In the proof of Theorem 1.1, the following assertion in line 4 of page 1725 is incorrect:
The kernel of the composition
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D70B}_{1}(G)^{\unicode[STIX]{x1D6E4}}$
is generated by the elements
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FD}^{\vee }$
, where
$\unicode[STIX]{x1D6FA}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}$
satisfies
$\unicode[STIX]{x1D6FA}\cap C\neq \emptyset$
(
$C$
defined as in Proposition 4.1.9).
The mistake consists of a misapplication of Proposition 4.1.9, which asserts only that an element in the kernel is a
$\mathbb{Z}$
-linear combination of elements in the Galois orbit of
$C$
. Although an element in the kernel is
$\unicode[STIX]{x1D6E4}$
-invariant, in general this is not enough to imply that it is a sum of elements of the form
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FD}^{\vee }$
, where one sums over a Galois orbit
$\unicode[STIX]{x1D6FA}$
of an element in
$C$
. More precisely, the assertion is incorrect in certain cases when there are Galois orbits of coroots of different orders. This was pointed out to the authors by Sian Nie.
We replace the above argument by the proposition below. All other assertions of the paper including the rest of the proof of Theorem 1.1 remain unchanged.
Proposition 0.0.1. There exists
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{0}\subseteq \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}$
such that:
-
(i) the kernel of the map
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D70B}_{1}(G)^{\unicode[STIX]{x1D6E4}}$
is generated by the elements
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{0}}\unicode[STIX]{x1D6FD}^{\vee }$
and
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FD}^{\vee }$
, where
$\unicode[STIX]{x1D6FA}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}$
satisfies
$\unicode[STIX]{x1D6FA}\cap C\neq \emptyset$
and
$\unicode[STIX]{x1D6FA}_{0}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{0}$
; -
(ii) for any element
$\unicode[STIX]{x1D6FA}_{0}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{0}$
, the element
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{0}}\unicode[STIX]{x1D6FD}^{\vee }$
is mapped to
$1$
by the composite
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\cong \unicode[STIX]{x1D70B}_{0}(X_{\unicode[STIX]{x1D707}_{x}}^{M}(b))\rightarrow \unicode[STIX]{x1D70B}_{0}(X_{\unicode[STIX]{x1D707}}^{G}(b))$
.
Lemma 0.0.2. Let
$x,x^{\prime }\in \bar{I}_{\unicode[STIX]{x1D707},b}^{M,G}$
be such that
$x^{\prime }=x+\unicode[STIX]{x1D6FC}^{\vee }-\unicode[STIX]{x1D6FC}^{m\vee }$
with
$\unicode[STIX]{x1D6FC}$
an adapted positive root in
$N$
. Let
$\unicode[STIX]{x1D6FA}$
be the Galois orbit of
$\unicode[STIX]{x1D6FC}$
. Then, for all
$g^{\prime }M({\mathcal{O}}_{L})\in X_{\unicode[STIX]{x1D707}_{x^{\prime }}}^{M}(b)$
, there exist
$gM({\mathcal{O}}_{L})\in X_{\unicode[STIX]{x1D707}_{x}}^{M}(b)$
and
$n\in \mathbb{Z}$
such that
$g\sim g^{\prime }$
and
$$\begin{eqnarray}w_{M}(g)-w_{M}(g^{\prime })=\mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}^{\vee }+n\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FD}^{\vee }\quad \text{in }\unicode[STIX]{x1D70B}_{1}(M).\end{eqnarray}$$
Proof. If
$x^{\prime }\rightarrow x$
is of immediate distance and
$x^{\prime }-x=\unicode[STIX]{x1D6FC}^{\vee }-\unicode[STIX]{x1D6FC}^{m\vee }$
is as in Definition 4.4.8, then for any
$g^{\prime }M({\mathcal{O}}_{L})\in X_{\unicode[STIX]{x1D707}_{x^{\prime }}}^{M}(b)$
, there exists
$gM({\mathcal{O}}_{L})\in X_{\unicode[STIX]{x1D707}_{x}}^{M}(b)$
such that
$g\sim g^{\prime }$
and
$$\begin{eqnarray}w_{M}(g)-w_{M}(g^{\prime })=\mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}^{i\vee }\quad \text{in }\unicode[STIX]{x1D70B}_{1}(M)\end{eqnarray}$$
by Proposition 4.5.4 (in which there is also a sign typo in (4.5.5)) and the fact that
$J_{b}^{M}(F)$
acts transitively on
$X_{\unicode[STIX]{x1D707}_{x}}^{M}(b)$
. Similarly, if
$x\rightarrow x^{\prime }$
, then for any
$g^{\prime }M({\mathcal{O}}_{L})\in X_{\unicode[STIX]{x1D707}_{x^{\prime }}}^{M}(b)$
, there exists
$gM({\mathcal{O}}_{L})\in X_{\unicode[STIX]{x1D707}_{x}}^{M}(b)$
such that
$g\sim g^{\prime }$
and
$$\begin{eqnarray}w_{M}(g^{\prime })-w_{M}(g)=\mathop{\sum }_{i=m}^{|\unicode[STIX]{x1D6FA}|-1}\unicode[STIX]{x1D6FC}^{i\vee }\quad \text{in }\unicode[STIX]{x1D70B}_{1}(M).\end{eqnarray}$$
Hence,
$$\begin{eqnarray}w_{M}(g)-w_{M}(g^{\prime })=\mathop{\sum }_{i=0}^{m-1}\unicode[STIX]{x1D6FC}^{i\vee }-\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FD}^{\vee }\quad \text{in }\unicode[STIX]{x1D70B}_{1}(M).\end{eqnarray}$$
The general case is reduced to the immediate distance case by the proof of Proposition 4.4.10.◻
Proof of Proposition 0.0.1. Let
$S:=\{\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FD}^{\vee }\in \unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}|\unicode[STIX]{x1D6FA}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}\text{ and }\unicode[STIX]{x1D6FA}\cap C\neq \emptyset \}$
.
If the kernel of the composition
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D70B}_{1}(G)^{\unicode[STIX]{x1D6E4}}$
is generated by the elements in
$S$
, then take
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{0}=\emptyset$
and we are done.
Therefore, it suffices to prove the proposition under the following hypothesis.
(HYP): the kernel of the composition
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D70B}_{1}(G)^{\unicode[STIX]{x1D6E4}}$
is not generated by the elements in
$S$
.
Then not all elements in
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}$
have the same cardinality and hence the Dynkin diagram of
$G$
is of type
$A_{2n+1}$
,
$D_{n}$
or
$E_{6}$
. Let
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{small}}$
be the subset of
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}$
consisting of the orbits of smallest cardinality, and let
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{large}}=\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}\backslash \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{small}}$
. Then the elements of
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{large}}$
are all of the same cardinality, which is
$n_{G}$
times the cardinality of the elements in
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{small}}$
with
$n_{G}=2$
or 3. Here,
$n_{G}=3$
only occurs when the Dynkin diagram of
$G$
is of type
$D_{4}$
. Moreover, by Proposition 4.1.9, for every
$\unicode[STIX]{x1D6FA}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{large}}$
,
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FD}^{\vee }$
is contained in the subgroup of
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}$
generated by
$S$
. Hence, we will define
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{0}$
as a subset of
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{small}}$
.
Case 1: the Dynkin diagram of
$G$
is of type
$A_{2n+1}$
or of type
$D_{4}$
with
$n_{G}=3$
. One can easily show that the kernel of the map
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D70B}_{1}(G)^{\unicode[STIX]{x1D6E4}}$
is generated by the elements in
$S$
and
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}^{\prime }}\unicode[STIX]{x1D6FD}^{\vee }$
for any
$\unicode[STIX]{x1D6FA}^{\prime }\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{small}}$
. Hence, it remains to find
$\unicode[STIX]{x1D6FA}_{0}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{small}}$
satisfying condition (2) in the statement and to define
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{0}=\{\unicode[STIX]{x1D6FA}_{0}\}$
.
Claim 1. Under hypothesis (HYP), there exist adapted positive roots
$\unicode[STIX]{x1D6FC}_{i}\in \unicode[STIX]{x1D6FA}_{i}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{large}}$
for
$i=1,2$
such that
$\unicode[STIX]{x1D6FC}_{1}-\unicode[STIX]{x1D6FC}_{2}=\unicode[STIX]{x1D6FC}_{1}^{d}-\unicode[STIX]{x1D6FC}_{2}^{d}$
is still a root in
$N$
, where
$d$
is the number of connected components of the Dynkin diagram of
$G$
and such that
$$\begin{eqnarray}x^{\prime }=x+\unicode[STIX]{x1D6FC}_{1}^{\vee }-\unicode[STIX]{x1D6FC}_{1}^{d\vee }=x+\unicode[STIX]{x1D6FC}_{2}^{\vee }-\unicode[STIX]{x1D6FC}_{2}^{d\vee }\in \bar{I}_{\unicode[STIX]{x1D707},b}^{M,G}.\end{eqnarray}$$
We only show Claim 1 when the Dynkin diagram is of type
$A_{2n+1}$
. The proof for the other case of type
$D_{4}$
is similar and much easier and is therefore omitted. By Proposition 4.1.9, we can choose a connected component of the Dynkin diagram of
$G$
with the following numbering of the simple roots.
such that there exists a pair
$(i_{0},j_{0})\in \mathbb{N}^{2}$
with
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D6FE}_{-i_{0}}+\cdots +\unicode[STIX]{x1D6FE}_{0}+\cdots +\unicode[STIX]{x1D6FE}_{j_{0}},\unicode[STIX]{x1D707}_{x}\rangle =-1. & & \displaystyle\end{eqnarray}$$
Moreover, by condition (HYP) for all
$0\leqslant i\leqslant n$
,
$\langle \unicode[STIX]{x1D6FE}_{-i}+\cdots +\unicode[STIX]{x1D6FE}_{0}+\cdots +\unicode[STIX]{x1D6FE}_{i},\unicode[STIX]{x1D707}_{x}\rangle \neq -1$
; therefore, we may assume that
$i_{0}>j_{0}$
(possibly exchanging the notations
$\unicode[STIX]{x1D6FE}_{i}$
and
$\unicode[STIX]{x1D6FE}_{-i}$
) and that
$$\begin{eqnarray}\langle \unicode[STIX]{x1D6FE}_{-i_{0}}+\cdots +\unicode[STIX]{x1D6FE}_{-j_{0}-1},\unicode[STIX]{x1D707}_{x}\rangle =-1,\quad \langle \unicode[STIX]{x1D6FE}_{j_{0}+1}+\cdots +\unicode[STIX]{x1D6FE}_{i_{0}},\unicode[STIX]{x1D707}_{x}\rangle =1.\end{eqnarray}$$
It follows that
$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{1}=\unicode[STIX]{x1D6FE}_{-i_{0}}+\cdots +\unicode[STIX]{x1D6FE}_{j_{0}},\quad \unicode[STIX]{x1D6FC}_{2}=\unicode[STIX]{x1D6FE}_{-i_{0}}+\cdots +\unicode[STIX]{x1D6FE}_{-j_{0}-1}\end{eqnarray}$$
are the desired elements in Claim 1.
Let
$\unicode[STIX]{x1D6FA}_{0}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{small}}$
be the Galois orbit of
$\unicode[STIX]{x1D6FC}_{1}-\unicode[STIX]{x1D6FC}_{2}$
, and
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{0}=\{\unicode[STIX]{x1D6FA}_{0}\}$
. We want to show that
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{0}}\unicode[STIX]{x1D6FD}^{\vee }\in \unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}$
is mapped to
$1$
under the composite
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\cong \unicode[STIX]{x1D70B}_{0}(X_{\unicode[STIX]{x1D707}_{x}}^{M}(b))\rightarrow \unicode[STIX]{x1D70B}_{0}(X_{\unicode[STIX]{x1D707}}^{G}(b))$
. By Lemma 0.0.2, for any
$g^{\prime }M({\mathcal{O}}_{L})\in X_{\unicode[STIX]{x1D707}_{x^{\prime }}}^{M}(b)$
, there exist
$g_{1}M({\mathcal{O}}_{L}),g_{2}M({\mathcal{O}}_{L})\in X_{\unicode[STIX]{x1D707}_{x}}^{M}(b)$
and
$m_{1},m_{2}\in \mathbb{Z}$
such that
$g_{1}\sim g^{\prime }\sim g_{2}$
and
$$\begin{eqnarray}\displaystyle w_{M}(g_{1})-w_{M}(g^{\prime }) & = & \displaystyle \mathop{\sum }_{i=0}^{d-1}\unicode[STIX]{x1D6FC}_{1}^{i}+m_{1}\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{1}}\unicode[STIX]{x1D6FD}^{\vee }\quad \text{in }\unicode[STIX]{x1D70B}_{1}(M),\nonumber\\ \displaystyle w_{M}(g_{2})-w_{M}(g^{\prime }) & = & \displaystyle \mathop{\sum }_{i=0}^{d-1}\unicode[STIX]{x1D6FC}_{2}^{i}+m_{2}\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{2}}\unicode[STIX]{x1D6FD}^{\vee }\quad \text{in }\unicode[STIX]{x1D70B}_{1}(M).\nonumber\end{eqnarray}$$
Taking the difference of the above two equalities, we get
$$\begin{eqnarray}w_{M}(g_{1})-w_{M}(g_{2})=\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{0}}\unicode[STIX]{x1D6FD}^{\vee }+(m_{1}-m_{2})\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{1}}\unicode[STIX]{x1D6FD}^{\vee }+m_{2}n_{G}\mathop{\sum }_{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{0}}\unicode[STIX]{x1D6FD}^{\vee }\quad \text{in }\unicode[STIX]{x1D70B}_{1}(M).\end{eqnarray}$$
Using
$g_{1}\sim g_{2}$
and the fact that
$J_{b}^{M}(F)$
acts transitively on
$X_{\unicode[STIX]{x1D707}_{x}}^{M}(b)$
, the element
$$\begin{eqnarray}w_{M}(g_{2})-w_{M}(g_{1})\in \unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\end{eqnarray}$$
is mapped to
$1$
by the composite
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\cong \unicode[STIX]{x1D70B}_{0}(X_{\unicode[STIX]{x1D707}_{x}}^{M}(b))\rightarrow \unicode[STIX]{x1D70B}_{0}(X_{\unicode[STIX]{x1D707}}^{G}(b))$
.
On the other hand, as
$\unicode[STIX]{x1D6FA}_{0}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{small}}$
and
$\unicode[STIX]{x1D6FA}_{1}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{large}}$
, by Proposition 4.1.9,
$n_{G}\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{0}}\unicode[STIX]{x1D6FD}^{\vee }\in \unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}$
and
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{1}}\unicode[STIX]{x1D6FD}^{\vee }\in \unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}$
are both contained in the subgroup generated by
$S$
and hence by the proof of Theorem 1.1 (more precisely the third paragraph on page 1725, which is not affected by the gap we are discussing here),
$n_{G}\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FD}^{\vee }$
and
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{1}}\unicode[STIX]{x1D6FD}^{\vee }$
are both mapped to
$1$
by the composite
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\cong \unicode[STIX]{x1D70B}_{0}(X_{\unicode[STIX]{x1D707}_{x}}^{M}(b))\rightarrow \unicode[STIX]{x1D70B}_{0}(X_{\unicode[STIX]{x1D707}}^{G}(b))$
. Therefore, so is
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{0}}\unicode[STIX]{x1D6FD}^{\vee }$
, i.e.
$\unicode[STIX]{x1D6FA}_{0}$
satisfies condition (2) of the proposition.
Case 2: the Dynkin diagram of
$G$
is of type
$D_{n}$
with
$n_{G}=2$
. Consider any connected component of the Dynkin diagram of
$G$
with the following numbering of the simple roots.
Let
$\unicode[STIX]{x1D6FA}_{i}$
be the Galois orbit of
$\unicode[STIX]{x1D6FC}_{i}$
for
$1\leqslant i\leqslant n-2$
. Then the kernel of
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D70B}_{1}(G)^{\unicode[STIX]{x1D6E4}}$
is generated by the elements in
$S$
and
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{i}}\unicode[STIX]{x1D6FD}^{\vee }$
for
$1\leqslant i\leqslant n-2$
. We will construct subsets
$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{1}\subseteq \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{2}\subseteq \cdots \subseteq \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{n-2}=:\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{0}\subseteq \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}\end{eqnarray}$$
such that for any
$\unicode[STIX]{x1D6FA}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{n-2}$
, the element
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{0}}\unicode[STIX]{x1D6FD}^{\vee }$
is mapped to
$1$
under the composite
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}\cong \unicode[STIX]{x1D70B}_{0}(X_{\unicode[STIX]{x1D707}_{x}}^{M}(b))\rightarrow \unicode[STIX]{x1D70B}_{0}(X_{\unicode[STIX]{x1D707}}^{G}(b))$
and, for any
$1\leqslant i\leqslant n-2$
,
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{i}}\unicode[STIX]{x1D6FD}^{\vee }$
is contained in the subgroup of
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}$
generated by the elements in
$S$
and
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}}\unicode[STIX]{x1D6FD}^{\vee }$
, where
$\unicode[STIX]{x1D6FA}\in \unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{i}$
. Note that the orbits of
$\unicode[STIX]{x1D6FC}_{-1}$
and
$\unicode[STIX]{x1D6FC}_{-1}^{\prime }$
are in
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{large}}$
and therefore need not be considered.
We first construct
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{1}$
. By Proposition 4.1.9, there exists
$\unicode[STIX]{x1D6FC}\in C$
such that
$\unicode[STIX]{x1D70E}^{l}(\unicode[STIX]{x1D6FC}_{1})\preccurlyeq \unicode[STIX]{x1D6FC}$
for some
$l$
. Without loss of generality, we assume that
$l=0$
. We may also assume that
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{1}}\unicode[STIX]{x1D6FD}^{\vee }$
is not contained in the subgroup of
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}$
generated by
$S$
, otherwise let
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{1}=\emptyset$
. By (HYP), the Galois orbit of
$\unicode[STIX]{x1D6FC}\in C$
is in
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{\text{large}}$
. In particular,
$\langle \unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D707}_{x}\rangle \geqslant 0$
and
$\langle \unicode[STIX]{x1D6FC}_{-1},\unicode[STIX]{x1D707}_{x}\rangle =-1$
(possibly exchanging
$\unicode[STIX]{x1D6FC}_{-1}$
and
$\unicode[STIX]{x1D6FC}_{-1}^{\prime }$
). By the existence of
$\unicode[STIX]{x1D6FC}$
and the minimality of
$\unicode[STIX]{x1D707}_{x}$
, there are four possibilities:
Case 2.1:
$\langle \unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D707}_{x}\rangle =0$
,
$\langle \unicode[STIX]{x1D6FC}_{-1}^{\prime },\unicode[STIX]{x1D707}_{x}\rangle =1$
;
Case 2.2:
$\langle \unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D707}_{x}\rangle =0$
,
$\langle \unicode[STIX]{x1D6FC}_{-1}^{\prime },\unicode[STIX]{x1D707}_{x}\rangle =0$
;
Case 2.3:
$\langle \unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D707}_{x}\rangle =1$
,
$\langle \unicode[STIX]{x1D6FC}_{-1}^{\prime },\unicode[STIX]{x1D707}_{x}\rangle =-1$
;
Case 2.4:
$\langle \unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D707}_{x}\rangle =1$
,
$\langle \unicode[STIX]{x1D6FC}_{-1}^{\prime },\unicode[STIX]{x1D707}_{x}\rangle =0$
and there exist
$2\leqslant i\leqslant n-2$
such that
$$\begin{eqnarray}\langle \unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D707}_{x}\rangle =\left\{\begin{array}{@{}ll@{}}-1,\quad & j=i,\\ 0,\quad & 2\leqslant j\leqslant i-1.\end{array}\right.\end{eqnarray}$$
In Cases 2.2 and 2.3, we have
$\unicode[STIX]{x1D6FC}_{1}+\unicode[STIX]{x1D6FC}_{-1}+\unicode[STIX]{x1D6FC}_{-1}^{\prime }\in C$
. In Case 2.4,
$\unicode[STIX]{x1D6FC}_{i}+\cdots +\unicode[STIX]{x1D6FC}_{1}+\unicode[STIX]{x1D6FC}_{-1}+\unicode[STIX]{x1D6FC}_{-1}^{\prime }\in C$
and
$\unicode[STIX]{x1D6FC}_{i}+\cdots +\unicode[STIX]{x1D6FC}_{2}\in C$
. Thus, for Cases 2.2–2.4,
$\sum _{\unicode[STIX]{x1D6FD}\in \unicode[STIX]{x1D6FA}_{1}}\unicode[STIX]{x1D6FD}^{\vee }$
is contained in the subgroup of
$\unicode[STIX]{x1D70B}_{1}(M)^{\unicode[STIX]{x1D6E4}}$
generated by
$S$
and hence these cases will not occur.
It remains to consider Case 2.1. In that case
$$\begin{eqnarray}x+\unicode[STIX]{x1D6FC}_{-1}^{\vee }-{\unicode[STIX]{x1D6FC}_{-1}^{\prime }}^{\vee }=x+(\unicode[STIX]{x1D6FC}_{-1}+\unicode[STIX]{x1D6FC}_{1})^{\vee }-(\unicode[STIX]{x1D6FC}_{-1}^{\prime }+\unicode[STIX]{x1D6FC}_{1})^{\vee }\in \bar{I}_{\unicode[STIX]{x1D707},b}^{M,G}.\end{eqnarray}$$
The same computation as in Case 1 shows that
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{1}:=\{\unicode[STIX]{x1D6FA}_{1}\}$
satisfies the desired properties.
For general
$i$
, we apply the same discussion as above. We obtain either
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{i}=\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{i-1}$
or
$\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{i}=\unicode[STIX]{x1D6F7}_{N,\unicode[STIX]{x1D6E4}}^{i-1}\cup \{\tilde{\unicode[STIX]{x1D6FA}}_{i}\}$
with
$\tilde{\unicode[STIX]{x1D6FA}}_{i}$
the Galois orbit of
$\unicode[STIX]{x1D6FC}_{1}+\cdots +\unicode[STIX]{x1D6FC}_{i}$
. Altogether, the proposition holds in Case 2.
Case 3: the Dynkin diagram of
$G$
is of type
$E_{6}$
. The discussion is very similar to Case 2 and is therefore omitted.◻
