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\begin{document}
%%\renewcommand{\baselinestretch}{2}

%%\

%%\vskip -10mm

%%\


\title[Cross-sections, quotients, and representation rings]{Cross-sections,
quotients, and \\
representation rings of\\ semisimple algebraic groups}

\author[Vladimir  L. Popov]{Vladimir  L. Popov${}^*$}
\address{Steklov Mathematical Institute,
Russian Academy of Sciences, Gubkina 8, Moscow\\
119991, Russia} \email{popovvl@orc.ru}

\thanks{
 ${}^*$\,Supported by
 grants {\rus RFFI
08--01--00095}, {\rus N{SH}--1987.2008.1}, and the
program {\it Contemporary Problems of Theoretical
Mathematics} of the Russian Academy of Sciences, Branch
of Mathematics. }

\date{August 20, 2009}

\subjclass[2000]{14M99, 14L30, 14R20, 14L24, 17B45 }



\keywords{Semisimple algebraic group, conjugating
action, cross-section, quotient, representation ring}




\dedicatory{ \hfill{
%%\parbox
\begin{minipage}[t]{2.4in}
{\raggedright {\rm``}Is Steinberg's theorem $[\ldots]$
%%(in the form that says
%%%%\\
%%that rational points exist on rational orbits,
%%\\
%%say)
only true for simply connected groups $[\ldots]$\hskip
.1mm{\rm ?}
%%, and is this also true of the corollary which %%states that the structural group can always be %%reduced to a maximal torus?
What hap\-pens for ${\rm GP}(1)$, for instance\,{\rm?}
Is there a rational section of $\;G$ over $I(G)$
{\rm(\!``}inva\-ri\-ants\,{\rm'')} in this case\,{\rm?}
$[\ldots]$ Is it true
that $I(G)$ is a rational variety $[\ldots]$\,{\rm ?''}\\
\hskip 10mm {\rm A. Grothendieck}, {\it Letter to J.-P.
Serre},\\ \hskip 10mm {\rm January} {\rm 15, 1969}, {\rm
\cite[pp.\,240--241]{GS}.}}
\end{minipage}
} }


\maketitle

%%\renewcommand{\baselinestretch}{1.0}

\begin{abstract}
Let $G$ be a connected semisimple algebraic group over
an algebraically closed field $k$. In 1965 {\sc
Steinberg} proved that if $G$ is simply connected, then
in $G$ there exists a closed irreducible cross-section
of the set of closures of regular conjugacy classes. We
prove that in arbitrary $G$ such a cross-section  exists
if and only if the universal covering isogeny
$\tau\colon \tG\to G$ is bijective. In particular, for
${\rm char}\,k=0$, the converse to {\sc Steinberg}'s
theorem holds. The existence of a cross-section in $G$
implies, at least for ${\rm char}\,k=0$, that the
algebra $k[G]^G$ of class functions on $G$ is generated
by ${\rm rk}\,G$ elements.  We describe, for arbitrary
$G$, a minimal generating set of
%%this algebra
$k[G]^G$ and that of the representation ring of $G$
%%. We
%%also
and answer two {\sc Grothendieck}'s questions on
constructing the generating sets of $k[G]^G$. We prove
the existence of a rational cross-section in any $G$
(for ${\rm char}\,k=0$, this has been proved earlier in
\cite{CTKPR}). We also prove that the existence of a
rational section of the quotient morphism for $G$ is
equivalent to the existence of a rational
$W$-equivariant map $T\dashrightarrow G/T$ where $T$ is
a maximal torus of $G$ and $W$ the Weyl group. We show
that both properties hold if the isogeny $\tau$ is
central.
%% for any connected semisimple group $G$. In %%particular, this extends the result of %%\cite[Theorem 0.3]{CTKPR} on the existence of %%rational cross-sections in any $G$ to arbitrary %%characteristic.
\end{abstract}


\section{Introduction}

Below all algebraic varieties are taken over an
algebraically closed field $k$. We use the standard
notation and conventions of \cite{Bor} and \cite{Sp}.
%%If $X$ is such a variety, $k[X]$ %%denotes the algebra of regular %%functions on $X$.

Let $G$ be a connected semisimple algebraic group,
$G\neq \{e\}$. Let $(\GG, \pG)$ be a categoricalRMR
quotient for the conjugating action of $G$ on itself,
i.e., $\GG$  is an affine variety and\ \vskip -5mm
\begin{equation*}
\pG\colon G\longrightarrow \GG
\end{equation*}
a surjective  morphism such that
$\pi^*_G\big(k[\GG]\big)$ is the algebra $k[G]^G$ of
class functions on $G$. Every fiber of $\pG$ is then the
closure of a regular conjugacy class  (i.e., that of the
maximal dimension) and such classes in general position
are closed  \cite[Theorem 6.11, Cor. 6.13, and Sect.
2.14]{St1}.


\begin{definition}
A closed irreducible subvariety $S$ of $G$ is called a
{\it cross-section} (of the collection of fibers of
$\pG$) in $G$ if $S$ intersects   at a single point
every fiber of $\pG$.
\end{definition}

 The elements of $S$
%%may be considered as
are the ``canonical forms'' of the elements of a dense
constructible subset of $G$ with respect to conjugation.
The image of any {\it section}
 of $\pG$ (i.e., a morphism $\;\sigma \colon \GG\to G$ such that $\pG\circ\sigma={\rm id}^{}_{\GG}$)
is the example of such $S$ and, for ${\rm char}\,k=0$,
every cross-section in $G$ is obtained in this manner
(see Remark 1 in Section \ref{re}).


In 1965 {\sc Steinberg} gave an explicit construction of
a section of $\pG$ for every simply connected semisimple
group $G$ (see his celebrated paper \cite{St1}).
%% (see  \cite[Theorem 1.4]{St1}).
%%, a cross-section in $G$ has been
%%explicitly
%%constructed by Steinberg.
Its image is the cross-section that intersects every
regular conjugacy class
%%\cite[Theorem 1.4]{St1}
 and does not intersect other conjugacy classes.
%% that if $G$ is simply connected, then a %%cross-section exists. Moreover, he %%explicitly constructed a cross (and its %%intersection with every regular conjugacy %%class automatically is a single point). %%Here

In this paper we explore what happens in the general
case, i.e., when  $G$ is not necessarily simply
connected. In this case   the following two facts about
cross-sections in $G$ for ${\rm char}\,k=0$ are known.

First, by \cite[Theorem 0.3]{CTKPR} in every connected
semisimple algebraic group $G$ there is a {\it rational
section} of $\pG$, i.e., a section over
%%closed irredu\-cib\-le subvariety that %%intersects at a single point
%%the fiber of $\pG$
%%over
%%every point of
a dense open subset of $\GG$ (local section).


Second, by {\sc Kostant}'s theorem \cite[Theorem
0.10]{K} there is an infinitesimal counterpart of {\sc
Steinberg}'s cross-section:
 for
 %%${\rm char}\,k=0$ and
 the adjoint action of $G$ on its Lie algebra
${\rm Lie}\,G$, there is a closed irreducible subvariety
in ${\rm Lie}\,G$ that intersects   every regular
$G$-orbit at a single point.


In order to formulate our result consider the universal
covering of $G$, i.e., an
%%central
isogeny
\begin{equation*}%%\label{isog}
\tau\colon \tG\longrightarrow G
\end{equation*}
such that $\tG$ is a simply connected semisimple
algebraic group and the composition of $\tau$ with every
projective rational representation of $G$ lifts to a
linear one of $\tG$.

We prove
%%if $G$ is not simply connected, then a %%cross-section in $G$ does not exist. So %%together with Steinberg's theorem this %%yields
the following
%%criterion.

\begin{theorem}\label{main} Let $G$ be a connected semisimple algebraic group.
\begin{enumerate}
\item[\rm(i)] The following properties are equivalent:
\begin{enumerate}
\item[\rm(a)] there is a cross-section in $G$;
\item[\rm(b)] the isogeny $\tau$ is bijective.
\end{enumerate}
\item[\rm(ii)] If  $\sigma\colon \GG\to G$ is a section
of $\pG$, then the cross-section $\sigma(\GG)$ in $G$
in\-ter\-sects every regular conjugacy class and does
not intersect other conjugacy classes.
\end{enumerate}
\end{theorem}

%%\begin{remark} {\rm The bijectivity %%condition (b) holds if and only if $\tau$ is %%either an isomorphism or purely %%inseparable.}

\begin{remark} {\rm
The isogeny $\tau$ is bijective if and only if it is
either an isomorphism or purely inseparable (radical).
The latter holds if and only if ${\rm char}\,k=p>0$ and
$p$ divides the order of the fundamental group of $G$.}
\end{remark}



Statement (ii) of the next corollary answers a question
posed in \cite[p.\,4]{CTKPR}.

\begin{corollary} Let $G$ be a connected semisimple algebraic group.

\begin{enumerate}
\item[\rm(i)] If a section of $\pG$ exists, then $\tau$
is bijective. \item[\rm(ii)] For ${\rm char}\,k=0$, the
following properties are equivalent:
\begin{enumerate}
\item[\rm(a)] there is a section of $\pG$; \item[\rm
(b)] there is a cross-section in $G$; \item[\rm (c)] $G$
is simply connected.
\end{enumerate}
\end{enumerate}
\end{corollary}

Theorem \ref{main} is proved in Section \ref{cs}.


One can show (see below Lemma \ref{smooth}) that if a
cross-section in $G$ exists, then, at least for ${\rm
char}\,k=0$, the variety $G/\!\!/G$ is smooth (the
converse is not true). The known criterion of smoothness
of $\GG$ (Theorem \ref{sing}) may be
%%regarded
interpreted as that of the existence of ${\rm rk}\,G$
generators of $k[G]^G$.
 In Section~\ref{singugen}  we consider the general case and describe
a minimal generating set of $k[G]^G$ and singularities
of $\GG$ for any $G$. This is based on the property that
actually $\GG$ is a toric variety of a maximal torus $T$
of $G$. In particular, it also implies
%%the following corollary that answers in
the affirmative answer to Grothendieck's question cited
in the epigraph:
%%\cite[p.\,241]{GS}:

\vskip 2mm

\noindent{\bf Corollary \ref{ratio}.} {\it $\GG$ and
$T/W$ are the rational varieties.}

\vskip 2mm

\noindent Here $W$ is the Weyl group of $G$, i.e., the
quotient of $T$ in its normalizer $N_G(T)$, acting on
$T$ via conjugation.

Parallel to this we describe a minimal generating set of
the representation ring $R(G)$ of $G$. Note that finding
generators of $R(G)$ attracted people's attention during
long time, in particular, because of the bearing on the
$K$-theory (cf., e.g., \cite[Chap.\;13]{Hus}  where the
generators of $R(G)$ are found for some classical $G$'s
utilizing the ad hoc bulky arguments; see
also~\cite{A}). Singularities of $\GG$ attracted the
attention as well
%%(see some sketchy
%%information in
(see \cite[Sects.\;3.15, 4.5]{Sl}).

 The precise
formulations of these results
%%on the minimal
%%generating sets of $k[G]^G$ and $R(G)$
are given below in Theorems \ref{toric},
\ref{generators} and Lemma \ref{explcenter}.


%%Theorem \ref{generators} describes a minimal generating
%%set of the algebra $k[G]^G$ of class functions on $G$.
Constructing the generating sets of $k[G]^G$ is the
topic of yet two {\sc Grothendieck}'s questions asked in
\cite[p.\;241]{GS}. In Section \ref{Yettwo} we answer
the first question
%%negatively
in the ne\-ga\-tive and the second
%%positively.
in the positive.


In Section \ref{rcs} we consider rational sections of
$\pG$ and {\it rational cross-sections} in $G$, i.e.,
irreducible closed subsets $S$ of $G$ that intersect at
a single point every fiber of $\pG$ over a point of a
dense open subset of $\GG$. The closure of the image of
a rational section of $\pG$ is the example of such $S$
and, for ${\rm char}\,k=0$, every rational cross-section
in $G$ is obtained in this way.


First we show that
%%find a property equivalent to
%%prove that
the existence of a rational section of\,$\pG$ is
equivalent to
%%is equivalent to
another property. Namely, %%
%%, i.e., rational maps $\sigma\colon \GG %%\dashrightarrow G$ such that %%$\pG\circ\sigma={\rm id}_{\GG}$.
%%
%%let $T$ be a maximal torus of $G$ and let $W$ be
%%the Weyl group of $G$, i.e., the quotient of $T$ %%in its normalizer $N_G(T)$, acting on $T$ via %%conjugation.
%%The group
$W$ also acts on $G/T$ as follows:
\begin{equation}\label{G/T}
w\cdot gT:=g\overset{.}{w}^{-1}T,
\end{equation} where $\overset{.}{w}\in N_G(T)$ is a representative of $w$.
We prove
%%the following

\begin{theorem}\label{rsrcs} Let $G$ be a connected semisimple algebraic group. The following pro\-perties are equivalent:
\begin{enumerate}
\item[\rm(i)] there is a rational section of $\;\pG$;
\item[\rm(ii)] there is a $W$-equivariant rational map
$T\dashrightarrow G/T$.
\end{enumerate}
\end{theorem}

Then we consider the existence problem and prove the
following.

Recall (see \cite[22.3]{Bor}) that the isogeny $\tau$ is
called {\,\it central\;} if ${\rm ker}\,\tau$ lies in
the cen\-ter of $G$ and ${\rm ker}\,d\tau_e$ lies in the
center of ${\rm Lie}\,G$.

The next theorem answers the other Grothendieck's
question cited in the epigraph.


%%
%%We prove that the assumption ${\rm char}\,k=0$ %%made in \cite[Theorem 0.3]{CTKPR}
%%is superfluous
%%can be dropped. More precisely, the following %%theorem holds:

\begin{theorem}\label{anychar}  Let $G$ be a connected semisimple algebraic group.
\begin{enumerate}
\item[\rm(i)] There is a rational cross-section in $G$.
\item[\rm(ii)] If the isogeny $\tau$ is central, then
there is a rational section of $\pG$.
\end{enumerate}
\end{theorem}

For
%%Statement (i) means that the assumption
${\rm char}\,k=0$, this theorem has been proved earlier
in
%%made in
\cite[Theorem 0.3]{CTKPR}. The strategy and %%significant
the essential part of our proof
%%goes along the same lines
are the same: we use the relevant characteristic free
results from \cite{CTKPR}, but bypass Theorem 2.12 from
this paper
%%\cite[Theorem 2.12]{CTKPR}
(whose proof is based on the assumption ${\rm
char}\,k=0$) by exploring properties of $\pG$ and
proving that versality of $G$ holds in any
characteristic; this permits us to use {\sc Steinberg}'s
section of $\ptG$ in place of {\sc Kostant}'s
cross-section in ${\rm Lie}\,G$ used in \cite{CTKPR}.

%%we only have to bypass
%%is superfluous
%%can be dropped.

%%The strategy of the proof is the same as in
%%\cite[Theorem 0.3]{CTKPR}, but, exploring
%%properties of
%%$\pG$, we show that versality of $G$ holds in %%any characteristic; this permits to use
%%Steinberg's section of $\ptG$ in place of
%%Kostant's cross-section in ${\rm Lie}\,G$.

Theorems \ref{rsrcs} and \ref{anychar} yield the
following


\begin{corollary}\label{ratcs}
Let $G$ be a connected semisimple algebraic group. If
\,the isogeny $\,\tau$ is central, then  there is a
$W$-equivariant rational map $T\dashrightarrow G/T$.
%%Let $G$ be a connected semisimple algebraic %%group.
%%The following pro\-per\-ties are equivalent:
%% and hold for any connected semisimple algebraic %%group $G$:
%%\begin{enumerate}
%%\item[\rm(i)] There exists a rational section of %%$\pG$.
%%\item[\rm(ii)] There exists a $W$-equivariant %%rational map $T\dashrightarrow G/T$.
%%\end{enumerate}
\end{corollary}

%%\begin{corollary} $G/T$ is a versal $(W, %%\{e\})$-variety in the sense of %%{\rm\cite[Sect.\,3]{CTKPR}}.
%%\end{corollary}

%%Thanks to \cite[Theorem 0.3]{CTKPR} this yield

%%\begin{corollary}\label{Wequi} If ${\rm %%char}\,k=0$, then for every connected semisimple %%algebraic group $G$,  there is a $W$-equivariant %%rational map $T\dashrightarrow G/T$.
%%\end{corollary}



Section \ref{re} contains some remarks, questions, and
%%one finds there
an example of a cross-section $S$ in $G$ such that %%the restriction of $\pG$ to $S$
$\pG|_S$ is not separable (hence $S$ is not the image of
a section of $\pG$).




\vskip 2mm


{\it Acknowledgements.} I am grateful to {\sc
Vik.\;Kulikov} for the inspiring remark, to {\sc A.
Parshin} for the useful discussion, and to {\sc J.-L.
Colliot-Th\'el\`ene} and {\sc Z. Rei\-ch\-stein} for the
comments.

\section{Cross-sections in $G$}\label{cs}\label{crssect}


Fix a choice of Borel subgroup $\widehat B$ of $\tG$ and
maximal torus $\widehat T\subset \widehat B$. Denote by
${X}(\widehat T)$ the
%%additive
character lattice
%%of characters
of $\widehat T$ in additive nonation. For $\lambda\in
X(\widehat T)$, denote by $t^\lambda$ the value of
$\lambda\colon \widehat T\to {\bf G}_m$ at $t\in
\widehat T$. Let $\varpi_1,\ldots,\varpi_r\in
{X}(\widehat T)$ be the system of fundamental weights of
$\widehat T$ with respect to~$\widehat B$.
%%Denote by $E_i$ a simple $G$-module of %%highest weight $\varpi_i$.

Let $\varrho_i\colon \tG\to {\bf GL}(V_i)$ be an
irreducible representation of $\tG$ with $\varpi_i$ as
the highest weight. Let $\chi_{\varpi_i}\in
k[\tG]^{\tG}$ be the character of $\varrho_i$.

%%Let $C$ be the center of $\tG$. Denote by %%$\chi_1,\ldots, \chi_r$ all fundamental %%characters of $\tG$.
%% Let $(\tGG, \ptG)$ be a categorical %%quotient for the conjugating action of %%$\tG$ on itself. This action


Let $\widehat C$ be the center of $\tG$; it is a finite
subgroup of $\widehat T$. The conjugating action of
$\tG$ on itself commutes with the action of  $\widehat
C$ on $\tG$ by left translations. Therefore the latter
action  descends to  $\tGG$ and\ \vskip -4mm
$$\ptG\colon \tG\longrightarrow \tG/\!\!/\tG$$
becomes a $\widehat C$-equivariant morphism.

Endow the $r$-dimensional affine space ${\mathbf A}\!^r$
with the linear action of $\tT$ by the formula
\begin{equation}\label{action}
t\cdot(a_1,\ldots,a_r):=(t^{\varpi_1}a_1,
\ldots,t^{\varpi_r}a_r),\qquad t\in \tT, \hskip 2mm
(a_1,\ldots, a_r)\in {\mathbf A}\!^r.
\end{equation}



\begin{lemma}\label{lin}
\

\begin{enumerate}
\item[\rm(i)] The $\tT$-stabilizer of the point
$(1,\ldots,1)\in {\bf A}^r$ is trivial. In particular,
the considered action of $\;\tT$ on ${\mathbf A}\!^r$ is
faithful. \item[\rm(ii)] There is a $\widehat
C$-equivariant isomorphism
$$\lambda\colon \tGG\overset{\simeq}\longrightarrow  {\mathbf A}\!^r.$$
\end{enumerate}
\end{lemma}
\begin{proof}
 Since $\varpi_1,\ldots,\varpi_r$
 generate $X(\widehat T)$, we have\ \vskip -4mm
\begin{equation}\label{kerpi}
%%\textstyle
\bigcap_{i=1}^{r} \{t\in T\mid t^{\varpi_i}=1\}=\{e\}.
\end{equation}
But \eqref{action} entails that the $\tT$-stabilizer of
the point $(1,\ldots,1)$ coincides with  the right-hand
side of equality \eqref{kerpi}. This proves (i).


 By \cite[Theorems 6.1, 6.16]{St1}  the $k$-algebra $k[\tG]^{\tG}$ is
 freely generated by $\chi_{\varpi_1},
 \ldots,\chi_{\varpi_r}$ and the morphism\ \vskip -4mm
\begin{equation*}
\theta\colon\tG\longrightarrow {\bf A}\!^r,\qquad
\theta(g)=(\chi_{\varpi_1}(g),\ldots,
\chi_{\varpi_r}(g)),
\end{equation*}
is surjective. Hence there is
an isomorphism %%\begin{equation*}
$\lambda\colon \tGG\longrightarrow {\bf A}\!^r $
%%\end{equation*}
such that the following diagram is commutative: \ \vskip
-5mm
\begin{equation}\label{commu}
\begin{matrix}
\xymatrix@C=6mm@R=5mm{&\widehat G\ar[dl]_{\ptG}\ar[dr]^\theta&\\
\tGG\ar[rr]^\lambda&&{\bf A}\!^r}
\end{matrix}\quad .
\end{equation}

%%If ${\bf A}\!^r$ is endowed with the %%$C$-action defined by formula %%\eqref{action}, then
The morphism $\theta$ is $\widehat C$-equivariant.
Indeed, let $c\in \widehat C$. Since $\varrho_i$ is
irreducible, {\sc Schur}'s lemma entails that
$\varrho_i(c)=\mu_{i, c}\,{\rm id}_{V_i}$
for some %%constant
$\mu_{i, c}\in k$. On the other hand, since
$c\in \widehat T$, any highest vector in $V_i$
with respect to $\widehat B$ is an eigenvector
of $c$ with the eigenvalue $c^{\varpi_i}$.
Hence $\mu_{i, c}= c^{\varpi_i}$. %%i.e.,
%%$$
%%$\varrho_i(c)=c^{\varpi_i}\,{\rm id}_{V_i}.$
%%$$
 Therefore, for every
$g\in\tG$, by \eqref{action} we have
\begin{align*}
\theta(cg)&=\big(\chi_{\varpi_1}(cg),\ldots,
\chi_{\varpi_r}(cg)\big)\\
&=\big({\rm trace}\,\big(\varrho_1(cg)\big),
\ldots,{\rm trace}\,\big(\varrho_r(cg)\big)\big)%%\\
\end{align*}
\begin{align*}
&=\big({\rm trace}\,\big(\varrho_1(c)\varrho_1(g)\big),
\ldots,{\rm
trace}\,\big(\varrho_1(c)\varrho_r(g)\big)\big)\\
%%\end{align*}
%%\begin{align*}
&=\big({\rm
trace}\,\big(c^{\varpi_1}\!\varrho_1(g)\big),
\ldots,{\rm trace}\,\big(c^{\varpi_r}\!\varrho_r(g)\big)\big)\\
&=\big(c^{\varpi_1}{\rm
trace}\,\big(\varrho_1(g)\big),\ldots, c^{\varpi_r}{\rm
trace}\,\big(\varrho_r(g)\big)\big)\\
&=\big(c^{\varpi_1}\chi_{\varpi_1}(g),\ldots, c^{\varpi_r}\chi_{\varpi_r}(g)\big)\\
&=c\cdot \theta(g),
\end{align*}
as claimed.

   Since both $\theta$ and $\ptG$ are $\widehat C$-equivariant and $\ptG$ is surjective,
commutativity of diagram \eqref{commu} entails that
$\lambda$ is $\widehat C$-equivariant as well. This
proves (ii). \quad $\square$
\renewcommand{\qed}{}\end{proof}

\begin{corollary}\label{no}
 Let  %%$\,Z$ be a nontrivial subgroup of
$g$ be a nonidentity element of $\,\widehat C$.
%% of order $\geqslant 2$, then
Then there is no %%$Z$
$g$-stable cross-section in $\tG$.
\end{corollary}
\begin{proof}  Assume the contrary and let $\tS$ be a %%$Z$
$g$-stable cross-section in $\tG$. Since $\ptG$ is
$\widehat C$-equivariant, $\ptG|_{\tS}\colon \tS\to \tGG
$
%%the definition of cross-section
%%entails that
%%the restriction of $\ptG$ to $\tS$
is a bijective
%%$Z$
$g$-equivariant morphism.
%%$\tS\to \tGG$.
As, by Lemma \ref{lin}(ii),
%%the action of $Z$ on
there is a point of $\tGG$ fixed by $\widehat C$, hence
by $g$,
%% has a fixed point,
this
%%entails
implies that there is a point of
%%the action of $Z$ on
$\widehat S$
%%has a
fixed by $g$. But  for the action of $\widehat C$ on
$\tG$ by left translations, the stabilizer of every
point is trivial, a contradiction with $g\neq e$.
%% This contradicts the condition that the order %%of $Z$ is $\geqslant 2$.
\quad $\square$
\renewcommand{\qed}{}\end{proof}


Given an element $h$ of an algebraic group $H$, we shall
denote its conjugacy class in $H$ by $H(h)$:
\begin{equation}\label{cc}
H(h):=\{shs^{-1}\mid s \in H\}.
\end{equation}


\begin{lemma}\label{prop}
Let $H$ and $\widetilde H$ be connected algebraic groups
and let $\sigma\colon \widetilde H\to H$ be an isogeny.
Then the following properties hold:
\begin{enumerate}
\item[\rm(i)] $\sigma$ is a finite morphism;
\item[\rm(ii)] $\sigma\big(\widetilde
H(h)\big)=H\big(\sigma(h)\big)$ and $\dim \widetilde
H(h)=\dim H\big(\sigma(h)\big)$
for every $h\in \widetilde H$; %%\begin{equation*}\nu(G'(x))=G(\nu(x))\quad %%\mbox{and}\quad
%%\dim\,\nu(G'(x))=\dim\,G(\nu(x));
%%\end{equation*}
\item[\rm(iii)] if $\,\widetilde H(h)$ is a regular
conjugacy class in $\widetilde H$ {\rm(}i.e., that of
the maximal dimension{\,\rm)}, then
$\,\sigma\big(\widetilde H(h)\big)$ is a regular
conjugacy class in $H$; \item[\rm(iv)] if $H$ and
$\widetilde H$ are semisimple, then for every $h\in
\widetilde H$, $$\sigma\big(\pi^{-1}_{\widetilde
H}\big(\pi^{}_{\widetilde H}(h)\big)\big)
=\pi^{-1}_{H}\big(\pi^{}_{H}\big(\sigma(h)\big)\big).$$
\end{enumerate}
%%Let $\widetilde G$ be a connected semisimple %%algebraic group and let $\sigma\colon \widetilde %%G\to G$ be an isogeny. Then
%%\begin{enumerate}
%%\item[\rm(i)] $\sigma$ is a finite morphism;
%%\item[\rm(ii)] $\sigma\big(\widetilde %%G(g)\big)=G\big(\sigma(g)\big)$ and $\dim %%\widetilde G(g)=\dim G\big(\sigma(g)\big)$
%%for every $g\in \widetilde G$;
%%\item[\rm(iii)] if $\,\widetilde G(g)$ is a %%regular conjugacy class in $\widetilde G$, then %%$\,\sigma\big(\widetilde G(g)\big)$ is a regular %%conjugacy class in $G$;
%%\item[\rm(iv)] $\sigma\big(\pi^{-1}_{\widetilde %%G}\big(\pi^{}_{\widetilde G}(g)\big)\big)
%%=\pi^{-1}_{G}\big(\pi^{}_{G}\big(\sigma(g)\big)\big)$
%%for every $g\in \widetilde G$.
%%\end{enumerate}
\end{lemma}

\begin{proof}
The varieties $H$ and $\widetilde H$ are normal (even
smooth) and  the fiber of $\sigma$ over every point of
$\,H$ is a finite set whose cardinality does not depend
on this point.
%%$g$.
Hence (cf.\;\cite[Sect.\,2, Cor.\,3]{G}) $\widetilde H$
is the normalization of $\,H$ in the field of rational
functions on $\widetilde H$ and $\sigma$ is the
normalization map. This proves (i).


The first equality in (ii) holds as $\sigma$ is an
epimorphism of groups. The second follows the first and
theorem on dimension of fibers, cf.,
e.g.,\;\cite[AG\,10.1]{Bor}. This proves~(ii).

%%Since
%%$\widetilde H$ and $H$ have the same dimensions %%and ranks, dimensions of regular conjugacy %%classes in $\widetilde G$ and $G$ are equal.
%%have the same dimension.
%%Hence
As $\sigma$ is surjective, (iii) follows from (ii).

%%By \cite[Theorem 6.11]{St1}
Since the fibers of $\pi^{}_{\widetilde H}$ and
$\pi^{}_{H}$ are the closures of regular conjugacy
classes and, by (i), the map $\sigma$ is closed, (iv)
follows from (iii).
%% because, by (i),
%%regular conjugacy classes
%%in $G'$ and $G$ have the same dimension %%and,
%% the map $\nu$ is closed.
\quad $\square$
\renewcommand{\qed}{}\end{proof}

\begin{corollary}\label{insep}
Let $\widetilde G$ be a connected semisimple algebraic
group and let $\sigma\colon \widetilde G\to G$ be a
bijective isogeny.
%% of separable degree $1$.
\begin{enumerate}
\item[\rm(i)] If $\,\widetilde S$ is a cross-section in
$\widetilde G$, then $\sigma (\widetilde S)$ is a
cross-section in $G$. \item[\rm(ii)] If $\,S$ is a
cross-section in $G$, then $\sigma^{-1}(S)$ is a
cross-section in $\widetilde G$.
\end{enumerate}
The same holds
%%is true
if {\rm``}\!cross-section{\rm''} is replaced with
{\rm``}rational cross-section{\rm''}.
\end{corollary}
\begin{proof} By Lemma \ref{prop}(i) the bijective map $\sigma$ is closed. Hence
it is a homeomorphism. Both claims follow from this, the
definitions of cross-section and rational cross-section,
and Lemma \ref{prop}(iv). \quad $\square$
\renewcommand{\qed}{}\end{proof}

\begin{lemma}\label{qZ} Assume that there is a subgroup $Z$ of $\,\widehat C$ such that $G=\tG/Z$ and $\tau$ is the quotient morphism $\tG\to \tG/Z$. Then there is a morphism
\begin{equation}\label{phi}
\varphi\colon \tGG\longrightarrow \GG\end{equation} such
that
\begin{enumerate}
\item[\rm(i)] $(\GG, \varphi)$ is
%%the
a categorical quotient
%%of $\,\tGG$ by
for the action of $Z$\,on $\tGG$;
%%\item[\rm(ii)] If $\varphi\colon \tGG\to %%\GG$ is the quotient morphism by the action %%of $Z$, then
\item[\rm(ii)] the following diagram is commutative:
\begin{equation}\label{dia}
\begin{matrix}
\xymatrix{\tG\ar[r]^\tau\ar[d]_{\ptG}&G\ar[d]^\pG\\
\tGG\ar[r]^\varphi&\GG}
\end{matrix}\quad ;
\end{equation}
\item[\rm(iii)] for every point $x\in \tGG$, the
following equality holds:
\begin{equation}\label{fibers}
\tau\big(\pi^{-1}_{\tG}(x)\big)=
\pi^{-1}_{G}\big(\varphi(x)\big).
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof} As $\tau^*$, $\pi_{\tG}^*$, and $\pi_G^*$
are injections, there is a unique morphism
%%$\varphi\colon \tGG\to \GG$
\eqref{phi} such that $\tau^*\circ\pi_G^*=
\pi_{\tG}^*\circ\varphi^*$, i.e., diagram \eqref{dia} is
commutative.


Consider the action of  $\tG$  on $G$ via the isogeny
$\tau$ and the conjugating action of $G$ on itself. The
isogeny $\tau$ is then $\tG$-equivariant and
$\tG$-orbits in $G$ are $G$-conjugacy classes, so we
have $k[G]^G=k[G]^{\tG}$.  Since the conjugating action
of $\tG$ on itself commutes with the action of $Z$ by
left translations, we have
\begin{align*} \pi_{\tG}^*\big(\varphi^*(k[\GG])\big)&=
\tau^*\big(\pi_G^*(k[\GG])\big)=\tau^*\big(k[G]^G\big)=
\tau^*\big(k[G]^{\tG}\big)=
\big(\tau^*(k[G])\big){}^{\tG}\\&=
\big(k[\tG]^Z\big){}^{\tG}=\big(k[\tG]^{\tG}\big){}^Z
=\big(\pi_{\tG}^*(k[\tGG])\big){}^Z=
\pi_{\tG}^*\big(k[\tGG]^Z\big).
\end{align*}
%%Since $\widehat C$ is central,  the subalgebra %%$\tau^*(k[G])$
%%of $k[\tG]$ is $\tG$-stable
%%and
%%\begin{equation}\label{Z}
%%\tau^*\big(\pi^*_G(k[\GG])\big)=
%%\big(\tau^*(k[G])\big){}^{\tG}=
%%\big(k[\tG]^Z)^{\tG}=
%%\big(k[\tG]^{\tG}\big){}^Z=
%%\big(\pi^*_{\tG}(k[\tGG])\big){}^Z.
%%\end{equation}
%%This shows that there is the unique morphism
%%$\varphi\colon \tGG\to \GG$
%%\eqref{phi} making diagram \eqref{dia} %%
%%commutative and that
Thus, $\varphi^*(k[\GG])=k[\tGG]^Z$. This proves (i) and
(ii). Lemma \ref{prop}(iv) and commutativity of diagram
\eqref{dia}
%%yield
imply (iii).\quad $\square$
\renewcommand{\qed}{}\end{proof}

Below, given a variety $Z$, we denote by ${\rm T}_{z,
Z}$ the tangent space of $Z$ at a point $z$.

%%\newpage

%%\vskip 2mm



%%\noindent {\it Proof of Theorem {\rm \ref{main}}}.

\begin{proof}[Proof of Theorem {\rm \ref{main}}]
First, we shall prove criterion (i).

\vskip 1mm $1$. By {\sc Steinberg}'s theorem, $\tG$ has
a cross-section. Hence, by Corollary \ref{insep}, if
%%the isogeny
$\tau$
%%\eqref{isog}
is bijective, then there exists a cross-section in $G$
as well.

So we may assume that %%the isogeny
%%\eqref{isog}
$\tau$ is not bijective and we then have to prove that
there
is no cross-section in $G$. Solving this problem, we may assume that %%the isogeny
%%\eqref{isog}
$\tau$ is separable. Indeed, if this is not the case,
then by  \cite[Prop.\,17.9]{Bor} there exist a connected
semisimple algebraic group $\widetilde G$ and a
commutative diagram of isogenies
%%factorization $\tau=\nu \circ\tau'$, where
\begin{equation}\label{triang}
\begin{matrix}
\xymatrix@C=6mm@R=5mm{\widehat G\ar[rr]^\tau\ar[dr]_\mu&&G\\
&\widetilde G\ar[ur]_\sigma&}
\end{matrix}\quad ,
\end{equation}
%%\begin{equation}
%%\tG\overset{\tau'}\longrightarrow G'\overset{\nu %%}\longrightarrow G,
%%\end{equation}
where %%$\tau'$
$\mu$ is %%a
separable
%%isogeny
and $\sigma $ is %%is a
purely inseparable.
%% one.
As $\sigma$ is bijective, Corollary \ref{insep} then
reduces the problem  to proving that there is no
cross-section in $\widetilde G$, i.e., we may replace
$G$ by $\widetilde G$ and $\tau$ by $\mu$.

So from now on we may (and shall) assume that %%\eqref{isog}
$\tau$ is a separable isogeny of degree $\geqslant 2$.
This means that  there is a nontrivial subgroup $Z$ of
$\widehat C$  such that $G=\tG/Z$ and $\tau$ is the
quotient morphism $\tG\to \tG/Z$.
%%}
%%\end{nothing}

%%\begin{nothing}{\rm

\vskip 2mm

%%$2$. Since $C$ is central,  the subalgebra %%$\tau^*(k[G])$
%%of $k[\tG]$ is $\tG$-stable
%%and
%%\begin{equation}\label{Z}
%%\tau^*\big(\pi^*_G(k[\GG])\big)=
%%\big(\tau^*(k[G])\big){}^{\tG}=
%%\big(k[\tG]^Z)^{\tG}=
%%\big(k[\tG]^{\tG}\big){}^Z=
%%\big(\pi^*_{\tG}(k[\tGG])\big){}^Z.
%%\end{equation}
%%\noindent
%%Hence $\GG$ is total space of the %%categorical (in fact, geometric) quotient %%of $\tGG$ by the action of $Z$ and if %%$\varphi\colon \tGG\to \GG$ is the quotient %%morphism, then
%%%%$\tau$, $\pG$, and $\ptG$ can be included %%%%in
%%we have the following commutative diagram:
%%\begin{equation}\label{dia}
%%\begin{matrix}
%%\xymatrix{\tG\ar[r]^\tau\ar[d]_{\ptG}&G\ar[d]^\pG\\
%%\tGG\ar[r]^\varphi&\GG}
%%\end{matrix}\quad.
%%\end{equation}
%%%%where $\varphi$ is the quotient morphism %%%%for the action of $Z$ on $\tGG$.
%%
%%From Lemma \ref{prop}(iv) we deduce that,
%%%%We claim that
%%%%\begin{proposition}\label{fib}
%%for every point $x\in \tGG$,
%%\begin{equation}\label{fibers}
%%\tau\big(\pi^{-1}_{\tG}(x)\big)=
%%\pi^{-1}_{G}\big(\varphi(x)\big).
%%\end{equation}
%%%%\end{proposition}
%%%%\begin{proof}
%%%%Commutativity of diagram \eqref{dia} %%%%implies that %%%%$\tau(\pi^{-1}_{\tG}(x))\subseteq
%%%%\pi^{-1}_{G}(\varphi(x))$. As $\tau$ is %%the %%quotient morphism for the action of a %%%%finite group, it is finite and hence %%%%closed. Therefore
%%%%$\tau(\pi^{-1}_{\tG}(x))$ is a closed %%%%subset
%%%%of $\pi^{-1}_{G}(\tau(x))$. Since fibers %%%%of %%$\tau$ are finite, %%%%$\dim\,\tau(\pi^{-1}_{\tG}(x)=\dim\,\pi^{-1}_{\tG}(x)$.
%%%%The claim now follows from the fact that %%%%all
%%%%fibers of $\ptG$ and $\pG$ are %%irreducible %%varieties of the same %%dimension %%\cite[Theorem 6.11]{St}.
%%%%\quad $\square$
%%%%\renewcommand{\qed}{}\end{proof}
%%%%}
%%%%\end{nothing}
%%%%\begin{nothing}{\rm



%%\smallskip

$2$. Now, arguing on the contrary, assume  that there is
a cross-section $S$ in $G$.

%%Let $x$ be a point of $\tGG$. Since $S$ %%intersects the fibre %%$\pi^{-1}_{G}\big(\varphi(x)\big)$ is a %%single point,
%%we deduce from \eqref{fibers} and Lemma %%\ref{prop}(iv) that the set
%%\begin{equation}\label{inters}
%%\pi^{-1}_{\tG}(x)
%%\cap \tau^{-1}(S)
%%\end{equation}
%%is contained in a fiber of $\tau$. As the %%fibers of $\tau$ are $Z$-orbits and


%%\vskip 2mm


%%\noindent {\bf Claim \boldmath$1$.}
\begin{claim}
%%\
%%
 %%
%%{\it The following properties hold}:
%%\item
%%\begin{enumerate}
%%\item[\rm(i)]
(i) {\it For every point $x\in \tGG$, the intersection
\begin{equation}\label{inters}
\pi^{-1}_{\tG}(x) \cap \tau^{-1}(S)\end{equation}
%%is a nonempty subset of a single $Z$-orbit; in %%particular, it is finite.
}
\begin{enumerate}
\item[] {\it is a nonempty subset of a single $Z$-orbit;
in particular, it is finite.}
\item[(ii)] {\it There is a nonempty open subset $\,U$\,of $\,\tGG$ such that,  for every $x\in U$, %%the
intersection \eqref{inters} is a single point.}
\end{enumerate}
\end{claim}
%%\vskip 2mm

\noindent {\it Proof of Claim $1$.} Consider diagram
\eqref{dia}. Since $S\cap
\pi^{-1}_{G}\big(\varphi(x)\big)$ is a single point $g$,
we deduce from \eqref{fibers}
%%and Lemma \ref{prop}(iv)
that
%%the
intersection \eqref{inters}
is contained in %%some
%%the fiber
$\tau^{-1}(g)$. This proves (i) as the fibers of $\tau$
are $Z$-orbits.



By Lemma \ref{lin}(i) there is a nonempty open subset
$U$ in $\tGG$ such that  the $\widehat C$-stabilizer of
every point of $\,U$ is trivial. Take a point $x\in U$.
Assume that
%%the
intersection \eqref{inters} contains two points $g_1$
and $g_2\neq g_1$. By (i) there exists an element $z\in
Z$ such that  $g_2=zg_1$. As $\ptG$ is $\widehat
C$-equivariant,
$x=\ptG(g_2)=\ptG(zg_1)=z\cdot\ptG(g_1)=z\cdot x$. Thus,
$z$ belongs to the $\widehat C$-stabilizer of $x$. The
definition of $\,U$ then implies that $z=e$. Hence
$g_1=g_2$, a contradiction. This proves (ii). \quad
$\square$

\vskip 2mm


$3$. Since all the fibers of $\tau$ are finite,
%%by theorem on dimension of fibers
every irreducible component of $\tau^{-1}(S)$ has
dimension $\leqslant \dim\,S=r$ and at least one of them
has dimension $r$.




%%\noindent{\bf Claim \boldmath$2$.}
\begin{claim}
%%{\it The following properties hold}:
%%\
%%\begin{enumerate}
%%\item[\rm (i)]
(i) {\it There is a unique $r$-dimensional irreducible
component $\tS$ of $\,\tau^{-1}(S)$.}
\begin{enumerate}
\item[(ii)]
 $\tau(\tS)=S$.
\end{enumerate}
\end{claim}

\noindent{\it Proof of Claim $2$.} Let $\tS$ be an
$r$-dimensional irreducible component of
$\,\tau^{-1}(S)$.
%%By theorem on dimension of fibers
Then $\tau(\tS)$ contains an open subset of $S$. Since
$\tau$ is closed, this proves (ii).

>From (ii) we conclude that
\begin{equation}\label{surrr} \pG\big(\tau(\widehat
S)\big)=\GG.
\end{equation} But by Lemma \ref{qZ} the fibers of $\varphi$ in commutative diagram \eqref{dia} are finite. This and \eqref{surrr} imply that $\ptG(\tS)$ contains a nonempty open subset of $\tGG$.

Let now $\tS'$ be another $r$-dimensional irreducible
components of $\,\tau^{-1}(S)$. Then, as above,
$\ptG(\tS')$ contains a nonempty open subset of $\tGG$
as well. Therefore, $\ptG(\tS)\cap\ptG(\tS')$ contains a
nonempty open subset $V$ of $\tGG$. We may assume that
$V\subseteq U$ for $U$ from Claim 1(ii). The latter then
yields that $\pi^{-1}_{\tG}(V)\cap
\tS=\pi^{-1}_{\tG}(V)\cap \tS'.$ As
%%the left- and right-hand
both sides of this equality are the open subsets of
respectively $\tS$ and $\tS'$, we infer that $\tS=\tS'$.
This proves (i). \quad $\square$

\vskip 2mm

4.  As $\tS$ is a unique $r$-dimensional irreducible
component of the $Z$-stable variety $\tau^{-1}(S)$, we
conclude that $\tS$ is $Z$-stable.
%%As $\dim \tS=\dim\tGG$,
%%by Claim 1(i) and theorem on dimension of %%fibers $\ptG(\tS)$ contains an open subset %%of $\tGG$.
We shall now show that $\tS$ is a cross-section in
$\tG$. As this property contradicts Corollary \ref{no},
the proof of
%%Theorem \ref{main}
(i) will be then completed.

\vskip 2mm

5. Let $x$ be a point of $\tGG$. As $S$ is a section of
$G$, the intersection $S\cap
\pi^{-1}_{G}\big(\varphi(x)\big)$ is a single point
$g\in G$. By Claim 2(ii) there is a point $\widehat
g\in\tS$ such that $\tau(\widehat g)=g$. Commutativity
of diagram \eqref{dia} then entails that $x$ and
$\widehat x:=\ptG(\widehat g)$ are in the same fiber of
$\varphi$. Since the fibers of $\varphi$ are $Z$-orbits,
there is an element $z\in Z$ such that $x=z\cdot
\widehat  x$. As $\ptG$ is $Z$-equivariant, this yields
$\ptG(z\widehat g)=x$. But $z\widehat  g\in\tS$ as $\tS$
is $Z$-stable and $\widehat  g\in \tS$. Hence
$\pi^{-1}_{\tG}(x)\cap \tS\neq \varnothing$, i.e.,
\begin{equation}\label{sur}
\ptG(\tS)=\tGG.
\end{equation}

%%\vskip 2mm

6. %%Let
%%Put $\rho:=\colon \tS\to \tGG$ be the %%restriction of $\ptG$ to $\tS$.
It follows from Claim 1(i),(ii) and \eqref{sur}
that %%$\rho$
$\ptG|_{\tS}$ is the surjective morphism with finite
fibers, bijective over an open subset of $\tGG$. As
%%the variety
$\tG$ is normal, $\tGG$ is normal as well. Let
$\nu\colon \widetilde S\to \tS$ be the normalization.
Then the surjective morphism $\ptG|_{\tS}\circ\nu\colon
\widetilde S\to \tGG$ of normal varieties has finite
fibers and is bijective over an open subset of $\tGG$.
 Hence  $\ptG|_{\tS}\circ\nu$ is bijective
 (see\;\cite[Sect.\,2, Cor.\,2]{G}). Whence $\ptG|_{\tS}$ is bijective as well, i.e., $\tS$ is a cross-section in $\tG$. This completes the proof of (i).

\vskip 1mm

We now turn to the proof of (ii).

Let $S:=\sigma(\GG)$.
%%and let $\rho$ be the
%%restriction of $\pG$ to $S$.
Take a point $x\in S$ and put $y:=\pG(x)$.
As $%%\rho
\pG|_S\colon S\to\GG$ is the isomorphism ($\sigma$ is
its inverse),
%%we have the isomorphism $d\rho_x\colon
%%{\rm T}_{x, S}\to {\rm T}_{y, \GG}$
$d(\pG|_S)_x$ is the isomorphism as well. Hence
$(d\pG)_x$ is surjective. As $\dim {\rm T}_{y,
\GG}\geqslant \dim \GG=r$, this implies that there are
functions $f_1,\ldots, f_r\in k[G]^G$ such that
$(df_1)_x,\ldots, (df_r)_x$ are linearly independent. By
\cite[Theorem 8.7]{St1} this yields that $x$ is regular.
As $S$ intersects every fiber of $\pG$ at a single point
and every such fiber contains a unique regular orbit,
this proves (ii). Thus, the proof of Theorem \ref{main}
comes to a close.
%%\quad $\square$
\quad $\square$
\renewcommand{\qed}{}\end{proof}





\section{
Singularities of  $\GG$ and generators of $k[G]^G$ and
$R(G)$ }\label{si}\label{singugen}

The following lemma  shows that there is a link between
the existence of a cross-section in $G$ and smoothness
of $\GG$.



\begin{lemma}\label{smooth} Let ${\rm char}\,k=0$. If a surjective morphism $\alpha\colon X\to Y$ of irreducible va\-ri\-eties admits a section $\sigma\colon Y\to X$, then smoothness of $X$ implies smoothness of~$\;Y$.
\end{lemma}

\begin{proof}
Arguing on the contrary, assume that $y$ is a singular
point of $Y$, i.e.,
\begin{equation}\label{>} \dim {\rm T}_{y, Y}>\dim
Y.
\end{equation}

Put $x=\sigma(y)\in X$. Since $\alpha\circ\sigma={\rm
id}_Y$, the composition $d\alpha_x\circ d\sigma_y$ is
the identity map of ${\rm T}_{y, Y}$. Hence $d\alpha_x$
is surjective, i.e., ${\rm rk}\,d\alpha_x=\dim {\rm
T}_{y, Y}$. By \eqref{>} this yields
\begin{equation}\label{rk}
{\rm rk}\,d\alpha_x>\dim Y.
\end{equation}

As ${\rm char}\,k=0$, there is a dense open subset $U$
of $X$ such that ${\rm rk}\,d\alpha_z=\dim Y$ for every
point $z\in U$, see \cite[14.4]{Har}. As $z\mapsto \dim
{\rm ker}\,d\alpha_z$ is the upper semi-continuous
function \cite[14.6]{Har}, we conclude that smoothness
of $X$ implies that ${\rm rk}\,d\alpha_z\leqslant\dim Y$
for every point $z\in X$. This contradicts \eqref{rk}.
\quad $\square$
\renewcommand{\qed}{}\end{proof}


This prompts to explore smoothness of $\GG$. The %%following criterion
answer is known:

\begin{theorem}[{{\rm \cite[\S3]{St4}, \cite[Prop.\;4.1]{R1}, \cite[Prop.\;13.3]{R2}}}]\label{sing} Let ${\rm char}\,k\neq 2$.
The following properties are equivalent:
\begin{enumerate}
\item[\rm(i)] $\GG$ is smooth; \item[\rm(ii)] $\GG$ is
isomorphic to the  affine space ${\mathbf A}^r$;
\item[\rm (iii)] $G=G_1\times\cdots\times G_s$ where
every $G_i$ is either a simply connected simple
algebraic group or isomorphic to ${\bf SO}_{n_i}$ for an
odd $n_i$.
\end{enumerate}
\end{theorem}



This criterion of smoothness of $\GG$ may be also
interpreted as that of the existence of $r$ generators
of the algebra of class functions on $G$. Below we
describe a minimal system of generators of this algebra
and singularities of $\GG$ in the general case. This
also yields a minimal system of generators of the
representation ring of $G$.



Let $B:=\tau(\widehat B)$ and $T:=\tau(\tT)$. This is
respectively a Borel subgroup and a maximal torus of
$G$. We consider the lattice ${\rm X}(T)$ of characters
of $T$ as the sublattice of ${\rm X}(\tT)$ identifying
$\mu\in {\rm X}(T)$ with $\tau^*(\mu)\in {\rm X}(\tT)$.
Then ${\rm X}(\tT)$ is the weight lattice of ${\rm
X}(T)$. The
%%additive
monoid of highest weights of simple $\tG$-modules (with
respect to $\widehat B$ and $\tT$) is
\begin{equation}\label{wD}
\widehat{\mathcal D}:={\bf N}\varpi_1+\cdots+ {\bf N}\varpi_r,\qquad\mbox{%%where
${\bf N}=\{0, 1, 2,\ldots\}$.}
\end{equation}
 and
that of simple $G$-modules (with respect to $B$ and $T$)
is
\begin{equation}\label{DDDD}
\mathcal D:=\widehat{\mathcal D}\cap {\rm X}(T).
\end{equation}


Let $W$ be the Weyl group of $\tG$, i.e., the quotient
of $\tT$ in its normalizer, acting on $\tT$ via
conjugation. The Weyl group of $T$ is naturally
identified with $W$ (see
%%\cite[Theorem 22.6(iii)]{Bor}
\cite[Prop.\,11.20 and Cor.\,2(d) in 13.17]{Bor}).
%%The Weyl group of $G$ is naturally isomorphic to %%$W$ and denoted by
%%the same letter.


If $\varpi\in{\mathcal D}$ and $E(\varpi)$ is a simple
$G$-module with $\varpi$ as the highest weight, we
denote by $\chi_{\varpi}\in k[G]^{G}$ the character of
$E(\varpi)$.

Given a nonzero commutative ring $A$ with identity
element and a commutative monoid $M$, we denote by
$A[M]$ the semigroup ring of $M$ over $A$. We
%%naturally
identify $A[M]$ with $A\otimes_{\bf Z}{\bf Z}[M]$ in the
natural way.
 If $S$ is a submonoid of the multiplicative monoid of $A[M]$ whose elements are linearly independent over $A$,
then the subring of $A[M]$ generated by $S$ is naturally
identified with $A[S]$. In particular, we consider
$A[{\rm X}(T)]$ and $A[\mathcal D]$ as the subrings of
$A[{\rm X}(\tT)]$. The former is stable with respect to
the natural action of $W$ on ${\bf Z}[{\rm X}(\tT)]$.
Using the notation and terminology of {\sc Bourbaki}
\cite[VI.3]{Bou}, we denote by $e^\mu$ the element of
${\bf Z}[{\rm X}(\tT)]$ corresponding to $\mu\in {\rm
X}(\tT)$  and put
\begin{equation}\label{S}
S(e^\mu):=\sum_{\nu\in W\cdot \mu} e^{\nu}\in  {\bf
Z}[{\rm X}(\tT)]^W.
\end{equation}

Given an algebraic group $H$, we denote by $R(H)$ the
{\it representation ring} of $H$: its additive group is
the Grothendieck group of the category of finite
dimensional algebraic $H$-modules with respect to exact
sequences and the multiplication is induced by tensor
product of modules. Using $\tau$, we  identify $R(G)$ in
the natural way with the subring of~$R(\tG)$.
%%Let
%%$E(\varpi)$ denote a simple algebraic $G$-module %%with
%%the highest weight $\varpi\in\mathcal D$.

If $E$ is a finite dimensional algebraic $G$-module and
$E_\mu$ is its weight space of a weight $\mu\in {\rm
X}(T)$, then the formal character of $E$,
\begin{equation}\label{ch}
{\rm ch}_{G}[E]:=\sum_{\mu\in {\rm X}(T)}(\dim E_\mu)
e^{\mu},
\end{equation}
is an element of ${\bf Z}[{\rm X}(T)]^W$ depending only
on the class $[E]$ of $E$ in $R(G)$. Clearly,
%%\vskip -2mm
\begin{equation}\label{tens}
{\rm ch}_{G}[E\otimes E']={\rm ch}_{G}[E]\,{\rm
ch}_{G}[E'].
\end{equation}

%%\vskip 1mm

According to \cite[3.6]{Se2}, the homomorphism of ${\bf
Z}$-modules
%%\vskip -2mm
\begin{equation}\label{chch}
{\rm ch}^{}_{G}\colon R(G)\longrightarrow {\bf Z}[{\rm
X}(T)]^W,\qquad [E]\mapsto {\rm ch}^{}_{G}[E],
\end{equation}

%%\vskip 1mm

\noindent is an isomorphism. By \eqref{tens} it is an
isomorphism of rings.



\begin{definition}
Let $\varpi\in \widehat{\mathcal D}$. We say that an
element $x\in {\bf Z}[{\rm X}(\tT)]^W$ is $\varpi$-{\it
sharp}  if the following property (M) holds:

\vskip 3mm

\begin{enumerate}
\item[(M)] $e^{\varpi}$ is the unique maximal term of
$x$.
\end{enumerate}
\end{definition}

\vskip -.5mm

\begin{example}\label{sharp} The elements
 $S(e^\varpi)$ and ${\rm ch}^{}_{\tG}[E(\varpi)]$
are $\varpi$-sharp (this follows, e.g., from
 \cite[VI.1.6, Prop.\;18]{Bou} and
\cite[31.3, Theorem]{Hum}). \quad $\square$
\end{example}

Property (M) implies that the support of a
$\varpi$-sharp element $x$ lies in $\varpi+{\rm X}(T)$.
This and \cite[VI.3.4, formula (6)]{Bou} yield
%%\vskip -1.5mm
\begin{equation}\label{SSS}
x=S(e^\varpi)+\mbox{sum of some $S(e^{\varpi'})$'s with
$\varpi'\in \widehat{\mathcal D}$, $\varpi'<\varpi$}.
\end{equation}

%%\vskip 1.5mm

\noindent By \cite[VI.3.2, Lemma 2]{Bou} if an element
$x'$ is a $\varpi'$-sharp, then $xx'$ is
$(\varpi+\varpi')$-sharp.




Now fix a $\varpi_i$-sharp element $x_{\varpi_i}\in {\bf
Z}[{\rm X}(\tT)]^W$, $i=1,\ldots, r$, and  put
%%\vskip -1.5mm
%%
\begin{equation*}
x_\varpi:= x_{\varpi_1}^{m_1}\cdots
x_{\varpi_r}^{m_r}\qquad\mbox{for}\qquad
\varpi=m_1\varpi_1+\cdots+m_r\varpi_r\in \widehat{
\mathcal D}.
\end{equation*}

%%\vskip 1.5mm

By \cite[VI.3.4, Theorem 1]{Bou} the set
$\{x_{\varpi}\mid \varpi\in \widehat{ \mathcal D}\}$ is
then a basis of the ${\bf Z}$-module ${\bf Z}[{\rm
X}(\tT)]^W\!.$ As $\{e^\mu\mid \mu\in {\rm X}(T)\}$ is a
basis of the ${\bf Z}$-module ${\bf Z}[{\rm X}(T)]$ and
the support of $x_\varpi$ lies in $\varpi+{\rm X}(T)$,
we deduce from this and \eqref{DDDD} that
$\{x_{\varpi}\mid \varpi\in\mathcal D\}$ is a basis of
the ${\bf Z}$-module ${\bf Z}[{\rm X}(T)]^W$. Hence the
homomorphism of the ${\bf Z}$-modules
%%\vskip -1.5mm
\begin{equation}\label{psi}
\vartheta\colon {\bf Z}[{\rm X}(T)]^W\to {\bf
Z}[\mathcal D], \qquad \vartheta(x_\varpi)=e^\varpi\quad
\mbox{for\quad $ \varpi\in \mathcal D$},
\end{equation}

%%\vskip 1.5mm

\noindent is an isomorphism. Since
$x_{\varpi+\varpi'}=x_\varpi x_{\varpi'}$, it is, in
fact, an isomorphism of rings.

As by Dedekind's theorem  $\{f_\mu\colon T\to k,
t\mapsto t^\mu\mid \mu\in {\rm X}(\tT)\}$ is a basis of
the vector space $k[\tT]$ over $k$, the $k$-linear map
$k[\tT]\to k[{\rm X}(\tT)]$,\ $ f_\mu\mapsto e^\mu$, is
the isomorphism of $k$-algebras. We identify them by
means of this isomorphism. So we have $k[T]=k[{\rm
X}(T)]$ and
\begin{equation}\label{T/W}
k[T/W]=k[T]^W=
%%k[{\rm X}(T)]^W=
%%k\otimes_{\bf Z} {\bf Z}
k[{\rm X}(T)]^W.
\end{equation}

%%\vskip 1.5mm

Finally, take into account that by \cite[6.4]{St1} the
restriction map
%%\vskip -1.5mm
\begin{equation}\label{G//G}
{\rm res}\colon k[\GG]=k[G]^G\longrightarrow
k[T]^W,\qquad {\rm res}(f)=f|_T,
\end{equation}

%%\vskip 1.5mm

\noindent is an isomorphism of $k$-algebras. Summing up,
we obtain

%%\newpage

\begin{theorem} \label{toric}
\
\begin{enumerate}
\item[\rm(i)] $\GG$ and $T/W$ are the affine toric
varieties of $\;T$ whose algebras of regular functions
are iso\-morphic to $k[\mathcal D]$. \item[\rm(ii)] In
the
%%following
diagram
%%\ \vskip -4mm
$$
k[\GG]\xrightarrow{\rm res} k[T/W] \xrightarrow{{\rm
id}\otimes \vartheta} k[\mathcal D]
$$
{\rm(}see {\rm \eqref{G//G}, \eqref{T/W},
\eqref{psi}}{\rm)} both maps are the isomorphisms of
$k$-algebras. \item[\rm(iii)]  Let $F$ be the simple
subring of $k$. Then the image of $F\otimes_{\bf Z}R(G)$
in $k[G]^G$ under the composition of the ring
isomorphisms
\begin{equation}
\label{composition} k\otimes_{\bf
Z}R(G)\xrightarrow{{\rm id}\otimes {\rm ch}^{}_G} k[{\rm
X}(T)]^W=k[T]^W \xrightarrow{{\rm res}^{-1}}k[G]^G
\end{equation}
%%under the map $(1\otimes {\rm id})
%%\circ{\rm ch}^{}_G$
is an $F$-form of $k[\GG]$ isomorphic to $F\otimes_{\bf
Z}R(G)$. In particular, if ${\rm char}\,k=0$, it is a
${\bf Z}$-form of $k[\GG]$ iso\-morphic to $R(G)$.
\end{enumerate}
%%, the semigroup algebra of $\mathcal D$ over %%$k$.
\end{theorem}

\begin{remark} {\rm The fact that ``multiplicative invariants'' of finite reflection groups are semigroup algebras is already in the literature, first implicitly, then explicitly, see
the historical account in
\cite[Int\-roduction]{L0}. Essentially, the main %%arguments
ingredients date back to  \cite[\hskip -.5mm\S6]{St1}
and \cite[\hskip -1mm VI~\S3]{Bou}.}
\end{remark}

Since toric varieties are rational,
Theorem \ref{toric}(i) yields %%the following

\begin{corollary}  \label{ratio}
$\GG$ and $T/W$ are rational varieties.
\end{corollary}



In the next statement Theorem \ref{toric} is applied to
finding a minimal system of ge\-ne\-rators of the
algebra $k[G]^G$ and that of the ring $R(G)$.




Let $\mathcal H$ be the Hilbert basis of the monoid
$\mathcal D$, i.e., the set of all its indecomposable
elements:
\begin{equation}\label{Hilbert}
\mathcal H=\mathcal D_+\setminus 2 \mathcal D_+
\quad\mbox{where}\quad \mathcal D_+:=\mathcal D\setminus
\{0\}, \quad 2\mathcal D_+:=\mathcal D_+ + \mathcal D_+.
\end{equation}
The set $\mathcal H$ is finite, generates $\mathcal D$,
and every generating set of $\mathcal D$ contains
$\mathcal H$ (see, e.g., \cite[3.4]{L}).

\begin{remark}
{\rm There is an algorithm for computing $\mathcal H$,
see \cite[13.2]{Stu} (cf.\;also Ex\-amp\-le \ref{singu}
below)}.
\end{remark}

%%Example ? explains
%%how this explicitly provides
%%one can use it for
%%this yields
%%leads to
%%explicit
%%finding
%%a finite set $\mathcal G\subset\mathcal D$ such that
%%$\{[E(\varpi)]\mid \varpi\in \}$ is



\begin{theorem}\label{generators} \

%%If $\tau$ is separable, then
%%Denote
%%$\mathcal D_+:=\mathcal D\setminus
%%\{0\}$ and
%%\qquad \mbox{and}\qquad
%%$\;2\mathcal D_+:=\mathcal D_++\mathcal D_+.$
%%Then
\begin{enumerate}
\item[\rm(i)] The cardinality of every generating set of
the algebra $k[G]^G$ of class functions on $G$ is not
less than the cardinality of $\;\mathcal H$. The same
holds for every generating set of the representation
ring $R(G)$ of $G$. \item[\rm(ii)] $\{[E(\varpi)] \mid
\varpi\in \mathcal H\}$ is a
%%{\rm(}minimal\,{\rm)}
generating set of the ring $R(G)$. \item[\rm(iii)]
$\{\chi_{\varpi}\mid \varpi\in \mathcal H\}$ is a
%%{\rm(}minimal\,{\rm)}
generating set of the algebra
$k[G]^G$.
\end{enumerate}
\end{theorem}
\begin{proof}
(i) Let $Y$ be the affine toric variety of $T$ with
$k[Y]=k[\mathcal D]$. The linear span $I$ of
$\{e^\varpi\mid \varpi\in\mathcal D_+\}$ over $k$ is a
maximal $T$-invariant ideal in $k[Y]$. Hence $I/I^2$ is
the cotangent space of $Y$ at the $T$-fixed point $v$
where $I$ vanishes. As $I^2$ is the linear span of
$\{e^\varpi\mid \varpi\in 2\mathcal D_+\}$ over $k$,
this and \eqref{Hilbert} yield
\begin{equation}\label{tange}
\dim {\rm T}_{v, Y}=\dim I/I^2= |\mathcal H|.
\end{equation}


Now take into account that, given an affine algebraic
variety $X$, the algebra $k[X]$ can be generated by $d$
elements if and only if $X$ admits a closed embedding in
${\bf A}\!^d$. Hence $d\geqslant \dim {\rm T}_{x, X}$
for every point $x\in X$. This, Theorem
\ref{toric}(i),(iii), and \eqref{tange} prove~(i).



\vskip 1mm

(ii) Let $\mu\in \mathcal D$. As $\mathcal H$ generates
$\mathcal D$, we deduce from Example \ref{sharp} that
there is a $\mu$-sharp monomial $M^\mu$  in the elements
of the set $\{{\rm ch}^{}_G[E(\varpi)]\mid \varpi\in
\mathcal D\}$. By \eqref{SSS} we have
%%\ \vskip -5.6mm
\begin{equation}\label{MM} M^\mu=S(e^\mu)+
\mbox{sum of some $S(e^{\mu'})$'s with $\mu'\in
{\mathcal D}$, $\mu'<\mu$}.
\end{equation}
%%\vskip 1.5mm
\noindent But $\{S(e^\mu)\mid \mu\in \mathcal D\}$ is a
basis of the ${\bf Z}$-module ${\bf Z}[{\rm X}(T)]^W$
(see \cite[VI.3.4, Lemma 3]{Bou}).  By \cite[VI.3.4,
Lemma 4]{Bou} and \eqref{MM}  we then conclude that the
set $\{M^\mu\mid \mu\in\mathcal D\}$ generates the ${\bf
Z}$-module ${\bf Z}[{\rm X}(T)]^W$. This means that the
ring ${\bf Z}[{\rm X}(T)]^W$ is generated by the set
$\{{\rm ch}^{}_{G}[E(\varpi)]\mid \varpi\in\mathcal
H\}$. As \eqref{chch} is an isomorphism of rings, this
proves (ii).

\vskip 1mm

(iii) It follows from (ii) that the set $\{1\otimes
[E(\varpi)]\mid \varpi\in\mathcal H\}$ generates the
ring $k\otimes_{\bf Z}R(G)$. But formula \eqref{ch}
shows that $\chi_\varpi$ is the image of $1\otimes
E[(\varpi)]$ under the composition of the ring
isomorphisms in diagram \eqref{composition}. This proves
(iii). \quad $\square$
\renewcommand{\qed}{}\end{proof}


Since the Weyl chambers are simplicial cones, Theorem
\ref{toric}(i) implies, at least for ${\rm char} \,k=0$,
that $\GG$ and $T/W$ are isomorphic to the quotient of
${\bf A}^r$ by a linear action of a certain finite
abelian group (see, e.g., \cite[Prop.\;1.25]{O}). In
particular, $\GG$ and $T/W$ may have only finite
quotient singularities. Below  this finite group and its
action on ${\bf A}^r$ are explicitly described  assuming
that $\tau$ is separable.

 The latter assumption means  that there is a subgroup
$Z$ of $\tC$ such that $G=\tG/Z$ and $\tau$ is the
quotient morphism $\tG\to \tG/Z$. In this situation we
have \begin{equation}\label{varpiX} {\rm X}(T)=\{\mu\in
{\rm X}(\tT)\mid c^\mu=1\ \mbox{for every $c\in
Z$}\}.\end{equation}


Consider the $\tT$-orbit map of the point
$(1,\ldots,1)\in{\bf A}^r$:
\begin{equation}\label{iot}
\iota\colon \tT\longrightarrow {\bf A}^r,\qquad
\iota(t)=t\cdot (1,\ldots, 1).
\end{equation}
The map $\iota^*\colon k[{\bf A}^r]\to k[\tT]=k[{\rm
X}(\tT)]$ is an embedding as $\iota$ is dominant by
Lemma \ref{lin}(i). Let $y_1,\ldots, y_r$ be the
standard coordinate functions on ${\bf A}^r$. Then
\eqref{action} and \eqref{iot} yield
%%\ \vskip -6mm
\begin{equation}\label{embed}
\iota^*(y_i)=e^{\varpi_i}.\end{equation}
%%\ \vskip -.5mm
\noindent From \eqref{action} we deduce that $k[{\bf
A}^r]^Z$ is the linear span over $k$ of all monomials
$y^{m_1}\cdots y^{m_r}$ with $m_1,\ldots, m_r\in {\bf
N}$ such that $c^{m_1\varpi_i+\cdots+m_r\varpi_r}=1$ for
every $c\in Z$. By \eqref{varpiX} the latter condition
is equivalent to the inclusion
$m_1\varpi_i+\cdots+m_r\varpi_r\in {\rm X}(T)$. This,
\eqref{embed}, \eqref{wD}, and \eqref{DDDD} imply that
$\iota^*(k[{\bf A}^r]^Z)=k[\mathcal D]$. Thus, taking
into account Theorem \ref{toric}, we obtain the
isomorphisms of $k$-algebras
%%\ \vskip .5mm
\begin{equation*}\label{isoms}
k[{\bf A}^r]^Z\xrightarrow{{\iota}^*} k[\mathcal
D]\xrightarrow{({\rm id}\otimes \vartheta)^{-1}} k[T/W]
\xrightarrow{({\rm res})^{-1}} k[\GG].
\end{equation*}%%\ \vskip .5mm\noindent
that, in turn, induce the isomorphisms of varieties
$\GG\to T/W\to{\bf A}^r/Z$.

%%Since $Z=\tC$ when $G$ is adjoint, formula
%%\eqref{varpiX} implies that $\tC=
By means of a special parametrization of $\tT$ one
obtains
%%the
%%more
an explicit description of the elements of $\tC$
%%as linear
%%transformations of ${\bf A}\!^r$
well adapted for computing $k[{\bf A}^r]^Z$. Since
$\tG=\tG_1\times\cdots\times \tG_s$ and $\widehat
C=\widehat C_1\times\cdots\times \widehat C_s$ where
every $\widehat G_i$ is a nontrivial  normal simply
connected simple subgroup of $\tG$ and $\widehat C_i$ is
the center of $\widehat G_i$, it suffices to describe
this parametrization for simple groups $\tG$. The answer
is given below in Lemma \ref{explcenter}.

Namely, let
${\widehat\alpha}_1,\ldots,{\widehat\alpha}_r\in
{X}(\widehat T)$ be the system of simple roots of
$\widehat T$ with respect to $\widehat  B$ and let
${\widehat\alpha}^\vee_i\colon {\bf G}_m\to \widehat T$
be the coroot corresponding to ${\widehat\alpha}_i$.
Then, for every $s\in {\bf G}_m$,
%%we have
\begin{equation}\label{pairr}
\big({\widehat\alpha}^\vee_i(s)\big){}^{\varpi_j}=
\begin{cases} s &\mbox{if $i=j$},\\
1 &\mbox{if $i\neq j$.}
\end{cases}
\end{equation}
%%\vskip 2mm

If $\langle\ {,}\ \rangle$ is the natural pairing
between the lattices of characters and cocharacters of
$\widehat T$, we put
$n_{ij}:=\langle{\widehat\alpha}_i,{\widehat\alpha}_j^\vee
\rangle$. So $(n_{ij})_{i,j=1}^r$ is the Cartan matrix
of $\tG$.

 By \cite[Lemma 28(b),(d) and its Cor.\,(a)]{St2} the map
\begin{equation}\label{toruss}
\nu \colon {\bf G}_m^{r} \longrightarrow \widehat T,
\qquad \nu(s_1,\ldots, s_r)=
{\widehat\alpha}^\vee_1(s_1)\cdots
{\widehat\alpha}^\vee_r(s_r),
\end{equation}%%\vskip .2mm\noindent
is an isomorphism of groups and %%\vskip .1mm
\begin{equation}\label{center}
\widehat C=\{{\widehat\alpha}^\vee_1(s_1)\cdots
{\widehat\alpha}^\vee_r(s_r)\mid s_1^{n_{i1}}\cdots
s_r^{n_{ir}}=1\hskip 2mm\mbox{for every $i=1,\ldots,
r$}\}.
\end{equation}

By \eqref{pairr} and \eqref{toruss}  we have
\begin{equation*}
\big(\nu(s_1,\ldots,s_r)\big)^{\varpi_i}=
\big({\widehat\alpha}^\vee_1 (s_1)\big){}^{\varpi_i}
\cdots
\big({\widehat\alpha}^\vee_r(s_r)\big){}^{\varpi_i}=s_i.
\end{equation*}
This and \eqref{action} imply that, for every
$s=(s_1,\ldots,s_r)\in {\bf G}_m^r$ and $(a_1, \ldots,
a_r)\in{\bf A}^r$, we have
%%\ \hskip -8mm
\begin{equation*}
\nu(s)\cdot(a_1,\ldots, a_r)=(s_1a_1,\ldots, s_ra_r).
\end{equation*}







\begin{lemma}\label{explcenter}
For every simple simply connected group $\tG$, the
subgroup $\nu^{-1}(\widehat C)$ of
%%$\tG$
the torus ${\bf G}_m^r$ is
%%given by
described in the following Table $1$ {\rm(}simple roots
in \eqref{toruss} are numerated as in \cite{Bou}{\rm)}:

\vskip 3mm \centerline{\sc Table $1$.}

\vskip 3mm
\begin{center}
\begin{tabular}{c|cc}
\text{\rm type of} $\tG$ && $\nu^{-1}(\widehat C)$
\\[2pt]
\hline \hline
&\\[-9pt]
${\sf A}_r$&& $\{(s, s^2, s^3,\ldots, s^r)\mid
s^{r+1}=1\}$
\\[3pt]
%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \hline
&\\[-8pt]
 ${\sf B}_r$ %%$l\geqslant 3$
 &&
 $\{(1,\ldots, 1, s^2)\mid s^2=1\}$
\\[3pt]
%%%%%%%%%%%%%%%%%%%%%%%%%%
\hline
&\\[-8pt]
${\sf C}_r$
%%$l\geqslant 2$
&& $\{(s, 1, s, 1,\ldots, s^{r\,{\rm mod}\,2})\mid
s^2=1\}$
\\[3pt]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hline
&\\[-8pt]
${\sf D}_r,$ $\mbox{$r${\fontsize{9pt}{5mm}
\selectfont\rm odd}}$ && $\{(s^2, 1, s^2, 1,\ldots, s^2,
s, s^{-1})\mid s^4=1\}$
\\[3pt]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hline
&\\[-8pt]
${\sf D}_r,$ $\mbox{$r${\fontsize{9pt}{5mm}
\selectfont\rm even}}$ && $\{(s, 1, s, 1,\ldots,s, 1,
st, t)\mid s^2=t^2=1\}$
%%$\{(1, 1, 1, 1,\ldots, 1, s, s)\mid s^2=1\}$
\\[3pt]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hline
&\\[-8pt]
${\sf E}_6$&& $\{(s, 1, s^{-1}, 1, s, s^{-1})\mid
s^3=1\}$
\\[3pt]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\hline
&\\[-8pt]
${\sf E}_7$&& $\{(1, s, 1, 1, s, 1, s)\mid s^2=1\}$
\\[3pt]
%%%%%%%%%%%%%%%%%%%%%%%%%%
\hline
&\\[-8pt]
${\sf E}_8$&& $\{(1,1,1,1,1,1,1,1)\}$
%%$\{e\}$
\\[3pt]
%%%%%%%%%%%%%%%%%%%%%%%%
\hline
&\\[-8pt]
${\sf F}_4$&& $\{(1,1,1,1)\}$
%%$\{e\}$
\\[3pt]
%%%%%%%%%%%%%%%%%%%%%%%%%
\hline
&\\[-8pt]
${\sf G}_2$&& $\{(1,1)\}$
%%$\{e\}$%%\\
%%\\[3pt]
%%%%%%%%%%%%%%%%%%%%%%%%%
%%\hline
%%&\\[-8pt]
\end{tabular}
%%\vskip 4mm
%% \centerline{\rm Table $2$}
\end{center}
 \end{lemma}


\vskip 1mm

\begin{proof} By \eqref{center} an element $(s_1,\ldots, s_r)\in {\bf G}_m^r$ lies in $\nu^{-1}(\widehat C)$ if and only if $(s_1,\ldots,s_r)$
%%$x_1=t_1, \ldots, x_r=t_r$
is a solution of the system of equations
\begin{equation}\label{systemm}
%%\left\{
\begin{split}
&x_1^{n_{11}}\cdots x_r^{n_{1r}}=1,\\[-2pt]
&\hskip 1mm .\ . \ . \ .\ .\ .\ .\ .\ .\ .\ . \\[1pt]
&x_1^{n_{r1}}\cdots x_r^{n_{rr}}=1,
\end{split}
\end{equation}
where $(n_{ij})_{i,j=1}^r$ is the Cartan matrix of
$\tG$.

Let, for instance, $\tG$ be of type ${\sf D}_r$ for even
$r$. Using the explicit form of the Cartan matrix
\cite[Planche IV]{Bou}, one immediately verified that
every element of $C':=\{(s, 1, s, 1,\ldots,s, 1, st,
t)\mid s^2=t^2=1\}$ is a solution of \eqref{systemm}.
%%, where $(n_{ij})_{i,j=1}^r$ is the Cartan matrix %%of type ${\sf D}_r$.
Hence, $C'\subseteq \nu^{-1}(\widehat C)$. On the other
hand, the fundamental group  of the root system of type
${\sf D}_r$ is isomorphic to ${\bf Z}/2{\bf Z}\oplus
{\bf Z}/2{\bf Z}$ (see \cite[8.1.11]{Sp} and
\cite[Planche IV]{Bou}). Hence, the {\sc Smith} normal
form of $(n_{ij})_{i,j=1}^r$ is ${\rm
diag}\,(1,\ldots,1,2,2)$. Therefore, there is a basis
$\beta_1,\ldots, \beta_r$ of the coroot lattice of $\tT$
such that, for $(s_1,\ldots,s_r)\in {\bf G}_m^r$, we
have $\beta_1(s_1)\cdots\beta_r(s_r)\in \widehat C$ if
and only if $(s_1,\ldots,s_r)$ is a solution of the
system
\begin{equation*}
x_1=1,\ldots, x_{r-2}=1, x_{r-1}^2=1, x_{r}^2=1.
\end{equation*}
This yields $|C'|=|\widehat C|$; whence $C'=\nu^{-1}
(\widehat C)$.

For the groups of the other types the proofs are
similar. \quad $\square$
\renewcommand{\qed}{}\end{proof}

The following examples illustrate how
%%Theorem \ref{explcenter}
this can be applied to  exploring  singularities of
$\GG$ and finding the minimal generating sets
$\{\chi_{\varpi}\mid \varpi\in \mathcal H\}$ and
$\{[E(\varpi)] \mid \varpi\in \mathcal H\}$ of,
respectively,  the algebra of class functions on $G$ and
 the representation ring
of $G$.



%%sets of $k[\GG]$ and $R(G)$.

\begin{examples}\label{singu}\


(1) Let $\tG$ be of type ${\sf B}_r$  (where ${\sf
B}_1\!:=\!{\sf A}_1$) and let ${\rm char}\,k\!\neq\!2$.
Table 1 imp\-lies that $\nu^{-1}(\tC)$ is generated by
$(1,\ldots,1,-1)$. Whence $k[{\bf A}\!^r]^{\widehat
C}\!\!=\!k[y_1,\ldots, y_{r-1}, y_r^2]$. Therefore, for
the adjoint $G$, i.e., for $G={\bf SO}_{2r+1}$, the
variety $\GG$ is isomorphic to ${\bf A}\!^r$ (this
agrees with Theorem  \ref{sing}) and
%%, the algebra
%%of class functions on $G$ is
%%minimally generated by $\{\chi_{\varpi}\mid %%\varpi\in \mathcal H\}$,
%%and the representation ring
%%of $G$ by
%%$\{[E(\varpi)]
%%\mid \varpi\in \mathcal H\}$, where
%%$$\chi_{\varpi_1},\;\ldots,\;\chi_{\varpi_{r-1}},\; %%\chi_{2\varpi_{r}},$$
%%and the representation ring
%%of $G$ is minimally generated by %%$$[E(\varpi_1)],\;\ldots,\;
%%[E(\varpi_{r-1})],\; [E(2\varpi_{r})].$$
$$\mathcal H=\{\varpi_1,\ldots,\varpi_{r-1}, 2\varpi_{r}
\}.
$$

(2) Let $\tG$ be of type ${\sf D}_r$, let ${\rm
char}\,k\neq 2$, and let $Z:=\{t\in \widehat C\mid
t^{\varpi_1}=1\}$. Then $G:=\tG/Z={\bf SO}_{2r}$.
 Table 1 implies that $\nu^{-1}(Z)$ is generated by
$(1,\ldots,1,-1,-1)$. Whence
$k[{\bf A}\!^r]^Z=k[y_1,\ldots,y_{r-2},y_{r-1}^2, y_{r}^2, y_{r-1}y_{r}]$. Therefore, $\GG$ is isomorphic to ${\bf A}^{r-2}\times X$ where $X$ is %%the affine cone over the Veronese variety
%%$\nu_2({\bf P^1})$ in ${\bf P}^2$ (that is
a nondegenerate quadratic cone in ${\bf A}^3$ and
$$\mathcal H=\{\varpi_1,\ldots,\varpi_{r-2}, 2\varpi_{r-1}, 2\varpi_{r}, \varpi_{r-1}+ \varpi_{r}
\}.
$$

%%\vskip 1.5mm

(3) Let $\tG$ be of type ${\sf E}_7$ and let ${\rm
char}\,k\neq 2$. Table 1 implies that  $k[{\bf
A}\!^7]^{\widehat C}$ is minimally generated by $y_1,
y_3, y_4, y_6$ and all the monomials of order $2$ in
$y_2, y_5, y_7$. Therefore, if $G$ is adjoint, then
$\GG$ is isomorphic to ${\bf A}^4\times Y$ where $Y$ is
the affine cone over the Veronese variety $\nu_2({\bf
P^2})$ in ${\bf P}^5$ (in particular, the tangent space
of the $7$-dimensional variety $\GG$ at the unique fixed
point of $T$, see Theorem \ref{toric}(i) and
\eqref{tange}, is $10$-dimensional) and
%%, the algebra
%%of class functions on $G$ is
%%minimally generated by $\{\chi_{\varpi}\mid %%\varpi\in \mathcal H\}$,
%%and the representation ring
%%of $G$ by
%%$\{[E(\varpi)]
%%\mid \varpi\in \mathcal H\}$, where
$$
\mathcal H=\{\varpi_1, \varpi_3, \varpi_4, \varpi_6,
2\varpi_{2}, 2\varpi_{5}, 2\varpi_{7},
\varpi_{2}+\varpi_{5}, \varpi_{2}+\varpi_{7},
\varpi_{5}+\varpi_{7}. \}. \quad \square
$$
%%the algebra of class functions on $G$ is %%minimally generated by
%%\begin{gather*}\chi_{\varpi_1},\; %%\chi_{\varpi_3},\; \chi_{\varpi_4},\; %%\chi_{\varpi_6},\;
%%\chi_{2\varpi_{2}},\; \chi_{2\varpi_{5}}, \; %%\chi_{2\varpi_{7}},\;
%%\chi_{\varpi_{2}+\varpi_{5}}, \;
%%\chi_{\varpi_{2}+\varpi_{7}},\;
%%\chi_{\varpi_{5}+\varpi_{7}},
%%\end{gather*}
%%and the representation ring of $G$ is minimally %%generated by
%%\begin{gather*}
%%[E(\varpi_1)],\; [E(\varpi_3)],\; %%[E(\varpi_4)],\; [E(\varpi_6)],\;
%%[E(2\varpi_2)],\; [E(2\varpi_5)],\; %%[E(2\varpi_7)],\\ [E(\varpi_2+\varpi_5)],\; %%[E(\varpi_2+\varpi_7)],\; %%[E(\varpi_5+\varpi_7)].
%%\end{gather*}
\end{examples}

\section{Yet two Grothendieck's questions}\label{Yettwo}

Theorem \ref{generators} describes a minimal generating
set of the algebra $k[G]^G$ of class functions on $G$.
Constructing the generating sets of $k[G]^G$ is the
topic of yet two {\sc Grothendieck}'s questions in
\cite[p.\;241]{GS}:
%%{\sc Grothendieck}:
%%asks the following two questions
%%about the gene\-ra\-ting sets of $k[G]^G$:
\begin{quote} ``$[\ldots]$ {\rm When $G$ is an adjoint group, is it
possible to generate the affine ring of $I(G)$ with
coefficients of the Killing polynomial{\rm?} In the
general case, is it enough to take the coefficients of
analogous polynomials for certain linear representations
{\rm(}perhaps arbitrary faithful representations})?
$[\ldots]$''
\end{quote}
Below we answer these questions.
%%the first question in the negative and
%%the second in the positive.

Let $\varrho\colon G\!\to\! {\bf GL}(V)$ be a finite
dimensional linear representation of $G$. Define the~set
\begin{equation*}
C_{\varrho}:=\{c_{\varrho, i}\in k[G]\mid i=1,\ldots,
%%n_{\varrho}:=
\dim V\}
\end{equation*}
by the equality
\begin{equation}\label{charpoly}
{\rm det}(xI-\varrho(g))=\sum_{i=0}^{\dim V}%%n_\varrho}
x^{\dim V-i}c_{\varrho, i}(g)\quad\mbox{for every}\hskip
2mm g\in G,
\end{equation}
where $x$ is a variable.
 If
$V\!=\!E(\varpi)$ and $\varrho$ determines the
$G$-module structure of $E(\varpi)$, we put
$C_{\varpi}:=C_{\varrho}$.
%%$f_{\varpi,
%%i}:=f_{\varrho, i}$.

Clearly, $c_{\varrho, i}\!\in\! k[G]^G$ and $c_{\varrho,
1}$ is the character of $\varrho$. Hence by Theorem
\ref{generators}(iii)
%%$\bigcup_{\varpi\in \mathcal H}
%%C_{\varpi}$
\begin{equation*}\label{Gro2}
\bigcup_{\varpi\in \mathcal H} C_{\varpi}
\end{equation*}
is a generating set of the algebra $k[G]^G$. This
answers  the second {\sc Gro\-then\-dieck}'s question in
the affirmative.

In order to answer the first one in the negative it is
sufficient to find an adjoint $G$ and two elements $z_1,
z_2\in T$ such that
\begin{enumerate}
\item[(i)] $z_1$ and $z_2$ are not in the same
$W$-orbit; \item[(ii)] the spectra
%%(i.e., the sets of eigenvalues
%%taken with their multiplicities)
of the linear operators ${\rm Ad}_{G}\,z_1$ and ${\rm
Ad}_{G}\,z_2$ on the vector space ${\rm Lie}\,G$
coincide.
\end{enumerate}
Indeed, property (i) implies that there is a function
$f\in k[T]^W$ such that $f(z_1)\neq f(z_2)$. Given
isomorphism \eqref{G//G}, this means that there is a
function ${\widetilde f}\in k[G]^G$ such that
${\widetilde f}(z_1)\neq {\widetilde f}(z_2)$. On the
other hand, \eqref{charpoly} and property (ii) imply
that
\begin{equation*}
c_{{\rm Ad}_G,i}(z_1)=c_{{\rm Ad}_G,i}(z_2)\qquad
\mbox{for every}\hskip 2mm i.
\end{equation*}
Therefore, $\widetilde f$ is not in the subalgebra of
$k[G]^G$ generated by $C_{\rm Ad_G}$, i.e., the latter
is not a generating set of $k[G]^G$.

The following two examples show that one indeed can find
$G$, $z_1$, and $z_2$ sharing properties (i) and (ii).
\begin{examples}\

(1) Let $G=H\times H$ where $H$ is a connected adjoint
semisimple algebraic group. Let $T=S\times S$ where $S$
is a maximal torus of $H$. Let $W_S$ be the Weyl group
of $H$ naturally acting on $S$. Take any two elements
$a, b\in S$ that are not in the same $W_S$-orbit and put
$z_1:=(a,b),\; z_2:=(b, a)\in T$. As $W=W_S\times W_S$,
property (i) holds. On the other hand,
%%as ${\rm
%%Lie}\,G={\rm Lie}\,H\oplus {\rm Lie}\,H$,
clearly, for every $i=1, 2$, the spectrum of ${\rm
Ad}_G\,z_i$ is the union of the spectra of ${\rm
Ad}_H\,a$ and ${\rm Ad}_H\,b$; whence property (ii)
holds.

\vskip 1mm

(2) In this example $G$ is simple, namely, $G={\bf
PGL}_3$. Let $\alpha_1, \alpha_2\in {X}(T)$ be the
simple roots of $T$ with respect to $B$. As the map
$T\to {\bf G}_{\rm m}^2$, $t\mapsto (t^{\alpha_1},
t^{\alpha_2})$, is surjective (in fact, an isomorphism),
%%$G$ is adjoint, $\alpha_1,
%%\alpha_2$ is a basis of ${X}(T)$. Hence
for every $u, v\in k$, $uv\neq 0$, there are $z_1,
z_2\in T$ such that $z_1^{\alpha_1}=u$,
$z_1^{\alpha_2}=v$ and $z_2^{\alpha_1}=v$,
$z_2^{\alpha_2}=u$. For  these $z_1, z_2$, property (ii)
holds as the set of roots of $G$ with respect to $T$ is
$\{\pm\alpha_1, \pm\alpha_2, \pm(\alpha_1+\alpha_2)\}$.
Now take $u$ and $v$ such that
%%there are no equal
all elements
%%among
$u$, $u^{-1}$, $v$, $v^{-1}$, $uv$, $u^{-1}v^{-1}$ are
pairwise %%distinct
different. Then property (i) holds as there are no $w\in
W$ such that $w(\alpha_1)=\alpha_2$ and
$w(\alpha_2)=\alpha_1$.
\end{examples}




\section{
Rational cross-sections }\label{si}\label{rcs}

Recall from \cite[2.14, 2.15]{St1} that an element $x\in
G$ is called {\it strongly regular} if its centralizer
$G_x$ is a maximal torus. Such $x$ is regular and
semisimple. Strongly regular
elements form a dense open subset $G_0$ of $G$ %%This subset
stable with respect to the conjugating action of $G$.
Every $G$-orbit in $G_0$ is regular and closed in $G$.
We put
\begin{equation*}\label{G0T0}
(\GG)_0:=\pG(G_0)\qquad\mbox{and}\qquad  T_0:=T\cap G_0.
\end{equation*}
Abusing the notation, we
%%shall
denote
%%the restriction of $\pG$ to $G_0$
$\pG|_{G_0}$ still by $\pG$:
\begin{equation}\label{pG0000}
\pG\colon G_0\longrightarrow (\GG)_0.
\end{equation}
\begin{lemma}\label{GG0}
\

\begin{enumerate}
\item[\rm(i)] $(\GG)_0$ is an open
%%nonsingular
smooth subset of $\GG$. \item[\rm(ii)] $\pG|_{T_0}\colon
T_0\to (\GG)_0$ is a surjective \'etale map.
\item[\rm(iii)] $((\GG)_0, \pG)$ is the geometric
quotient for the action of $\;G$ on $G_0$.
\end{enumerate}
\end{lemma}
\begin{proof} Since $\GG$ is normal and all
fibers of $\pG$ are of constant dimension and
irreducible,
%%dimension $\dim G-\dim \GG$,
$\pG$ is an open map (see \cite[AG.18.4]{Bor}). Hence
$(\GG)_0$ is open in $\GG$.

As every element of $G_0$ is semisimple, it is conjugate
to an element of $T_0$; whence $\pG|_{T_0}$ is
surjective.

The set
%%\begin{equation*}
$T_0$
%%:=T\cap G_0
%%\end{equation*}
is open in $T$ and $W$-stable. For every point $t\in
T_0$, we have $G_t=T$, hence the $W$-stabilizer of $t$
is trivial. Thus, the action of $W$ on $T_0$ is set
theoretically free.
%%the action of $W$ on $T_0$ is free (i.e., with %%trivial stabilizers).
Since $T$ is
%%nonsingular
smooth, $\GG$ is normal, and $(\GG, \pG|_T)$ is the
quotient for the action of $W$ on $T$ (see
\cite[6.4]{St1}),
%%it follows
we deduce from \cite[Exp.\;I, Th\'eor\`eme 9.5(ii)]{G3}
and \cite[V.2.3, Cor.\,4]{Bour} that $\pG|_{T_0}$ is
\'etale and hence
%%, for every point $t\in T_0$, the restriction of
%%$(d\pG)_t$ to ${\rm T}_{t, T_0}$ is the %%isomorphism with ${\rm T}_{\pG(t),\GG}$.
%%In particular, the variety
%%$(\GG)_0=\pG(T_0)$ is
%%Hence
$(\GG)_0$ is %%nonsingular
smooth.
%% and $\pG|_{T_0}$ is \'etale.
This proves (i) and (ii).

By (ii) the map $\pG\colon G_0\to (\GG)_0$ is separable
and surjective. As its fibers are $G$-orbits and
$(\GG)_0$ is normal, (iii) follows from \cite[6.6]{Bor}.
%%This proves (iii)
\quad $\square$
\renewcommand{\qed}{}\end{proof}

%%\begin{corollary}\label{geoquo}
%%\end{corollary}






The group $W$ acts on $G/T\times T_0$  diagonally with
the action  on the first factor defined by formula
\eqref{G/T}. The group $G$ acts on $G/T\times T_0$  via
left translations of the first factor. These two actions
commute with each other.

Consider the $G$-equivariant morphism
\begin{equation}\label{psss}
\psi\colon G/T\times T_0\longrightarrow G_0,\qquad (gT,
t)\mapsto gtg^{-1}.
\end{equation}

%%The proof of the following lemma is taken from %%my letter \cite{P}.
%% The proofs of Lemmas \ref{dpsi} and %%\ref{vartheta}
%%below reproduce
%% my proofs
The proofs of Lemma \ref{dpsi} and Corollary
\ref{vartheta} reproduce that from my letter \cite{P}.

\begin{lemma}\label{dpsi}
$\psi$ is a surjective \'etale map.
%%$d\psi_x$ is the isomorphism for every point %%$x\in G/T\times T_0$.
\end{lemma}
\begin{proof}
As every $G$-orbit in $G_0$ intersects $T_0$,
surjectivity of $\psi$ follows from \eqref{psss}.

Take a point $z\in G/T\times T_0$. We shall prove that $d\psi_z$ is an isomorphism. As $G/T\times T_0$ and $G_0$ are %%nonsingular
smooth, this is equivalent to proving that $\psi$ is
\'etale at $z$. Using that $\psi$ is $G$-equivariant, we
may assume that $z=(eT, s)$, $s\in T_0$.


Let $U_\alpha$ be the one-dimensional unipotent root
subgroup of $G$ corresponding to a root $\alpha$ with
respect to $T$ and let
%%\begin{equation*}
$\theta_\alpha\colon {\bf G}_a\!\to\! U_\alpha$
%%\end{equation*}
be the isomorphism of groups such that
%%\ \vskip -5mm
\begin{equation*}
t\theta_\alpha(x)t^{-1}=\theta_\alpha(t^{\alpha}x)
\qquad\mbox{ for all $\ t\in T, \ x\in {\bf G}_a$},
\end{equation*} see \cite[IV.13.18]{Bor}. Put
\begin{gather*}
C_\alpha:=\{(\theta_\alpha(x)T, s)\in G/T\times T_0\mid x\in {\bf G}_a\},\\%%\qquad \mbox{and}\qquad
D:=\{(eT, t)\in G/T\times T_0\mid t\in T_0\}.
\end{gather*}


%%The tangent space of $G/T\times T_0$ at $z$  is
The linear span of all ${\rm T}_{z, C_{\alpha}}$\hskip
-1mm's
%%for %%the curve
%%the tangent spaces of the curves
%%\begin{equation*}
%%C_\alpha:=\{(\theta_\alpha(x)T, t_0\mid x\in %%{\bf G}_a\}
%%\end{equation*}
%%at $z$
and
%%the tangent space of the subvariety
${\rm T}_{z, D}$ is ${\rm T}_{z, G/T\times T_{0}}$.
%% for
%%\begin{equation*}
%%S:=\{(eT, t)\mid t\in T_0\}.
%%\end{equation*}
%%at $z$.
We have
\begin{equation}\label{alllg}
\begin{split}
\psi(\theta_\alpha(x)T,
s)&=\theta_\alpha(x)s\theta_\alpha(x)^{-1}=
\theta_\alpha(x)s\theta_\alpha(-x)\\
&=\theta_\alpha(x)\theta_\alpha(-s^\alpha x)s=
\theta_\alpha((1-s^\alpha)x)s.
\end{split}
\end{equation}
%%{align}
Since $s$ is
%%strongly
regular, $s^\alpha\neq 1$.
%%for every
%%root
%%$\alpha$.
%% of $G$ with respect to $T$.
Hence \eqref{alllg} shows that $\psi$ maps the curve
$C_\alpha$ isomorphically onto the curve
\begin{equation*}
\psi(C_\alpha)=\{\theta_\alpha((1-s^\alpha)x)s\mid x\in
{\bf G}_\alpha\}.
\end{equation*}

Clearly, $\psi(D)=T_0$ and
%%the restriction of
$\psi|_D\colon D\to T_0$
%%to $S$
is the isomorphism.
%% $S\to T_0$.
But ${\rm T}_{e, G}$ is the linear span of ${\rm T}_{e,
T}$ and the tangent spaces of the curves
$\{\theta_\alpha(x)\mid x\in {\bf G}_\alpha\}$
%%and the torus $T$
at $e$.
%%generate
%%${\rm T}_{e, G}$.
%%the tangent space of $G$ at $e$.
Hence ${\rm T}_{s, G}$ is the linear span of ${\rm
T}_{s, T}$ and the tangent spaces  at $s$ of the right
translations of these curves by $s$.
%%and $T$  by $t_0$ at $t_0$ generate ${\rm T}_{t_0, G}$.
%%the tangent space  at $t_0$
%%to $G$.
This implies the claim of the lemma.
%%completes the proof.
\quad $\square$
\renewcommand{\qed}{}\end{proof}
\begin{corollary} \label{psisep}
$\psi$ is separable.
\end{corollary}


\begin{corollary}\label{vartheta}
%%\
%%\begin{enumerate}
%%\item[\rm(i)]
%%$d\psi_x$ is an isomorphism for every point %%$x\in G/T\times T_0$.
%%\item[\rm(ii)]
$(G_0, \psi)$ is the quotient for the action of $\;W$ on
$G/T\times T_0$.
%%\end{enumerate}
\end{corollary}

%%Below is reproduced my proof from letter %%\cite{P}.


\begin{proof}
By \cite[Prop.\,II.6.6]{Bor}, as $G_0$ is normal and
$\psi$ is surjective and separable, it suffices to prove
that the fibers of $\psi$ are $W$-orbits.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%\begin{enumerate}
%%\item[(a)] the fibers of $\psi$ are $W$-orbits;
%%\item[(b)] $\psi$ is separable.
%%\end{enumerate}
%%If this is done, the claim follows from %%\cite[Prop.\,II.6.6]{Bor} as $G_0$
%%%%is nonsingular and hence
%%is normal.
%%(even nonsingular).

%%(a)
Using \eqref{G/T} and \eqref{psss} one immediately
verifies that the fibers of $\psi$ are $W$-stable. On
the other hand, let $\psi(g_1T, t_1)\!=\!\psi(g_2T,
t_2)$. By \eqref{psss} this equality is equivalent to
$(g_1^{-1}g_2)t_2(g_1^{-1}g_2)^{-1} \!\!=t_1$. By
\cite[6.1]{St1} the latter, in turn, implies that there
is an element $w\in W$ such that
\begin{equation*}\label{wg}
\overset{.}{w}t_2\overset{.}{w}^{-1}=
(g_1^{-1}g_2)t_2(g_1^{-1}g_2)^{-1}.
\end{equation*}
Hence $g_1^{-1}g_2=\overset{.}{w}z$ for $z\in G_{t_2}$.
As $t_2\in T$ is strongly regular, this yields that
$z\in T$.
%%Hence
Therefore,
\begin{equation*}
(g_2T, t_2)=(g_1\overset{.}{w}T,
\overset{.}{w}^{-1}t_1\overset{.}{w})= w^{-1}\cdot(g_1T,
t_1).
\end{equation*}
Thus, $(g_1T, t_1)$ and $(g_2T, t_2)$ are in the same
$W$-orbit.  This completes the proof. \quad $\square$
\renewcommand{\qed}{}\end{proof}

Let $\pi^{}_2\colon G/T\times T_0\to T_0$ be the second
projection.
%%As the action of $G$ on $G/T$ is transitive,
Clearly, $(T_0, \pi^{}_2)$ is the %%categorical
geometric quotient for the action of $G$ on $G/T\times
T_0$. As $\psi$ is $G$-equivariant, this implies that
there is a morphism $\phi\colon T_0\to \GG$ such that
the following diagram is commutative:
\begin{equation}\label{cddd}
\begin{matrix}
\xymatrix{G/T\times T_0\ar[d]_{\pi^{}_2}\ar[r]^{\hskip 4mm\psi} & G_0\ar[d]^{\pG}\\
T_0\ar[r]^{\phi}&(\GG)_0 }
\end{matrix}\quad .
\end{equation}
%%(we denote
%%%%the restriction of $\pG$ to $G_0$
%%$\pG|_{G_0}$
%%still by $\pG$).

\begin{lemma}\label{commutdia}
\
\begin{enumerate}
\item[\rm(i)] $\phi=\pG|_{T_{0}}$. \item[\rm(ii)]
%%$\psi$
%%is the isomorphism on the fibers of~$\pi^{}_2$, %%i.e.,
For every point $t\in T_0$, the restriction of $\psi$ to
$\pi_2^{-1}(t)$ is a $G$-equivariant isomorphism
$\pi_2^{-1}(t)\to \pi_G^{-1}(\phi(t))$.
\end{enumerate}
\end{lemma}
\begin{proof} Take a point $t\in T_0$. Commutativity of diagram \eqref{cddd} and formula \eqref{psss} yield that
$\pG(t)=\pG(\psi(eT, t))=\phi(\pi^{}_2(eT, t))=
\phi(t)$. This proves (i).

Commutativity of diagram \eqref{cddd} implies that the
restriction of $\psi$ to $\pi_2^{-1}(t)$  is a
$G$-equivariant morphism $\pi_2^{-1}(t)\to
\pi_G^{-1}(\phi(t))$. As both
%%fibers
$\pi_2^{-1}(t)$ and  $\pi_G^{-1}(\phi(t))$ are the
$G$-orbits
%%with
and the stabilizers of their points are conjugate to
$T$, this morphism is bijective. By Lemma \ref{dpsi} it
is separable. Then, as $\pi_G^{-1}(\phi(t))$ is normal,
it is an isomorphism. This proves (ii). \quad $\square$
\renewcommand{\qed}{}\end{proof}

%%\begin{theorem} \label{eWm}
%%Let $G$ be a connected semisimple algebraic %%group. The following pro\-perties are %%equivalent:
%%\begin{enumerate}
%%\item[\rm(i)] there is a rational section of %%$\;\pG$;
%%\item[\rm(ii)] there is a $W$-equivariant
%%rational map $T\dashrightarrow G/T$.
%%\end{enumerate}
%%\end{theorem}

\begin{proof}[Proof of Theorem {\rm \ref{rsrcs}}]
Assume that (i) holds. Let $\sigma\colon
\GG\dashrightarrow G$ be a rational section of $\pG$,
i.e., a section of $\pG$ over a dense open subset $U$ of
$(\GG)_0$. Let $S$ be the closure of $\sigma(U)$. Put
$\rho:=\pG|_S\colon S\to (\GG)_0$. Since
$\pG\circ\sigma={\rm id}$, shrinking $U$ if necessary,
we may
%%(and shall)
assume that, for every point $x\in U$, the following
properties hold:%%\vskip 2mm
\begin{enumerate}
\item[(a)] $S\cap \pi_G^{-1}(x)$ is a single point $s$;
\item[(b)] $d\rho_s$ is an isomorphism.
%%the restriction of $(d\pG)_s$ to ${\rm T}_{s, %%S}$ is the isomorphism with ${\rm T}_{x, %%(\GG)_0}$.
\end{enumerate}
%%\vskip 1mm

Since $\psi$ is an isomorphism on the fibers of $\pi_2$,
property (a) implies that, for every point $t\in
\phi^{-1}(U)$, the $W$-stable closed set $\psi^{-1}(S)$
intersects $\pi_2^{-1}(t)$ at a single point. From this
we infer that $\psi^{-1}(S)$ has a unique irreducible
component $\widetilde S$ whose image under $\pi_2$ is
dense in $T_0$\,---\,the urgument is the same as that in
the proof of Claim 2(i) in Section~\ref{crssect}. Due to
the uniqueness, this $\widetilde S$ is $W$-stable.

Let $V\subseteq \pi^{}_2(\widetilde S)\cap \phi^{-1}(U)$
be an open subset of $T_0$. Replacing it, if necessary,
by $\bigcap_{w\in W} w(V)$, we may assume that $V$ is
$W$-stable. Let ${\widetilde\rho}\colon
\pi_2^{-1}(V)\cap \widetilde S\to V$ be the restriction
of $\pi_2$ to $\pi_2^{-1}(V)\cap \widetilde S$. Then
$\widetilde\rho$ is a bijective $W$-equivariant
morphism. We claim that it is separable and hence, by
{\sc Zariski}'s Main Theorem, an isomorphism  (as $V$ is
normal). Indeed, take a point $\widetilde s\in
\pi_2^{-1}(V)\cap \widetilde S$ and put
$\pi^{}_2(\widetilde s)=t$. Then property (b), Lemma
\ref{dpsi}, and commutativity of diagram \eqref{cddd}
imply that $d{\widetilde\rho}_s\colon {\rm
T}_{\widetilde s, \widetilde S}\to {\rm T}_{t, V}$ is an
isomorphism; whence the claim by \cite[AG.17.3]{Bor}.

Thus, ${\widetilde\rho}^{-1}\colon
%%T_0\dashrightarrow G/T\times T_0
V\to  \pi_2^{-1}(V)\cap \widetilde S$ is a rational
$W$-equivariant section of $\pi_2$. Its composition with
the first projection $G/T\times T_0\to G/T$ is then a
$W$-equivariant rational map $T\dashrightarrow G/T$.
This proves (i)$\Rightarrow$(ii).

\vskip 1mm

Conversely, assume that (ii) holds. Let $\gamma\colon
T\dashrightarrow G/T$ be a $W$-equivariant rational map.
Then $\varsigma:=(\gamma, {\rm id})\colon
T_0\dashrightarrow G/T\times T_0$ is a $W$-equivariant
rational section of $\pi^{}_2$, i.e., a section of
$\pi^{}_2$ over a dense open subset $V$ of $T_0$. We may
assume that $\varsigma(V)$ and $S:=\psi(\varsigma(V))$
are open in their closures, $\varsigma\colon V\to
\varsigma(V)$ is an isomorphism, and the subsets
$\phi(V)$, $\pG(S)$ of $\GG$ are open and coincide.
%% and equal.
As above, we may also assume that $V$ is $W$-stable.

Taking into account that $\varsigma$ is $W$-equivariant,
$\varsigma(V)\cap \pi_2^{-1}(t)$ is a single point for
every $t\in V$, and $\psi$ is an isomorphism on the
fibers of $\pi^{}_2$, we conclude that
%%$S\cap \pi_G^{-1}(x)$ is a single point
property (a) holds for every $x\in \varsigma(V)$. Thus,
$\rho:=\pG|_S\colon S\to \phi(V)$ is a bijection.

We claim that $\rho$ is separable, hence an isomorphism
as $\phi(V)$ is normal by Lemma \ref{GG0}(i).
%%Indeed, for every $t\in T_0$, commutativity of %%diagram \eqref{cdd} and formula \eqref{psss} %%imply
%%that $\pG(t)=\pG(\psi(eT, t))=\phi(\pi^{}_2(eT, %%t))=
%%\phi(t)$. Hence $\phi=\pG|_{T_{0}}$.
Indeed, $d\phi_t$ is an isomorphism by Lemma
\ref{commutdia}(i) and Lemma \ref{GG0}(ii).  Let
$s=\psi(\varsigma(t))\in S$. %%
%%
%%Therefore, $d\phi_t$
%%is the isomorphism (see the beginning of this %%section). Now take a point %%$s=\psi(\varsigma(t))\in S$ for $t\in V$.
Since the restriction of $(d\pi^{}_2)_{\varsigma(t)}$ to
${\rm T}_{\varsigma(t), \varsigma(V)}$ is an isomorphism
with ${\rm T}_{t, V}$, commutativity of diagram
\eqref{cddd} and
%%Corollary \ref{ssepp}
Lemma \ref{dpsi} imply that
%%$d\rho_s$
%%\colon
%%{\rm T}_{s, S}\to{\rm T}_{\pG(s), \phi(V)}$
%%is the isomorphism
property (b) holds; whence the claim.

Thus, the composition of $\rho^{-1}\colon \phi(V)\to S$
and the identical embedding $S\hookrightarrow G$ is a
rational section of $\pG$. This proves
(ii)$\Rightarrow$(i)
 and completes the proof of the theorem.
%% \ref{eWm}.
\quad $\square$
%%\quad $\square$
\renewcommand{\qed}{}\end{proof}

\vskip -1mm

Recall some definitions from \cite[Sects.\;2.2, 2.3, and
3]{CTKPR}.

%%\vskip 1mm

Let $P$ be a linear algebraic group acting on a variety
$X$ and let $Q$ be its closed subgroup. $X$ is called a
$(P, Q)$-{\it variety} if in $X$ there is a dense open
$P$-stable subset $U$, called {\it a friendly subset},
such that a geometric quotient $\pi^{}_U\colon U\to U/P$
exists and $\pi^{}_U$
%%that
%%is a $(H, D)$-fibration, i.e., $\pi$
becomes the second projection $P/Q\times \widehat{U/P}\to \widehat{U/P}$ %%trivial
after a surjective \'etale base change $\widehat{U/P}\to
U/P$. If there is a rational section of $\pi^{}_U$, one
says that $X$ {\it admits a rational section}. $X$ is
called a {\it versal $(P, Q)$-variety} \;if $U/P$ is
irreducible and, for every its dense open subset
$(U/P)_0$ and $(P, Q)$-variety $Y$, there is a friendly
subset $V$ of $Y$ such that $\pi^{}_V$
%%\colon V\to V/H$
is induced from $\pi^{}_U$
%%\colon
%%U\to U/H$
by a base change $V\to (U/H)_0$.


%%\begin{enumerate}
%%\item[(i)]
%%a variety $X$ endowed with an action of $G$ is %%called $(G, H)$-{\it variety}, where $H$ is a %%closed subgroup of $G$, if there is a dense open %%$G$-stable subset $U$ of $X$ such that
%%there is a geometric quotient $\pi\colon U\to Y$ %%that is a $(G, H)$-fibration, i.e., becomes trivial with fiber $G/H$ after an surjective %%\'etale base change ($\pi\colon U\to Y$;
%%\item[(ii)] a $(G,H)$-variety is called {\it %%versal} if
%%\end{enumerate}

\vskip 1.5mm

Now we shall give the characteristic free proofs of the
following two statements proved  in \cite{CTKPR} for
${\rm char}=0$.

\begin{lemma}\label{fingrou}
Let $X$ be an irreducible  variety endowed with a
faithful action of a finite algebraic group $H$. Then
\begin{enumerate}
\item[\rm(i)] $X$ is an $(H,\{e\})$-variety;
\item[\rm(ii)] $X$ is a versal $(H,\{e\})$-variety in
each of the following cases:
\begin{enumerate}
\item[\rm(a)] $X$ is a free $H$-module; \item[\rm(b)]
$X$ is a linear algebraic torus and $H$ acts by its
automorphisms.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
 (i) Replacing $X$ by its smooth locus, we may assume that $X$ is smooth.

As $H$ is finite, for any nonempty open affine subset
$U$ of $X$, the set   $\bigcap_{h \in H}h(U)$ is
$H$-stable, affine, and open in $X$. So, replacing $X$
by it, we may assume that $X$ is affine. Then, as is
well known, for the action of $H$ on $X$ there is a
geometric quotient $\pi\colon X\to X/H$ (see,
e.g.,\;\cite[Prop.\;6.15]{Bor}).  As $X$ is normal,
$X/H$ is normal as well.



Since $H$ is finite and the action is faithful, the
points with trivial stabilizer form an open subset of
$X$. Replacing $X$ by it, we may also assume that the
action is set-theoretically free, i.e., the
$H$-stabilizer of every point of $X$ is trivial. As $X$
and $X/G$ are normal, by \cite[Exp.\;I, Th\'eor\`eme
9.5(ii)]{G3} and \cite[V.2.3, Cor.\,4]{Bour} this
property implies  that $\pi$ is \'etale and hence $X/H$
is smooth.

For every base change $\beta\colon Y\to X/H$ of $\pi$,
the group $H$ acts on $X\times_{X/H} Y$ via $X$. As the
action of $H$ on $X$ is set-theoretically free, taking
%%For the \'etale base change with
$Y=X$ and $\beta=\pi$, we obtain
\begin{equation*}\label{XXX}
X\times_{X/H}X=\bigsqcup_{h\in H}
h(D)\qquad\mbox{where}\quad D:=\{(x,x)\mid x\in X\}.
\end{equation*}
%%because $H$ acts on $X$ set-theoretically free.
%%This yields the commutative diagram
>From this we deduce that  in the commutative diagram

\begin{equation*}\label{trianggg}
\begin{matrix}
\xymatrix@C=3mm@R=5mm{H\times X\ar[rr]^{\hskip
-3mm\alpha} \ar[dr]
%%_\mu
&&X\times_{X/H}X\ar[dl]\\
&X
%%_\sigma
& }
\end{matrix}\quad,
\end{equation*}
%%
%%
%%
%%\begin{equation*}
%%H\times X\longrightarrow X\times_{X/H}X,\qquad %%(h, x)\mapsto (h(x), x),
%%\end{equation*}
where $\alpha(h,x):=(h(x), x)$
%%is the $H$-equivariant isomorphism
%%$(h, x)\mapsto (h(x), x)$
and two other maps are the second projections, $\alpha$
is an $H$-equivariant isomorphism. This proves (i).

The proofs of (ii)(a) and (ii)(b) are the same as that
of (b) and (d) in \cite[Lemma 3.3]{CTKPR} if one
replaces in them the references to \cite[Theorem
2.12]{CTKPR} (whose proof is based on the assumption
${\rm char}\,k=0$) by the references to statement (i) of
the present lemma. \quad $\square$
\renewcommand{\qed}{}\end{proof}
\begin{remark} {\rm The proof of (i) shows that, for finite group actions, set-theoretical freeness coincides with that in the sense of GIT,
%%algebraic transformation groups theory
\cite[Def.\;0.8]{MF}.}
\end{remark}


\begin{lemma}\label{G/Tversal}
%%\
%% \begin{enumerate}
%%\item[\rm(i)]
$G$ is a versal $(G,T)$-variety.
%%\item[\rm(ii)] $T$ is a versal %%$(W,\{e\})$-variety.
%%\end{enumerate}
\end{lemma}

\begin{proof} First we shall give
%%have
%%to give the
a characteristic free proof of the fact that $G$ is a
$(G, T)$-variety
%%and $T$ is a $(W, \{e\})$-variety
(the proof given in \cite{CTKPR} is based on the
assumption ${\rm char}\,k=0$).
 %%
%%For ${\rm char}\,=0$,
%%the properties that
%%$G$ is a $(G, T)$-variety and $T$ is a $(W, %%\{e\})$-variety
%%are deduced in \cite{CTKPR}
%%from Theorem 2.14 of this paper
%%\cite[Theorem 2.14]{CTKPR}
%% whose proof is based on the assumption that %%${\rm char}\,k=0$.
%%As we do not make this assumption,
%%Hence we have to give a characteristic free %%proof of
%%them.
%%these properties.
%%valid for any ${\rm char}\,k$.
%%If this is done,
%%then the characteristic free arguments in %%\cite[proofs of Prop.\;4.3 and Lemma 3.2]{CTKPR} %%(that are characteristic free)
%%valid for any ${\rm char}\,k$,
%%show that (i)$\Leftrightarrow$(ii).
%%
%%\vskip 1.5mm
%%
%%
%%2. First we shall prove that $G$ is a %%$(G,T)$-variety.
By Lemma \ref{GG0}(iii) this is equivalent to proving
the existence of a dense open subset $U$ of $(\GG)_0$
such that after a surjective \'etale base change $U'\to
U$
%%transforms
morphism \eqref{pG0000}
%%into
becomes the second projection $G/T\times U'\to U'$.

Consider the base change of $\pG$ in \eqref{cddd} by
means of $\phi$. Lemma \ref{commutdia}(i) implies that
%%regarding the notation)
\begin{equation}
\label{FP} F:=G_0\times_{(\GG)_0} T_0=\{(g, t)\in
G_0\times T_0 \mid G(g)=G(t)\}
\end{equation}
(see \eqref{cc}).
%%The set $F$ is $G$-stable with respect to the %%action of $G$ on $G_0\times T_0$ via the first %%factor.
We have the canonical map corresponding to commutative
diagram \eqref{cddd}:
\begin{equation}\label{cmap}
\gamma:=\psi\times {\rm id}\colon G/T\times
T_0\longrightarrow F,\qquad (gT, t)\mapsto (gtg^{-1},
t).
\end{equation}
It follows from \eqref{FP} that $\gamma$ is surjective;
whence $F$ is irreducible. But if for $t\in T_0$ and
$g_1, g_2\in G$ we have $g_1tg_1^{-1}=g_2tg_2^{-1}$,
then $g_1T=g_2T$ since $G_t=T$. Therefore, $\gamma$ is
bijective. Lemma \ref{dpsi} and \eqref{cmap} show that
$d\gamma_x$ is injective for every $x\in G/T\times T_0$.
Hence if $\gamma(x)$ lies
%%lays
in  the smooth locus $F_{\rm sm}$ of $F$, then
$d\gamma_x$ is the isomorphism. This implies that
$\gamma$ is separable and then, by {\sc Zariski}'a Main
Theorem, that the restriction of $\gamma$
%%|_{
to $\gamma^{-1}(F_{\rm sm})$
%%\colon
is an isomorphism $\gamma^{-1}(F_{\rm sm})\to F_{\rm
sm}$.
%% is an %%isomorphism.

As $F_{\rm sm}$ is $G$-stable and $\gamma$ is
$G$-equivariant,  $\gamma^{-1}(F_{\rm sm})$ is a
$G$-stable open subset of $G/T\times T_0$. Hence
%%$\gamma^{-1}(F_{\rm sm})=
it is of the form $G/T\times U'$ for
%%the
an open subset $U'$
%%=\pi_2(\gamma^{-1}(F_{\rm sm}))$
of $T_0$. But Lemmas \ref{G0T0}(ii) and
\ref{commutdia}(i) imply that $U:=\phi(U')$ is open in
$(\GG)_0$ and $\phi|_{U'}\colon U'\to U$ is \'etale.
This proves that after the \'etale base change
$\phi|_{U'}\colon U'\to U$ morphism \eqref{pG0000}
becomes the second projection $G/T\times U'\to U'$.
Hence $G$ is a $(G, T)$-variety.

By Lemma \ref{fingrou}(b), $T$ is a versal
$(W,\{e\})$-variety. The characteristic free arguments
from \cite[proof of Prop.\;4.3(c)]{CTKPR} then show that
this fact implies versality of the $(G, T)$-variety $G$.
This completes the proof of the lemma. \quad $\square$
\renewcommand{\qed}{}\end{proof}

%%\noindent {\it
\begin{proof}[Proof of
Theorem {\rm \ref{anychar}}] Let us first show how to
deduce (i) from (ii). Consider commutative diagram
\eqref{triang}. As $\mu$ is separable, it is central.
Therefore, if (ii) holds, there is a rational section of
$\pi^{}_{\widetilde G}$; whence there is a rational
cross-section in $\widetilde G$. Then (i) follows from
Corollary \ref{insep}.


 Now we shall prove (ii). Assume that $\tau$ is central. Then the natural morphism
$\tG/\tT\to G/T$ is an isomorphism by
\cite[Props.\;6.13, 22.4]{Bor}.

Using $\tau$, every action of $G$ naturally lifts to an
action of $\tG$ on the same variety. In particular, $G$
is endowed with
%%the
an action of $\tG$.  But $G$ is a $(G, T)$-variety by
Lemma \ref{G/Tversal}(i).
%%Therefore,
As $\tG/\tT$ and $G/T$ are isomorphic, this means that
$G$ is a $(\tG,\tT)$-variety. But $\tG$ is a versal
$(\tG,\tT)$-variety (by Lemma \ref{G/Tversal}(i)) that
admits a rational section (by Lemma \ref{GG0}(iii) and
\cite{St1}). Hence by \cite[Theorem 3.6(a)]{CTKPR} (the
proof of this result is characteristic free) every
$(\tG, \tT)$-variety admits a rational section. In
particular, this is so for $G$.  This proves (ii) and
completes the proof of the theorem. \quad $\square$
\renewcommand{\qed}{}\end{proof}



  \section{Remarks} \label{re}
\subsection*{1.\;Cross-sections versus sections}
%%1.
If there is a section $\sigma\colon \GG\to G$ of $\pG$,
then $\sigma(\GG)$ is a cross-section in $G$. Indeed, as
${\rm id}_{k[\GG]}$ is the composition of the
homomorphisms
%%\ \vskip -4mm
$$ k[\GG]\xrightarrow{\pi_G^*} k[G]\xrightarrow{\sigma^*} k[\GG],
$$
$\pi_G^*$ is surjective; by \cite[Cor.\;4.2.3]{G2} this
means that $\sigma$ is
%%the
a closed embedding.
%%(see
%%, e.g.,  \cite[1.5]{St3}).
%%let $\overline{\sigma(\GG)}$ be the closure of %%$\sigma(\GG)$ in $G$. Then
%%we have the sequence of embeddings
%%
%%Since $\pG\circ\sigma=\{\rm id^{}_{\GG}}$ and %%$\sigma(\GG)$ is dense in %%$\overline{\sigma(\GG)}$,
%%we have the sequence of inclusions whose c

The cross-section $\sigma(\GG)$ has the property that
the restriction of $\pG$
%%|_{
to $\sigma(\GG)$ is an isomorphism
%%\colon
$\sigma(\GG)\to \GG$.
%%  is
%%to it is an
%%the isomorphism.
%% with $\GG$.
Conversely, let $S$ be a cross-section in $G$. If
%%the restriction $\rho$ of
$\pG|_S\colon S\to \GG$
%%to $S$
is separable, then, since
%%$\rho$
$\pG|_S$ is bijective and $\GG$ is normal, {\sc
Zariski}'s Main Theorem implies that $\pG|_S$ is an
%%the
isomorphism
%%with
%%$\GG$
(cf.\;\cite[AG\,18.2]{Bor}). So in this case the
composition of $(\pG|_S)^{-1}$ with the identity
embedding $S\hookrightarrow G$ is a section of $\pG$
whose image is $S$. In particular, if ${\rm char}\,k=0$,
then every cross-section in $G$ is the image of a
section of $\pG$. If ${\rm char}\,k>0$,
then in the general case this %%this %%property does not hold
is not true.
%%so.
%%then,  in general, it is not so.



\begin{example}\label{exm}  Let $G={\bf SL}_3$ and ${\rm char}\,k=p>0$. Then for every integer $d>0$,
%%$S:=\{s(a_1, a_2)\mid a_1, a_2\in k\}$, %%where
%%\
%%
%%\vskip -5.2mm
%%
\begin{equation*}
S:=\{s(a_1, a_2)\mid a_1, a_2\in k\}, \qquad
\mbox{where}\quad
%%S
s(a_1, a_2):=
%%\Biggl\{
\begin{pmatrix}a_1 & a_2 & 1\\
1& a^{p^d}_1-a_1 &0\\
0& 1 & 0
\end{pmatrix},
%%\;\biggl\vert\;a_1, a_2\in k\Biggr\}
\end{equation*}
%%\vskip -1.6mm
\noindent is a cross-section in $G$ such that $\rho$ is
not separable. Indeed, as $\chi_{\varpi_i}(g)$ is the
sum of principal $i$-minors of $g\in G$, we have (see
Lemma \ref{lin}(ii))
%%\
%%\vskip -3mm
$$(\lambda\circ\rho)\big(s(a_1, a_2)\big)=\big(a^{p^d}_1, a_1\big(a_1^{p^d}-a_1\big)-a_2\big).\qquad \square$$
\end{example}


Similarly, if $\sigma\colon \GG\dashrightarrow G$ is a
rational section of $\pG$ and $S$  the closure $S$ of
its image, then $S$ is a rational cross-section in $G$
such that the restriction of $\pG$ to it is a birational
isomorphism with $\GG$.

%%\vskip 1mm

\subsection*{2.\;Group action on the set of cross-sections}  Let ${\rm Mor}(\GG, G)$ be the group of
morphisms $\GG\to G$. If $S$ is a cross-section in $G$
and $\gamma\in {\rm Mor}(\GG, G)$, then
%%\ \vskip
%%-4mm
$$
\gamma(S):=\{\gamma(s)s\gamma(s)^{-1}\mid s\in S\}
$$
\noindent is a cross-section in $G$. This defines an
action of ${\rm Mor}(\GG, G)$ on the set of
cross-sections in $G$. %%
%%This group acts
%%The conjugating action of $G$ on itself %%induces the action of $G$
%%on the set of cross-sections in $G$ by the %%rule
%%$$
%%
%%$$
If ${\rm char}\,k=0$, then by
%%Corollary \ref{cor} and
\cite{FM} this action is transitive. If ${\rm
char}\,k>0$, then in the general case
%%it
this %%this %%property does not hold
is not
%%so
true: in Example \ref{exm}, {\sc Steinberg}'s section
and $S$ are not in the same ${\rm Mor}(\GG, G)$-orbit
since, for the former,
%%$\rho$
the restriction of $\pG$ is separable \cite[Theorem
1.5]{St1}, but, for the latter, it is not.


%%\vskip 1mm

\subsection*{3.\;Lifting \boldmath$T$-action} By Theorem \ref{toric} there is an
action of $T$ on $T/W$ determining a structure of a
toric variety. This action cannot be lifted to $T$
making $\pi^{}_T\colon T\to T/W$ equivariant. This
follows from the fact that the automorphism group of the
underlying variety of $\,T$ is ${\bf GL}_r({\bf
Z})\ltimes T$.



\subsection*{4.\;Questions} Given Theorem \ref{rsrcs} and Corollary
\ref{ratcs}, it would be interesting to construct
explicitly an example of a $W$-equivariant rational map
$T\dashrightarrow G/T$ for central~$\tau $.

Is there such a map defined on $T_0$?

Is there a rational section of $\pG$ defined on
$(\GG)_0$?

%%\begin{questions}\
%%
%%\begin{enumerate}
%%\item[(i)] Is there a $W$-equivariant rational %%map $T\dashrightarrow G/T$ whose domain of %%definition
%%contains $T_0$? If yes, does this property hold
%%for every such rational map?

%%it true that the domain of definition of every %%$W$-equivariant rational map $T\dashrightarrow %%G/T$ contains $T_0$?
%%\item[(ii)] Is
%%it true that the domain of definition of every %%rational section of $\pG$ contains $(\GG)_0$?
%%\end{enumerate}
%%\end{questions}



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\end{thebibliography}





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