# Normality of Schubert varieties
in affine Grassmannians

Patrick Bieker and Timo Richarz  Fakultät für Mathematik, Universität
Bielefeld, Postfach 100 131, 33501 Bielefeld, Germany<EMAIL_ADDRESS>bielefeld.de Fachbereich Mathematik, TU Darmstadt, Schlossgartenstrasse 7,
64289 Darmstadt, Germany<EMAIL_ADDRESS>
###### Abstract.

We classify all normal Schubert varieties in the affine Grassmannian of a
semisimple group over an arbitrary field with special attention to small
positive characteristic. The proof is elementary and relies on tangent space
calculations for quasi-minuscule Schubert varieties, a refined Levi lemma in
positive characteristic and the classification of minimal degenerations.

The first named author P.B. acknowledges support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) TRR 358 Integral
Structures in Geometry and Representation Theory, project number 49139240. The
second named author T.R. is funded by the European Research Council (ERC)
under Horizon Europe (grant agreement nº 101040935), by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and
Arithmetic of Uniformized Structures, project number 444845124 and the LOEWE
professorship in Algebra.

##  1\. Introduction

Schubert varieties in classical flag varieties are known to be normal by work
of Ramanan–Ramanathan [RR85], Seshadri [Ses87], Andersen [And85] and
Mehta–Srinivas [MS87]. The normality of Schubert varieties in Kac–Moody affine
flag varieties is due to Kumar [Kum87], Mathieu [Mat88] and Littelmann
[Lit98]. For affine flag varieties, interpreted as moduli spaces of torsors
over an algebraic disc together with a trivialization over the punctured one,
Faltings [Fal03] proves normality of Schubert varieties for all split, simply
connected groups. More generally, Pappas–Rapoport [PR08] and Lourenço [Lou]
show normality whenever the characteristic of the ground field is either zero
or big enough in comparison to the group.

For a semisimple group $G$ over an algebraically closed field $k$, a special
case of [PR08, Theorem 0.3] shows normality of Schubert varieties in the
affine Grassmannian $\mathop{\rm Gr}\nolimits_{G}$ whenever ${\rm
char}(k)\nmid\\#\pi_{1}(G)$ where $\pi_{1}(G)$ is the algebraic fundamental
group of $G$. For almost simple groups $G$ the number $\\#\pi_{1}(G)$ divides
the connection index of the Dynkin type of $G$ from [Bou68, Tables], and
agrees with it whenever $G$ is simple. The normality results along with other
techniques are the basis for finer geometric properties such as having
rational singularities, and are used in geometric representation theory or in
the reduction to positive characteristic of Shimura varieties.

When ${\rm char}(k)\mid\\#\pi_{1}(G)$ the work of Haines, Lourenço and the
second named author [HLR22] shows that almost all Schubert varieties in
$\mathop{\rm Gr}\nolimits_{G}$ are non-normal. For example, if ${\rm
char}(k)\mid n$, then only finitely many Schubert varieties in the affine
Grassmannian for $G=\mathrm{PGL}_{n,k}$ are normal. The results of [HLR22]
allow to classify all normal Schubert varieties for groups of small rank such
as $G=\mathrm{PGL}_{2,k}$ for ${\rm char}(k)=2$, or $G=\mathrm{PGL}_{3,k}$ for
${\rm char}(k)=3$. In the present manuscript we classify normal Schubert
varieties in $\mathop{\rm Gr}\nolimits_{G}$ for all semisimple groups $G$.

###  1.1. Results

Let $G$ be an almost simple, semisimple group over an algebraically closed
field $k$. Fix a maximal torus $T$ contained in a Borel subgroup $B$ in $G$.
Let $X_{*}(T)^{+}$ be the monoid of dominant cocharacters equipped with the
Bruhat partial order. For each $\mu\in X_{*}(T)^{+}$ the Schubert variety

$\mathop{\rm Gr}\nolimits_{G,\leq\mu}\hskip
2.84544pt\stackrel{{\scriptstyle\text{\tiny\rm def}}}{{=}}\hskip
2.84544pt\overline{L^{+}G\cdot\mu(\varpi)\cdot L^{+}G/L^{+}G},$

is an orbit closure in the affine Grassmannian $\mathop{\rm Gr}\nolimits_{G}$,
see Section 2.1. By [Fal03, PR08] the $k$-variety $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}$ is normal for all $\mu\in X_{*}(T)^{+}$ whenever
${\rm char}(k)\nmid\\#\pi_{1}(G)$. On the other hand if ${\rm
char}(k)\mid\\#\pi_{1}(G)$, then $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is
normal for only finitely many elements $\mu\in X_{*}(T)^{+}$ by [HLR22]. Note
that $X_{*}(T)^{+}$ is an infinite set whenever $G$ is nontrivial.

To state the classification consider the inclusion $X_{*}(T)\subset
P_{G}^{\vee}$ into the coweight lattice for $G$. Let $n\in{\mathbb{Z}}_{\geq
0}$ be the rank of $G$, and denote by
$\omega_{1}^{\vee},\ldots,\omega_{n}^{\vee}$ the ${\mathbb{Z}}$-basis of
$P_{G}^{\vee}$ of fundamental coweights in the notation of [Bou68, Tables].
The inclusion restricts to the partially ordered monoids
$X_{*}(T)^{+}\subset(P_{G}^{\vee})^{+}$ and
$\omega_{1}^{\vee},\ldots,\omega_{n}^{\vee}$ form a monoid basis of
$(P_{G}^{\vee})^{+}$. Together with the normality result when ${\rm
char}(k)\nmid\\#\pi_{1}(G)$ the following theorem classifies all normal
Schubert varieties in $\mathop{\rm Gr}\nolimits_{G}$:

######  Theorem 1.1 (Section 7).

classification-normal-schubert-varieties Assume ${\rm
char}(k)\mid\\#\pi_{1}(G)$. Let $\mu\in X_{*}(T)\backslash\\{0\\}$ be
dominant. Then, the $k$-variety $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is
normal if and only if the pair $(G,\mu)$ belongs to the following list:

  * •

any $G$, and $\mu$ minuscule (for a list, see [HPR20, Section 5, Table 1]);

  * •

$G$ of type $A_{n}$ for $n\geq 2$ (so ${\rm char}(k)\mid n+1$), and $\mu\leq
d\omega_{1}^{\vee}$ or $\mu\leq d\omega_{n}^{\vee}$ for some
$d\in\\{2,\ldots,n\\}$;

  * •

$G\cong\mathrm{PSO}_{2n}$ for $n\geq 5$ odd (so ${\rm char}(k)=2$), and
$\mu\in\\{\omega_{1}^{\vee}+\omega_{n-1}^{\vee},\omega_{1}^{\vee}+\omega_{n}^{\vee}\\}$;

  * •

$G$ adjoint of type $E_{6}$ (so ${\rm char}(k)=3$), and
$\mu\in\\{2\omega_{1}^{\vee},\omega_{3}^{\vee},\omega_{5}^{\vee},2\omega_{6}^{\vee}\\}$;

  * •

$G$ adjoint of type $E_{7}$ (so ${\rm char}(k)=2$), and
$\mu=\omega_{2}^{\vee}$.

In particular, the classification is independent of the isogeny class of $G$
and the characteristic of $k$ with ${\rm char}(k)\mid\\#\pi_{1}(G)$. For a
list of all simple groups $G$ with ${\rm char}(k)\mid\\#\pi_{1}(G)$ the reader
is referred to [HLR22, Section 6] or Section 7.

The proof of classification-normal-schubert-varieties is elementary in that no
cohomological and representation-theoretic methods are used. In fact, most
cases are handled by the following proposition which extends [HLR22, Corollary
6.2] relying on tangent space calculations. Recall that the quasi-minuscule
element $\mu_{\rm qm}\in X_{*}(T)^{+}$ is the unique element $\mu\in
X_{*}(T)^{+}$ that is minimal with the property $\mu>0$.

######  Proposition 1.2 (Section 3).

prop:intro Assume ${\rm char}(k)\mid\\#\pi_{1}(G)$. Let $\lambda,\mu\in
X_{*}(T)^{+}$ with $\lambda$ minuscule and $\lambda+\mu_{\rm qm}\leq\mu$.
Then, $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is non-normal.

In other words each connected component of $\mathop{\rm Gr}\nolimits_{G}$
contains small, non-normal Schubert varieties and all bigger Schubert
varieties are non-normal as well. The remaining cases are classified using the
transversal slice and a refinement in positive characteristic of the Levi
lemma from Malkin–Ostrik–Vybornov [MOV05], see Sections 4 and 5. The
refinement in levi-lemma-derived is necessarily more complicated as normality
of Schubert varieties is sensitive to the isogeny type of $G$. The methods
lead in mindeg-nonnormal to the classification of non-normal minimal
degeneration singularities, which is of independent interest to us. Even
though many Schubert varieties are non-normal when ${\rm
char}(k)\mid\\#\pi_{1}(G)$ it turns out that most minimal degeneration
singularities are still normal, see mindeg-nonnormal. So, somewhat
surprisingly the combination of prop:intro with the study of minimal
degeneration singularities is sufficient to prove classification-normal-
schubert-varieties in Section 7.

Regarding the normal locus of Schubert varieties in Dynkin type $A$ we have
the following result where we denote by $\Delta$ the set of $B$-simple roots
in $G$ with respect to $T$:

######  Corollary 1.3 (Section 7.2).

coro:intro Let $G$ be of type $A_{n}$ with ${\rm char}(k)\mid\\#\pi_{1}(G)$.
For $\mu\in X_{*}(T)^{+}$ let $J_{\mu}$ be the subset of all $\lambda\in
X_{*}(T)^{+}$ such that
$\mu-\lambda=\sum_{\alpha\in\Delta}n_{\alpha}\alpha^{\vee}$ for some
$n_{\alpha}\in{\mathbb{Z}}_{\geq 0}$ and with at least one $n_{\alpha}=0$.
Then, the open subvariety $\cup_{\lambda\in J_{\mu}}\mathop{\rm
Gr}\nolimits_{G,\lambda}$ inside $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is
normal.

The exact determination of the normal locus seems challenging. In fact, the
open locus in coro:intro agrees with the normal locus in all examples known to
us.

Acknowledgements. It is our pleasure to thank Tom Haines, João Lourenço and
Michael Rapoport for comments on an early draft.

##  2\. Schubert varieties

Let $G$ be a connected reductive group over an algebraically closed field $k$.
For a $k$-algebra $R$ we denote by $R[[{\varpi}]]$, respectively
$R(({\varpi}))$ the ring of formal power series, respectively Laurent series
in the formal variable $\varpi$. The loop group $LG$ (respectively, positive
loop group or jet group $L^{+}G$) is the group-valued functor on the category
of $k$-algebras $R$ defined by $LG(R)=G(R(({\varpi})))$ (respectively,
$L^{+}G(R)=G(R[[{\varpi}]])$). Then, $L^{+}G\subset LG$ is a subgroup functor.
The affine Grassmannian is the étale quotient

$\mathop{\rm Gr}\nolimits_{G}\hskip
2.84544pt\stackrel{{\scriptstyle\text{\tiny\rm def}}}{{=}}\hskip
2.84544ptLG/L^{+}G,$ (2.1)

which is representable by an ind-projective ind-scheme over $k$. The affine
Grassmannian is equipped with a left action by $L^{+}G$. One has
$L^{+}G=\lim_{i\geq 0}G_{i}$ where $G_{i}(R)=G(R[\varpi]/(\varpi^{i+1}))$ is
the group of $i$-jets. Each $G_{i}$ is a smooth, affine $k$-group scheme with
connected fibers.

###  2.1. Definition

Fix a maximal torus $T$ contained in a Borel subgroup $B$ in $G$. We denote by
$\langle\operatorname{-},\operatorname{-}\rangle$ the natural paring between
the characters $X^{*}(T)$ and cocharacters $X_{*}(T)$. The sum of the
$B$-positive roots is denoted $2\rho\in X^{*}(T)$. Let $(X_{*}(T)^{+},\leq)$
be the partially ordered monoid of $B$-dominant cocharacters equipped with the
Bruhat partial order: for $\lambda,\mu\in X_{*}(T)^{+}$ one has
$\lambda\leq\mu$ if and only if $\mu-\lambda$ is a sum of $B$-positive coroots
with ${\mathbb{Z}}_{\geq 0}$-coefficients.

The map $\mu\mapsto\mu(\varpi)$ identifies $X_{*}(T)^{+}$ with a subset of
$\mathop{\rm Gr}\nolimits_{G}(k)$. For each $\mu\in X_{*}(T)^{+}$, the
Schubert variety $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ (respectively,
$L^{+}G$-orbit $\mathop{\rm Gr}\nolimits_{G,\mu}$) is defined as the scheme-
theoretic image (respectively, sheaf-theoretic image) of the orbit map
$L^{+}G\to\mathop{\rm Gr}\nolimits_{G}$, $g\mapsto g\cdot\mu(\varpi)$. Then,
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is representable by a projective
$k$-variety of relative dimension $\langle 2\rho,\mu\rangle$. Each orbit
$\mathop{\rm Gr}\nolimits_{G,\mu}$ is representable by a smooth $k$-variety.
The inclusion $\mathop{\rm Gr}\nolimits_{G,\mu}\subset\mathop{\rm
Gr}\nolimits_{G,\leq\mu}$ is open. On the underlying sets one has $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}=\sqcup_{\lambda\in
X_{*}(T)^{+}:\lambda\leq\mu}\mathop{\rm Gr}\nolimits_{G,\lambda}$. The
left-$L^{+}G$-action on each Schubert variety factors through the group of
$i$-jets $L^{+}G\to G_{i}$ for some $i\geq 0$.

###  2.2. Normality

We denote by $\pi_{1}(G)$ the algebraic fundamental group of $G$ which is
isomorphic to the quotient of $X_{*}(T)$ by the coroot lattice. In particular,
$\pi_{1}(G)$ is a finite group whenever $G$ is semisimple. The following
theorem shows that most Schubert varieties are normal:

######  Theorem 2.1 ([PR08, Theorem 0.3]).

theo:normal Assume either ${\rm char}(k)=0$, or ${\rm char}(k)>0$ and ${\rm
char}(k)\nmid\\#\pi_{1}(G_{\rm der})$ where $G_{\rm der}$ is the derived group
of $G$. Then, the Schubert variety $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is
normal for all $\mu\in X_{*}(T)^{+}$.

In addition, normal Schubert varieties are known to be Cohen–Macaulay with
only rational singularities. The proof of theo:normal is by reduction to the
simply connected cover $G_{\rm sc}\to G_{\rm der}$, which is an étale isogeny
if and only if ${\rm char}(k)\nmid\\#\pi_{1}(G_{\rm der})$. The remaining
cases where ${\rm char}(k)\mid\\#\pi_{1}(G_{\rm der})$ reduce to almost
simple, semisimple groups. In contrast to theo:normal almost all Schubert
varieties are non-normal in these cases:

######  Theorem 2.2 ([HLR22, Theorem 1.1]).

theo:not-normal Assume $G$ is almost simple, semisimple and ${\rm
char}(k)\mid\\#\pi_{1}(G)$. Then, the Schubert variety $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}$ is normal for only finitely many $\mu\in
X_{*}(T)^{+}$.

The non-normal Schubert varieties are known to be geometrically unibranch and
regular in codimension $1$, but neither S2 nor seminormal. The methods in
[HLR22] allow to give an explicit bound on the number of normal Schubert
varieties in $\mathop{\rm Gr}\nolimits_{G}$ based on the following result:

######  Proposition 2.3 ([HLR22, Corollary 6.2]).

prop:quasi-minuscule Assume $G$ is almost simple, semisimple and
$p\mid\\#\pi_{1}(G)$. Then, the Schubert variety $\mathop{\rm
Gr}\nolimits_{G,\leq\mu_{\rm qm}}$ is non-normal where $\mu_{\rm qm}\in
X_{*}(T)^{+}$ is quasi-minuscule.

##  3\. Minuscule plus quasi-minuscule

In this section we prove prop:intro. We continue to fix $T\subset B\subset G$
as in Section 2.1. In what follows the elements in $X_{*}(T)^{+}$ that are
minimal with respect to the Bruhat partial order $\leq$ are called minuscule.
In particular, $\mu=0$ is minuscule by this definition.

Let $I\subset L^{+}G$ be the Iwahori subgroup defined as the preimage of
$B\subset G$ under the map $L^{+}G\to G$ induced by $\varpi\mapsto 0$. To
handle the $I$-orbit stratification on $\mathop{\rm Gr}\nolimits_{G}$ we refer
to Haines–Rapoport [PR08, Appendix] or [Ric16] for the following properties.
Let $N_{G}(T)$ be the normalizer of $T$ in $G$. The Iwahori-Weyl group (or,
extended affine Weyl group) $W:=N_{G}(T)(k(({\varpi})))/T(k[[{\varpi}]])$ is
equipped with the structure of a quasi-Coxeter group. The inclusion
$N_{G}(T)(k(({\varpi})))\subset G(k(({\varpi})))$ induces a bijection

$W\overset{\cong}{\longrightarrow}I(k)\backslash LG(k)/I(k).$ (3.1)

One has a semidirect product decomposition
$W=T(k(({\varpi})))/T(k[[{\varpi}]])\rtimes W_{0}$ where $W_{0}$ is the Weyl
group of $(G,T)$. Furthermore, the inclusion $X_{*}(T)^{+}\subset
T(k(({\varpi})))$, $\mu\mapsto\mu(\varpi)$ induces a bijection

$X_{*}(T)^{+}\overset{\cong}{\longrightarrow}W_{0}\backslash W/W_{0}.$ (3.2)

######  Lemma 3.1.

lem:length-zero-minuscule Let $\Omega\subset W$ be the subset of length zero
elements. Then, the map $\omega\mapsto W_{0}\omega W_{0}$ induces under the
inverse of (3.2) a bijection between $\Omega$ and the minuscule elements in
$X_{*}(T)^{+}$.

###### Proof.

By [Ric13, Corollary 1.8] the bijection (3.2) is compatible with the Bruhat
partial orders. This immediately implies the lemma. ∎

For a minuscule element $\lambda\in X_{*}(T)^{+}$ we denote by
$\omega_{\lambda}\in\Omega$ the associated element under lem:length-zero-
minuscule. For any element $w\in W$ we fix a lift $\dot{w}\in
N_{G}(T)(k(({\varpi})))$ and denote by

$\mathop{\rm
Gr}\nolimits_{G}\overset{\dot{w}\cdot}{\longrightarrow}\mathop{\rm
Gr}\nolimits_{G}$ (3.3)

the automorphism induced by left multiplication with $\dot{w}$ on the loop
group $LG$. For a closed subscheme $X\subset\mathop{\rm Gr}\nolimits_{G_{k}}$
let $\dot{w}(X)$ be the image of $X$ under (3.3). Clearly, if $X$ is
$I$-invariant, then $\dot{w}(X)$ is independent of the choice of lift
$\dot{w}\in N_{G}(T)(k(({\varpi})))$ in $w$.

######  Lemma 3.2.

lem:translation-by-minuscule Let $\lambda,\mu\in X_{*}(T)^{+}$ with $\lambda$
minuscule. Then, $\dot{\omega}_{\lambda}^{-1}(\mathop{\rm
Gr}\nolimits_{G,\leq\mu+\lambda})$ is an $I$-invariant, closed subvariety of
$\mathop{\rm Gr}\nolimits_{G}$ containing $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}$.

###### Proof.

The variety $\dot{\omega}_{\lambda}^{-1}(\mathop{\rm
Gr}\nolimits_{G,\leq\mu+\lambda})$ is $I$-invariant because
$\dot{\omega}I\dot{\omega}^{-1}=I$ for all $\omega\in\Omega$. It contains
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ if and only if one has

$\dot{\omega}_{\lambda}(\mathop{\rm
Gr}\nolimits_{G,\leq\mu})\subset\mathop{\rm Gr}\nolimits_{G,\leq\mu+\lambda}.$
(3.4)

As $\omega_{\lambda}$ and $\lambda(\varpi)$ define the same double coset by
$W_{0}$ there exist elements $w,w^{\prime}\in W_{0}$ such that
$w\omega_{\lambda}w^{\prime}=\lambda(\varpi)$ inside $W$. The Schubert
varieties $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ and $\mathop{\rm
Gr}\nolimits_{G,\leq\mu+\lambda}$ are $L^{+}G$-invariant by definition. So
(3.4) holds if and only if $\lambda(\varpi)(\mathop{\rm
Gr}\nolimits_{G,\leq\mu})\subset\mathop{\rm Gr}\nolimits_{G,\leq\mu+\lambda}$.
Clearly, this containment holds true because $\lambda$ and $\mu$ are dominant.
∎

For the proof we also need the following result from [HLR22, Corollary 2.2]:

######  Lemma 3.3.

lemm:continuity Let $X\subset Y$ be $I$-invariant, closed subvarieties in
$\mathop{\rm Gr}\nolimits_{G}$. If $X$ is non-normal, so is $Y$.

###### Proof.

Any $I$-invariant, closed subvariety in $\mathop{\rm Gr}\nolimits_{G}$ is an
$I$-orbit closure of some element in $W/W_{0}\subset\mathop{\rm
Gr}\nolimits_{G}(k)$. So, the lemma is a special case of [HLR22, Corollary
2.2]. ∎

###### Proof of prop:intro.

Since $G$ is almost simple, semisimple and ${\rm char}(k)\mid\\#\pi_{1}(G)$
the Schubert variety $\mathop{\rm Gr}\nolimits_{G,\leq\mu_{\rm qm}}$ is non-
normal by prop:quasi-minuscule. It is contained in
$\dot{\omega}_{\lambda}^{-1}(\mathop{\rm Gr}\nolimits_{G,\leq\lambda+\mu_{\rm
qm}})$ by lem:translation-by-minuscule. The latter is therefore non-normal as
well by lemm:continuity and so is $\mathop{\rm
Gr}\nolimits_{G,\leq\lambda+\mu_{\rm qm}}$. Since any $L^{+}G$-invariant,
closed subvariety of $\mathop{\rm Gr}\nolimits_{G}$ is also $I$-invariant, the
Schubert variety $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is non-normal for all
$\mu\in X_{*}(T)^{+}$ with $\lambda+\mu_{\rm qm}\leq\mu$. This proves the
proposition. ∎

##  4\. Transversal slices

The technique is useful for studying the singularities arising when passing
from a stratum to another one inside a Schubert variety. It is used in [MOV05,
JW14, KWWY14] over ${\mathbb{C}}$, [Zhu17, Section 2.3] over $k$ and [Lou23,
Section 4] over ${\mathbb{Z}}$. Due to the existence of non-normal Schubert
varieties some care is needed. For example, the analogue of [MOV05, Lemma 2.6]
may fail in positive characteristic.

Fix $T\subset B\subset G$ as in Section 2.1. The negative loop group $L^{-}G$
is the group functor on the category of $k$-algebras $R$ defined by
$L^{-}G(R)=G(R[\varpi^{-1}])$. The strictly negative loop group is the group
valued functor

$L^{--}G\hskip 2.84544pt\stackrel{{\scriptstyle\text{\tiny\rm
def}}}{{=}}\hskip 2.84544pt\ker(L^{-}G\overset{\varpi^{-1}\mapsto
0}{\longrightarrow}G).$ (4.1)

By [Fal03, Corollary 3] and [Lou23, Corollaire 4.2.11] (see also [dCHL18,
Theorem 2.3.1]) the composition $L^{--}G\subset LG\to\mathop{\rm
Gr}\nolimits_{G}$ is representable, affine and an open immersion. There is an
open cover

$\mathop{\rm Gr}\nolimits_{G}=\bigcup_{\mu\in X_{*}(T)}\mu(\varpi)\cdot
L^{--}G,$ (4.2)

see [Fal03, Definition 5 ff.], [GH07, Lemma 3.1] or [Lou23, Equation
(4.2.24)].

######  Definition 4.1.

transversal-slice-defi For $\lambda,\mu\in X_{*}(T)^{+}$ the transversal slice
of $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ at $\lambda(\varpi)$ is the étale
subsheaf of $\mathop{\rm Gr}\nolimits_{G}$ defined as

$\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}\hskip
2.84544pt\stackrel{{\scriptstyle\text{\tiny\rm def}}}{{=}}\hskip
2.84544ptL^{--}G\cdot\lambda(\varpi)\cap\mathop{\rm Gr}\nolimits_{G,\leq\mu}.$
(4.3)

The following lemma is analogous to [MOV05, Section 2.5] over the complex
numbers. It is essentially contained in [Mül08, Lemma 6.7], [Zhu17, Section
2.3], [Lou23, Section 4] or [dCHL18, Lemma 6.1.3].

######  Proposition 4.2.

transversal-slice-lemm For $\lambda,\mu\in X_{*}(T)^{+}$ the sheaf
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is representable by a locally
closed subscheme of $\mathop{\rm Gr}\nolimits_{G}$. It is non-empty if and
only if $\lambda\leq\mu$. In this case, the following properties hold:

  1. (1)

The scheme $\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is smoothly
equivalent to the open subvariety $\cup_{\nu\in
X_{*}(T)^{+}:\lambda\leq\nu\leq\mu}\mathop{\rm Gr}\nolimits_{G,\nu}$ in
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}$. More precisely, for each $i\geq 0$
such that the $L^{+}G$-action on $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$
factors through the group of $i$-jets $G_{i}$ the maps in the diagram

$\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}\overset{\;{\rm
pr}}{\longleftarrow}G_{i}\times\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}\overset{{\rm
act}\;}{\longrightarrow}\cup_{\nu\in
X_{*}(T)^{+}:\lambda\leq\nu\leq\mu}\mathop{\rm Gr}\nolimits_{G,\nu}$ (4.4)

are smooth and surjective. Here ${\rm pr}\colon(g,x)\mapsto x$ and ${\rm
act}\colon(g,x)\mapsto g\cdot x$ are the projection and the action,
respectively.

  2. (2)

The scheme $\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is affine,
integral, geometrically unibranch, regular in codimension $1$ and of dimension
$\langle 2\rho,\mu-\lambda\rangle$. It is normal if and only if $\cup_{\nu\in
X_{*}(T)^{+}:\lambda\leq\nu\leq\mu}\mathop{\rm Gr}\nolimits_{G,\nu}$ is so.
This happens, for example, if ${\rm char}(k)\nmid\\#\pi_{1}(G_{\rm der})$ by
theo:normal.

###### Proof.

As étale sheaves
$L^{--}G\cdot\lambda(\varpi)=L^{--}G/(L^{--}G\cap\lambda(\varpi)L^{+}G\lambda(\varpi)^{-1})$,
which is representable by a closed sub-ind-scheme of $L^{--}G$, see [Zhu17,
Proposition 2.3.3 (1)] and [Lou23, Equation (4.2.26)]. Using that $L^{--}G$ is
ind-affine the transversal slice (4.3) is representable by an affine, locally
closed subscheme of $\mathop{\rm Gr}\nolimits_{G}$.

The argument for the non-emptiness criterion is as follows. If
$\lambda\leq\mu$, then clearly $\lambda(\varpi)\in\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}(k)$. Conversely, if $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is non-empty, then
$L^{--}G\cdot\lambda(\varpi)\cap\mathop{\rm Gr}\nolimits_{G,\nu}$ is non-empty
for some $\nu\in X_{*}(T)^{+}$ with $\nu\leq\mu$. By [Lou23, Equation
(4.2.31)] we get $\lambda\leq\nu\leq\mu$. For the rest of the proof we assume
$\lambda\leq\mu$.

(1): The maps in (4.4) are surjective using the non-emptiness of
$L^{--}G\cdot\lambda(\varpi)\cap\mathop{\rm Gr}\nolimits_{G,\nu}$ for all
$\nu\in X_{*}(T)^{+}$ with $\lambda\leq\nu\leq\mu$ to see the surjectivity of
the map labelled “act”. Since $G_{i}$ is $k$-smooth, the smoothness of “pr”
follows by base change. The map “act” arises by base change along the
inclusion $\cup_{\nu\in X_{*}(T)^{+}:\lambda\leq\nu\leq\mu}\mathop{\rm
Gr}\nolimits_{G,\nu}\subset\mathop{\rm Gr}\nolimits_{G}$ from the bottom map
in the following commutative diagram

${L^{+}G\times L^{--}G}$${LG}$${LG}$${L^{+}G\times
L^{--}G\cdot\lambda(\varpi)}$${\mathop{\rm
Gr}\nolimits_{G}}$act$\cdot\lambda(\varpi)$$\cong$act (4.5)

where the vertical maps are the quotient maps. As all schemes in (4.4) are of
finite presentation over $k$, it suffices to show that the bottom arrow
satisfies the lifting property for formal smoothness and for first order nil
thickenings of Artinian local rings, see [Sta, Tag 0DYF]. On such rings we can
lift points along the vertical quotient maps by their Zariski local
triviality, see (4.2). The composition of the top arrows in (4.5) is an open
immersion by the discussion above (4.2). So it is formally smooth. The right
vertical arrow in (4.5) is formally smooth because it is an $L^{+}G$-torsor.
So the bottom arrow has the same lifting property for nil thickenings of
Artinian local rings.

(2): We have already seen that $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is an affine, locally closed subscheme of
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}$. As the properties “normal”, “reduced”
and “regular in codimension $1$” are local in the smooth topology [Sta, Tag
034D], it follows from (1) that $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is normal if and only if $\cup_{\nu\in
X_{*}(T)^{+}:\lambda\leq\nu\leq\mu}\mathop{\rm Gr}\nolimits_{G,\nu}$ is so,
and from [HLR22, Theorem 1.1] that $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is reduced and regular in codimension $1$.
Next, we claim that $\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is
connected of dimension $\langle 2\rho,\mu-\lambda\rangle$. For this we
consider the $\mathbb{G}_{m,k}$-action on $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}$ induced by loop rotation $\mathbb{G}_{m,k}\times
LG\to LG$, $(x,g(\varpi))\mapsto g(x\cdot\varpi)$. Then, on topological spaces
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ identifies with the fiber of
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{-}\to\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{0}$ above $\lambda(\varpi)$ where $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{-}$ is the repeller and $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{0}$ the fixed points, and thus is connected by
[Ric19, Corollary 1.12]. For the dimension we note that “act” is a surjective,
smooth map between connected $k$-schemes of finite type. So we can compute the
dimension of the source as the dimension of the target plus the dimension of
any fiber. Since $\mathop{\rm
Gr}\nolimits_{G,\leq\lambda}^{\lambda}=\\{\lambda(\varpi)\\}$ the fiber ${\rm
act}^{-1}(\lambda(\varpi))\simeq{\rm Stab}_{G_{i}}(\lambda(\varpi))$ is the
stabilizer of $G_{i}$ at $\lambda(\varpi)$. Using $G_{i}/{\rm
Stab}_{G_{i}}(\lambda(\varpi))\cong\mathop{\rm Gr}\nolimits_{G,\lambda}$ one
easily gets the desired dimension formula. It remains to show that
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is irreducible and
geometrically unibranch.

Now, if $G_{\rm der}$ is simply connected, then $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is normal by the above discussion and
hence integral by connectedness. To treat the general case we lift
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ along the central extension
$\tilde{G}\to G$ from [Lou23, Section 2.4] as follows. Let $T_{\rm sc}$ and
$B_{\rm sc}$ be the preimage of $T\cap G_{\rm der}$ respectively $B\cap G_{\rm
der}$ under the simply connected cover $G_{\rm sc}\to G_{\rm der}$. The action
of $T$ by conjugation lifts to $G_{\rm sc}$, and there is the central
extension

$1\to T_{\rm sc}\to\tilde{G}:=G_{\rm sc}\rtimes T\to G\to 1$

with $\tilde{G}_{\rm der}=G_{\rm sc}$. The group $\tilde{T}=T_{\rm sc}\times
T$ is a split maximal torus in $\tilde{G}$ and $\tilde{B}=B_{\rm sc}\rtimes T$
a Borel subgroup. Clearly, $\lambda\leq\mu$ in $X_{*}(T)^{+}$ lifts to
$\tilde{\lambda}\leq\tilde{\mu}$ in $X_{*}(\tilde{T})^{+}$. The induced map on
Schubert varieties $\mathop{\rm
Gr}\nolimits_{\tilde{G},\leq\tilde{\mu}}\to\mathop{\rm
Gr}\nolimits_{G,\leq\mu}$ is a finite, birational, universal homeomorphism by
[HR23, Proposition 3.5] which induces isomorphisms on all residue fields by
[FHLR22, Proof of Lemma 4.4]. In particular, the map agrees with the
seminormalization [Sta, Tag 0H3G]. Thus, $\mathop{\rm
Gr}\nolimits_{\tilde{G},\leq\tilde{\mu}}^{\tilde{\lambda}}\to\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is a finite, birational, homeomorphism
inducing isomorphisms on all residue fields, so agrees with the
seminormalization as $\mathop{\rm
Gr}\nolimits_{\tilde{G},\leq\tilde{\mu}}^{\tilde{\lambda}}$ is integral and
normal because $\tilde{G}_{\rm der}$ is simply connected. This implies that
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is integral and geometrically
unibranch. ∎

##  5\. A Levi lemma

In this section we prove a refinement of the Levi lemma from [MOV05, Section
3.3] adapted to our purposes. The main difference is that due to the
phenomenon of non-normal Schubert varieties we need to work with derived
groups as opposed to adjoint groups, which complicates the situation.

Fix $T\subset B\subset G$ as in Section 2.1. Let $M\subset G$ be a Levi
subgroup containing $T$. The closed immersion on affine Grassmannians
$\mathop{\rm Gr}\nolimits_{M}\to\mathop{\rm Gr}\nolimits_{G}$ induces a
locally closed immersion

$\mathop{\rm Gr}\nolimits_{M,\leq\mu}^{\lambda}\longrightarrow\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ (5.1)

for any $\lambda,\mu\in X_{*}(T)^{+}$ with $\lambda\leq\mu$. Note that every
$B$-dominant cocharacter in $X_{*}(T)$ is $B\cap M$-dominant, so the
transversal slice for the Levi subgroup is well-defined, see transversal-
slice-defi.

######  Lemma 5.1.

levi-lemma If $\mu-\lambda$ lies in the coroot lattice of $M$, then the map
(5.1) is an isomorphism.

###### Proof.

The inclusion $M\subset G$ induces a closed immersion $L^{--}M\to L^{--}G$,
hence a closed immersion on orbits $L^{--}M\cdot\lambda(\varpi)\to
L^{--}G\cdot\lambda(\varpi)$ and so on transversal slices, compare with the
proof of transversal-slice-lemm. By assumption on $\mu-\lambda$ we have
$\langle 2\rho_{M},\mu-\lambda\rangle=\langle 2\rho_{G},\mu-\lambda\rangle$
where $2\rho_{M}$ and $2\rho_{G}$ denotes the sum of the roots in $B\cap M$
and $B$ respectively. So, (5.1) is a closed immersion of $k$-varieties of the
same dimension and hence an isomorphism. ∎

Let $M_{\rm der}$ denote the derived group of $M$, and $M_{\operatorname{ad}}$
its adjoint group. The map $M\to M_{\operatorname{ad}}$ induces a map on
transversal slices that, however, might not be an isomorphism in positive
characteristic. Therefore, we are only able to pass from $M$ to $M_{\rm der}$
under certain restrictions on the cocharacters as follows. Let $Z$ be the
maximal torus in the center of $M$. Denote by
${\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}}:=Z\cap M_{\rm der}$ the intersection.

######  Lemma 5.2.

central-isog-adjoint-lemm The map $Z\times M_{\rm der}\to M$, $(z,m)\mapsto
z\cdot m$ induces an isomorphism

$Z\times^{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}}M_{\rm der}\overset{\cong}{\longrightarrow}M,$ (5.2)

and
${\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}}=Z\cap M_{\rm der}$ is a finite $k$-group scheme of
multiplicative type. In particular, if
$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$ is trivial (for example, if $M_{\rm der}$ is adjoint), then
$M=Z\times M_{\rm der}$.

###### Proof.

The map $Z\times M_{\rm der}\to M$ is a central isogeny. As its kernel is
clearly
$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$, the map (5.2) is an isomorphism. Further,
$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$ is a closed $k$-subgroup of the center $Z_{M_{\rm der}}$. As
$Z_{M_{\rm der}}$ is finite over $k$ of multiplicative type, so is
$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$. ∎

The intersection $T\cap M_{\rm der}$ is a split maximal torus of $M_{\rm
der}$. Then,
$Z\times^{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}(T\cap M_{\rm der})=T$ under (5.2). In particular, the induced
map on cocharacter lattices

$X_{*}(T\cap M_{\rm der})\oplus X_{*}(Z)\longrightarrow X_{*}(T).$ (5.3)

is injective with cokernel isomorphic to
$X^{*}({\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}})$, a finite abelian group of order
$\textrm{dim}_{k}\Gamma({\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}},{\mathcal{O}}_{{\mbox{$\raisebox{-1.77817pt}{$l$}\mu\raisebox{-2.95358pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{2.65222pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>3.22005pt$}{}}})$. So (5.3) is an isomorphism after rationalization. For every
$\lambda\in X_{*}(T)_{\mathbb{Q}}$, let $\lambda_{M_{\rm der}}$ be its
projection to $X_{*}(T\cap M_{\rm der})_{\mathbb{Q}}$.

######  Lemma 5.3.

cochars-derivedgroup A cocharacter $\lambda\in X_{*}(T)$ is in the image of
(5.3) if and only if $\lambda_{M_{\rm der}}\in X_{*}(T\cap M_{\rm der})$. This
holds, for example, if $\lambda_{M_{\rm der}}=0$ or if $M_{\rm der}$ is
adjoint. Further, one has

$X_{*}(T\cap M_{\rm der})=X_{*}(T)\cap{\mathbb{Q}}\Phi^{\vee}(M,T),$ (5.4)

where $\Phi^{\vee}(M,T)\subset X_{*}(T)$ is the set of coroots of $M$ with
respect to $T$.

###### Proof.

First off, for any subtorus $T^{\prime}\subset T$ its cocharacter lattice
$X_{*}(T^{\prime})$ is saturated in $X_{*}(T)$. So (5.4) follows from
$X_{*}(T\cap M_{\rm der})_{\mathbb{Q}}={\mathbb{Q}}\Phi^{\vee}(M,T)$ applying
the saturation property to $T^{\prime}=T\cap M_{\rm der}$. Now let $\lambda\in
X_{*}(T)$. If $\lambda$ lies in the image of (5.3), then clearly
$\lambda_{M_{\rm der}}\in X_{*}(T\cap M_{\rm der})$. Conversely, if
$\lambda_{M_{\rm der}}\in X_{*}(T\cap M_{\rm der})$, then
$\lambda-\lambda_{M_{\rm der}}\in X_{*}(Z)_{\mathbb{Q}}$ and so
$\lambda-\lambda_{M_{\rm der}}\in X_{*}(Z)$ applying the saturation property
to $T^{\prime}=Z$. ∎

######  Corollary 5.4 (Levi lemma).

levi-lemma-derived Let $\lambda,\mu\in X_{*}(T)^{+}$ with $\lambda\leq\mu$. If
$\mu-\lambda$ lies in the coroot lattice of $M$ and if $\lambda_{M_{\rm
der}}\in X_{*}(T\cap M_{\rm der})$, then the maps

$\mathop{\rm Gr}\nolimits_{M_{{\rm der}},\leq\mu_{\rm der}}^{\lambda_{\rm
der}}\longrightarrow\mathop{\rm Gr}\nolimits_{M,\leq\mu_{\rm
der}}^{\lambda_{{\rm der}}}\xrightarrow{(\lambda-\lambda_{{\rm
der}})(\varpi)\cdot}\mathop{\rm
Gr}\nolimits_{M,\leq\mu}^{\lambda}\xrightarrow{\eqref{immersion-Levi-
slice}}\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ (5.5)

are isomorphisms where $\mu_{\rm der}:=\mu_{M_{\rm der}}$ and $\lambda_{\rm
der}:=\lambda_{M_{\rm der}}$.

###### Proof.

By assumption one has $\mu-\lambda=(\mu-\lambda)_{\rm der}=\mu_{\rm
der}-\lambda_{{\rm der}}$, so $\lambda_{{\rm der}}\in X_{*}(T\cap M_{\rm
der})$ implies $\mu_{{\rm der}}\in X_{*}(T\cap M_{\rm der})$ and
$\lambda_{{\rm der}}\leq\mu_{{\rm der}}$. Hence, all maps are well-defined.
Next, the last map in (5.5) is an isomorphism by levi-lemma, so is the first
map by an analogous argument. By the proof of cochars-derivedgroup, the
cocharacter $\kappa:=\lambda-\lambda_{{\rm der}}=\mu-\mu_{{\rm der}}$ is
central for $M$. So multiplication by $\kappa(\varpi)$ induces the middle
isomorphism in (5.5). ∎

##  6\. Minimal degenerations

In this section we classify the minimal degeneration singularities that are
normal. We continue to fix $T\subset B\subset G$ as in Section 2.1.

######  Definition 6.1.

A pair $\lambda,\mu\in X_{*}(T)^{+}$ with $\lambda\leq\mu$ is called a minimal
degeneration if $\mu$ lies directly above $\lambda$, that is, $\lambda<\mu$
and for all $\nu\in X_{*}(T)^{+}$ with $\lambda\leq\nu\leq\mu$ one has either
$\lambda=\nu$ or $\nu=\mu$. The singularities arising in the transversal slice
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ are called minimal
degeneration singularities.

######  Remark 6.2.

If $\lambda<\mu$ is a minimal degeneration in $X_{*}(T)^{+}$, then
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is smoothly equivalent to
$\mathop{\rm Gr}\nolimits_{G,\mu}\cup\mathop{\rm Gr}\nolimits_{G,\lambda}$,
see transversal-slice-lemm. So, the minimal degeneration singularities capture
the singularities arising at the boundary of $\mathop{\rm
Gr}\nolimits_{G,\mu}$ inside $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$.

For an almost simple, semisimple group $G$ one has an inclusion
$X_{*}(T)\subset P_{G}^{\vee}$ into the coweight lattice of $G$. Let $n$ be
the rank of $G$, and denote by $\omega_{1}^{\vee},\ldots,\omega_{n}^{\vee}$
the ${\mathbb{Z}}$-basis of $P_{G}^{\vee}$ of fundamental coweights in the
notation of [Bou68, Tables]. The inclusion restricts to the partially ordered
monoids $X_{*}(T)^{+}\subset(P_{G}^{\vee})^{+}$ and
$\omega_{1}^{\vee},\ldots,\omega_{n}^{\vee}$ form a monoid basis of
$(P_{G}^{\vee})^{+}$. For $\lambda<\mu$ in $X_{*}(T)^{+}$ we denote by
$I_{\lambda,\mu}$ the set of simple coroots that appear in $\mu-\lambda$ with
non-zero coefficients. Then, $I_{\lambda,\mu}$ corresponds to a subset of the
Dynkin diagram for $G$. We denote by $M_{\lambda,\mu}\subseteq G$ the Levi
subgroup corresponding to $I_{\lambda,\mu}$. As in cochars-derivedgroup for
$\lambda\in X_{*}(T)$ let $\lambda_{M_{\rm der}}$ be its projection to
$X_{*}(M_{\rm der}\cap T)_{\mathbb{Q}}$ where $M=M_{\lambda,\mu}$. The
following is a special case of Stembridge’s classification of minimal
degenerations:

######  Theorem 6.3.

Assume $G$ is almost simple, semisimple and not of Dynkin type $G_{2}$. Let
$\lambda,\mu\in X_{*}(T)^{+}$ with $\lambda<\mu$, and denote by
$M=M_{\lambda,\mu}$ the associated Levi subgroup of $G$. Then, $\lambda<\mu$
is a minimal degeneration if and only if one of the following conditions is
satisfied:

  1. (1)

$\mu-\lambda$ a simple coroot;

  2. (2)

$\mu-\lambda$ quasi-minuscule for $M_{\rm der}$ and $\lambda_{M_{\rm der}}=0$;

  3. (3)

$\mu-\lambda$ quasi-minuscule for $M_{\rm der}$, which is of type $C_{n}$ for
some $n\geq 2$, and $\lambda_{M_{\rm der}}=\omega_{n}^{\vee}$ is the $n$-th
fundamental coweight of $M_{\rm der}$.

thmStembridge

###### Proof.

This is [Ste98, Theorem 2.8]. Namely, the Dynkin diagram of $G_{2}$ does not
arise as a subdiagram of a different Dynkin type in [Bou68, Tables]. Hence,
the Levi subgroup $M$ has no factor of type $G_{2}$, so this case is excluded
from the classification. ∎

Using thmStembridge together with levi-lemma-derived we obtain the following
criterion for non-normal minimal degeneration singularities:

######  Proposition 6.4.

mindeg-nonnormal Assume $G$ is almost simple and semisimple. Let $\lambda<\mu$
be a minimal degeneration in $X_{*}(T)^{+}$, and denote by
$M:=M_{\lambda,\mu}$ the associated Levi subgroup. Then, the minimal
degeneration singularities in $\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$
are non-normal precisely in the following cases:

  1. (1)

$\mu-\lambda$ a simple coroot, $M_{\rm der}\simeq\mathrm{PGL}_{2,k}$ and ${\rm
char}(k)=2$;

  2. (2)

$\mu-\lambda$ quasi-minuscule for $M_{\rm der}$, $\lambda_{M_{\rm der}}=0$ and
${\rm char}(k)\mid\\#\pi_{1}(M_{\rm der})$;

  3. (3)

$\mu-\lambda$ quasi-minuscule for $M_{\rm der}$, which is of type $C_{n}$ for
some $n\geq 2$, $\lambda_{M_{\rm der}}=\omega_{n}^{\vee}$ and ${\rm
char}(k)=2$.

###### Proof.

First off, if $G$ is of type $G_{2}$, then its connection index is $1$. So,
any minimal degeneration singularity is normal by levi-lemma and transversal-
slice-lemm. Therefore, thmStembridge applies. We now go through the different
cases:

  1. (1)

As the corresponding Levi subgroup $M$ has semisimple rank 1, the assertion in
this case follows from levi-lemma and semi-simple-rank-1.

  2. (2)

By levi-lemma-derived and transversal-slice-lemm the minimal degeneration
singularity is smoothly equivalent to the quasi-minuscule Schubert variety for
$M_{\rm der}$. This variety is non-normal if and only if ${\rm
char}(k)\mid\pi_{1}(M_{\rm der})$ by theo:normal and prop:quasi-minuscule.

  3. (3)

As $M_{\rm der}$ is of type $C_{n}$, it is either simply connected or adjoint,
in which case it has connection index $2$. If $M_{\rm der}$ is simply
connected or ${\rm char}(k)\neq 2$, then the minimal degeneration singularity
is normal by levi-lemma and transversal-slice-lemm. If $M_{\rm der}$ is
adjoint and ${\rm char}(k)=2$, levi-lemma-derived and transversal-slice-lemm
reduce us to the case of the Schubert variety for
$\mu=\omega_{1}^{\vee}+\omega_{n}^{\vee}$ in the affine Grassmmannian for
$M_{\rm der}\simeq\mathrm{PSP}_{2n,k}$. As $\omega_{1}^{\vee}$ is quasi-
minuscule and $\omega_{n}^{\vee}$ is minuscule this variety is non-normal by
prop:intro.

∎

######  Remark 6.5.

Section 7 shows that the first case in mindeg-nonnormal only appears for
$G\simeq\mathrm{SO}_{2n+1,k}$ with $n\geq 1$, including
$\mathrm{SO}_{3,k}\simeq\mathrm{PGL}_{2,k}$.

###  6.1. Minimal degeneration singularities in Dynkin type $A_{1}$

The singularities are calculated in [MOV05, Lemma 5.1] and [KWWY14, Example
2.2] over $\mathbb{C}$. The manuscript [Mül08] works over $k$ of
characteristic different from $2$. Based on [HLR22, Appendix B] the result is
generalized in [Kum23] to all $k$. To explain the result we denote by

$A_{G,\leq\mu}^{\lambda}\hskip 2.84544pt\stackrel{{\scriptstyle\text{\tiny\rm
def}}}{{=}}\hskip 2.84544pt\Gamma(\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda},{\mathcal{O}}_{\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}})$ (6.1)

the ring of functions of the transversal slice $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ which is an affine scheme. The following
is a slight generalization of [Kum23]:

######  Proposition 6.6.

semi-simple-rank-1 Assume $G$ has semi-simple rank $1$. Let $\lambda,\mu\in
X_{*}(T)^{+}$ with $\lambda<\mu$ be a minimal degeneration. Then one of the
following cases holds:

  1. (1)

One has $G_{\rm der}\simeq\mathrm{SL}_{2,k}$. Then, the ring
$A_{G,\leq\mu}^{\lambda}$ is isomorphic to

$k[x,y,z]/(z^{\langle 2\rho,\mu\rangle}+xy),$ (6.2)

which is a normal domain of dimension $2$.

  2. (2)

One has $G_{\rm der}\simeq\mathrm{PGL}_{2,k}$. Then, the ring
$A_{G,\leq\mu}^{\lambda}$ is isomorphic to the subring of (6.2) generated by
the elements $x,y,xz,yz,z^{2},2z$. In particular, if ${\rm char}(k)\neq 2$ the
rings $A_{G,\leq\mu}^{\lambda}$ and (6.2) are isomorphic whereas the domain
$A_{G,\leq\mu}^{\lambda}$ is non-normal if ${\rm char}(k)=2$.

###### Proof.

It is well-known that $G\simeq Z\times G^{\prime}$ where $Z$ is a torus and
$G^{\prime}=\mathrm{GL}_{2,k}$ or $G^{\prime}=\mathrm{SL}_{2,k}$ in case (1),
and $G^{\prime}=\mathrm{PGL}_{2,k}$ in case (2). Thus, we may reduce to the
two cases $G=\mathrm{GL}_{2,k}$ and $G=\mathrm{PGL}_{2,k}$ which are treated
in [Kum23]. For convenience we give a different argument here.

(1): $G=\mathrm{GL}_{2,k}$. Fix $T$ to be the subgroup of diagonal matrices
and $B$ the subgroup of upper triangular ones. Then
$X_{*}(T)^{+}\subset{\mathbb{Z}}^{2}$ is the subset of elements
$(a,b)\in{\mathbb{Z}}^{2}$ with $a\geq b$. Since
$\lambda=(\lambda_{1},\lambda_{2})<\mu=(\mu_{1},\mu_{2})$ in $X_{*}(T)^{+}$ is
a minimal degeneration we have $\mu_{1}=\lambda_{1}+1$ and
$\mu_{2}=\lambda_{2}+1$. Further, $\langle 2\rho,\mu\rangle=\mu_{1}-\mu_{2}$
and similarly for $\lambda$, so $\langle 2\rho,\mu-\lambda\rangle=2$. Then,
one checks that the map

$(x,y,z)\mapsto\left(1+\varpi^{-1}\begin{pmatrix}\sum_{i=0}^{\lambda_{1}-\lambda_{2}}(-z)^{i+1}\varpi^{-i}&(-1)^{\lambda_{2}-\lambda_{1}}x\\\
y\varpi^{\lambda_{2}-\lambda_{1}}&z\\\
\end{pmatrix}\right)\cdot\lambda(\varpi)$ (6.3)

induces a closed immersion

$\mathop{\rm
Spec}\left(k[x,y,z]/(z^{\mu_{1}-\mu_{2}}+xy)\right)\longrightarrow\mathop{\rm
Gr}\nolimits_{\mathrm{GL}_{2,k},\leq\mu}^{\lambda}.$ (6.4)

As both schemes are integral of dimension $2$ by transversal-slice-lemm, the
map (6.4) is an isomorphism, see also [HLR22, Lemma B.3] for a similar
argument.

(2): $G=\mathrm{PGL}_{2,k}$. Under the quotient map
$\mathrm{GL}_{2,k}\to\mathrm{PGL}_{2,k}$ the minimal degeneration
$\lambda<\mu$ in $\mathrm{PGL}_{2,k}$ lifts to a minimal degeneration
$\tilde{\lambda}<\tilde{\mu}$ in $\mathrm{GL}_{2,k}$ as in (1). This induces a
map on Schubert varieties $\mathop{\rm
Gr}\nolimits_{\mathrm{GL}_{2,k},\leq\tilde{\mu}}\to\mathop{\rm
Gr}\nolimits_{\mathrm{PGL}_{2,k},\leq\mu}$, so a map on transversal slices

$\mathop{\rm Spec}(A_{\leq\tilde{\mu}}^{\tilde{\lambda}}):=\mathop{\rm
Gr}\nolimits_{\mathrm{GL}_{2,k},\leq\tilde{\mu}}^{\tilde{\lambda}}\longrightarrow\mathop{\rm
Gr}\nolimits_{\mathrm{PGL}_{2,k},\leq\mu}^{\lambda}=:\mathop{\rm
Spec}(A_{\leq\mu}^{\lambda}).$ (6.5)

The map (6.5) identifies $A_{\leq\tilde{\mu}}^{\tilde{\lambda}}$ with the
normalization of the domain $A_{\leq\mu}^{\lambda}$ by the proof of
transversal-slice-lemm (2). It remains to identify the subring
$A_{\leq\mu}^{\lambda}$ under the isomorphism
$A_{\leq\tilde{\mu}}^{\tilde{\lambda}}\cong k[x,y,z]/(z^{m}+xy)$ from (6.4)
where $m:=\langle 2\rho,\mu\rangle$. For this consider the adjoint
representation
$\mathrm{GL}_{2,k}\to\mathrm{GL}(\mathfrak{gl}_{2,k})=\mathrm{GL}_{4,k}$
explicitly given by the formula

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\mapsto\frac{1}{ad-
bc}\begin{pmatrix}ad-bc&0&0&0\\\ 0&ad+bc&-ac&bd\\\ 0&-2ab&a^{2}&-b^{2}\\\
0&2cd&-c^{2}&d^{2}\\\ \end{pmatrix}.$ (6.6)

This induces a closed immersion $\mathrm{PGL}_{2,k}\to\mathrm{GL}_{4,k}$,
hence a closed immersion $\mathop{\rm
Gr}\nolimits_{\mathrm{PGL}_{2,k}}\to\mathop{\rm
Gr}\nolimits_{\mathrm{GL}_{4,k}}$ of affine Grassmannians. In particular, the
image of $\mathop{\rm Gr}\nolimits_{\mathrm{PGL}_{2,k},\leq\mu}$ in
$\mathop{\rm Gr}\nolimits_{\mathrm{GL}_{4,k}}$ agrees with the scheme
theoretic image of $\mathop{\rm
Gr}\nolimits_{\mathrm{GL}_{2,k},\leq\tilde{\mu}}$ under $\mathop{\rm
Gr}\nolimits_{\mathrm{GL}_{2,k}}\to\mathop{\rm
Gr}\nolimits_{\mathrm{GL}_{4,k}}$. In order to describe
$A_{\leq\mu}^{\lambda}$ as a subring of
$A_{\leq\tilde{\mu}}^{\tilde{\lambda}}$ one computes the image of the matrix
in (6.3) under (6.6). Then $A_{\leq\mu}^{\lambda}$ is the subring of
${\mathbb{Z}}[x,y,z]/(z^{m}+xy)$ generated by the coefficients, which lie in
${\mathbb{Z}}[x,y,z]$, as polynomials in $\varpi^{\pm 1}$ of all entries. The
diagonal matrix $\tilde{\lambda}(\varpi)$ in (6.3) maps under the adjoint
representation to $\mathrm{diag}(1,1,\varpi^{n},\varpi^{-n})$ where
$n:=m-2=\langle 2\rho,\lambda\rangle$. As multiplication by this matrix does
not effect the coefficients as polynomials in $\varpi^{\pm 1}$ we only have to
compute the image of

$1+\varpi^{-1}\begin{pmatrix}\sum_{i=0}^{n}(-z)^{i+1}\varpi^{-i}&(-1)^{-n}x\\\
y\varpi^{-n}&z\\\
\end{pmatrix}=\begin{pmatrix}1+\sum_{i=0}^{n}(-\frac{z}{\varpi})^{i+1}&(-1)^{-n}x\varpi^{-1}\\\
y\varpi^{-n-1}&1+z\varpi^{-1}\\\ \end{pmatrix}$ (6.7)

under the adjoint representation. From the entries $bd$, $-ac$ and $d^{2}$ in
(6.6) we see that $x,xz$, $y,yz$ and $2z,z^{2}$ respectively lie in
$A_{\leq\mu}^{\lambda}$, so do $x^{2}$ and $y^{2}$ covering the entries
$-b^{2}$ and $-c^{2}$. As $2A_{\leq\tilde{\mu}}^{\tilde{\lambda}}$ lies in the
subring generated by $x,xz,y,yz,2z,z^{2}$ the coefficients of the remaining
entries in (6.6) lie in it as well. ∎

##  7\. The classification

In this section we finish the proof of classification-normal-schubert-
varieties, that is, we classify all normal Schubert varieties in
characteristic ${\rm char}(k)\mid\\#\pi_{1}(G)$. As the root systems of type
$E_{8}$, $F_{4}$ and $G_{2}$ have connection index $1$, these cases are
excluded from our list.

Let $T\subset B\subset G$ be as in Section 2.1 with $G$ almost simple,
semisimple and ${\rm char}(k)\mid\\#\pi_{1}(G)$. We use the notation of the
previous sections and [Bou68, Tables]: Let $n$ be the rank of $G$. We denote
by $\Phi^{\vee}(G,T)$ the set of coroots for $G$ with respect to $T$, by
$\Delta=\\{\alpha_{1},\ldots,\alpha_{n}\\}$ the set of $B$-simple roots and by
$\Delta^{\vee}=\\{\alpha_{1}^{\vee},\ldots,\alpha_{n}^{\vee}\\}$ the
corresponding simple coroots. The fundamental coweights
$\omega_{1}^{\vee},\ldots,\omega_{n}^{\vee}$ form the ${\mathbb{Z}}$-basis of
the coweight lattice $P_{G}^{\vee}$ dual to $\alpha_{1},\ldots,\alpha_{n}$.
For a subset $I\subset\Delta^{\vee}$ we denote by $M_{I}$ the corresponding
Levi subgroup of $G$. For $\lambda,\mu\in X_{*}(T)^{+}$ with $\lambda\leq\mu$
let $I_{\lambda,\mu}\subset\Delta^{\vee}$ be the subset of simple coroots that
appear in $\mu-\lambda$ with a non-zero coefficient. We use the notation

${\mathbb{Z}}^{n}_{\text{even}}=\big{\\{}(a_{1},\ldots,a_{n})\in{\mathbb{Z}}^{n}\colon\textstyle{\sum_{i=1}^{n}}a_{i}\equiv
0\text{ mod
}2\big{\\}}\quad\text{and}\quad{\mathbb{Z}}^{n}_{\Sigma=0}=\big{\\{}(a_{1},\ldots,a_{n})\in{\mathbb{Z}}^{n}\colon\textstyle{\sum_{i=1}^{n}}a_{i}=0\big{\\}}.$

###  7.1. Strategy of proof

Let $\mu_{\rm qm}\in X_{*}(T)^{+}$ be the quasi-minuscule element. We freely
use the following results during the proof:

  1. (1)

The Schubert variety $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is non-normal
whenever $\lambda+\mu_{\rm qm}\leq\mu$ for some minuscule $\lambda\in
X_{*}(T)^{+}$ , see prop:intro.

  2. (2)

One has $\mathop{\rm Gr}\nolimits_{M,\leq\mu}^{\lambda}\cong\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ for all $\lambda,\mu\in X_{*}(T)^{+}$ with
$\lambda\leq\mu$ and $M=M_{I_{\lambda,\mu}}$, see levi-lemma. Under the
condition and notation in levi-lemma-derived, one also has $\mathop{\rm
Gr}\nolimits_{M_{\rm der},\leq\mu_{\rm der}}^{\lambda_{\rm
der}}\cong\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$.

  3. (3)

If ${\rm char}(k)\nmid\\#\pi_{1}(M_{\rm der})$ with $M$ as in (2) (for
example, $M_{\rm der}$ simply connected), then $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is normal and so is $\cup_{\nu\in
X_{*}(T)^{+}:\lambda\leq\nu\leq\mu}\mathop{\rm Gr}\nolimits_{G,\nu}$, see
theo:normal and transversal-slice-lemm.

With principles (1)–(3) in hand we use a case by case analysis relying on the
classification of groups [Bou68, Tables]. In many cases $M_{\rm der}$ turns
out to be simply connected. The remaining cases are handled by (1) applied to
$M$ and mindeg-nonnormal.

###  7.2. Type $A_{n}$, $n\geq 1$

The case $G=\mathrm{PGL}_{2,k}$ is treated in [HLR22, Corollary 6.8], see also
semi-simple-rank-1. So, we consider $n\geq 2$. Let
$G\simeq\mathrm{SL}_{n+1,k}/{\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}}_{\ell,k}$ for some $1<\ell\mid(n+1)$ with ${\rm
char}(k)\mid\ell$.

######  Lemma 7.1.

lemm:simply-connected-type-A Let $I$ be a proper subset of $\Delta^{\vee}$.
Then, the derived subgroup of $M=M_{I}$ is simply connected. In particular,
when $I$ is connected $M_{\rm der}\simeq\mathrm{SL}_{m+1,k}$, where
$m:=\\#I<n$. lemAnLevi

###### Proof.

It clearly suffices to check the case $G=\mathrm{PGL}_{n+1,k}$. Using cochars-
derivedgroup we compute

$X_{\ast}(T\cap M_{\rm
der})=X_{\ast}(T)\cap{\mathbb{Q}}\Phi^{\vee}(M,T)={\mathbb{Z}}^{n+1}/\langle(1,\ldots,1)\rangle\cap{\mathbb{Q}}_{\Sigma=0}^{I}={\mathbb{Z}}_{\Sigma=0}^{I}={\mathbb{Z}}\Phi^{\vee}(M,T).$

Hence, $M_{\rm der}$ is simply connected and thus isomorphic to
$\mathrm{SL}_{m+1,k}$ when $I$ is connected. ∎

We use lemm:simply-connected-type-A to prove coro:intro:

###### Proof of coro:intro.

Let $\lambda,\mu\in X_{*}(T)^{+}$ with $\lambda<\mu$ such that
$\mu-\lambda=\sum_{\alpha\in\Delta}n_{\alpha}\alpha^{\vee}$ for some
$n_{\alpha}\in{\mathbb{Z}}_{\geq 0}$ and with at least one $n_{\alpha}=0$. In
particular, $I_{\lambda,\mu}$ is a proper subset of $\Delta^{\vee}$. So by
lemAnLevi the derived subgroup of $M=M_{I_{\lambda,\mu}}$ is simply connected.
Hence, $\mathop{\rm Gr}\nolimits_{M,\leq\mu}^{\lambda}\cong\mathop{\rm
Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is normal. As the subset $J_{\mu}$ is the
union of all $\lambda$ as above the open subvariety $\cup_{\lambda\in
J_{\mu}}\mathop{\rm Gr}\nolimits_{G,\lambda}$ in $\mathop{\rm
Gr}\nolimits_{G,\leq\mu}$ is normal. ∎

######  Corollary 7.2.

For $\mu\in\\{d\omega_{1}^{\vee},d\omega_{n}^{\vee}\\}$ the variety
$\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is normal for all
$d\in\\{1,\ldots,n\\}$.

###### Proof.

It suffices to treat the case $G=\mathrm{PGL}_{n+1,k}$. Let
$\lambda=\omega_{d}^{\vee}$ which is the unique minuscule coweight with
$\lambda\leq\mu$. It is quickly verified that
$I_{\lambda,d\omega_{1}^{\vee}}=\\{\alpha_{1}^{\vee},\ldots,\alpha_{d-1}^{\vee}\\}$
and
$I_{\lambda,d\omega_{n}^{\vee}}=\\{\alpha_{d+1}^{\vee},\ldots,\alpha_{n}^{\vee}\\}$.
With the notation of coro:intro one has $\cup_{\lambda\in J_{\mu}}\mathop{\rm
Gr}\nolimits_{G,\lambda}=\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ which is
normal. ∎

The following lemma finishes the proof of classification-normal-schubert-
varieties in Dynkin type $A_{n}$:

######  Lemma 7.3.

Let $\mu\in X_{\ast}(T)\setminus\\{0\\}$ be dominant. Then, $\mu$ satisfies
precisely one of the following two properties:

  * •

$\mu\leq d\omega_{1}^{\vee}$ or $\mu\leq d\omega_{n}^{\vee}$ for some
$d\in\\{1,\ldots,n\\}$;

  * •

$\mu\geq\lambda+\mu_{\rm qm}$ for some $\lambda\in X_{*}(T)^{+}$ minuscule.

###### Proof.

It is easy to see that the two cases are mutually exclusive. Let $\lambda\in
X_{*}(T)^{+}$ be the unique minuscule cocharacter with $\lambda<\mu$. When
$\lambda=0$, the claim follows from the fact that the quasi-minuscule
cocharacter $\mu_{\rm qm}$ is the unique minimal degeneration of $\lambda$.
Let us now assume that $\lambda=\omega_{d}^{\vee}$ for some
$d\in\\{1,\ldots,n\\}$ and $\mu\not\geq\lambda+\mu_{\rm qm}$. As $\mu_{\rm
qm}=\sum_{i=1}^{n}\alpha_{i}^{\vee}$, this means that $I_{\lambda,\mu}$ is a
proper subset of $\Delta^{\vee}$ and hence a subset of
$\\{\alpha_{1}^{\vee},\ldots,\alpha_{d-1}^{\vee}\\}$ or
$\\{\alpha_{d+1}^{\vee},\ldots,\alpha_{n}^{\vee}\\}$. In the first case we get
$\mu\leq d\omega_{1}^{\vee}$ and in the second case $\mu\leq
d\omega_{n}^{\vee}$. ∎

###  7.3. Type $B_{n}$, $n\geq 3$

The case $B_{2}=C_{2}$ is treated in the next subsection. The root system of
type $B_{n}$ has connection index $2$. So, we consider a simple group
$G\simeq\mathrm{SO}_{2n+1,k}$ with ${\rm char}(k)=2$. classification-normal-
schubert-varieties in Dynkin type $B_{n}$ follows from the next lemma:

######  Lemma 7.4.

lem:classification-Bn The unique minuscule $\lambda:=\omega_{1}^{\vee}\in
X_{*}(T)\backslash\\{0\\}$ admits $\lambda<\mu:=\omega_{3}^{\vee}$ as the
unique minimal degeneration, and $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is
non-normal. In particular, every non-minuscule $\nu\in X_{*}(T)^{+}$ satisfies
$\nu\geq\mu_{\rm qm}$ or $\nu\geq\mu$, so the normal Schubert varieties in
$\mathop{\rm Gr}\nolimits_{G}$ are precisely the minuscule ones.

###### Proof.

The fact that $\lambda<\mu$ is the unique minimal degeneration of $\lambda$
follows from thmStembridge. Namely, it is case (2) of the theorem. Further, we
note that
$\omega_{3}^{\vee}-\omega_{1}^{\vee}=\sum_{i=2}^{n}\alpha_{i}^{\vee}$, so
$I_{\lambda,\mu}=\\{\alpha_{2}^{\vee},\ldots,\alpha_{n}^{\vee}\\}$. Hence, the
corresponding Levi subgroup $M=M_{I_{\lambda,\mu}}$ is of type $B_{n-1}$. By
cochars-derivedgroup together with the explicit description of coroots and
coweights in [Bou68], we see that

$X_{*}(T\cap M_{{\rm
der}})=X_{*}(T)\cap{\mathbb{Q}}\Phi^{\vee}(M,T)=\bigoplus_{i\in
I_{\lambda,\mu}}{\mathbb{Z}}(\omega_{M})_{i}^{\vee}={\mathbb{Z}}^{n-1}$

contains ${\mathbb{Z}}\Phi^{\vee}(M_{\rm der},T\cap M_{\rm
der})={\mathbb{Z}}^{n-1}_{\rm{even}}$ as an index 2 subgroup. In particular,
$M_{\rm der}\cong\mathrm{SO}_{2n-1,k}$ is adjoint, and so $\mathop{\rm
Gr}\nolimits_{M_{\rm der},\leq\mu_{\rm der}}^{\lambda_{\rm
der}}\cong\mathop{\rm Gr}\nolimits_{G,\leq\mu}^{\lambda}$ is non-normal by
mindeg-nonnormal (2). ∎

###  7.4. Type $C_{n}$, $n\geq 2$

The root system of type $C_{n}$ has connection index 2. So, we consider an
adjoint group
$G\simeq\mathrm{PSP}_{2n,k}=\mathrm{Sp}_{2n,k}/{\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}}_{2,k}$ with ${\rm char}(k)=2$. We include the case $n=2$, that
is, $\mathrm{SO}_{5,k}\simeq\mathrm{PSP}_{4,k}$. classification-normal-
schubert-varieties in Dynkin type $C_{n}$ follows from the next lemma:

######  Lemma 7.5.

Both minuscule cocharacters $\lambda$ of $G$ admit $\lambda<\lambda+\mu_{\rm
qm}$ as the unique minimal degeneration. In particular, the normal Schubert
varieties in $\mathop{\rm Gr}\nolimits_{G}$ are precisely the minuscule ones.

###### Proof.

When $\lambda=0$, the assertion is obvious. If $\lambda\neq 0$, then
$\lambda+\mu_{\rm qm}$ is the unique minimal degeneration by thmStembridge.
Namely, it corresponds to case (3) of the theorem. ∎

###  7.5. Type $D_{n}$ for $n\geq 4$

The root system $D_{n}$ has connection index $4$, so ${\rm char}(k)=2$. Recall
that the quotient of the weight lattice by the root lattice is isomorphic to
${\mathbb{Z}}/2{\mathbb{Z}}\times{\mathbb{Z}}/2{\mathbb{Z}}$ when $n$ is even
and ${\mathbb{Z}}/4{\mathbb{Z}}$ when $n$ is odd. Thus, when $n$ is odd,
$G\simeq\mathrm{SO}_{2n,k}$ or
$G\simeq\mathrm{PSO}_{2n,k}=\mathrm{SO}_{2n,k}/{\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}}_{2,k}$. When $n$ is even there is a third possibility.
classification-normal-schubert-varieties in Dynkin type $D_{n}$ follows from
the next lemma:

######  Lemma 7.6.

For $\mu\in
X_{*}(T)\cap\\{\omega_{2}^{\vee},\omega_{3}^{\vee},\omega_{2}^{\vee}+\omega_{n-1}^{\vee},\omega_{2}^{\vee}+\omega_{n}^{\vee}\\}$
the Schubert variety $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ is non-normal.
Moreover, for $\lambda\in
X_{*}(T)\cap\\{\omega_{1}^{\vee}+\omega_{n-1}^{\vee},\omega_{1}^{\vee}+\omega_{n}^{\vee}\\}$
the Schubert variety $\mathop{\rm Gr}\nolimits_{G,\leq\lambda}$ is normal if
and only if $G\simeq\mathrm{PSO}_{2n,k}$ for $n$ odd.

###### Proof.

First off, one can quickly verify the description of the Bruhat partial order
in Figure 1 using thmStembridge. So, the non-normality of the Schubert
varieties for $\omega_{2}^{\vee},\omega_{2}^{\vee}+\omega_{n}^{\vee}$ and
$\omega_{2}^{\vee}+\omega_{n-1}^{\vee}$ is clear.

The coweight $\omega_{3}^{\vee}$ is a dominant cocharacter for
$\mathrm{SO}_{2n,k}$ and $\mathrm{PSO}_{2n,k}$. By thmStembridge the minimal
degeneration $\omega_{1}^{\vee}<\omega_{3}^{\vee}$ is quasi-minuscule for
$M_{\rm der}$ where $M=M_{I}$ with
$I=\\{\alpha_{2}^{\vee},\ldots,\alpha_{n}^{\vee}\\}$. We have
${\mathbb{Z}}\Phi^{\vee}(M,T)={\mathbb{Z}}^{n}_{\rm{even}}$ and
$X_{*}(T)={\mathbb{Z}}^{n}$ for $G\cong\mathrm{SO}_{2n,k}$ or
$X_{*}(T)={\mathbb{Z}}^{n}+{\mathbb{Z}}(1/2,\ldots,1/2)$ for
$G\simeq\mathrm{PSO}_{2n,k}$. One has $X_{\ast}(T\cap M_{\rm
der})={\mathbb{Q}}\Phi^{\vee}(M,T)\cap
X_{\ast}(T)\simeq{\mathbb{Q}}^{n}\cap{\mathbb{Z}}^{n}={\mathbb{Z}}^{n}$ by
cochars-derivedgroup. Thus, $\pi_{1}(M_{\rm
der})\cong{\mathbb{Z}}/2{\mathbb{Z}}$ and the minimal degeneration singularity
$\mathop{\rm Gr}\nolimits^{\omega_{1}^{\vee}}_{G,\leq\omega_{3}^{\vee}}$ is
non-normal by mindeg-nonnormal. Note that $M_{\rm
der}\simeq\mathrm{SO}_{2n-2,k}$ for $n\geq 5$ and $M_{\rm
der}\simeq\mathrm{SL}_{4,k}/{\mbox{$\raisebox{-2.54025pt}{$l$}\mu\raisebox{-4.2194pt}{\scalebox{2.0}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}.$}}\raisebox{3.7889pt}{$\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>4.60007pt$}{}}_{2,k}$ for $n=4$.

The minimal degeneration
$\omega_{n}^{\vee}<\omega_{1}^{\vee}+\omega_{n-1}^{\vee}$ is quasi-minuscule
for the Levi subgroup $M=M_{I}$ with
$I=\\{\alpha_{1}^{\vee},\ldots,\alpha_{n-1}^{\vee}\\}$. In this case
${\mathbb{Z}}\Phi^{\vee}(M,T)={\mathbb{Z}}^{n}_{\Sigma=0}$. By cochars-
derivedgroup, $X_{*}(T\cap M_{\rm
der})={\mathbb{Z}}^{n}_{\Sigma=0}+{\mathbb{Z}}(1/2,\ldots,1/2,-1/2,\ldots,-1/2)$,
so $\\#\pi_{1}(M_{\rm der})=2$, when $n$ is even and $X_{*}(T\cap M_{\rm
der})={\mathbb{Z}}^{n}_{\Sigma=0}$, so $\\#\pi_{1}(M_{\rm der})=1$, when $n$
is odd. The claim now follows from mindeg-nonnormal. ∎

${2\omega_{1}^{\vee}}$${0}$${\omega_{2}^{\vee}}$${\cdots}$${\omega_{4}^{\vee}}$${(\text{only
for }n\geq
6)}$$\scriptstyle{\\{1,\ldots,n\\}}$$\scriptstyle{\\{3,\ldots,n\\}}$$\scriptstyle{\\{1\\}}$

${\omega_{1}^{\vee}+\omega_{2}^{\vee}}$${\omega_{1}^{\vee}}$${\omega_{3}^{\vee}}$${\ldots}$${\omega_{5}^{\vee}}$${(\text{only
for }n\geq
7)}$$\scriptstyle{\\{2,\ldots,n\\}}$$\scriptstyle{\\{1,2\\}}$$\scriptstyle{\\{1,\ldots,4\\}}$

${\omega_{n-1}^{\vee}}$${\omega_{1}^{\vee}+\omega_{n}^{\vee}}$${\omega_{2}^{\vee}+\omega_{n-1}^{\vee}}$${\ldots}$$\scriptstyle{\\{1,\ldots,n-2,n\\}}$$\scriptstyle{\\{2,\ldots,n-1\\}}$

${\omega_{n}^{\vee}}$${\omega_{1}^{\vee}+\omega_{n-1}^{\vee}}$${\omega_{2}^{\vee}+\omega_{n}^{\vee}}$${\ldots}$$\scriptstyle{\\{1,\ldots,n-1\\}}$$\scriptstyle{\\{2,\ldots,n-2,n\\}}$

Figure 1. The Bruhat partial order on dominant cocharacters for
$\mathrm{PSO}_{2n,k}$. Each arrow is labelled by the subset of indices of
simple coroots appearing in the associated minimal degeneration.

###  7.6. Type $E_{6}$

The root system of type $E_{6}$ has connection index 3. So, we consider an
adjoint group $G$ of type $E_{6}$ with ${\rm char}(k)=3$. classification-
normal-schubert-varieties in Dynkin type $E_{6}$ follows from the next lemma:

${0}$${\omega_{2}^{\vee}}$${\omega_{1}^{\vee}+\omega_{6}^{\vee}}$${\omega_{4}^{\vee}}$${\ldots}$$\scriptstyle{\\{1,\ldots,6\\}}$$\scriptstyle{\\{1,3,\ldots,6\\}}$$\scriptstyle{\\{2,\ldots,5\\}}$

${2\omega_{6}^{\vee}}$${\omega_{1}^{\vee}}$${\omega_{5}^{\vee}}$${\omega_{3}^{\vee}+\omega_{6}^{\vee}}$${\ldots}$${\omega_{1}^{\vee}+\omega_{2}^{\vee}}$$\scriptstyle{\\{1,\ldots,5\\}}$$\scriptstyle{\\{2,\ldots,6\\}}$$\scriptstyle{\\{6\\}}$$\scriptstyle{\\{1,\ldots,4\\}}$$\scriptstyle{\\{3,\ldots,6\\}}$

${2\omega_{1}^{\vee}}$${\omega_{6}^{\vee}}$${\omega_{3}^{\vee}}$${\omega_{5}^{\vee}+\omega_{1}^{\vee}}$${\ldots}$${\omega_{6}^{\vee}+\omega_{2}^{\vee}}$$\scriptstyle{\\{2,\ldots,6\\}}$$\scriptstyle{\\{1,\ldots,5\\}}$$\scriptstyle{\\{1\\}}$$\scriptstyle{\\{2,4,5,6\\}}$$\scriptstyle{\\{1,3,4,5\\}}$

Figure 2. The Bruhat partial order on dominant cocharacters for $G$ of type
$E_{6}$. The boxes indicate the normal Schubert varieties.

######  Lemma 7.7.

The normal Schubert varieties in $\mathop{\rm Gr}\nolimits_{G}$ are precisely
the minuscule ones and $\mathop{\rm Gr}\nolimits_{G,\leq\mu}$ for all
$\mu\in\\{2\omega_{1}^{\vee},\omega_{3}^{\vee},\omega_{5}^{\vee},2\omega_{6}^{\vee}\\}$.

###### Proof.

As in case $D$ above one checks the Bruhat partial order in Figure 2 using the
Stembridge classification. So, the Schubert varieties for all dominant
cocharacters not listed in the lemma are non-normal.

Now let $\mu\in\\{\omega_{5}^{\vee},2\omega_{6}^{\vee}\\}$ and
$\lambda=\omega_{1}^{\vee}$. Then,
$I_{\lambda,\mu}=\\{\alpha_{2}^{\vee},\ldots,\alpha_{6}^{\vee}\\}$. Hence, the
Levi subgroup $M_{I_{\lambda,\mu}}$ is of type $D_{5}$. In particular,
$3\nmid\\#\pi_{1}(M_{\rm der})$ and so the transversal slice $\mathop{\rm
Gr}\nolimits^{\lambda}_{G,\leq\mu}$ is normal. The assertion for
$\mu\in\\{\omega_{3}^{\vee},2\omega_{1}^{\vee}\\}$ follows similarly. ∎

###  7.7. Type $E_{7}$

The root system of type $E_{7}$ has connection index 2. So, we consider an
adjoint group $G$ of type $E_{7}$ with ${\rm char}(k)=2$. classification-
normal-schubert-varieties in Dynkin type $E_{7}$ follows from the next lemma:

${0}$${\omega_{1}^{\vee}}$${\omega_{6}^{\vee}}$${\ldots}$$\scriptstyle{\\{1,..,7\\}}$$\scriptstyle{\\{2,..,7\\}}$

${\omega_{7}^{\vee}}$${\omega_{2}^{\vee}}$${\omega_{1}^{\vee}+\omega_{7}^{\vee}}$${\dots}$$\scriptstyle{\\{1,..,6\\}}$$\scriptstyle{\\{1,3,..,7\\}}$

Figure 3. Same description as in Figure 2 but for $G$ of type $E_{7}$.

######  Lemma 7.8.

The normal Schubert varieties in $\mathop{\rm Gr}\nolimits_{G}$ are precisely
the minuscule ones as well as $\mathop{\rm
Gr}\nolimits_{G,\leq\omega_{2}^{\vee}}$.

###### Proof.

We proceed as for type $E_{6}$. One checks Figure 3 using the Stembridge
classification. So, the Schubert varieties for all dominant cocharacters not
listed in the lemma are non-normal.

Now let $\mu=\omega_{2}^{\vee}$ and $\lambda=\omega_{7}^{\vee}$. Then,
$I_{\lambda,\mu}=\\{\alpha_{1}^{\vee},\ldots,\alpha_{6}^{\vee}\\}$. Hence, the
Levi subgroup $M_{I_{\lambda,\mu}}$ is of type $E_{6}$. In particular,
$2\nmid\\#\pi_{1}(M_{\rm der})$ and so the transversal slice $\mathop{\rm
Gr}\nolimits^{\lambda}_{G,\leq\mu}$ is normal. ∎

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