where (R, m) is a Noetherian local ring and D = V (f ) for some non-zero divisor f ∈ m. By [StaPro, 0AVZ], we have to show that the local cohomology H i m (R) vanishes for i = 0, . . . , d − 1, d := dim(R). Our finiteness assumptions on X imply that R admits a dualizing complex (cf. [StaPro,0BFR]) so that Lemma 5.6 below applies. Hence, there exists an N > > 0 with f N · H i m (R) = 0 for all i = 0, . . . , d − 1. By [BS13, Lem. 5.2.3], there exists for any e ∈ Z ≥1 a splitting of the composition where q e := 1 + p + . . . + p e−1 , i.e., Spec(R) is F e * -split relative to q e · V (f ). Now choose e > > 0 such that q e ≥ N , i.e., f qe kills the local cohomologies as above. Finally, consider the sequence of Lemma 5.6. Let (R, m, k) be a Noetherian local ring of dimension d which admits a dualizing complex in the sense of [StaPro,0A7B]. Let f ∈ R be a non-zero divisor such that the localization R f is Cohen-Macaulay. For any finite R-module M whose localization M f is a projective R f -module, there exists an integer N > > 0 such that the local cohomology vanishes Proof. By the local duality theorem [StaPro, 0AAK], we have , where (·) ∧ denotes the m-adic completion, ω • R the normalized dualizing complex (cf. [StaPro,0A7B]) and E the injective hull of k. As R f is Cohen-Macaulay, the localized complex (ω Lemma 5.7. Let X be a flat scheme of finite type over a discrete valuation ring. i) Assume that the generic fiber X η is normal and the special fiber X s is reduced. Then X is normal. ii) Assume that the generic fiber X η is Cohen-Macaulay. Now let F be a non-archimedean local field (either of mixed or equal characteristic), and fix a triple (G, {µ}, G f ) as in §2 where G is defined over F . Let M := M (G,{µ},G f ) be the associated local model over O E , where E is the reflex field. As the geometric generic fiber MF is, by definition, a Schubert variety inside an affine Grassmannian which we assume to be normal, it is Cohen-Macaulay by Proposition 3.1 i). By [StaPro,0380, 045V] the generic fiber M E is normal and Cohen-Macaulay as well. In view of Theorem 2.1 i) and Lemma 5.7 this implies that M is normal. Here we are assuming that each maximal Schubert variety inside the admissible locus A(G, {µ}) is normal. It remains to show that if p > 2 then M is also Cohen-Macaulay. By Lemma 5.7 ii) this is equivalent to the Cohen-Macaulayness of the geometric special fiber Mk = A(G, {µ}), cf. Theorem 2.1 i). As the combinatorics of Iwahori-Weyl groups are the same in mixed and equal characteristic (cf. [HRb,Lem. 4.11] for a precise statement), we may and do assume that F k((t)) is of equal characteristic, i.e., M is a scheme in characteristic p > 0. For this, we must remark that the group G in (4.4) arising from the mixed characteristic situation is tamely ramified so that the tame ramification hypotheses on G sc used in Theorem 2.1 is satisfied; this holds by the description of the maximal torus T 1 ⊂ G 1 given in [HRb,Ex. 4.14]. Also we may and do assume that k =k is algebraically closed because the formation of local models commutes with unramified base change. Further, by where G ad = j∈J Res Fj /F (G j ) for absolutely simple F j -groups G j and m j := [F j : F ur j ], l j := [F ur j : F ] for the maximal unramified subextension F j /F ur j /F . As products of locally Noetherian flat Cohen-Macaulay schemes are Cohen-Macaulay (cf. [StaPro, 0C0W, 045J]), we see that it is enough to proof the Cohen-Macaulayness of each A(G j , {µ i,mj }) o , i.e., we may and do assume that G is absolutely almost simple. Also note that under (5.2) the Schubert varieties in each factor are still normal by Proposition 3.1 ii) so that our normality assumption still holds for the Schubert varieties in each absolutely almost simple factor.
Summarizing the discussion, we have an equal characteristic local model M attached with some absolutely almost simple group (so that Lemma 5.8 below is available) which satisfies the normality assumptions of Theorem 2.1. We know that M is a flat projective scheme over O = O E which is normal. Its generic fiber M E is Cohen-Macaulay, and its special fiber M k (=the admissible locus) is an effective Cartier divisor on M . We aim to show that M is, as a whole, Cohen-Macaulay.
The key to the proof is now the following lemma which is a direct consequence of [Zhu14, Thm. 6.5] (also this is the key step in the proof of the Coherence Conjecture).
Lemma 5.8. Let p > 2. Then the local model M is Frobenius split compatibly with its special fiber M k ⊂ M viewed as a closed subscheme.
Proof. In [Zhu14, Thm. 6.5], a O-scheme X together with a closed immersion M ⊂ X is constructed such that X is Frobenius split compatibly with both M and its special fiber X k . Hence, Lemma 5.4 implies that M is Frobenius split compatibly with M ∩ X k = M k .
Corollary 5.9. For p > 2, the local model M splits relative to its special fiber M k ⊂ M viewed as an effective Cartier divisor.
Proof. This is follows from Lemmas 5.8 and 5.2.
As we already know that M \M k = M E is Cohen-Macaulay, we can now apply Proposition 5.5 to conclude that M (and hence M k = A(G, {µ})) is Cohen-Macaulay. It remains to identify the dualizing sheaf on the local model.