# On the smooth locus of affine Schubert varieties

Georgios Pappas  Dept. of Mathematics, Michigan State University, E. Lansing,
MI 48824, USA<EMAIL_ADDRESS>and  Rong Zhou  Department of Pure Mathematics
and Mathematical Statistics, University of Cambridge, Cambridge, UK, CB3 0WA
<EMAIL_ADDRESS>
###### Abstract.

We give a simple and uniform proof of a conjecture of Haines–Richarz
characterizing the smooth locus of Schubert varieties in twisted affine
Grassmannians. Our method is elementary and avoids any representation
theoretic techniques, instead relying on a combinatorial analysis of tangent
spaces of Schubert varieties.

###### Contents

  1. 1 Introduction
  2. 2 Tangent spaces of partial affine flag varieties
  3. 3 Root curves in twisted affine Grassmannians
  4. 4 Tangent spaces for minimal degenerations

##  1\. Introduction

Schubert subvarieties of flag varieties, classical or affine, and their
singularities have been a subject of considerable interest for some time. A
basic problem is to characterize the Schubert varieties which are smooth or,
more generally, identify their smooth locus; see, for example, [BL00] for many
results in the classical case. It turns out that this problem is simpler for
Schubert varieties in affine Grassmannians. Indeed, it was first shown by
Evens–Mirkovic [EM99] that, for Schubert varieties in affine Grassmannians of
split groups, the smooth locus is simply the ‘obvious’ open orbit. This result
follows from their calculation of the characteristic cycle of the intersection
cohomology sheaf of the Schubert variety. Affine Grassmannians are also
objects of Kac-Moody theory: The split case above is obtained from ‘untwisted’
affine Kac–Moody Lie algebras but one can equally consider the general
(twisted) affine case, cf. [Kum02, Ch. 13]. The corresponding (twisted) affine
Grassmannians can also be constructed using loop groups obtained from
reductive groups over the field $\mathbb{C}(\\!(t)\\!)$ of Laurent power
series, see [PR08], also for the comparison between the two points of view. In
this group-theoretic set-up, a generalization of the Evens–Mirkovic result was
conjectured by Haines–Richarz [HR20, Conjecture 5.4].

In this paper, we give a simple combinatorial proof of both the original
result of Evens–Mirkovic and of its extension conjectured by Haines–Richarz.
Let us now give some details. We formulate our results using the theory of
reductive groups over the discretely valued field $\mathbb{C}(\\!(t)\\!)$. Let
$G$ be a connected reductive group over $F=\mathbb{C}(\\!(t)\\!)$. Associated
to any special vertex $\mathfrak{s}$ in the Bruhat–Tits building for $G$, we
have an associated group scheme $\mathcal{G}$ over $\mathbb{C}[\\![t]\\!]$. We
let $\mathrm{Gr}_{\mathcal{G}}:=LG/L^{+}\mathcal{G}$ denote the twisted affine
Grassmannian associated to $\mathcal{G}$, see [PR08]. We fix $T\subset
B\subset G$ a maximal torus and Borel pair in $G$ defined over $F$ which are
in good position relative to $\mathfrak{s}$, and let
$I=\mathrm{Gal}(\overline{F}/F)$. Then to each dominant element $\mu\in
X_{*}(T)_{I}^{+}$, we have an associated point
$t^{\mu}\in\mathrm{Gr}_{\mathcal{G}}(\mathbb{C})$ and the associated (affine)
Schubert variety $\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$. By definition
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ is the closure of the
$L^{+}\mathcal{G}$-orbit $\mathrm{Gr}_{\mathcal{G},\mu}$ of $t^{\mu}$. Our
main theorem is the following, which verifies [HR20, Conjecture 5.4].

######  Theorem 1.1.

Let $\mathfrak{s}$ be an absolutely special vertex. Then the smooth locus of
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ is
$\mathrm{Gr}_{\mathcal{G},\mu}$.

As we mentioned above, the result is well-known in the case of split groups
due to work of Evens–Mirkovic [EM99], see also Malkin–Ostrik–Vybornov [MOV05,
Cor. B]. When the group $G$ is not split, the picture is complicated by a
phenomenon which was first observed in the theory of local models for integral
models of Shimura varieties by Richarz ([Arz09, Proposition 4.6]): If the
vertex $\mathfrak{s}$ is special but not absolutely special, it can happen
that the smooth locus of a Schubert variety is larger than the open stratum.
This implies the existence of certain Shimura varieties with ‘exotic good
reduction’ (i.e. Shimura varieties with non-hyperspecial level at a prime
number $p$ which, nevertheless, have good reduction at $p$, see [HPR20, §5]).
Our result verifies the expectation that this does not happen in any other
case. Let us also mention here that the related problem of classifying all
Schubert varieties in twisted affine Grassmannians which are smooth has
already been settled by Haines and Richarz in [HR20].

This conjecture has also been considered in work of Besson–Hong [BH], who
verified it when $G_{\mathrm{ad}}$ does not have factors of type $C-BC_{n}$.
Their method relies on difficult representation theoretic techniques. In
particular, it crucially uses an extension of a result of Zhu [Zhu09] which
gives a geometric interpretation of the Frenkel–Kac–Segal isomorphism between
basic representations of affine Lie algebras and lattice vertex algebras.
Furthermore, in the case of $E_{6}$, the proof requires computer assistance.
Combined with Richarz’s thesis, which analyses the $C-BC_{n}$ case explicitly,
this can be used to deduce the conjecture in general. In contrast, our method
is simpler, avoiding any use of representation theory and computer aided
computations, and can be applied uniformly to all cases. Specialized to the
case of split groups, it gives a new proof of the result in [EM99] and
[MOV05]. Moreover, Theorem 1.1, combined with now standard techniques, can be
used to classify the smooth locus of twisted affine Grassmannians when
$\mathbb{C}$ is replaced by an algebraically closed field of any
characteristic (cf. [HR20, Corollary 6.1], [Lou23]).

We now give some details of the proof. Since the smooth locus is open, it
suffices to show that when $\mathfrak{s}$ is absolutely special,
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ is singular along the irreducible
components of the boundary. These are given by
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\lambda}$, where
$\mu\rightsquigarrow\lambda$ is a minimal degeneration (cf. §4.1). To prove
this, we consider a lower bound for the dimension of the tangent space to
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ at $t^{\lambda}$. We obtain such a
bound by considering the tangent vectors arising from smooth curves in the
Schubert variety coming from affine root groups; we call these root curves. We
show that for a minimal degeneration, the lower bound we obtain is equal to
the dimension of the Schubert variety; here we use Stembridge’s classification
of minimal degenerations and some simple computations using the échelonnage
root system. When $\mathfrak{s}$ is absolutely special, conjugating one of the
root curves by an element of the opposite unipotent gives an extra tangent
vector which makes the point non-smooth; this extra tangent direction has a
component ‘along the Cartan direction’ (cf. Proposition 2.9). The assumption
that $\mathfrak{s}$ is absolutely special is crucial here; in the case of
exotic good reduction the shape of the neighbouring affine root groups for the
special but not absolutely vertex prevents us from producing extra tangent
directions. From this point of view, one can think of the singularities of
minimal degenerations as being caused by extra tangent directions along the
Cartan. Roughly speaking, the moral of the main Theorem of [Zhu09] is similar:
Properties of $\mathrm{Gr}_{\mathcal{G}}$ can be studied by restriction to the
$0$-dimensional ind-subscheme
$\mathrm{Gr}_{\mathcal{T}}\subset\mathrm{Gr}_{\mathcal{G}}$, where
$\mathcal{T}\subset\mathcal{G}$ is a maximal torus.

We now give an outline of the paper. In section 2, we give an explicit
description of the tangent space of partial affine flag varieties at the base
point in terms of affine roots. This should be well-known to experts, but we
provide the details here as they are needed for later. In section 3, we use
the tangent vectors from root curves to give a lower bound on the dimension of
tangent spaces of Schubert varieties, and in section 4 we apply the bound to
the boundary components of minimal degenerations to prove the main result. At
the end of the section, we give an alternative, less combinatorial, argument
and finish with an example of the method applied to the case of the quasi-
miniscule Schubert variety in the ramified triality, which gives a short proof
of a result of Haines–Richarz [HR20, §8].

Acknowledgements: G. P. is supported by NSF grant #DMS-2100743. R. Z. is
supported by EPSRC grant ref. EP/Y030648/1 as part of the ERC Starting Grant
guarantee scheme. The authors would like to thank Michael Rapoport and Timo
Richarz for comments on an earlier version of this draft.

##  2\. Tangent spaces of partial affine flag varieties

We give a description of the tangent space of partial affine flag varieties at
the base point in terms of the associated affine root system. Such a
description can also be deduced from the relationship between the affine flag
varieties and Kac–Moody algebras as in [HLR18, Proposition 4.5]. However our
construction of the isomorphism in Proposition 2.9 and its relation with
affine root groups will be useful for our purposes.

### 2.1.

Let $F=\mathbb{C}(\\!(t)\\!)$ and $G/F$ be an adjoint, absolutely simple
reductive group. Let $H$ be the split Chevalley form of $G$ and let
$F^{\prime}:=\mathbb{C}(\\!(u)\\!)/F$ be the minimal splitting field for $G$
of degree $e$ with $u^{e}=t$. Let $\tau$ be a generator of
$I:=\mathrm{Gal}(F^{\prime}/F)$; then $\tau(u)=\zeta u$ where $\zeta$ is an
$e^{\mathrm{th}}$ root of unity. Since $G$ is absolutely simple, we have
$e=1,2$ or $3$. For any affine group scheme $X$ over $F$, we write $LX$ for
the loop group scheme over $\mathbb{C}$ associated to $X$. Similarly for
$\mathcal{X}$ an affine group scheme over
$\mathcal{O}_{F}:=\mathbb{C}[\\![t]\\!]$, we write $L^{+}\mathcal{X}$ for the
positive loop group scheme over $\mathbb{C}$ associated to $\mathcal{X}$, see
[PR08].

Let $H$ denote the split Chevalley form over $\mathbb{C}$ of $G$. Upon fixing
a Chevalley pinning $(B_{H},T_{H},\\{X_{\alpha^{\prime}}\\})$ of $H$, we may
realize $G$ as the functor whose value on an $F$-algebra $R$ is given by

$G(R):=\\{g\in H(R\otimes_{F}F^{\prime})\,|\,\sigma(g)=g\\}$

where $\sigma=\sigma_{0}\circ\tau$, and $\sigma_{0}$ is a diagram automorphism
of the Dynkin diagram of $H$, which we consider as an element of
$\mathrm{Aut}(H)$ via the pinning.

The torus $T_{H}$ determines a maximal torus $T$ of $G$, and we let $S$ denote
the maximal $F$-split subtorus of $T$, which is a maximal $F$-split torus of
$G$. We let $B\supset T$ be the Borel subgroup defined over $F$ determined by
$B_{H}$ and let $X_{*}(T)_{I}^{+}\subset X_{*}(T)_{I}$ denote the set of
dominant elements with respect to the choice of $B$.

### 2.2.

We recall the description of the affine root system for $G$ and $H$ following
[PR09]. Let $\Phi$ denote the set of relative roots for $G$ and
$\Phi_{\mathrm{af}}$ the set of affine roots. Similarly, we let $\Phi_{H}$
denote the set of absolute roots for $H$, and $\Phi_{H,\mathrm{af}}$ the set
of affine roots. Let $\mathcal{A}(G,S,F)$ (resp.
$\mathcal{A}(H,T_{H},F^{\prime})$) denote the apartment for $S$ over $F$
(resp. $T_{H}$ over $F^{\prime}$). Then the affine roots
$\Phi_{\mathrm{af}},\Phi_{H,\mathrm{af}}$ are affine functions on
$\mathcal{A}(G,S,F)$, $\mathcal{A}(H,T_{H},F^{\prime})$ respectively.

######  Definition 2.2.1.

A vertex $\mathfrak{s}\in\mathcal{A}(G,S,F)$ is said to be absolutely special
if its image under the simplicial embedding
$\mathcal{A}(G,S,F)\rightarrow\mathcal{A}(H,T_{H},F^{\prime})$ is a special
(equivalently hyperspecial) vertex.

######  Remark 2.2.2.

This is a modification of [HR20, Definition 5.1]. It is necessary to take
$F^{\prime}$ the minimal splitting field of $G$ since for $G={\rm PU}_{2n+1}$,
the vertex labelled ‘s’ in [HR20, equation (5.1)] becomes special over a
degree 4 ramified extension of $F$.

The action of $\sigma$ on $H_{F^{\prime}}$ induces an action on
$\mathcal{A}(H,T_{H},F^{\prime})$ which fixes the hyperspecial vertex
corresponding to $H_{\mathcal{O}_{F^{\prime}}}$, and hence descends to a point
$\mathfrak{s}_{0}\in\mathcal{A}(G,S,F)$. Then $\mathfrak{s}_{0}$ is an
absolutely special vertex in the sense above. The choice of $\mathfrak{s}_{0}$
induces identifications

$\mathcal{A}(G,S,F)\cong X_{*}(T)_{I}\otimes_{\mathbb{Z}}\mathbb{R}\cong
X_{*}(S)\otimes_{\mathbb{Z}}\mathbb{R},$ $\mathcal{A}(H,T_{H},F^{\prime})\cong
X_{*}(T_{H})\otimes_{\mathbb{Z}}\mathbb{R}.$

Using these identifications, we can write affine roots
$a\in\Phi_{\mathrm{af}},a^{\prime}\in\Phi_{H,\mathrm{af}}$ as $a=\alpha+m$,
$a^{\prime}=\alpha^{\prime}+m^{\prime}$, where $\alpha\in\Phi$ and
$\alpha^{\prime}\in\Phi_{H}$ and $m,m^{\prime}\in\mathbb{Q}$. We normalize the
valuation $v$ on $F^{\prime}$ so that $v(u)=\frac{1}{e}$. Then we have

$\Phi_{H,\mathrm{af}}=\\{\alpha^{\prime}+m^{\prime}\,|\,\alpha^{\prime}\in\Phi_{H},m^{\prime}\in\frac{1}{e}\mathbb{Z}\\}.$

For $a^{\prime}=\alpha^{\prime}+m^{\prime}\in\Phi_{H,\mathrm{af}}$, the
corresponding root subgroup $\mathcal{U}^{\prime}_{a^{\prime}}\subset LH$ over
$\mathbb{C}$ corresponds to the subspace $\mathbb{C}X_{\alpha^{\prime}}u^{m}$
of the Lie algebra of the loop group $LH$, where
$X_{\alpha^{\prime}}\in\mathrm{Lie}H$ is our fixed Chevalley generator.

Let $\alpha\in\Phi$ be a relative root, then $\alpha$ determines a
$\sigma$-orbit $C_{\alpha}\subset\Phi_{H}$ of absolute roots. We write
$d_{\alpha}=|C_{\alpha}|$. We define

$R_{\alpha}=\begin{cases}\frac{1}{d_{\alpha}}\mathbb{Z}&\text{if
$\frac{1}{2}\alpha\notin\Phi$,}\\\
\frac{1}{2d_{\alpha}}+\frac{1}{d_{\alpha}}\mathbb{Z}&\text{if
$\frac{1}{2}\alpha\in\Phi$}.\end{cases}$

Then by [PR09, Proposition 2.2], we have

$\Phi_{\mathrm{af}}=\\{\alpha+m\,|\,\alpha\in\Phi,m\in R_{\alpha}\\}.$

### 2.3.

For $\alpha\in\Phi$ we write $U_{\alpha}\subset G$ for the corresponding
relative root subgroup over $F$, and for $a=\alpha+m\in\Phi_{\mathrm{af}}$, we
let $\mathcal{U}_{a}\subset LU_{\alpha}$ denote the corresponding root
subgroup over $\mathbb{C}$ defined in [PR08]. For each
$a\in\Phi_{\mathrm{af}}$ we recall the description of $\mathcal{U}_{a}$ below.
We then use this to define a morphism
$\tilde{f}_{a}:\mathbb{A}^{1}\rightarrow\mathcal{U}_{a}$ and a corresponding
tangent vector $e_{a}$ in the Lie algebra of $LH$ fixed by $\sigma$.

Case (1): $\frac{1}{2}{\alpha},2\alpha\notin\Phi$. Then all elements of
$C_{\alpha}$ are orthogonal to one another, and our choice of
$X_{\alpha^{\prime}}$ determines an isomorphism
$x_{\alpha}:U_{\alpha}\cong\mathrm{Res}_{K/F}\mathbb{G}_{a}$ for $K/F$ of
degree $d=d_{\alpha}=[K:F]$. Let $a=\alpha+m\in\Phi_{\mathrm{af}}$. Then
$m\in\frac{1}{d}\mathbb{Z}$, and

$x_{\alpha}(\mathcal{U}_{a}(\mathbb{C}))=\mathbb{C}u^{em}.$

We define an isomorphism
$\tilde{f}_{a}:\mathbb{A}^{1}\xrightarrow{\sim}\mathcal{U}_{a}$ such that its
composition with $x_{\alpha}$ is given by $r\mapsto ru^{em}.$ The
corresponding subspace of the Lie algebra of $LH$ is generated by

$e_{a}=\sum_{i=1}^{d}X_{\sigma_{0}^{i}(\alpha^{\prime})}(\zeta^{i}u)^{em},$

where $\alpha^{\prime}\in C_{\alpha}$.

Case (2): $\alpha,2\alpha\in\Phi$. In this case $F^{\prime}/F$ is a degree 2
extension, and we have an isomorphism $x_{\alpha}:U_{\alpha}\cong
M_{F^{\prime},F}$ where $M_{F^{\prime},F}$ is the group scheme with $F$-points
given by

$M_{F^{\prime},F}(F)=\\{(c,d)\,|\,c\sigma(c)+d+\sigma(d)=0\\}.$

Moreover, $U_{2\alpha}$ is the subgroup of $U_{\alpha}$ which corresponds
under $x_{\alpha}$ to

$\\{(0,d)\,|\,d+\sigma(d)=0\\}\subset M_{F^{\prime},F}.$

(2a) Let $a=\alpha+m\in\Phi_{\mathrm{af}}$. Then $m\in\frac{1}{2}\mathbb{Z}$,
and we have

$x_{\alpha}(\mathcal{U}_{\alpha}(\mathbb{C}))=\\{(ru^{2m},\frac{(-1)^{2m+1}}{2}r^{2}u^{4m})|r\in\mathbb{C}\\}.$

We let $\tilde{f}_{a}:\mathbb{A}^{1}\xrightarrow{\sim}\mathcal{U}_{a}$ be such
that

$x_{\alpha}\circ\tilde{f}_{a}(r)=(ru^{2m},\frac{(-1)^{2m+1}}{2}r^{2}u^{4m}).$

The corresponding subspace of the Lie algebra of $LH$ is generated by

$e_{a}=u^{2m}(X_{\alpha^{\prime}}+(-1)^{2m}X_{\sigma_{0}(\alpha^{\prime})}),$

where $\alpha^{\prime}\in C_{\alpha}$.

(2b) Let $a=2\alpha+m$. Then $m\in\frac{1}{2}+\mathbb{Z}$, and we have

$x_{2\alpha}(\mathcal{U}_{a}(\mathbb{C}))=\\{(0,ru^{2m})|r\in\mathbb{C}\\}.$

We let $\tilde{f}_{a}:\mathbb{A}^{1}\xrightarrow{\sim}\mathcal{U}_{a}$ be such
that

$x_{\alpha}\circ\tilde{f}_{a}(r)=(0,ru^{2m}).$

The corresponding element of the Lie algebra of $LH$ is

$e_{a}=u^{2m}X_{\alpha^{\prime}}$

for $\alpha^{\prime}\in C_{2\alpha}$.

### 2.4.

Let $\mathfrak{h}$ denote the Lie algebra of $H$ which decomposes as

$\mathfrak{h}\cong(\bigoplus_{\alpha^{\prime}\in\Phi_{H}}\mathbb{C}X_{\alpha^{\prime}})\oplus\mathfrak{t}_{H},$

where $\mathfrak{t}_{H}$ is the Lie algebra of $T_{H}$. This gives a
corresponding decomposition of the vector space

$\mathfrak{h}\otimes_{\mathbb{C}}\mathbb{C}[u,u^{-1}]=\left(\bigoplus_{\alpha^{\prime}\in\Phi_{H}}\mathbb{C}[u,u^{-1}]X_{\alpha^{\prime}}\right)\oplus(\mathfrak{t}_{H}\otimes_{\mathbb{C}}\mathbb{C}[u,u^{-1}])\subset\mathrm{Lie}LH.$

This space is equipped with an action of $\sigma$, which preserves
$\bigoplus_{\alpha^{\prime}\in\Phi_{H}}\mathbb{C}[u,u^{-1}]X_{\alpha^{\prime}}$
and $\mathfrak{t}_{H}\otimes_{\mathbb{C}}\mathbb{C}[u,u^{-1}]$.

######  Proposition 2.5.

The vector space
$(\bigoplus_{\alpha^{\prime}\in\Phi_{H}}\mathbb{C}[u,u^{-1}]X_{\alpha^{\prime}})^{\sigma}$
has a basis given by the set
$\mathcal{B}=\\{e_{a}\\}_{a\in\Phi_{\mathrm{af}}}$.

###### Proof.

Clearly, the $e_{a}$ are $\sigma$-invariant and linearly independent.

Note that the vector space $\mathbb{C}[u,u^{-1}]X_{\alpha^{\prime}}$ has a
basis given by

$\\{X_{\alpha^{\prime}}u^{m}|\alpha^{\prime}\in\Phi_{H},m\in\mathbb{Z}\\}.$

Suppose
$v\in(\bigoplus_{\alpha^{\prime}\in\Phi_{H}}X_{\alpha^{\prime}}\otimes\mathbb{C}[u,u^{-1}])^{\sigma}$
has a non-zero component $X_{\alpha^{\prime}}u^{m}$; upon scaling we may
assume this coefficient is 1. Let $\alpha\in\Phi_{\mathrm{af}}$ denote the
associated relative root. We show $v$ is spanned by the $e_{a}$ by induction
on the number of non-zero coefficients.

Case (1): $\frac{1}{2}{\alpha},2\alpha\notin\Phi$. Let $d=|C_{\alpha}|$ as
above. Then by $\sigma^{d}$-invariance, we have $\frac{e}{d}|m$, and the
coefficient of $X_{\sigma_{0}^{i}(\alpha^{\prime})}u^{m}$ in $v$ is
$\zeta^{im}$ for $i=1,\dotsc,d$. Set
$a=\alpha+\frac{1}{e}m\in\Phi_{\mathrm{af}}$. Then $v-e_{a}$ is spanned by
$\mathcal{B}$ by induction.

Case (2a): $2\alpha\in\Phi$. In this case
$C_{\alpha}=\\{\alpha^{\prime},\sigma_{0}(\alpha^{\prime})\\}$, and the
coefficient of $X_{\sigma_{0}(\alpha^{\prime})}u^{m}$ is $(-1)^{m}u^{m}$. Then
for $a=\alpha+\frac{1}{2}m$, $v-e_{a}$ is spanned by $\mathcal{B}$ by
induction.

Case (2b): $\frac{1}{2}\alpha\in\Phi$. In this case
$C_{\alpha}=\\{\alpha^{\prime}\\}$, and the $\sigma$-invariance implies $m$ is
odd. Set $a=\alpha+m$; then $v-e_{a}$ is spanned by $\mathcal{B}$ by
induction. ∎

### 2.6.

The following computation will be used in §4 to produce extra tangent
directions along the Cartan.

######  Proposition 2.7.

  1. (1)

Let $a=\alpha+m$ with $\frac{1}{2}\alpha,2\alpha\notin\Phi$. Set
$b=-\alpha-m-\frac{1}{d_{\alpha}}\in\Phi_{\mathrm{af}}$ and
$h=\tilde{f}_{b}(1)$. Then $\mathrm{Ad}(h)e_{a}$ has a non-zero component in
$(\mathfrak{t}_{H}\otimes u^{-1}\mathbb{C}[u^{-1}])^{\sigma}$.

  2. (2)

Let $a=2\alpha+m$ with $\alpha\in\Phi$ so that $2m$ is an odd integer. Set
$b=-\alpha-\frac{m}{2}-\frac{1}{2}\in\Phi_{\mathrm{af}}$ and
$h=\tilde{f}_{b}(1)$. Then $\mathrm{Ad}(h)e_{a}$ has a non-zero component in
$(\mathfrak{t}_{H}\otimes u^{-1}\mathbb{C}[u^{-1}])^{\sigma}$.

###### Proof.

In each case let $G_{\alpha}$ denote the simply-connected cover of the
subgroup of $G$ generated by $U_{\alpha}$ and $U_{-\alpha}$. Then
$G_{\alpha}\cong\mathrm{Res}_{K/F}\mathrm{SL}_{2}$ for some $K/F$ finite in
(1) and $G_{\alpha}\cong\mathrm{SU}_{3}$ the special unitary group for a
degree 2 ramified extension $F^{\prime}/F$ in (2). Since the affine root
groups contained in $U_{\alpha},U_{-\alpha}$ arise from $G_{\alpha}$, and the
natural map $LG_{\alpha}\rightarrow LG$ induces an injection on tangent spaces
preserving Cartan spaces, it suffices to prove (1) and (2) for the group
$G_{\alpha}$. In these cases, we compute using the standard matrix
representations of these groups. Moreover, upon possibly changing the choice
of absolutely special vertex $\mathfrak{s}_{0}$, we may assume $m=0$ in (1)
and $m=-\frac{1}{2}$ in (2).

(1) Suppose $K=F(u^{\prime})$, so that $u^{\frac{e}{d_{\alpha}}}=u^{\prime}$.
We have

$e_{a}=\left(\begin{matrix}0&1\\\ 0&0\end{matrix}\right)\text{ and
}h=\left(\begin{matrix}1&0\\\ u^{\prime-1}&1\end{matrix}\right).$

Then we have

$\displaystyle\mathrm{Ad}(h)e_{a}$ $\displaystyle=\left(\begin{matrix}1&0\\\
u^{\prime-1}&1\end{matrix}\right)\left(\begin{matrix}0&1\\\
0&0\end{matrix}\right)\left(\begin{matrix}1&0\\\
-u^{\prime-1}&1\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}-u^{\prime-1}&1\\\
-u^{\prime-2}&u^{\prime-1}\end{matrix}\right).$

The diagonal component of $\mathrm{Ad}(h)e_{a}$ gives a non-zero component in
$(\mathfrak{t}_{H}\otimes u^{-1}\mathbb{C}[u^{-1}])^{\sigma}$.

(2) In this case, $e=d_{\alpha}=2$. Then we have

$e_{a}=\left(\begin{matrix}0&0&u^{-1}\\\ 0&0&0\\\
0&0&0\end{matrix}\right)\text{ and }h=\left(\begin{matrix}1&0&0\\\ 1&1&0\\\
-\frac{1}{2}&-1&1\end{matrix}\right).$

Then we compute

$\displaystyle\mathrm{Ad}(h)e_{a}$ $\displaystyle=\left(\begin{matrix}1&0&0\\\
1&1&0\\\ -\frac{1}{2}&-1&1\end{matrix}\right)\left(\begin{matrix}0&0&u^{-1}\\\
0&0&0\\\ 0&0&0\end{matrix}\right)\left(\begin{matrix}1&0&0\\\ -1&1&0\\\
-\frac{1}{2}&1&1\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}-\frac{u^{-1}}{2}&u^{-1}&u^{-1}\\\
-\frac{u^{-1}}{2}&u^{-1}&u^{-1}\\\
-\frac{u^{-1}}{4}&-\frac{u^{-1}}{2}&-\frac{u^{-1}}{2}\end{matrix}\right).$

The diagonal of $\mathrm{Ad}(h)e_{a}$ gives a non-zero component in
$(\mathfrak{t}_{H}\otimes u^{-1}\mathbb{C}[u^{-1}])^{\sigma}$. ∎

### 2.8.

Now let $\mathfrak{f}\subset\mathcal{A}(G,S,F)$ be a facet. Then
$\mathfrak{f}$ determines a facet in $\mathcal{A}(H,T_{H},F^{\prime})$ also
denoted $\mathfrak{f}$. We let $\mathcal{G}_{\mathfrak{f}}$ and
$\mathcal{H}_{\mathfrak{f}}$ denote the parahoric group schemes over
$\mathcal{O}_{F}$ and $\mathcal{O}_{F^{\prime}}$ respectively corresponding to
$\mathfrak{f}$. We let
$\mathrm{Fl}_{G,\mathfrak{f}}:=LG/L^{+}\mathcal{G}_{\mathfrak{f}}$ denote the
partial affine flag variety associated to $\mathcal{G}_{\mathfrak{f}}$, and
$e\in\mathrm{Fl}_{G,\mathfrak{f}}(\mathbb{C})$ the base-point. We have the
following description of the tangent space $T_{e}\mathrm{Fl}_{G,\mathfrak{f}}$
to $\mathrm{Fl}_{G,\mathfrak{f}}$ at the base-point.

######  Proposition 2.9.

There is an isomorphism

$T_{e}\mathrm{Fl}_{G,\mathfrak{f}}\cong\left(\bigoplus_{a\in\Phi_{\mathrm{af}},a(\mathfrak{f})<0}\mathbb{C}e_{a}\right)\oplus(\mathfrak{t}_{H}\otimes
u^{-1}\mathbb{C}[u^{-1}])^{\sigma}.$

Here, we write $a(\mathfrak{f})<0$ if $a(x)<0$ for some (equivalently) all
$x\in\mathfrak{f}$.

###### Proof.

By [HLR18, Corollary 3.9], we have an isomorphism
$T_{e}{\mathrm{Fl}_{G,\mathfrak{f}}}\cong(T_{e}L^{--}\mathcal{H}_{\mathfrak{f}})^{\sigma}$,
where $L^{--}\mathcal{H}_{\mathfrak{f}}$ denotes the negative loop group as
defined in [HLR18, §3.3]. Let
$\mathfrak{a}^{\prime}\subset\mathcal{A}(H,T_{H},F^{\prime})$ be an alcove
whose closure contains $\mathfrak{f}$ with associated Iwahori group scheme
$\mathcal{H}_{\mathfrak{a}^{\prime}}$. Then by definition of
$L^{--}\mathcal{H}_{\mathfrak{f}}$ there is an isomorphism

$T_{e}L^{--}\mathcal{H}_{\mathfrak{f}}\cong\bigcap_{w\in
W_{H,\mathfrak{f}}}{{}^{w}(T_{e}L^{--}\mathcal{H}_{\mathfrak{a}^{\prime}}}).$

Here $W_{H,\mathfrak{f}}$ denotes the subgroup of the Iwahori Weyl group
$W_{H}$ for $H_{F^{\prime}}$ generated by the simple reflections fixing
$\mathfrak{f}$. Recall we identify $\Phi_{H,\mathrm{af}}$ with the set of
pairs $(a^{\prime},m),a^{\prime}\in\Phi_{H},m\in\mathbb{Z}$ using the choice
of absolutely special vertex $\mathfrak{s}_{0}$. We have an isomorphism

$T_{e}L^{--}\mathcal{H}_{\mathfrak{a}^{\prime}}\cong\left(\bigoplus_{a^{\prime}=\alpha^{\prime}+m\in\Phi_{H,\mathrm{af}},a^{\prime}(\mathfrak{a}^{\prime})<0}\mathbb{C}X_{\alpha^{\prime}+m}\right)\oplus(\mathfrak{t}_{H}\otimes
u^{-1}\mathbb{C}[u^{-1}]).$

Since

$\\{a^{\prime}\in\Phi_{H,\mathrm{af}},a^{\prime}(\mathfrak{f})<0\\}=\bigcap_{w\in
W_{H,\mathfrak{f}}}\\{wa^{\prime}\in\Phi_{H,\mathrm{af}},a^{\prime}(\mathfrak{a}^{\prime})<0\\},$

we have an isomorphism

$T_{e}\mathrm{Gr}_{\mathcal{G}}\cong\left(\bigoplus_{a^{\prime}=\alpha^{\prime}+m\in\Phi_{H,\mathrm{af}},a^{\prime}(\mathfrak{f})<0}\mathbb{C}X_{\alpha^{\prime}}u^{m}\right)^{\sigma}\oplus(\mathfrak{t}_{H}\otimes
u^{-1}\mathbb{C}[u^{-1}])^{\sigma}.$

As in Proposition 2.5, we find that
$\\{e_{a}\\}_{a\in\Phi_{\mathrm{af}},a(\mathfrak{f})<0}$ is a basis for the
vector space
$(\bigoplus_{a^{\prime}=\alpha^{\prime}+m\in\Phi_{H,\mathrm{af}},a^{\prime}(\mathfrak{f})<0}\mathbb{C}X_{\alpha^{\prime}}u^{m})^{\sigma}$.
Here we use the fact that $\mathfrak{f}$ is $\sigma$-invariant in order to
show that the condition $a(\mathfrak{f})<0$ for $a\in\Phi_{\mathrm{af}}$ is
equivalent to the condition that $a^{\prime}(\mathfrak{f})<0$, for
$a^{\prime}=\alpha^{\prime}+m\in\Phi_{H,\mathrm{af}}$ attached to $a^{\prime}$
as in §2.2. The result follows. ∎

##  3\. Root curves in twisted affine Grassmannians

### 3.1.

We keep the notations and assumptions of the previous section so that $G$ is
an adjoint absolutely simple reductive group, but we will specialize the
discussion to the case of twisted affine Grassmannians. Thus let
$\mathfrak{s}\in\mathcal{A}(G,S,F)$ be a fixed special vertex (note that
$\mathfrak{s}$ is absolutely special unless $G$ is of type $C-BC_{n}$). We
write $\mathcal{G}=\mathcal{G}_{\mathfrak{s}}$ for the parahoric group scheme
over $\mathcal{O}_{F}$, and let
$\mathrm{Gr}_{\mathcal{G}}=\mathrm{Fl}_{G,\mathfrak{f}}$ be the twisted affine
Grassmannian associate to $\mathfrak{s}$. The choice of special vertex induces
an identification $\mathcal{A}(G,S,F)\cong
X_{*}(T)_{I}\otimes_{\mathbb{Z}}\mathbb{R}$ taking $\mathfrak{s}$ to 0. We fix
$\mathfrak{a}\subset\mathcal{A}(G,S,F)$ the alcove which is contained in the
anti-dominant chamber under this identification.

Let $N$ denote the centralizer of $T$ and let $\mathcal{T}$ denote the
connected Néron model of $T$ over $\mathcal{O}_{F}$. The Iwahori Weyl group is
defined to be the quotient

$W:=N(F)/\mathcal{T}(\mathcal{O}_{F}).$

There is a natural short exact sequence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X_{*}(T)_{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
89.28098pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
89.28098pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
130.11432pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
130.11432pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
173.74765pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
173.74765pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{1}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where $W_{0}:=N(F)/T(F)$ is the spherical Weyl group. For each $\lambda\in
X_{*}(T)_{I}$, we write $t^{\lambda}$ for the corresponding translation
element in $W$. We also write
$t^{\lambda}\in\mathrm{Gr}_{\mathcal{G}}(\mathbb{C})$ for the image of an
element $\tilde{t}^{\lambda}\in N(F)$ lifting $t^{\lambda}$; this point is
independent of the choice of $\tilde{t}^{\lambda}$.

The affine Weyl group $W_{a}\subset W$ is defined to be the subgroup generated
by the set of simple reflection in the walls of $\mathfrak{a}$. Then we have
exact sequences

$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{*}(T_{\mathrm{sc}})_{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$

where $T_{\mathrm{sc}}$ is the preimage of $T$ in the simply connected cover
of the derived group of $G$. The choice of alcove $\mathfrak{a}$ determines a
length function and Bruhat order on $W$. The choice of special vertex
$\mathfrak{s}$ determines splittings of the above exact sequences, and hence
semi-direct product decompositions

$W\cong X_{*}(T)_{I}\rtimes W_{0},\ \ \ \ \ W_{a}\cong
X_{*}(T_{\mathrm{sc}})_{I}\rtimes W_{0}.$

There is a reduced root system $\Sigma$ (the échelonnage root system) such
that

$W_{a}\cong Q(\Sigma)\times W(\Sigma)$

where $Q(\Sigma)$ (resp. $W(\Sigma)$) denotes the coroot lattice (resp. Weyl
group) for $\Sigma$.

### 3.2.

By [Hai18, Theorem B], $\Sigma$ has the following description. We write
$N_{I}:X^{*}(T_{H})\rightarrow X^{*}(T_{H})$ for the norm map
$\alpha\mapsto\sum_{\alpha^{\prime}\in I\alpha}\alpha^{\prime}$ and let
$\Delta_{H}$ denote the set of simple roots for $H$ determined by $B_{H}$. For
$\alpha\in\Delta_{H}$, we let $N_{I}^{\prime}(\alpha)$ be the modified norm,
defined as

$N_{I}^{\prime}(\alpha)=\begin{cases}N_{I}(\alpha)&\text{if all elements of
$I\alpha$ are orthogonal,}\\\ 2N_{I}(\alpha)&\text{otherwise.}\end{cases}$

Then $\Sigma$ has a basis given by the image of $\Delta_{H}$ under
$N_{I}^{\prime}$.

For $\alpha\in\Sigma$ and $k\in\mathbb{Z}$, we have the affine function
$\alpha+k$ on $\mathcal{A}(G,S,F)$; here by convention we have
$\alpha(\mathfrak{s})=0$ for $\alpha\in\Sigma$. The decomposition of
$\mathcal{A}(G,S,F)$ into facets is given by the vanishing hyperplanes of
these functions. We write
$\Phi_{\mathrm{af}}(\Sigma)=\\{\alpha+k|\alpha\in\Sigma,k\in\mathbb{Z}\\}$;
then we obtain an identification

$\psi_{\mathfrak{s}}:\Phi_{\mathrm{af}}(\Sigma)\xrightarrow{\sim}\Phi_{\mathrm{af}}$

which takes $\alpha+k$ to the affine root which vanishes on the affine
hyperplane $\alpha+k=0$. For $a=\alpha+k\in\Phi_{\mathrm{af}}(\Sigma)$, we
have $\mathcal{U}_{\psi_{\mathfrak{s}}(a)}\subset L^{+}\mathcal{G}$ if and
only if $k\geq 0$.

If $\mathfrak{s}=\mathfrak{s}_{0}$, we have the following relation between the
description of $\Phi_{\mathrm{af}}$ given in terms of $\Sigma$ and the
description given in terms of the relative roots $\Phi$. Note that for each
$\alpha\in\Phi$, there is a unique element $\nu(\alpha)$ of $\Sigma$
proportional to it.

Case (1): Let $\alpha\in\Phi$ with $\frac{1}{2}\alpha,2\alpha\notin\Phi$. For
$a=\alpha+m\in\Phi_{\mathrm{af}}$, we have $\nu(\alpha)=d_{\alpha}\alpha$ and
$\psi_{\mathfrak{s}_{0}}^{-1}(a)=\nu(\alpha)+d_{\alpha}m$.

Case (2): Let $\alpha,2\alpha\in\Phi$. For $a=2\alpha+m$, we have $2m$ is an
odd integer, and $\psi_{\mathfrak{s}_{0}}^{-1}(a)=\nu(\alpha)+2m.$ For
$a=\alpha+m$, $m\in\frac{1}{2}\mathbb{Z}$, we have
$\psi_{\mathfrak{s}_{0}}^{-1}(a)=\nu(\alpha)+4m.$

### 3.3.

For $a=\alpha+k\in\Phi_{\mathrm{af}}(\Sigma)$, we write $\mathcal{U}_{a}$ for
the affine root subgroup $\mathcal{U}_{\psi_{\mathfrak{s}}(a)}$. Then by our
assumptions $\mathcal{U}_{a}\subset L^{+}\mathcal{G}$ if and only if $k\geq
0$. We write $\langle\ ,\ \rangle$ for the pairing between
$X_{*}(T)_{I}\otimes_{\mathbb{Z}}\mathbb{R}$ and the root lattice associated
to $\Sigma$. Then for $\lambda\in X_{*}(T)_{I}$ and
$a=\alpha+k\in\Phi_{\mathrm{af}}(\Sigma)$, we have

(3.3.1)
$t^{\lambda}\mathcal{U}_{a}t^{-\lambda}=\mathcal{U}_{a+\langle\lambda,\alpha\rangle.}$

For each $a\in\Phi_{\mathrm{af}}(\Sigma)$, we define a morphism
$f_{a}:\mathbb{A}^{1}\rightarrow\mathrm{Gr}_{\mathcal{G}}$ by taking the
composition

$\mathbb{A}^{1}\xrightarrow{\tilde{f}_{\psi_{\mathfrak{s}}(a)}}\mathcal{U}_{\psi_{\mathfrak{s}(a)}}\rightarrow
LG\rightarrow\mathrm{Gr}_{\mathcal{G}}.$

We will call (a translate of) $f_{a}$ a _root curve_. A simple calculation in
$\mathrm{SL}_{2}$ or $\mathrm{SU}_{3}$ using the explicit description of the
affine root groups $\mathcal{U}_{a}$ gives the following proposition (cf. [KZ,
Prop. 6.2.5]).

######  Proposition 3.4.

Let $a=\alpha-k\in\Phi_{\mathrm{af}}$ with $\alpha\in\Sigma,k\in\mathbb{Z}$.
Then $f(0)=e$, where $e$ is the base point of $\mathrm{Gr}_{\mathcal{G}}$, and
we have

$f_{a}(x)\in\mathcal{U}_{-a}t^{-k\alpha^{\vee}}L^{+}\mathcal{G}/L^{+}\mathcal{G},$

for $x\in\mathbb{A}^{1}\setminus\\{0\\}$. Here $\alpha^{\vee}\in X_{*}(T)_{I}$
corresponds to the coroot for $\alpha$.∎

### 3.5.

Now let $X_{*}(T)_{I}^{+}$ denote the set of dominant elements in
$X_{*}(T)_{I}$ with respect to $B$. For $\mu\in X_{*}(T)_{I}^{+}$, we write
$\mathrm{Gr}_{\mathcal{G},\mu}$ for the open Schubert cell given by the
$L^{+}\mathcal{G}$-orbit of $t^{\mu}$ in $\mathrm{Gr}_{\mathcal{G}}$ , and we
write $\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ for its closure. Let
$\preccurlyeq$ denote the dominance order on $X_{*}(T)_{I}^{+}$ with respect
to $\Sigma$, i.e. $\lambda\preccurlyeq\mu$ if and only if $\mu-\lambda$ is an
integral linear combination of positive coroots for $\Sigma$ with non-negative
coefficients. By [Ric13, Corollary 1.8], this coincides with the Bruhat order
under the identification $W_{K}\backslash W/W_{K}\cong X_{*}(T)_{I}^{+}$; here
$W_{K}\subset W$ is the subgroup generated by the simple reflections fixing
$\mathfrak{s}$. In particular
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\lambda}\subset\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$
if and only if $\lambda\preccurlyeq\mu$.

Fix $\mu,\lambda\in X_{*}(T)_{I}^{+}$ with $\mu\succcurlyeq\lambda$. For an
element $\nu\in X_{*}(T)_{I}$, we write $\nu_{\mathrm{dom}}\in
X_{*}(T)_{I}^{+}$ for the dominant representative of $\nu$. Then for
$\alpha\in\Sigma$, we define

$k_{\alpha}=k_{\alpha}^{(\lambda,\mu)}:=\max\\{k\in\mathbb{Z}\,|\,(\lambda-k\alpha^{\vee})_{\mathrm{dom}}\preccurlyeq\mu\\}.$

######  Proposition 3.6.

Let $\alpha\in\Sigma$.

  1. (1)

We have $k_{\alpha}=k_{-\alpha}+\langle\lambda,\alpha\rangle$.

  2. (2)

For $1\leq k\leq k_{\alpha}$, the root curve
$t^{\lambda}f_{\alpha-k}:\mathbb{A}^{1}\rightarrow\mathrm{Gr}_{\mathcal{G}}$
has image contained in $\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ and maps
$0\in\mathbb{A}^{1}$ to the point $t^{\lambda}$.

###### Proof.

(1) Let $s_{\alpha}\in W_{0}$ denote the simple reflection corresponding to
$\alpha$ which acts on $X_{*}(T)_{I}\otimes_{\mathbb{Z}}\mathbb{R}$. Then we
have

$\displaystyle s_{\alpha}(\lambda-k\alpha^{\vee})$
$\displaystyle=\lambda+k\alpha^{\vee}-\langle\lambda,\alpha\rangle\alpha^{\vee}$
$\displaystyle=\lambda-(k-\langle\lambda,\alpha\rangle)(-\alpha^{\vee})$

for any $k\in\mathbb{Z}$.

Since
$s_{\alpha}(\lambda-k\alpha^{\vee})_{\mathrm{dom}}=(\lambda-k\alpha^{\vee})_{\mathrm{dom}}$,
we have $(\lambda-k\alpha^{\vee})_{\mathrm{dom}}\leq\mu$ if and only if
$(\lambda-(k-\langle\lambda,\alpha\rangle)(-\alpha^{\vee}))_{\mathrm{dom}}\leq\mu$,
hence (1) follows.

(2) By Proposition 3.4, we have $t^{\lambda}f_{\alpha-k}(0)=t^{\lambda}$. For
the first statement, we first assume $1\leq
k\leq\langle\lambda,\alpha\rangle$; note this only occurs when $\alpha$ is a
positive root. Then

$t^{\lambda}f_{\alpha-k}(x)\in\mathcal{U}_{\alpha-k+\langle\lambda,\alpha\rangle}t^{\lambda}L^{+}\mathcal{G}/L^{+}\mathcal{G}\subset\mathrm{Gr}_{\mathcal{G},\preccurlyeq\lambda}$

and hence $t^{\lambda}f_{\alpha-k}$ has image in
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ since $\lambda\preccurlyeq\mu$.

Now assume $\langle\lambda,\alpha\rangle\leq k\leq k_{\alpha}$. By Proposition
3.4, we have $t^{\lambda}f_{\alpha-k}(x)\in
t^{\lambda}\mathcal{U}_{-\alpha+k}t^{-k\alpha^{\vee}}L^{+}\mathcal{G}/L^{+}\mathcal{G}$
for $x\in\mathbb{A}^{1}\setminus\\{0\\}$, and hence by (3.3.1)

$t^{\lambda}f_{\alpha-k}(x)\in\mathcal{U}_{-\alpha+k-\langle\lambda,\alpha\rangle}t^{\lambda-k\alpha^{\vee}}L^{+}\mathcal{G}/L^{+}G.$

By assumption $k-\langle\lambda,\alpha\rangle\geq 0$. Thus
$\mathcal{U}_{-\alpha+k-\langle\lambda,\alpha\rangle}\subset L^{+}\mathcal{G}$
and we have

$t^{\lambda}f_{\alpha-k}(x)\in\mathrm{Gr}_{\mathcal{G},\preccurlyeq(\lambda-k\alpha^{\vee})_{\mathrm{dom}}}$

for $x\in\mathbb{A}^{1}\setminus\\{0\\}$. Now $\lambda-k\alpha^{\vee}$ lies in
the interval $[\lambda-k_{\alpha}\alpha^{\vee},s_{\alpha}(\lambda-
k_{\alpha}\alpha^{\vee})]$, and hence is contained the convex hull of the
$W_{0}$-orbit of $(\lambda-k_{\alpha}\alpha^{\vee})_{\mathrm{dom}}$. Thus we
have

$(\lambda-k\alpha^{\vee})_{\mathrm{dom}}\preccurlyeq(\lambda-
k_{\alpha}\alpha^{\vee})_{\mathrm{dom}}\preccurlyeq\mu$

and the result follows. ∎

### 3.7.

The root curves above allow us to obtain the following bound on dimensions of
tangent spaces to Schubert varieties. Using the isomorphism
$\mathrm{Gr}_{\mathcal{G}}\xrightarrow{\sim}\mathrm{Gr}_{\mathcal{G}}$ given
by multiplication by $t^{\lambda}$ which takes $e$ to $t^{\lambda}$, we obtain
an identification of tangent spaces

$T_{e}\mathrm{Gr}_{\mathcal{G}}\cong
T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G}}.$

Thus by Proposition 2.9, we have an identification

(3.7.1)
$T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G}}\cong\left(\bigoplus_{a\in\Phi_{\mathrm{af}},a(\mathfrak{s})<0}\mathbb{C}e_{a}\right)\oplus(\mathfrak{t}_{H}\otimes
u^{-1}\mathbb{C}[u^{-1}])^{\sigma}.$

For a subspace $V\subset T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G}}$, we write
$V^{\mathrm{root}}$ for the intersection of $V$ with the subspace
$\bigoplus_{a\in\Phi_{\mathrm{af}},a(\mathfrak{s})<0}\mathbb{C}e_{a}\subset
T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G}}$.

######  Corollary 3.8.

Let $\lambda,\mu\in X_{*}(T)^{+}_{I}$ with $\lambda\preccurlyeq\mu$.

$\dim(T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu})^{\mathrm{root}}\geq\sum_{\alpha\in\Sigma}k_{\alpha}^{(\lambda,\mu)}.$

###### Proof.

Using the bijection
$\psi_{\mathfrak{s}}:\Phi_{\mathrm{af}}(\Sigma)\cong\Phi_{\mathrm{af}}$, we
can rewrite (3.7.1) as

$T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G}}\cong\left(\bigoplus_{\alpha-k\in\Phi_{\mathrm{af}}(\Sigma),k>0}\mathbb{C}e_{\psi_{\mathfrak{s}}{(\alpha-k)}}\right)\oplus(\mathfrak{t}_{H}\otimes
u^{-1}\mathbb{C}[u^{-1}])^{\sigma}.$

By the construction of the isomorphism in Proposition 2.9 and the computations
in §2.2, the one-dimensional subspace of
$T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G}}$ spanned by the tangent space of the
root curve $t^{\lambda}f_{\alpha-k}$ is
$\mathbb{C}e_{\psi_{\mathfrak{s}}(\alpha-k)}$. By Proposition 3.6, for $1\leq
k\leq k_{\alpha}$, this lies in the subspace
$(T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu})^{\mathrm{root}}$.
Summing over all $\alpha\in\Sigma$ gives the result.

∎

##  4\. Tangent spaces for minimal degenerations

### 4.1.

The bound obtained in Corollary 3.8 can be made more explicit when applied to
boundary components of minimal degenerations. We keep the notation of the
previous section so that $\mathrm{Gr}_{\mathcal{G}}$ is the twisted affine
Grassmannian associated to an adjoint absolutely simple reductive group over
$F$ and a choice of special parahoric corresponding to a point
$\mathfrak{s}\in\mathcal{A}(G,S,F)$.

For $\mu,\lambda\in X_{*}(T)^{+}_{I}$, we say that
$\mu\rightsquigarrow\lambda$ is a minimal degeneration if

  1. (1)

$\mu\succcurlyeq\lambda$ and $\mu\neq\lambda$.

  2. (2)

If $\nu\in X_{*}(T)_{I}^{+}$ is such that
$\mu\succcurlyeq\nu\succcurlyeq\lambda$, then $\mu=\nu$ or $\lambda=\nu$.

For $\lambda\preccurlyeq\mu$, we may write $\mu-\lambda$ uniquely as a linear
combination of simple coroots $\alpha^{\vee}\in\Sigma^{\vee}\subset
X_{*}(T_{\mathrm{sc}})_{I}$ with non-negative integral coefficients. We write
$\mathrm{supp}(\mu-\lambda)$ for the sub-root system of $\Sigma$ generated by
the roots associated to the set of simple coroots which appear with positive
coefficients.

We have the following classification of minimal degenerations due to
Stembridge [Ste98, Theorem 2.8].

######  Proposition 4.2.

Let $\mu\rightsquigarrow\lambda$ be a minimal degeneration. Then one of
following holds:

  1. (1)

$\mu-\lambda$ is a simple coroot.

  2. (2)

$\mu-\lambda$ is the short dominant coroot of $\mathrm{supp}(\mu-\lambda)$,
and $\lambda=0$ on $\mu-\lambda$.

  3. (3)

$\mu-\lambda$ is the short dominant coroot of $\mathrm{supp}(\mu-\lambda)$;
$\mathrm{supp}(\mu-\lambda)$ is of type $C_{n}$ and $\lambda$ on
$\mathrm{supp}(\mu-\lambda)$ is

$0\,\rule[2.84544pt]{11.38092pt}{0.5pt}\,0\,\rule[2.84544pt]{11.38092pt}{0.5pt}\,\cdots\,\rule[2.84544pt]{11.38092pt}{0.5pt}\,0\xLeftarrow{\
\ \ }1\ .$

  4. (4)

$\mathrm{supp}(\mu-\lambda)$ is of type $G_{2}$, and $\lambda=(2\ext@arrow
0359\arrowfill@\equiv\equiv\Rrightarrow{}{\ \ \ }0)$, $\mu=(1\ext@arrow
0359\arrowfill@\equiv\equiv\Rrightarrow{}{\ \ \ }1)$.

  5. (5)

$\mathrm{supp}(\mu-\lambda)$ is of type $G_{2}$, and $\lambda=(1\ext@arrow
0359\arrowfill@\equiv\equiv\Rrightarrow{}{\ \ \ }0)$, $\mu=(0\ext@arrow
0359\arrowfill@\equiv\equiv\Rrightarrow{}{\ \ \ }1)$.

Here the integer at a vertex of the Dynkin diagram represents the pairing with
the simple root corresponding to that vertex.

### 4.3.

Let $\Sigma^{+}$ denote the subset of positive roots, and we let $\rho$ denote
the half sum of positive roots in $\Sigma$.

######  Proposition 4.4.

Let $\mu\rightsquigarrow\lambda$ be a minimal degeneration. Then

$\dim(T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu})^{\mathrm{root}}\geq\dim\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}.$

###### Proof.

By [Ric13, Corollary 2.10], we have
$\dim\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}=\langle\mu,2\rho\rangle$. Thus
by Corollary 3.8, it suffices to show

(4.4.1) $\sum_{\alpha\in\Sigma}k_{\alpha}\geq\langle\mu,2\rho\rangle.$

By Proposition 3.6, we have
$k_{\alpha}=k_{-\alpha}+\langle\lambda,\alpha\rangle$ for any
$\alpha\in\Sigma$. Substituting this in for each $\alpha\in\Sigma^{+}$,
equation (4.4.1) becomes

(4.4.2)
$\sum_{\alpha\in\Sigma^{+}}2k_{-\alpha}+\langle\lambda,2\rho\rangle\geq\langle\mu,2\rho\rangle.$

We consider the fives cases in Proposition 4.2 separately. Note that in each
case $\mu-\lambda$ is a positive coroot in $\Sigma^{\vee}$ which we call
$\beta^{\vee}$. Hence it suffices to show that

$\sum_{\alpha\in\Sigma^{+}}2k_{-\alpha}\geq\langle\beta^{\vee},2\rho\rangle.$

  1. (1)

In this case, $\beta$ is a simple root. Then
$\langle\beta^{\vee},2\rho\rangle=2$. To see this, let $\alpha\in\Sigma^{+}$.
Then if $\langle\beta^{\vee},\alpha\rangle\neq 0$ and $\alpha\neq\beta$, we
have $s_{\beta}(\alpha)\in\Sigma^{+}$ and
$\langle\beta^{\vee},s_{\beta}(\alpha)\rangle=-\langle\beta^{\vee},\alpha\rangle$.
Pairing such roots together gives

$\langle\beta^{\vee},2\rho\rangle=\left(\sum_{\alpha\in\Sigma^{+},\alpha\neq\beta}\langle\beta^{\vee},\alpha\rangle\right)+2=2.$

Thus it suffices to find some $\alpha\in\Sigma^{+}$ with $k_{-\alpha}\geq 1$.
It is easy to see that $k_{-\beta}\geq 1$ works.

  2. (2)

Let $\Sigma_{J}$ denote the sub-root system $\mathrm{supp}(\mu-\lambda)$, and
$W_{J}\subset W(\Sigma)$ its Weyl group. If $\gamma^{\vee}$ is a short
positive coroot in $\Sigma_{J}^{\vee}$, then there is a $w\in W_{J}$ such that
$w(\gamma^{\vee})=\beta^{\vee}$. Then
$w(\lambda+\gamma^{\vee})=\lambda+\beta^{\vee}=\mu$ and hence
$(\lambda+\gamma^{\vee})_{\mathrm{dom}}\preccurlyeq\mu$, i.e. $k_{-\gamma}\geq
1$. We write $\Sigma_{J,\mathrm{short}}^{\vee,+}$ for the set of short
positive coroots in $\Sigma_{J}^{\vee}$. Then we have

$(\sum_{\alpha\in\Sigma^{+}}2k_{-\alpha})\geq
2|\Sigma_{J,\mathrm{short}}^{\vee,+}|$

and hence it suffices to show that

$2|\Sigma_{J,\mathrm{short}}^{\vee,+}|\geq\langle\beta^{\vee},2\rho\rangle.$

Recall that $\beta^{\vee}$ is the short dominant coroot in $\Sigma_{J}$. We
write $\beta^{\vee}$ as a sum of simple coroots which necessarily lie in
$\mathrm{supp}(\mu-\lambda)$, and we let $N_{\beta}$ be the number of simple
coroots appearing. Then $\langle\beta^{\vee},2\rho\rangle=2N_{\beta}$. We thus
reduce to showing

$|\Sigma_{J,\mathrm{short}}^{\vee,+}|\geq N_{\beta}.$

Note that this only depends on the root system $\Sigma_{J}$.

To show this last inequality, let $\gamma^{\vee}$ be a short simple coroot and
$v$ a minimal length element in $W_{J}$ such that
$v(\gamma^{\vee})=\beta^{\vee}$. We let $s_{n}s_{n-1}\dotsc s_{1}$ be a
reduced word decomposition for $v$ and write $v_{i}=s_{i}\dotsc s_{1}$ for
$i=1,\dotsc,n$. Then $n+1=N_{\beta}$, and
$\gamma^{\vee},v_{1}(\gamma^{\vee}),\dotsc,v_{n}(\gamma^{\vee})$ give $n+1$
distinct elements of $\Sigma_{J,\mathrm{short}}^{\vee,+}$.

  3. (3)

Let $\Sigma_{J}$ denote the sub-root system $\mathrm{supp}(\mu-\lambda)$ which
is of type $C_{n}$. Let $\gamma^{\vee}$ be a positive short root. Then there
is a $w\in W_{J}$ such that $w(\lambda)=\lambda$ and
$w(\gamma^{\vee})=\beta^{\vee}$. We can prove this explicitly as follows.

We identify the coweight lattice of $\Sigma_{J}$ with a submodule of
$\frac{1}{2}\mathbb{Z}^{n}$. Let $e_{1},\dotsc,e_{n}$ be the standard basis of
$\mathbb{Z}^{n}$. Then the coroots are given by

$\pm e_{i},i=1,\dotsc,n\text{ and }\pm e_{i}\pm e_{j},1\leq i<j\leq n.$

We choose the ordering so that the positive coroots are given by
$e_{i},i=1,\dotsc,n$ and $e_{i}\pm e_{j},1\leq i<j\leq n$. Thus the short
positive coroots are given by $e_{1},\dotsc,e_{n}$ with $\beta^{\vee}=e_{1}$,
and each can be made dominant using $w\in S_{n}\subset W_{J}$. By the
condition on $\lambda$, the subgroup $S_{n}$ fixes $\lambda$. Thus
$w(\lambda+\gamma^{\vee})=\lambda+\beta^{\vee}=\mu$ and hence
$(\lambda+\gamma^{\vee})_{\mathrm{dom}}\preccurlyeq\mu$, i.e. $k_{-\gamma}\geq
1$. The same argument as case (2) proves the desired inequality.

  4. (4)+(5)

Let $\mathrm{supp}(\mu-\lambda)=\\{\alpha_{1},\alpha_{2}\\}$ with $\alpha_{1}$
the short root. Then in both cases, we have

$\mu-\lambda=\beta^{\vee}=\alpha_{1}^{\vee}+\alpha_{2}^{\vee}.$

Thus $\langle\beta^{\vee},2\rho\rangle=4$ and it suffices to show that there
exist two positive coroots $\gamma^{\vee}_{1}$ and $\gamma^{\vee}_{2}$ such
that
$(\lambda+\gamma_{1}^{\vee})_{\mathrm{dom}}=(\lambda+\gamma_{2}^{\vee})_{\mathrm{dom}}=\mu$,
in which case we have $k_{-\gamma_{1}}\geq 1$ and $k_{-\gamma_{2}}\geq 1$.
Clearly we can take $\gamma^{\vee}_{1}=\beta^{\vee}$. Note that
$s_{\alpha_{1}}(\lambda)=\lambda$, and
$s_{\alpha_{1}}(\alpha_{2}^{\vee})=\alpha_{1}^{\vee}+\alpha_{2}^{\vee}$. Thus
$(\lambda+\alpha_{2}^{\vee})_{\mathrm{dom}}=\mu$ and we can take
$\gamma_{2}^{\vee}=\alpha_{2}^{\vee}$.

∎

### 4.5.

We now prove our main theorem which verifies a conjecture of Haines–Richarz
[HR20, Conjecture 5.4].

######  Theorem 4.6.

Let $\mathrm{Gr}_{\mathcal{G}}$ be the twisted affine Grassmannian associated
to an absolutely special vertex $\mathfrak{s}_{0}$. Let $\mu\in
X_{*}(T)^{+}_{I}$. Then the smooth locus of
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ is
$\mathrm{Gr}_{\mathcal{G},\mu}$.

###### Proof.

Clearly
$\mathrm{Gr}_{\mathcal{G},\mu}\subset\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$
is smooth. For the converse, note that the smooth locus is open, hence if
$\lambda\preccurlyeq\mu$ is such that
$\mathrm{Gr}_{\mathcal{G},\lambda}\subset\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}^{\mathrm{sm}}$,
then
$\mathrm{Gr}_{\mathcal{G},\nu}\subset\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}^{\mathrm{sm}}$
for $\lambda\preccurlyeq\nu\preccurlyeq\mu$. Thus it suffices to show for
$\mu\rightsquigarrow\lambda$ a minimal degeneration that
$\mathrm{Gr}_{\mathcal{G},\lambda}$ does not lie in the smooth locus of
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$. Using the
$L^{+}\mathcal{G}$-action, it suffices to show
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ is singular at $t^{\lambda}$, i.e.
that

$\dim
T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}>\dim\mathrm{Gr}_{\preccurlyeq\mu}=\langle\mu,2\rho\rangle.$

By Proposition 4.4, we have

$\dim(T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu})^{\mathrm{root}}\geq\dim\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}.$

Thus it suffices to show there exists $X\in
T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ with non-zero
component in $(u^{-1}\mathbb{C}[u^{-1}]\mathfrak{t}_{H})^{\sigma}$.

Note that the restriction of the $L^{+}\mathcal{G}$ action on
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ to the subgroup

$\mathcal{G}^{\lambda}:=\mathcal{G}(\mathbb{C}[[t]])\cap
t^{\lambda}\mathcal{G}(\mathbb{C}[[t]])t^{-\lambda}$

preserves the point $t^{\lambda}$. Thus we obtain an action $\star$ of
$\mathcal{G}^{\lambda}$ on
$T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$. Under the
identification (3.7.1), the action on the right is given by composing the
natural adjoint action with the map $g\mapsto t^{-\lambda}gt^{\lambda}$. Now
since $\langle\mu,2\rho\rangle>\langle\lambda,2\rho\rangle$, we have
$k_{-\alpha}\geq 1$ for some $\alpha\in\Sigma^{+}$ and hence $e_{a}\in
T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ for
$a=\psi_{\mathfrak{s}_{0}}(-\alpha-1)$.We set
$h^{\prime}:=\tilde{f}_{\psi_{\mathfrak{s}_{0}}(\alpha+\langle\lambda,\alpha\rangle)}(1)\in\mathcal{G}^{\lambda}$
and
$h:=t^{-\lambda}h^{\prime}t^{\lambda}=\tilde{f}_{\psi_{\mathfrak{s}_{0}}(\alpha)}(1)$.
We claim that

$X:=h^{\prime}\star
e_{\psi_{\mathfrak{s}_{0}}(-\alpha-1)}=\mathrm{Ad}(h)e_{\psi_{\mathfrak{s}_{0}}(-\alpha-1)}\in
T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$

has non-zero component along
$(u^{-1}\mathbb{C}[u^{-1}]\mathfrak{t}_{H})^{\sigma}$.

Note that since $\mathfrak{s}_{0}$ is absolutely special, we have
$\psi_{\mathfrak{s}_{0}}(\alpha)=\alpha^{\prime}\in\Phi_{\mathrm{af}}$ with
$\alpha^{\prime}\in\Phi$ a relative root and
$\frac{1}{2}\alpha^{\prime}\notin\Phi$ (cf. §3.2). Then

$\psi_{\mathfrak{s}_{0}}(-\alpha-1)=\begin{cases}\alpha^{\prime}-\frac{1}{d_{\alpha^{\prime}}}&\text{if
$2\alpha^{\prime}\notin\Phi$,}\\\
2\alpha^{\prime}-\frac{1}{2}&\text{otherwise.}\end{cases}$

Then by Proposition 2.7, $X$ has a non-zero component in
$(u^{-1}\mathbb{C}[u^{-1}]\mathfrak{t}_{H})^{\sigma}$ as desired. ∎

######  Remark 4.6.1.

A standard argument can be used to extend the theorem to a general reductive
group $G$ over $\mathbb{C}(\\!(t)\\!)$. We write
$G_{\mathrm{ad}}=\prod_{i=1}^{r}\mathrm{Res}_{F_{i}/F}G_{i}$, where $G_{i}$ is
an adjoint absolutely simple group over a finite extension $F_{i}$ of $F$. In
this case a vertex $\mathfrak{s}_{0}$ in the building $\mathcal{B}(G,F)$ for
$G$ over $F$ is said to be absolutely special if it maps to an absolutely
special vertex in
$\mathcal{B}(\mathrm{Res}_{F_{i}/F}G_{i},F)\cong\mathcal{B}(G_{i},F_{i})$ for
all $i$. Then if $\mathcal{G}/\mathbb{C}[\\![t]\\!]$ corresponds to an
absolutely special vertex $\mathfrak{s}_{0}$, any Schubert variety of
$\mathrm{Gr}_{\mathcal{G}}$ is isomorphic to a product of Schubert varieties
in $\mathrm{Gr}_{\mathcal{G}_{i}}$, where $\mathcal{G}_{i}$ corresponds to the
image of $\mathfrak{s}_{0}$ in $\mathcal{B}(G_{i},F_{i})$. It follows that the
smooth locus is the open stratum.

### 4.7.

Here we explain an alternative proof of Theorem 4.6 that uses some extra
algebro-geometric inputs, but which avoids the combinatorics in Proposition
4.4.

Let $\mu\rightsquigarrow\lambda$ be a minimal degeneration and set
$Y:=\mathrm{Gr}_{\mathcal{G},\mu}\cup\mathrm{Gr}_{\mathcal{G},\lambda}\subset\mathrm{Gr}_{\mathcal{G}}$.
Let $T^{\prime}:=T_{H}^{\sigma}$ be the fixed point scheme of $T_{H}$ over
$\mathbb{C}$ under the action of $\sigma$. Then $T^{\prime}\subset
L^{+}\mathcal{G}$ and hence acts on $Y$. By [BH, Theorem 4.2, Corollary 4.3],
the fixed point scheme $Y^{T^{\prime}}$ has reduced locus
$(Y^{T^{\prime}})_{\rm red}$ which is $0$-dimensional and consists of just the
points $t^{\mu}$ and $t^{\lambda}$.

Suppose now that $Y$ is smooth. Then $Y^{T^{\prime}}$ is smooth by [Fog73],
and hence reduced and equal to $\\{t^{\mu},t^{\lambda}\\}$, so $t^{\lambda}$
is an isolated $T^{\prime}$-fixed point in $Y$. Then, the tangent space
$T_{t^{\lambda}}Y=T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ is
a $T^{\prime}$-module whose weight zero direct summand
$(T_{t^{\lambda}}Y)_{0}$ is trivial. This implies that
$T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}=(T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu})^{\rm
root}$. Now $\mu-\lambda=\beta^{\vee}$, for some $\beta\in\Sigma^{+}$, and
hence $k_{-\beta}^{(\lambda,\mu)}\geq 1$. Then as in the proof of Theorem 4.6,
taking
$h^{\prime}=\tilde{f}_{\psi_{\mathfrak{s}_{0}}(\beta+\langle\lambda,\beta\rangle)}(1)$,
we obtain an element

$X=h^{\prime}\star e_{\psi_{\mathfrak{s}_{0}}(-\beta-1)}\in
T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$

whose component in $(u^{-1}\mathbb{C}[u^{-1}]\mathfrak{t}_{H})^{\sigma}$ is
non-trivial. Then
$X\not\in(T_{t^{\lambda}}\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu})^{\rm
root}$ which gives a contradiction.

###  4.8. Example: The ramified triality

We explain how our computations can be used to give a lower bound on the
dimension of the tangent space at the worst stratum of the quasi-minuscule
Schubert variety for the ramified triality. This recovers a result of
Haines–Richarz [HR20, Theorem 8.1].

In our notation we can take $H=\mathrm{Spin}_{8}$. The absolute root system is
of type $D_{4}$ with simple roots $\alpha_{i},i=1,2,3,4$; here we use the
notation from Bourbaki [Bou68]. Then $\sigma_{0}$ fixes $\alpha_{2}$ and acts
as the cycle $1\rightarrow 3\rightarrow 4\rightarrow 1$ on the other roots.
The $\sigma_{0}$-orbits on positive roots can be listed as follows:

$\\{\alpha_{2}\\},\\{\alpha_{1},\alpha_{3},\alpha_{4}\\},\\{\alpha_{1}+\alpha_{2},\alpha_{2}+\alpha_{3},\alpha_{2}+\alpha_{4}\\},$
$\\{\alpha_{1}+\alpha_{2}+\alpha_{3},\alpha_{2}+\alpha_{3}+\alpha_{4},\alpha_{1}+\alpha_{2}+\alpha_{4}\\},$
$\\{\alpha_{1}+\alpha_{2}+\alpha_{3}+\alpha_{4}\\},\\{\alpha_{1}+2\alpha_{2}+\alpha_{3}+\alpha_{4}\\}.$

The simple roots for the échelonnage root system $\Sigma$ are the modified
norms of the simple absolute roots, so a basis for $\Sigma$ is the pair
$\alpha=\alpha_{2},\beta:=\alpha_{1}+\alpha_{3}+\alpha_{4}$ which is of type
$G_{2}$. The highest root is $\tilde{\alpha}=3\alpha+2\beta$.

The coroots are given by

$\alpha^{\vee}=\alpha_{2}^{\vee},\quad\beta^{\vee}=\frac{\alpha_{1}^{\vee}+\alpha_{3}^{\vee}+\alpha_{4}^{\vee}}{3}$

and we have $\tilde{\alpha}^{\vee}=\alpha^{\vee}+\beta^{\vee}$. Now take
$\mu=\tilde{\alpha}^{\vee}$ and $\lambda=0$. We obtain 6 tangent directions of
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ at the base point
$\mathrm{Gr}_{\mathcal{G},\lambda}=\\{e\\}$ by using the root curves

$x\mapsto f_{\pm\beta-1}(x):=U_{\alpha_{1}}(u^{-1}x)U_{\alpha_{3}}(u^{-1}\zeta
x)U_{\alpha_{4}}(u^{-1}\zeta^{2}x),$ $x\mapsto
f_{\pm(\beta+3\alpha)-1}(x):=U_{\alpha_{1}+\alpha_{2}}(u^{-1}x)U_{\alpha_{2}+\alpha_{3}}(u^{-1}\zeta
x)U_{\alpha_{2}+\alpha_{4}}(u^{-1}\zeta^{2}x),$ $x\mapsto
f_{\pm(2\beta+3\alpha)-1}(x):=U_{\alpha_{1}+\alpha_{2}+\alpha_{3}}(u^{-1}x)U_{\alpha_{2}+\alpha_{3}+\alpha_{4}}(u^{-1}\zeta
x)U_{\alpha_{1}+\alpha_{2}+\alpha_{4}}(u^{-1}\zeta^{2}x),$

where $\zeta$ is a third root of unity.

Let $e_{1},e_{2},e_{3},e_{4}$ be a basis for the Cartan $\mathfrak{t}_{H}$ so
that $e_{i}-e_{i+1}$ corresponds to the root $\alpha_{i}$. Then conjugating
the root curve $x\mapsto f_{-\beta-1}(x)$ by $h:=\tilde{f}_{\beta}(1)$ gives
an extra tangent direction along the Cartan given by

$u^{-1}((e_{1}-e_{2})+\zeta(e_{3}-e_{4})+\zeta^{2}(e_{3}+e_{4}))\in(u^{-1}\mathfrak{t}_{H})^{\sigma}.$

In fact $(u^{-1}\mathfrak{t}_{H})^{\sigma}$ is 1-dimensional and the tangent
direction along the Cartan given by conjugating the other two root curves
above will be the same. Thus we get a lower bound of 7 for the dimension of
the tangent space. On the other hand $\langle\mu,2\rho\rangle=6$, hence
$\dim\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}=6$. So,
$\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ is singular at the point
$\mathrm{Gr}_{\mathcal{G},\lambda}=\\{e\\}$.

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