of type $D_{n}$ with $n$ even. Otherwise,
$m({\breve{y}})=n_{{\breve{z}}_{0}}^{j}$ for a suitable $0\leqslant
j<|(\Omega_{{\breve{\mathbcal{a}}},{\mathrm{ad}}}|$.

We show that ${\mathrm{Ad}}(g_{\breve{z}})(m({\breve{y}}))=m({\breve{y}})$.

When $G_{{\breve{F}},{\mathrm{ad}}}$ is of type $D_{n},n$ even, this is a
consequence of Corollary 6.6(4). Otherwise, $g_{{\breve{z}}}$ and
$m({\breve{y}})$ are both powers of $n_{{\breve{z}}_{0}}$ and the claim is
obvious.

We show that $\sigma(m({\breve{y}}))=m({\breve{y}})$.

The proof involves a detailed case-by-case analysis. Recall that the
representatives $\\{n_{s_{\breve{a}}}\;|\;{\breve{a}}\in\breve{\Delta}\\}$
satisfies $H(\breve{\Delta},\sigma)$. When $G_{{\breve{F}},{\mathrm{ad}}}$ is
of type $D_{n},n$ even, we have
$\sigma(m({\breve{y}}))=\sigma(n_{\breve{y}})=n_{\breve{y}}=m({\breve{y}})$.
Next we consider the case where
$\Omega_{{\breve{\mathbcal{a}}},{\mathrm{ad}}}$ is cyclic. If the action of
$\sigma$ on $\Omega_{{\breve{\mathbcal{a}}},ad}$ is trivial, then
$\sigma({\breve{z}}_{0})={\breve{z}}_{0}$ and
$\sigma(n_{{\breve{z}}_{0}})=n_{{\breve{z}}_{0}}$. In this case,
$\sigma(m({\breve{y}}))=\sigma(n_{{\breve{z}}_{0}}^{j})=\sigma(n_{{\breve{z}}_{0}})^{j}=n_{{\breve{z}}_{0}}^{j}=m({\breve{y}})$.
It remains to prove the claim when
$\Omega_{{\breve{\mathbcal{a}}},{\mathrm{ad}}}$ is cyclic and the action of
$\sigma$ on $\Omega_{{\breve{\mathbcal{a}}},{\mathrm{ad}}}$ is non-trivial.

Recall that $j:G_{\breve{F}}\rightarrow G_{{\breve{F}},{\mathrm{ad}}}$ is the
adjoint quotient. Let
${\breve{\tau}}\in\Omega_{{\breve{\mathbcal{a}}}}^{\sigma^{*}}$. If
$j({\breve{\tau}})=1$, then ${\breve{y}}=1$ and $m({\breve{y}})=1$. In this
case, $\sigma(m({\breve{y}}))=1=m({\breve{y}})$.

Now we assume that $j({\breve{\tau}})\neq 1$. This happens when
$G_{{\breve{F}},{\mathrm{ad}}}$ is of type $A_{2n+1}$ and $j=n+1$ or
$G_{{\breve{F}},{\mathrm{ad}}}$ is of type $D_{2n+1}$ and $j=2$. In both these
cases
$\sigma({{\breve{\nu}}_{{\mathrm{ad}},0}})={{\breve{\nu}}_{{\mathrm{ad}},0}}^{-1}$
and $\sigma({\breve{z}}_{0})={\breve{z}}_{0}^{-1}$.

If $G_{{\breve{F}},{\mathrm{ad}}}$ is of type $A_{2n+1}$, then
$m({\breve{y}})=n_{{\breve{z}}_{0}}^{n+1}$ and
${\breve{z}}_{0}^{n+1}={\breve{y}}_{(n+1)}$. By Corollary 6.6(2), we have

$n_{{\breve{z}}_{0}}^{n+1}=\begin{cases}n_{{\breve{z}}_{0}^{n+1}},&\text{ if
}n\text{ is even},\\\
(a_{1}^{\vee}+a_{3}^{\vee}+\cdots+a_{n}^{\vee})(-1)n_{{\breve{z}}_{0}^{n+1}},&\text{
if }n\text{ is odd}.\end{cases}$

We have $\sigma({\breve{y}}_{(n+1)})={\breve{y}}_{(n+1)}$. Thus

$\sigma(n_{{\breve{z}}_{0}}^{n+1})=\begin{cases}n_{{\breve{z}}_{0}}^{n+1},&\text{
if }n\text{ is even},\\\
(a_{1}^{\vee}+a_{3}^{\vee}+\cdots+a_{2n+1}^{\vee})(-1)n_{{\breve{z}}_{0}}^{n+1},&\text{
if }n\text{ is odd}.\end{cases}$

We identify $\Omega_{{\breve{\mathbcal{a}}},{\mathrm{ad}}}$ with
$X_{*}(T_{{\mathrm{ad}}})/X_{*}(T_{sc})$. Under this identification,

$\displaystyle j({\breve{\tau}})$
$\displaystyle=(n+1){{\breve{\nu}}_{{\mathrm{ad}},0}}=(n+1)\frac{{\breve{a}}_{1}^{\vee}+2{\breve{a}}_{2}^{\vee}+3{\breve{a}}_{3}^{\vee}+\cdots+(2n+1){\breve{a}}_{2n+1}^{\vee}}{2n+2}$
$\displaystyle=\frac{{\breve{a}}_{1}^{\vee}+3{\breve{a}}_{3}^{\vee}+5{\breve{a}}_{5}^{\vee}+\cdots+(2n+1){\breve{a}}_{2n+1}^{\vee}}{2}+{\breve{a}}_{2}^{\vee}+2{\breve{a}}_{4}^{\vee}+\cdots+n{\breve{a}}_{2n}^{\vee}$
$\displaystyle\equiv\frac{{\breve{a}}_{1}^{\vee}+{\breve{a}}_{3}^{\vee}+{\breve{a}}_{5}^{\vee}+\cdots+{\breve{a}}_{2n+1}^{\vee}}{2}\mod
X_{*}(T_{sc}).$

Since ${\breve{\tau}}\in X_{*}(T)/X_{*}(T_{sc})$ and $j$ acts as identity on
$X_{*}(T_{sc})$,
$\frac{{\breve{a}}_{1}^{\vee}+{\breve{a}}_{3}^{\vee}+{\breve{a}}_{5}^{\vee}+\cdots+{\breve{a}}_{2n+1}^{\vee}}{2}\in
X_{*}(T)$. Hence

$\displaystyle({\breve{a}}_{1}^{\vee}+{\breve{a}}_{3}^{\vee}+\cdots+{\breve{a}}_{2n+1}^{\vee})(-1)=\left(\left(\frac{{\breve{a}}_{1}^{\vee}+{\breve{a}}_{3}^{\vee}+{\breve{a}}_{5}^{\vee}+\cdots+{\breve{a}}_{2n+1}^{\vee}}{2}\right)(-1)\right)^{2}=1\in
G(\breve{F}).$

Therefore we have
$\sigma(m({\breve{y}}))=\sigma(n_{{\breve{z}}_{0}}^{n+1})=n_{{\breve{z}}_{0}}^{n+1}=m({\breve{y}})$.

If $G_{{\breve{F}},{\mathrm{ad}}}$ is of type $D_{2n+1}$, we have
$m({\breve{y}})=n_{{\breve{z}}_{0}}^{2}$, and by Corollary 6.6(3),
$\sigma(m({\breve{y}}))=tm({\breve{y}})$ where
$t=({\breve{a}}_{2n}^{\vee}+{\breve{a}}_{2n+1}^{\vee})(-1)$. We claim that
$t=1$ in $G({\breve{F}})$. The argument is similar to type $A_{2n+1}$.
Consider the element
$j({\breve{\tau}})\in\Omega_{{\breve{\mathbcal{a}}},{\mathrm{ad}}}=X_{*}(T_{{\mathrm{ad}}})/X_{*}(T_{sc})$.
Then

${{\breve{\nu}}_{{\mathrm{ad}},0}}=\frac{{\breve{a}}_{1}^{\vee}+2{\breve{a}}_{2}^{\vee}+\cdots+(2n-1){\breve{a}}_{2n-1}^{\vee}+\frac{1}{2}(2n-1){\breve{a}}_{2n}^{\vee}+\frac{1}{2}(2n+1){\breve{a}}_{2n+1}^{\vee}}{2}\mod
X_{*}(T_{sc}).$

Then $j({\breve{\tau}})\equiv
2{{\breve{\nu}}_{{\mathrm{ad}},0}}\equiv\frac{1}{2}{\breve{a}}_{2n}^{\vee}+\frac{1}{2}{\breve{a}}_{2n+1}^{\vee}\mod
X_{*}(T_{sc})$. Since $2{\breve{\tau}}\in X_{*}(T_{sc})$ and $j$ acts as
identity on $X_{*}(T_{sc})$, we see that

$\displaystyle{\breve{\tau}}$
$\displaystyle\equiv\frac{1}{2}{\breve{a}}_{2n}^{\vee}+\frac{1}{2}{\breve{a}}_{2n+1}^{\vee}\mod
X_{*}(T_{sc}).$

Since ${\breve{\tau}}\in X_{*}(T)/X_{*}(T_{sc})$, we see that
$\frac{1}{2}{\breve{a}}_{2n}^{\vee}+\frac{1}{2}{\breve{a}}_{2n+1}^{\vee}\in
X_{*}(T)$. Hence

$\displaystyle t$
$\displaystyle=({\breve{a}}_{2n}^{\vee}+{\breve{a}}_{2n+1}^{\vee})(-1)=\left(\left(\frac{1}{2}{\breve{a}}_{2n}^{\vee}+\frac{1}{2}{\breve{a}}_{2n+1}^{\vee}\right)(-1)\right)^{2}=1.$

Hence $\sigma(m({\breve{y}}))=m({\breve{y}})$.

This finishes the proof of Proposition 6.9.

###  6.7. Proof of Theorem 6.4

For ${\breve{s}}\in\breve{\mathbb{S}}$, let
$m({\breve{s}})\in\breve{}\mathcal{T}$ be the lifting of ${\breve{s}}$ in
Corollary 6.8. For any
$\breve{\tau}\in\Omega_{{\breve{\mathbcal{a}}}}^{\sigma^{*}}$, let
$m(\breve{\tau})\in\breve{}\mathcal{T}$ be the lifting of $\breve{\tau}$
constructed in §6.6. Then $\sigma^{*}(m(\breve{\tau}))=m(\breve{\tau})$ for
all $\breve{\tau}\in\Omega_{{\breve{\mathbcal{a}}}}^{\sigma^{*}}$ and
$\sigma^{*}(m({\breve{s}}))=m(\sigma^{*}({\breve{s}}))$ for all
${\breve{s}}\in\breve{\mathbb{S}}$.

We set $\mathcal{T}=\breve{}\mathcal{T}^{\sigma^{*}}$.

For any $s\in\mathbb{S}$, we have $s={\breve{w}}_{\mathcal{X}}$ for some
$\sigma$-orbit $\mathcal{X}$ in $\breve{\Delta}$ with
${\breve{W}}_{\mathcal{X}}$ finite (see §2.3). Let
${\breve{w}}_{\mathcal{X}}={\breve{s}}_{i_{1}}\cdots{\breve{s}}_{i_{n}}$ be a
reduced expression of ${\breve{w}}_{\mathcal{X}}$ in
${\breve{W}}_{\text{af}}$. Then
${\breve{w}}_{\mathcal{X}}=\sigma({\breve{s}}_{i_{1}})\cdots\sigma({\breve{s}}_{i_{n}})$
is again a reduced expression of ${\breve{w}}_{\mathcal{X}}$ in
${\breve{W}}_{\text{af}}$. We have

$\displaystyle m({\breve{w}}_{\mathcal{X}})$
$\displaystyle=m({\breve{s}}_{i_{1}})\cdots
m({\breve{s}}_{i_{n}})=m(\sigma({\breve{s}}_{i_{1}}))\cdots
m(\sigma({\breve{s}}_{i_{n}}))=\sigma(m({\breve{s}}_{i_{1}}))\cdots\sigma(m({\breve{s}}_{i_{n}}))$
$\displaystyle=\sigma(m({\breve{w}}_{\mathcal{X}})).$

In particular,
$m(s)=m({\breve{w}}_{\mathcal{X}})\in\mathcal{T}=\breve{}\mathcal{T}^{\sigma^{*}}$.

Let $w\in W_{\text{af}}$ and $s_{i_{1}}\cdots s_{i_{n}}$ be a reduced
expression of $w$ in $W$. We set $m(w)=m(s_{i_{1}})\cdots m(s_{i_{n}})$. Then
$m(w)\in\mathcal{T}$. Suppose that $s^{\prime}_{i_{1}}\cdots
s^{\prime}_{i_{n}}$ is another reduced expression of $w$ in $W$. By §2.3 (a),
$\breve{\ell}(w)=\breve{\ell}(s_{i_{1}})+\cdots+\breve{\ell}(s_{i_{n}})=\breve{\ell}(s^{\prime}_{i_{1}})+\cdots+\breve{\ell}(s^{\prime}_{i_{n}})$.
Since $\\{m({\breve{s}})\;|\;{\breve{s}}\in\breve{\mathbb{S}}\\}$ satisfies
the Coxeter relations, by condition (2)(b) † in §3.2, $m(s_{i_{1}})\cdots
m(s_{i_{n}})=m(s^{\prime}_{i_{1}})\cdots m(s^{\prime}_{i_{n}})$. In other
words, $m(w)$ is independent of the choice of reduced expression in $W$.
Finally for $w\in W$, we have $w=w_{1}\tau$ for a unique $w_{1}\in
W_{\text{af}}$ and
$\tau\in\Omega_{{{\mathbcal{a}}}}=\Omega_{{\breve{\mathbcal{a}}}}^{\sigma^{*}}$.
We set $m(w)=m(w_{1})m(\tau)$. Then $m(w)\in\mathcal{T}$. In other words, the
map $\phi:\mathcal{T}W$ is surjective. We have

$\ker(\phi)=\ker(\breve{\phi})\cap\mathcal{T}_{\text{af}}=\breve{S}_{2}\cap\mathcal{T}_{\text{af}}=S_{2}.$

It remains to show that for each $a\in\Delta$, we have
$m(s_{a})^{2}=b^{\vee}(-1)$ where $b$ is the gradient of $a$. By [8, §5.1], we
know that the elements of $\Delta$ and in bijection with $\sigma^{*}$-orbits
$\mathcal{X}$ of $\breve{\Delta}$ with ${\breve{a}}|_{\mathcal{A}(A,F)}$ non-
constant for ${\breve{a}}\in\mathcal{X}$. Let $\mathcal{F}$ denote the
$\mathbb{R}$-vector space of affine linear functions on $\mathcal{A}(A,F)$.
Then $\mathcal{F}$ may be identified with the $\sigma^{*}$-invariants of
$\breve{}\mathcal{F}$ where $\breve{}\mathcal{F}$ is the $\mathbb{R}$-vector
space of affine functions on $\mathcal{A}(T,{\breve{F}})$ (which we have
identified with $V=X_{*}(T)\otimes\mathbb{R}$ after choosing a special point).
Under this identification, we have for $a\in\Delta$,

$a\frac{1}{|\mathcal{X}|}\sum_{{\breve{a}}\in\mathcal{X}}{\breve{a}},$

where $\mathcal{X}$ is the $\sigma^{*}$-orbit on $\breve{\Delta}$ that
corresponds to $a$.

Fix a $\sigma^{*}$-invariant scalar product $\langle\cdot,\cdot\rangle$ on
$V=X_{*}(T)\otimes\mathbb{R}$ and we identify $V$ with $V^{*}$ via this inner
product. We may extend this to a scalar product on $\breve{}\mathcal{F}$ by
setting $\langle\breve{f},\breve{g}\rangle=\langle
D\breve{f},D\breve{g}\rangle$, where $D\breve{f}\in V$ is the gradient of
$\breve{f}$. For any $\breve{f}\in\breve{}\mathcal{F}$ with $D\breve{f}\neq
0$, let
$\breve{f}^{\vee}=\frac{2\breve{f}}{\langle\breve{f},\breve{f}\rangle}$.

Then for $a\in\Delta$, we have $a^{\vee}=\frac{2a}{\langle a,a\rangle}$. Fix
${\breve{a}}\in\breve{\Delta}$ whose restriction to $\mathcal{A}(A,F)$ is the
affine root $a$. Then

$\langle
a,a\rangle=\frac{1}{|\mathcal{X}|}\sum_{{\breve{a}}^{\prime}\in\mathcal{X}}\langle{\breve{a}},{\breve{a}}^{\prime}\rangle.$

This implies that

$a^{\vee}=c_{a}\sum_{{\breve{a}}^{\prime}\in\mathcal{X}}{\breve{a}}^{\prime\vee},\;\;b^{\vee}=c_{a}\sum_{{\breve{a}}^{\prime}\in\mathcal{X}}{\breve{b}}^{\prime\vee}$
(6.10)

where

$c_{a}=\frac{\langle{\breve{a}},{\breve{a}}\rangle}{\sum_{{\breve{a}}^{\prime}\in\mathcal{X}}\langle{\breve{a}},{\breve{a}}^{\prime}\rangle}=\frac{\langle{\breve{b}},{\breve{b}}\rangle}{\sum_{{\breve{a}}^{\prime}\in\mathcal{X}}\langle{\breve{b}},{\breve{b}}^{\prime}\rangle}.$

Now, let us prove that $m(s_{a})^{2}=b^{\vee}(-1)$. We may easily reduce
ourselves to the case where $G_{\breve{F}}$ is ${\breve{F}}$-simple and simply
connected. Via a simple case-by-case analysis, each $\sigma^{*}$-orbit
$\mathcal{X}$ consists of simple roots in $\breve{\Delta}$ whose corresponding
Dynkin diagram is either a product of $A_{1}$’s or is a single copy of
$A_{2}$.

In the former case, $c_{a}=1$ and by (6.10),

$m(s_{a})^{2}=m(s_{{\breve{a}}_{1}})^{2}m(s_{{\breve{a}}_{2}}^{2})\cdots
m(s_{{\breve{a}}_{k}})^{2}={\breve{b}}_{1}^{\vee}(-1){\breve{b}}_{2}^{\vee}(-1)\cdots{\breve{b}}_{k}^{\vee}(-1)=b^{\vee}(-1),$

where
$\mathcal{X}=\\{{\breve{a}}_{1},{\breve{a}}_{2},\cdots,{\breve{a}}_{k}\\}$.

In the latter case, $\mathcal{X}=\\{{\breve{a}}_{1},{\breve{a}}_{2}\\}$ and
${\breve{a}}_{1}+{\breve{a}}_{2}$ is an affine root. In this case, $c_{a}=2$.
By (6.10),

$m(s_{a})^{2}=(m(s_{{\breve{a}}_{1}})m(s_{{\breve{a}}_{2}})m(s_{\breve{a}}))^{2}=1=b^{\vee}(-1).$

Thus $\mathcal{T}$ is a Tits group of $W$ and $\\{m(w)\;|\;w\in W\\}$ is a
Tits cross-section of $W$ in $\mathcal{T}$.

## References

  * [1] M. Adrian, _A remark on the Kottwitz homomorphism_ Manuscripta Math. 155 (2018), 1–14.
  * [2] Anne-Marie Aubert, Paul Baum, Roger Plymen, and Maarten Solleveld, _The local Langlands correspondence for inner forms of $SL_{n}$_, Res. Math. Sci. 3 (2016), Paper No. 32, 34.
  * [3] Jeffrey Adams and Xuhua He, _Lifting of elements of Weyl groups_ , J. Algebra 485 (2017), 142–165.
  * [4] N. Abe, G. Henniart, F. Herzig, and M.-F. Vignéras, _A classification of irreducible admissible ${\rm mod}\,p$ representations of $p$-adic reductive groups_, J. Amer. Math. Soc. 30 (2017), no. 2, 495–559.
  * [5] Alexandru Ioan Badulescu, _Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle_ , Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 5, 695–747.
  * [6] Nicolas Bourbaki, _Lie groups and Lie algebras. Chapters 4–6_ , Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002, Translated from the 1968 French original by Andrew Pressley.
  * [7] F. Bruhat and J. Tits, _Groupes réductifs sur un corps local_ , Inst. Hautes Études Sci. Publ. Math. (1972), no. 41, 5–251.
  * [8] F. Bruhat and J. Tits: Groupes réductifs sur un corps local:II. Schéma en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376.
  * [9] Stephen DeBacker and Mark Reeder, _Depth-zero supercuspidal $L$-packets and their stability_, Ann. of Math. (2) 169 (2009), no. 3, 795–901.
  * [10] Radhika Ganapathy, _The local Langlands correspondence for $\rm GSp_{4}$ over local function fields_, Amer. J. Math. 137 (2015), no. 6, 1441–1534.
  * [11] Radhika Ganapathy and Sandeep Varma, _On the local Langlands correspondence for split classical groups over local function fields_ , J. Inst. Math. Jussieu 16 (2017), no. 5, 987–1074.
  * [12] X. He, _Cocenters of $p$-adic groups, I: Newton decomposition_, Forum Math. Pi 6 (2018), e2, 27 pp.
  * [13] Roger Howe.  Harish-Chandra homomorphisms for ${p}$-adic groups, with the collaboration of A. Moy, CBMS Regional Conf. Ser. in Math. 59, Amer. Math. Soc., Providence, RI, 1986.
  * [14] Robert E. Kottwitz, _Isocrystals with additional structure. II_ , Compositio Math. 109 (1997), no. 3, 255–339.
  * [15] Bertrand Lemaire, _Représentations génériques de ${\rm GL}_{N}$ et corps locaux proches_, J. Algebra 236 (2001), no. 2, 549–574.
  * [16] Lusztig, G. _Hecke algebras with unequal parameters_ , CRM Monograph Series, 18. American Mathematical Society, Providence, RI, 2003. vi+136 pp. Enlarged and updated version at arxiv.org/0208154.
  * [17] Mark Reeder, Paul Levy, Jiu-Kang Yu, and Benedict H. Gross, _Gradings of positive rank on simple Lie algebras_ , Transform. Groups 17 (2012), no. 4, 1123–1190.
  * [18] T. Richarz, _On the Iwahori-Weyl group_ , Bull. Soc. Math. France 144 (2016), no. 1, 117–124.
  * [19] S. Rostami, _On the canonical representatives of a finite Weyl group_ , arXiv:1505.07442.
  * [20] T. A. Springer, _Linear algebraic groups_ , second ed., Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2009.
  * [21] J. -P. Serre.  Local fields, volume 67 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1979.  Translated from the French by Marvin Jay Greenberg.
  * [22] Robert Steinberg, _Regular elements of semisimple algebraic groups_ , Inst. Hautes Études Sci. Publ. Math. (1965), no. 25, 49–80.
  * [23] J. Tits, _Normalisateurs de tores, I. Groupes de Coxeter étendus_ , J. Algebra 4 (1966), 96–116.