# Torus actions on quotients of affine spaces

Ana-Maria Brecan<EMAIL_ADDRESS>University of Bayreuth,
Faculty of Mathematics, Physics, and Computer Science, Universitätsstraße 30,
95447 Bayreuth, Germany  and  Hans Franzen<EMAIL_ADDRESS>Paderborn
University, Institute of Mathematics, Warburger Straße 100, 33098 Paderborn,
Germany

  *     * scAbstract. We study the locus of fixed points of a torus action on a GIT quotient of a complex vector space by a reductive complex algebraic group which acts linearly. We show that, under the assumption that $G$ acts freely on the stable locus, the components of the fixed point locus are again GIT quotients of linear subspaces by Levi subgroups.

scKeywords. Geometric quotients, torus actions, torus fixed points,
destabilizing one-parameter subgroups

sc2020 Mathematics Subject Classification. 14L30

  *
cMarch 6, 2023Received by the Editors on September 22, 2022.
Accepted on June 6, 2023.

sce-mail<EMAIL_ADDRESS>
University of Bayreuth, Faculty of Mathematics, Physics, and Computer Science,
Universitätsstrascße 30, 95447 Bayreuth, Germany

sce-mail<EMAIL_ADDRESS>
Paderborn University, Institute of Mathematics, Warburger Strascße 100, 33098
Paderborn, Germany

While this research was conducted, H.F. was supported by the DFG SFB TR 191
“Symplektische Strukturen in Geometrie, Algebra und Dynamik”.

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction
  2. 2 Setup
  3. 3 Optimal destabilizing one-parameter subgroups
  4. 4 Lifts of fixed points
  5. 5 An embedding of quotients
  6. 6 Analysis of the fixed point locus
  7. 7 A necessary condition for a non-empty fixed point component
  8. 8 Application: Quiver moduli
  9. 9 Application: Torus quotients

## 1\. Introduction

In this paper, we are concerned with actions of tori on geometric quotients of
linear actions of reductive complex algebraic groups on vector spaces. Such
geometric quotients appear very often. One prominent class of examples are
moduli spaces of quivers without relations; see e.g. [Rei08] for an overview
of this theory. Geometric quotients of vector spaces arise in other contexts
as well. For example, every toric variety whose fan satisfies some mild
hypotheses can be realized as such a quotient; see [Cox97, Theorem 2.1]. Other
instances where GIT quotients of vector spaces are investigated are [ES89,
Hal10, Hos14].

We study actions of a torus $\mathscr{T}$ on a GIT quotient of the form
$V^{\operatorname{st}}(G,\theta)/G$, where $G$ is a reductive complex
algebraic group, $V$ is a finite-dimensional representation of $G$, and
$\theta$ is a character of $G$. We assume that this torus action comes from a
linear $\mathscr{T}$-action on $V$ that commutes with the action of $G$. Our
main objective in this paper is to determine the locus of fixed points of this
torus action.

In the case of quiver moduli, this question has been studied by Weist in
[Wei13]. More precisely, he considers torus actions on moduli spaces of stable
quiver representations which are given by scaling of the maps along the arrows
of the quiver. He shows that the fixed point locus of such a torus action
decomposes into irreducible components, each of which is isomorphic to a
moduli space of stable representations of a covering quiver.

We generalize this result to GIT quotients of the form
$V^{\operatorname{st}}(G,\theta)/G$. We need to make the assumption that the
action of $G$ on the stable locus is free. This is automatically satisfied in
the context of quiver moduli spaces.

Our main result, Theorem 6.3, asserts that, under the assumption from the
previous paragraph, the fixed point locus of the action of a torus on
$V^{\operatorname{st}}(G,\theta)/G$ decomposes into irreducible components,
each of which is again a stable GIT quotient of a linear subspace of $V$ by a
Levi subgroup of $G$. More precisely, we fix a maximal torus $T$ in $G$ and
let $W$ be its Weyl group. For every morphism $\rho\colon\mathscr{T}\to T$ of
algebraic groups, we set $V_{\rho}=\\{v\in V\mid t.v=\rho(t)v\text{ (all
$t\in\mathscr{T}$)}\\}$ and define $G_{\rho}$ to be the centralizer in $G$ of
$\operatorname{im}\rho$.

######  Theorem (Theorem 6.3).

The $\mathscr{T}\\!$-fixed point locus
$(V^{\operatorname{st}}(G,\theta)/G)^{\mathscr{T}}$ decomposes into connected
components $\smash{\bigsqcupop\displaylimits_{\rho}}\smash{F_{\rho}}$ indexed
by a full set of representatives of morphisms of tori
$\rho\colon\mathscr{T}\to T$ up to conjugation with $W$. The component
$F_{\rho}$ equals the image of $V_{\rho}\cap V^{\operatorname{st}}(G,\theta)$
in $V^{\operatorname{st}}(G,\theta)/G$, it is irreducible, and it is
isomorphic to

$V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho}.$

The index set of this disjoint union is infinite, but there are only finitely
many $\rho$ for which $F_{\rho}$ is non-empty. This is discussed in Section 7.

There are two main ingredients of the proof of the above main result. The
first is concerned with the description of the intersection of the
(semi-)stable locus with $V_{\rho}$.

######  Theorem (Theorem 4.6).

Let $\rho\colon\mathscr{T}\to G$ be a morphism of algebraic groups. Then

  1. (1)

$V_{\rho}\cap
V^{\operatorname{sst}}(G,\theta)=V_{\rho}^{\operatorname{sst}}(G_{\rho},\theta)$;

  2. (2)

$V_{\rho}\cap
V^{\operatorname{st}}(G,\theta)=V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)$.

The proof of the first statement uses Kempf’s theory of optimal destabilizing
one-parameter subgroups [Kem78]. For the second statement, we need to define,
associated with a one-parameter subgroup $\lambda$ of $G$ and a vector $v\in
V$, a Zariski-closed subset $Y_{[\lambda]}(v)$ of $G/P_{\lambda}$ and find a
$\mathscr{T}\\!$-fixed point of it. The definition of $Y_{[\lambda]}(v)$ can
be found in Section 3.

The second major auxiliary result is Theorem 5.1.

######  Theorem (Theorem 5.1).

The induced morphism $i_{\rho}\colon
V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho}\to
V^{\operatorname{st}}(G,\theta)/G$ is a closed immersion.

To show this, it is enough to prove that $i_{\rho}$ is proper, injective (on
$\mathbb{C}$-valued points) and that it induces an injective map on tangent
spaces. We show all three assertions in Section 5. Properness is shown using a
result of Luna.

Throughout the paper, we work over the field of complex numbers. All results
hold over an algebraically closed field of characteristic zero, though. In
positive characteristic, Lemma 4.1, which is used to assign to every component
of the fixed point locus a morphism $\rho\colon\mathscr{T}\to G$ of algebraic
groups, fails; see Example 4.2. Therefore, we do not know how to describe the
fixed point locus over a field of positive characteristic.

The paper is organized as follows. In Section 2, we fix the setup. Section 3
recalls Kempf’s theory of optimal destabilizing one-parameter subgroups; we
define the sets $Y_{[\lambda]}(v)$. In Section 4, we show that lifts of fixed
points induce a morphism $\rho\colon\mathscr{T}\to G$, and we prove Theorem
4.6. Section 5 is devoted to the proof of Theorem 5.1. In Section 6, the main
result is stated and proved. Section 7 is concerned with finiteness conditions
on the index set of the irreducible components of the fixed point locus. In
Sections 8 and 9, we apply our theory to some classes of quotients. We discuss
quiver moduli and show how to derive Weist’s result [Wei13, Theorem 3.8] from
ours in Section 8. In Section 9, we discuss the case where $G$ is a torus and
compare our description of the fixed point locus with the characterization in
terms of the toric fan.

### Acknowledgments

We are grateful to Michel Brion, Alan Huckleberry, and Markus Reineke for
interesting conversations and insightful remarks on this subject. We would
like to thank Sergey Mozgovoy, who suggested an argument that made it possible
to get rid of an extra assumption. We thank the referee for various helpful
comments that improved the exposition and for interesting questions.

## 2\. Setup

Let us introduce the setup with which we are going to work throughout the
article. We are interested in GIT quotients of complex vector spaces by
actions of reductive complex algebraic groups. As references for actions of
algebraic groups and (geometric) invariant theory, the reader may consult, for
instance, [Bri10, Dol03, MFK94].

We work over the complex numbers. Let $G$ be a connected reductive algebraic
group. Let $V$ be a finite-dimensional representation of $G$. For the action
of an element $g\in G$ on a vector $v\in V$, we write $gv$. We consider $V$ as
an affine space, and we are interested in GIT quotients for this action. Let
$X^{*}(G)$ be the group of characters of $G$. For $\chi\in X^{*}(G)$, we
define

$\mathbb{C}[V]^{G,\chi}=\\{f\in\mathbb{C}[V]\mid f(gv)=\chi(g)f(v)\text{ all
$g\in G$, $v\in V$}\\}$

and call $f\in\mathbb{C}[V]^{G,\chi}$ a $\chi$-semi-invariant function. Fix
$\theta\in X^{*}(G)$.

######  Definition 2.1.

Let $v\in V$.

  1. (1)

We call $v$ a $\theta$-_semi-stable_ point if there exist $n>0$ and
$f\in\mathbb{C}[V]^{G,n\theta}$ such that $f(v)\neq 0$. Let
$V^{\operatorname{sst}}(G,\theta)\subseteq V$ be the subset of $\theta$-semi-
stable points.

  2. (2)

The point $v$ is called $\theta$-_stable_ if $v$ is $\theta$-semi-stable and
has finite stabilizer and $G\cdot v$ is closed in
$V^{\operatorname{sst}}(G,\theta)$. Let $V^{\operatorname{st}}(G,\theta)$ be
the set of $\theta$-stable points.

  3. (3)

We call $v$ a $\theta$-_unstable_ point if $v$ is not $\theta$-semi-stable.
Let $V^{\operatorname{unst}}(G,\theta)$ be the set of $\theta$-unstable
points.

######  Remark 2.2.

For a character $\chi\in X^{*}(G)$, we define the $G$-linearized line bundle
$L(\chi)$ as the trivial line bundle together with the $G$-action on the total
space $G\times V\times\mathbb{C}\to V\times\mathbb{C}$ given by
$g\cdot(v,z)=(gv,\chi(g)z)$. Then

$H^{0}(V,L(\chi))^{G}=\mathbb{C}[V]^{G,\chi};$

i.e. $G$-invariant sections of $L(\chi)$ are the same as $\chi$-semi-invariant
functions. This shows that a point $v\in V$ is $\theta$-semi-stable if and
only if it is semi-stable with respect to $L(\theta)$ in the sense of Mumford
(see [MFK94, Definition 1.7]). Our notion of $\theta$-stability agrees with
proper stability with respect to $L(\theta)$, as defined in [MFK94, Definition
1.8].

We define the graded ring
$\mathbb{C}[V]_{\theta}^{G}:=\bigoplusop\displaylimits_{n\geq
0}\mathbb{C}[V]^{G,n\theta}$. We obtain a good categorical quotient

$\pi\colon V^{\operatorname{sst}}(G,\theta)\longrightarrow
V^{\operatorname{sst}}(G,\theta)/\\!\\!/G:=V/\\!\\!/_{L(\theta)}G=\operatorname{Proj}\left(\mathbb{C}[V]_{\theta}^{G}\right).$

We also define
$V^{\operatorname{st}}(G,\theta)/G:=\pi(V^{\operatorname{st}}(G,\theta))$. The
restriction of $\pi$ gives a geometric quotient.

Let us recall the Hilbert–Mumford criterion for (semi-)stability from [MFK94].
We are using the form stated in King’s paper [Kin94, Proposition 2.5].

######  Theorem 2.3 (Hilbert–Mumford criterion).

Let $v\in V$.

  1. (1)

The point $v$ is $\theta$-semi-stable if and only if
$\langle\theta,\eta\rangle\geq 0$ for every one-parameter subgroup
$\eta\colon\mathbb{C}^{\times}\to G$ for which $\lim_{z\to 0}\eta(z)v$ exists.

  2. (2)

The point $v$ is $\theta$-stable if and only if $\langle\theta,\eta\rangle>0$
for every non-trivial one-parameter subgroup $\eta\colon\mathbb{C}^{\times}\to
G$ for which $\lim_{z\to 0}\eta(z)v$ exists.

We will assume the following to hold throughout the text.

######  Assumption 2.4.

We suppose that $G$ acts freely on $V^{\operatorname{st}}(G,\theta)$.

Assumption 2.4 guarantees that $\pi\colon V^{\operatorname{st}}(G,\theta)\to
V^{\operatorname{st}}(G,\theta)/G$ is a principal $G$-bundle in the étale
topology. This follows from Luna’s slice theorem [Lun73, Section III.1]. Étale
local triviality implies smoothness of $V^{\operatorname{st}}(G,\theta)/G$.

Consider a torus $\mathscr{T}:=(\mathbb{C}^{\times})^{r}$. Suppose that
$\mathscr{T}$ acts linearly on $V$ in such a way that the actions of $G$ and
$\mathscr{T}$ commute.

######  Lemma 2.5.

The subsets $V^{\operatorname{st}}(G,\theta)$ and
$V^{\operatorname{sst}}(G,\theta)$ are $\mathscr{T}\\!$-invariant.

###### Proof.

Both assertions are easily proved using the Hilbert–Mumford criterion. We will
only show the $\mathscr{T}\\!$-invariance of the stable locus. The invariance
of the semi-stable locus is entirely analogous.

Let $v\in V^{\operatorname{st}}(G,\theta)$ and $t\in\mathscr{T}$. Suppose that
$\eta\colon\mathbb{C}^{\times}\to G$ is a one-parameter subgroup for which
$\lim_{z\to 0}\eta(z)t.v$ exists. As the actions of $\mathscr{T}$ and $G$
commute, we get $\eta(z)t.v=t.\eta(z)v$, and hence

$\lim_{z\to 0}\eta(z)v=t^{-1}.\lim_{z\to 0}\eta(z)t.v$

exists. The stability of $v$ implies $\langle\theta,\eta\rangle>0$. ∎

This implies that the $\mathscr{T}\\!$-action descends to an action on the
quotient $V^{\operatorname{sst}}(G,\theta)/\\!\\!/G$ and to
$V^{\operatorname{st}}(G,\theta)/G$. We are going to describe the fixed point
locus of the latter action.

######  Remark 2.6.

As we work in characteristic zero, the group $G$ is linearly reductive, so the
finite-dimensional representation $V$ decomposes into irreducible
representations, say $V\cong V_{1}^{n_{1}}\oplus\ldots\oplus V_{l}^{n_{l}}$,
where $V_{1},\ldots,V_{l}$ are pairwise non-isomorphic irreducible
representations of $G$. An application of Schur’s lemma shows that
$t\in\mathscr{T}$ leaves the summand $V_{i}^{n_{i}}$ invariant. Moreover, as
$\mathbb{C}$ is algebraically closed, Schur’s lemma also tells us that $t$
acts on $V_{i}^{n_{i}}\cong V_{i}\otimes\mathbb{C}^{n_{i}}$ as an element of
$\operatorname{GL}_{n_{i}}(\mathbb{C})$. The thus obtained
$\mathscr{T}$-action on $\mathbb{C}^{n_{i}}$ can be diagonalized: we find a
basis $e_{i,1},\ldots,e_{i,n_{i}}$ of $\mathbb{C}^{n_{i}}$ such that
$\mathscr{T}$ operates on $e_{i,a}$ with weight $w_{i,a}\in
X^{*}(\mathscr{T})\cong\mathbb{Z}^{r}$. This means that every vector $v\in V$
can be decomposed in a unique way as
$v=\smash{\sumop\displaylimits_{i=1}^{l}}\sumop\displaylimits_{a=1}^{n_{i}}v_{i,a}\otimes
e_{i,a}$ with $v_{i,a}\in V_{i}$ and a torus element $t\in\mathscr{T}$ acts on
$v$ by

$t.v=\sumop\displaylimits_{i=1}^{l}\sumop\displaylimits_{a=1}^{n_{i}}w_{i,a}(t)v_{i,a}\otimes
e_{i,a}.$

The irreducible representations of $G$ are completely classified. Choose a
maximal torus $T$ of $G$. Let $D\subseteq X^{*}(T)$ be the set of dominant
weights. For every $\lambda\in D$, there exists a unique representation
$M_{G}(\lambda)$ of $G$ whose highest weight is $\lambda$. For a weight
$\mu\in X^{*}(T)$, the dimension $d_{\lambda,\mu}$ of the weight space
$M_{G}(\lambda)_{\mu}$ is combinatorially determined as the number of ways to
write $\lambda-\mu$ as a sum of positive roots.

Let $T$ be a maximal torus of $G$. Let $\mu\in X^{*}(T)$. By $V_{\mu}(T)$ we
denote the weight space of weight $\mu$. In the same vein, for $w\in
X^{*}(\mathscr{T})$, we let $V_{w}(\mathscr{T})$ be the weight space of weight
$w$ with respect to the $\mathscr{T}\\!$-action. As the actions commute, we
obtain an action of $T\times\mathscr{T}$. The weight space
$V_{(\mu,w)}(T\times\mathscr{T})$ agrees with $V_{\mu}(T)\cap
V_{w}(\mathscr{T})$. Let $N_{T}(V)=\\{\mu\in X^{*}(T)\mid V_{\mu}(T)\neq 0\\}$
be the set of weights of the $T$-action on $V$. Define $N_{\mathscr{T}}(V)$
and $N_{T\times\mathscr{T}}(V)$ in just the same way.

## 3\. Optimal destabilizing one-parameter subgroups

The theory of optimal destabilizing one-parameter subgroups is developed by
Kempf in [Kem78]; see also [Kem18] for a transcript of a preliminary version
of the article which contains simpler, more accessible versions of the main
results. Kempf’s theory is used by Hesselink [Hes78], Kirwan [Kir84], and Ness
[Nes84] to give a stratification of the unstable locus. A treatment of Kempf’s
theory for instability with respect to a character can be found in Hoskin’s
paper [Hos14]. We are going to recall the basics of this theory for the
convenience of the reader.

Assume that $v$ is unstable with respect to the $G$-action and the stability
condition $\theta$. Then, by the Hilbert–Mumford criterion, there exists a
one-parameter subgroup $\eta$ of $G$ such that $\lim_{z\to 0}\eta(z)v$ exists
and $\langle\theta,\eta\rangle<0$.

Kempf shows in [Kem78] how to find a one-parameter subgroup which is “most
responsible” for the instability of $v$. To this end, we again fix a maximal
torus $T\subseteq G$ and choose an inner product $(\underline{\ \
},\underline{\ \ })$ on the vector space $X_{*}(T){R}$ which is $W$-invariant
(i.e. $(w\lambda w^{-1},w\mu w^{-1})=(\lambda,\mu)$ for all $\lambda,\mu\in
X_{*}(T){R}$) and for which $(\lambda,\lambda)\in\mathbb{Z}$ provided that
$\lambda\in X_{*}(T)$. We thus obtain a $W$-invariant norm $\|\underline{\ \
}\|$ on $X_{*}(T){R}$ whose square takes integral values on $X_{*}(T)$. For
any $\eta\in X_{*}(G)$, there exists a $g\in G$ such that $g\eta g^{-1}\in
X_{*}(T)$. By the Weyl group invariance of the norm, setting

$\|\eta\|:=\|g\eta g^{-1}\|$

is well defined. By definition, the value of $\|\eta\|$ is invariant under
conjugation by $G$.

Now for any $v\in V$, we set

$m_{G}^{\theta}(v):=\inf\left\\{\frac{\langle\theta,\eta\rangle}{\|\eta\|}\
\middle|\ 1\neq\eta\in X_{*}(G)\text{ such that }\lim_{z\to 0}\eta(z)v\text{
exists}\right\\}.$

Obviously, $v\in V^{\operatorname{unst}}(G,\theta)$ if and only if
$m_{G}^{\theta}(v)<0$.

######  Definition 3.1.

Let $v\in V^{\operatorname{unst}}(G,\theta)$. A one-parameter subgroup
$\lambda\in X_{*}(G)$ is called _adapted_ to $v$ if

  * •

$\lim_{z\to 0}\lambda(z)v$ exists, and

  * •

$\frac{\langle\theta,\lambda\rangle}{\|\lambda\|}=m_{G}^{\theta}(v)$.

We call a one-parameter subgroup $\lambda$ _primitive_ if it is not divisible
by any integer $d\geq 2$. Let

${}_{G}^{\theta}(v):=\\{\lambda\in X_{*}(G)\mid\lambda\text{ is primitive and
adapted to $v$}\\}.$

Let $G_{\lambda}$ be the centralizer of the image of $\lambda$ inside $G$, and
let $P_{\lambda}$ be the closed subgroup

$P_{\lambda}=\left\\{g\in G\ \middle|\ \lim_{z\to
0}\lambda(z)g\lambda(z)^{-1}\text{ exists}\right\\}.$

Then $P_{\lambda}$ is a parabolic subgroup of $G$ with Levi factor
$G_{\lambda}$. It can also be described as the closed subgroup whose Lie
algebra $\mathfrak{p}_{\lambda}$ is the sum of the non-negative weight spaces
$\bigoplusop\displaylimits_{n\geq 0}\mathfrak{g}^{\lambda}_{n}$ with respect
to the adjoint action of $\lambda$ on the Lie algebra $\mathfrak{g}$. The
following result is [Kem78, Theorem 2.2]; we use the form stated in [Hos14,
Theorem 2.15].

######  Theorem 3.2 (Kempf).

Let $v\in V^{\operatorname{unst}}(G,\theta)$. Then:

  1. (1)

We have ${}_{G}^{\theta}(v)\neq\emptyset$.

  2. (2)

We have ${}_{G}^{\theta}(gv)=g{}_{G}^{\theta}(v)g^{-1}$.

  3. (3)

For any two $\lambda,\lambda^{\prime}\in{}_{G}^{\theta}(v)$, the parabolic
subgroups $P_{\lambda}$ and $P_{\lambda^{\prime}}$ agree. We denote this
subgroup by $P_{v}$.

  4. (4)

The subset ${}_{G}^{\theta}(v)\subseteq X_{*}(G)$ is a $P_{v}$-conjugacy
class.

  5. (5)

For every maximal torus $H\subseteq P_{v}$, there exists a unique
$\lambda\in{}_{G}^{\theta}(v)\cap X_{*}(H)$.

######  Remark 3.3.

If the vector $v$ is $\theta$-semi-stable, then we have
${}_{G}^{\theta}(v)\neq\emptyset$ as well, and for a given torus $H$, there is
still a primitive one-parameter subgroup $\lambda$ of $H$ which is adapted to
$v$, but $\lambda$ need not be unique. This is because [Kem18, Lemma 1]
requires a certain function to attain a positive value somewhere, which is not
necessarily true if $v$ is semi-stable. Hence, we cannot conclude that
$P_{\lambda}=P_{\lambda^{\prime}}$, nor will ${}_{G}^{\theta}(v)$ be a
conjugacy class.

For our purposes, we need some further results of this kind. Fix a one-
parameter subgroup $\lambda_{0}\in X_{*}(G)$. Denote by
$P_{0}:=P_{\lambda_{0}}$ its associated parabolic subgroup of $G$. Let
$[\lambda_{0}]:=[\lambda_{0}]_{G}$ be the $G$-conjugacy class inside
$X_{*}(G)$. Define

$f\colon[\lambda_{0}]\longrightarrow G/P_{0}$

by $f(\lambda)=P_{\lambda}$. Note that we have identified the closed points of
$G/P_{0}$ with the set of all parabolic subgroups of $G$ which are conjugate
to $P_{0}$. The map $f$ is well defined as for $\lambda=g\lambda_{0}g^{-1}$,
we have $f(\lambda)=P_{\lambda}=gP_{0}g^{-1}$. The above argument also shows
that $f$ is $G$-equivariant and surjective.

######  Lemma 3.4.

The fiber of $f$ in $P\in G/P_{0}$ is a single $P$-conjugacy class.

###### Proof.

Let $\lambda,\mu\in[\lambda_{0}]$ be such that $f(\lambda)=f(\mu)=P$. Choose
$g,h\in G$ such that $\lambda=g\lambda_{0}g^{-1}$ and
$\mu=h\lambda_{0}h^{-1}$. Then $P=gP_{0}g^{-1}=hP_{0}h^{-1}$ and thus
$p_{0}:=g^{-1}h\in P_{0}$. We see that

$\mu=h\lambda_{0}h^{-1}=gp_{0}\lambda_{0}p_{0}^{-1}g^{-1}=\bigl{(}\underbrace{gp_{0}g^{-1}}_{=:p\in
P}\bigr{)}\,g\lambda_{0}g^{-1}\left(gp_{0}g^{-1}\right)^{-1}=p\lambda p^{-1}.$

Conversely, if $f(\lambda)=P$ and $\mu=p\lambda p^{-1}$ for some $p\in P$,
then it is easy to see that $f(\mu)=P$. ∎

Now let $v\in V$ and consider the set

$L_{G}(v):=\left\\{\lambda\in X_{*}(G)\ \middle|\ \lim_{z\to
0}\lambda(z)v\text{ exists}\right\\}.$

This set can be described as follows. Consider $V$ as a representation
$V(\lambda)$ of $\mathbb{C}^{\times}$ via $\lambda$, and consider the weight
space decomposition
$V(\lambda)=\bigoplusop\displaylimits_{n\in\mathbb{Z}}V_{n}(\lambda)$, i.e.

$V_{n}(\lambda):=\\{w\in V\mid\lambda(z)w=z^{n}w\text{ (all
$z\in\mathbb{C}^{\times}$)}\\}.$

Decompose the vector $v$ as
$v=\sumop\displaylimits_{n\in\mathbb{Z}}v_{n}(\lambda)$ with
$v_{n}(\lambda)\in V_{n}(\lambda)$. We obtain $L_{G}(v)=\\{\lambda\in
X_{*}(G)\mid v_{n}(\lambda)=0\text{ (all $n<0$)}\\}$.

######  Remark 3.5.

For $\lambda\in L_{G}(v)$ and $p\in P_{\lambda}$, we have $p\lambda p^{-1}\in
L_{G}(v)$.

######  Definition 3.6.

We define $Y_{[\lambda_{0}]}(v):=f([\lambda_{0}]\cap L_{G}(v))$.

######  Proposition 3.7.

The set $Y_{[\lambda_{0}]}(v)$ is a Zariski-closed subset of $G/P_{0}$.

###### Proof.

To show this proposition, we consider the subset

$X_{[\lambda_{0}]}(v):=\left\\{g\in G\ \middle|\ \lim_{z\to
0}g\lambda_{0}(z)g^{-1}v\text{ exists}\right\\}.$

Evidently, $X_{[\lambda_{0}]}(v)$ is the inverse image of
$Y_{[\lambda_{0}]}(v)$ under the quotient map $\pi\colon G\to G/P_{0}$. This
shows that $X_{[\lambda_{0}]}(v)$ is $P_{0}$-invariant. As $\pi$ maps
$P_{0}$-invariant closed subsets of $G$ to closed subsets of $G/P_{0}$, the
proposition will follow from the lemma below. ∎

######  Lemma 3.8.

The subset $X_{[\lambda_{0}]}(v)$ is Zariski closed in $G$.

###### Proof.

We introduce some notation. Let $V_{n}:=V_{n}(\lambda_{0})$ and $V_{\geq
0}:=\bigoplusop\displaylimits_{m\geq 0}V_{m}$. For $g\in G$, we let

$V_{n}^{(g)}:=V_{n}\left(g\lambda_{0}g^{-1}\right)=gV_{n}$

and $\smash{v_{n}^{(g)}}=v_{n}(g\lambda_{0}g^{-1})\in V_{n}^{(g)}$, so
$v=\sumop\displaylimits_{n\in\mathbb{Z}}v_{n}^{(g)}$. Then

$\displaystyle X_{[\lambda_{0}]}(v)$ $\displaystyle=\left\\{g\in G\mid
v_{n}^{(g)}=0\text{ (all $n<0$)}\right\\}$ $\displaystyle=\left\\{g\in G\mid
g^{-1}v\in V_{\geq 0}\right\\}.$

This can be translated into a fiber product diagram

${X_{[\lambda_{0}]}(v)}$${G}$${g}$${V_{\geq 0}}$${V}$${g^{-1}v}$

which shows that $X_{[\lambda_{0}]}(v)$ is indeed closed. ∎

## 4\. Lifts of fixed points

Let $y\in(V^{\operatorname{st}}(G,\theta)/G)^{\mathscr{T}}$, and let $x\in
V^{\operatorname{st}}(G,\theta)$ be such that $\pi(x)=y$. Then for every
$t\in\mathscr{T}$ there exists a unique $g\in G$ such that $t.x=gx$ (recall
that the $G$-action on the stable locus is assumed to be free). This yields a
map $\rho=\rho_{x}\colon\mathscr{T}\to G$. Note that $\rho$ depends on $x$ and
not just on $y$. The following argument can also be found in [Wei13, Lemma
3.1]. We give it for completeness.

######  Lemma 4.1.

The map $\rho$ is a morphism of algebraic groups.

###### Proof.

Let $H=\\{(g,t)\in G\times\mathscr{T}\mid t.x=gx\\}$. This is a closed
subgroup of $G\times\mathscr{T}$. The projection $p_{2}\colon H\to\mathscr{T}$
is surjective, and it is also injective by Assumption 2.4. As we work in
characteristic 0, this means that $p_{2}$ is an isomorphism. Let $p_{1}\colon
H\to G$ be the projection to the first component. Then $\rho$ is given by
$\rho=p_{1}\circ p_{2}^{-1}$. ∎

######  Example 4.2.

The following example shows that Lemma 4.1 fails in positive characteristic.
Let $k$ be an algebraically closed field of characteristic $p>0$. Let
$V=\mathbb{A}^{2}$ and $G=\mathbb{G}_{m}$, which acts on $V$ via
$g\cdot(v_{1},v_{2})=(g^{p}v_{1},g^{p}v_{2})$. For the character
$\theta=\operatorname{id}$, the stable locus is
$V^{\operatorname{st}}(G,\theta)=\mathbb{A}^{2}\setminus\\{(0,0)\\}$. The
group $G$ acts freely on the stable locus because the characteristic is $p$.
The quotient $V^{\operatorname{st}}(G,\theta)/G$ is isomorphic to
$\mathbb{P}^{1}$.

Let $\mathscr{T}=\mathbb{G}_{m}$ act on $V$ by
$t.(v_{1},v_{2})=(v_{1},tv_{2})$. This action commutes with the $G$-action,
and the fixed points of the induced action on $\mathbb{P}^{1}$ are $[1:0]$ and
$[0:1]$. For the lift $x=(0,1)$, we consider the group

$H:=\\{(g,t)\in G\times\mathscr{T}\mid t.x=g\cdot x\\}=\left\\{(g,t)\in
G\times\mathscr{T}\mid t=g^{p}\right\\}$

and the projections $p_{1}\colon H\to G$ and $p_{2}\colon H\to\mathscr{T}$. We
argue that $p_{2}$ is not an isomorphism. We observe that $p_{1}$ is an
isomorphism; its inverse is given by $p_{1}^{-1}(g)=(g,g^{p})$. The
composition $p_{2}\circ p_{1}^{-1}\colon G\to\mathscr{T}$ maps $g$ to $g^{p}$.
This is a bijective morphism which is not an isomorphism. Therefore, $p_{2}$
is not an isomorphism either.

Via $\rho$, we obtain an action of $\mathscr{T}$ on every $G$-representation;
for a representation $W$ of $G$, we denote the representation of $\mathscr{T}$
obtained via $\rho$ by ${}_{\rho}{W}$. We may in particular consider
${}_{\rho}{V}$. This new $\mathscr{T}\\!$-action by $\rho$ differs in general
from the $\mathscr{T}\\!$-action on $V$ that we started with. We look at the
subset

$V_{\rho}:=\left\\{v\in V\mid t.v=\rho(t)v\text{ for all
}t\in\mathscr{T}\right\\}$

on which these two $\mathscr{T}$-actions agree. The following properties of
$V_{\rho}$ are obvious.

######  Lemma 4.3.

Let $y$, $x$, and $\rho=\rho_{x}$ be as above.

  1. (1)

$V_{\rho}$ is a linear subspace of $V$.

  2. (2)

$V_{\rho}$ is a $\mathscr{T}\\!$-subrepresentation of both $V$ $($with the
initial $\mathscr{T}\\!$-action$)$ and ${}_{\rho}{V}$.

  3. (3)

We have $x\in V_{\rho}$.

  4. (4)

We have $y\in\pi(V_{\rho}\cap
V^{\operatorname{st}}(G,\theta))\subseteq(V^{\operatorname{st}}(G,\theta)/G)^{\mathscr{T}}$.

The last statement of Lemma 4.3 is the reason to consider $V_{\rho}$ in the
first place. Let us define

$F_{\rho}:=\pi\left(V_{\rho}\cap V^{\operatorname{st}}(G,\theta)\right).$

We will see in the following that $F_{\rho}$ is actually a closed subset of
$V^{\operatorname{st}}(G,\theta)/G$ and is moreover the connected/irreducible
component of $y$ inside $(V^{\operatorname{st}}(G,\theta)/G)^{\mathscr{T}}$.
We will also give a more intrinsic description of $F_{\rho}$. For this
description, it will not be important that $\rho$ comes from the lift of a
fixed point.

Fix a morphism of algebraic groups $\rho\colon\mathscr{T}\to G$. Let
$G_{\rho}$ be the centralizer of $\operatorname{im}(\rho)$.

######  Lemma 4.4.

The subspace $V_{\rho}$ is $G_{\rho}$-invariant.

###### Proof.

Let $v\in V_{\rho}$ and $g\in G_{\rho}$. To show that $gv\in V_{\rho}$,
consider $t\in\mathscr{T}$. We compute

$t.gv=gt.v=g\rho(t)v=\rho(t)gv,$

which shows the claim. ∎

######  Remark 4.5.

Let $T$ be a maximal torus which contains the image of $\rho$. Recall the
notation from the last paragraph of Section 2. Regarding $V_{\rho}$ as a
representation of $T$, we obtain $(V_{\rho})_{\mu}(T)=V_{\mu}(T)\cap
V_{\rho^{*}(\mu)}(\mathscr{T})$. Consequently,

$N_{T}(V_{\rho})=\left\\{\mu\in X^{*}(T)\mid(\mu,\rho^{*}(\mu))\in
N_{T\times\mathscr{T}}(V)\right\\}.$

We now move to the central result of this section.

######  Theorem 4.6.

Let $\rho\colon\mathscr{T}\to G$ be a morphism of algebraic groups. Then

  1. (1)

$V_{\rho}\cap
V^{\operatorname{sst}}(G,\theta)=V_{\rho}^{\operatorname{sst}}(G_{\rho},\theta)$;

  2. (2)

$V_{\rho}\cap
V^{\operatorname{st}}(G,\theta)=V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)$.

###### Proof.

(1) Suppose that $v\in V_{\rho}\cap V^{\operatorname{sst}}(G,\theta)$. Then
the Hilbert–Mumford criterion shows at once that $v$ belongs to
$V_{\rho}^{\operatorname{sst}}(G_{\rho},\theta)$.

For the converse inclusion, we employ Kempf’s theory of optimal one-parameter
subgroups; see Section 3. Let
$v\in\smash{V_{\rho}^{\operatorname{sst}}(G_{\rho},\theta)}$. Assume that
$v\in\smash{V^{\operatorname{unst}}(G,\theta)}$. Choose a primitive adapted
one-parameter subgroup $\lambda\in\smash{{}_{G}^{\theta}(v)}$. Let
$t\in\mathscr{T}$, and consider $\rho(t)\lambda\rho(t)^{-1}\in X_{*}(G)$. It
is primitive, and we get

$\displaystyle\rho(t)\lambda(z)\underbrace{\rho(t)^{-1}v}_{=t^{-1}.v}=\rho(t)t^{-1}.(\lambda(z)v)\xrightarrow[]{z\to
0}\rho(t)t^{-1}.\left(\lim_{z\to 0}\lambda(z)v\right).$

Moreover,

$\frac{\langle\theta,\rho(t)\lambda\rho(t)^{-1}\rangle}{\|\rho(t)\lambda\rho(t)^{-1}\|}=\frac{\langle\theta,\lambda\rangle}{\|\lambda\|}=m_{G}^{\theta}(v),$

which shows $\rho(t)\lambda\rho(t)^{-1}\in{}_{G}^{\theta}(v)$. By Theorem 3.2,
there exists a $p\in P_{\lambda}$ such that

(*) $p\lambda p^{-1}=\rho(t)\lambda\rho(t)^{-1}.$

We re-write this identity as

$\lambda(z)\rho(t)^{-1}\lambda(z)^{-1}=\rho(t)^{-1}p\lambda(z)p^{-1}\lambda(z)^{-1}$

which, as $p^{-1}$ lies in $P_{\lambda}$, converges as $z\to 0$. This shows
that $\rho(t)$ itself belongs to $P_{\lambda}$. As $\rho(t)$ is a semi-simple
element, it is contained in a maximal torus $T^{\prime}$ of $P_{\lambda}$; in
fact, the whole image of $\rho$ lies in some maximal torus $T^{\prime}$. For
the same reason, $\lambda$ is contained in a maximal torus
$T^{\prime\prime}\subseteq G_{\lambda}\subseteq P_{\lambda}$. The two maximal
tori are $P_{\lambda}$-conjugate, so we find a $q\in P_{\lambda}$ such that
$qT^{\prime}q^{-1}=T^{\prime\prime}$. In particular, $q\rho(t)q^{-1}\in
T^{\prime\prime}\subseteq G_{\lambda}$, which implies

$q\rho(t)q^{-1}\lambda(z)=\lambda(z)q\rho(t)q^{-1}$

for all $t\in\mathscr{T}$ and all $z\in\mathbb{C}^{\times}$. As this identity
holds for every $t$, we find $q^{-1}\lambda(z)q\in G_{\rho}$ for every $z$. As
$q\in P_{\lambda}$, we obtain

$\lambda^{\prime}:=q^{-1}\lambda q\in{}_{G}^{\theta}(v)\cap X_{*}(G_{\rho}).$

This says $\lambda^{\prime}$ is a one-parameter subgroup of $G_{\rho}$ for
which the limit $\lim_{z\to 0}\lambda^{\prime}(z)v$ exists and
$\langle\theta,\lambda^{\prime}\rangle=\left\lVert\lambda^{\prime}\right\rVert
m_{G}^{\theta}(v)<0$. This gives a contradiction to $v\in
V_{\rho}^{\operatorname{sst}}(G_{\rho},\theta)$.

(2) Again, as in the first item, the inclusion from left to right follows
readily from the Hilbert–Mumford criterion.

Now to the converse inclusion. Let $v\in
V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)$, and assume $v$ is not
$\theta$-stable for the $G$-action on $V$. By (1), we know that $v$ is at
least $\theta$-semi-stable for the $G$-action. So there must be a non-trivial
one-parameter subgroup $\lambda\in X_{*}(G)$ for which $\lim_{z\to
0}\lambda(z)v$ exists—so $\lambda\in L_{G}(v)$—and such that
$\langle\theta,\lambda\rangle=0$. Then $\langle\theta,g\lambda
g^{-1}\rangle=0$ for every $g\in G$, and as in (1) we see that
$\rho(t)\lambda\rho(t)^{-1}\in L_{G}(v)$ for every $t\in\mathscr{T}$.
Therefore, $\mathscr{T}$ acts via $\rho$ on $[\lambda]\cap L_{G}(v)$, and by
the $G$-equivariance of the map $f\colon[\lambda]\cap L_{G}(v)\to
G/P_{\lambda}$, the Zariski-closed subset $Y_{[\lambda]}(v)\subseteq
G/P_{\lambda}$ (see Proposition 3.7) is $\mathscr{T}$-invariant. By Borel’s
fixed point theorem, there is a fixed point for the $\mathscr{T}$-action on
$Y_{[\lambda]}(v)$. Let $P$ be a $\mathscr{T}$-fixed point in
$Y_{[\lambda]}(v)$. Then there exists a $\mu\in[\lambda]\cap L_{G}(v)$ such
that $P=P_{\mu}$. By the fixed point property, we get

$f(\rho(t)\mu\rho(t)^{-1})=f(\mu).$

As the fiber $f^{-1}(P_{\mu})$ is a $P_{\mu}$ conjugacy class (see Lemma 3.4),
we find a $p\in P_{\mu}$ such that $p\mu p^{-1}=\rho(t)\mu\rho(t)^{-1}$. Now
we are in equation (* ‣ 4) of the proof of (1). We may conclude in the same
way that $q^{-1}\mu q\in L_{G}(v)\cap X_{*}(G_{\rho})=L_{G_{\rho}}(v)$ for
some $q\in P_{\mu}$. But $\langle\theta,q^{-1}\mu
q\rangle=\langle\theta,\mu\rangle=\langle\theta,\lambda\rangle=0$, which
contradicts $v\in V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)$. ∎

## 5\. An embedding of quotients

We now look at the inclusion of the linear subspace $V_{\rho}$ into $V$. The
intersection with $V^{\operatorname{st}}(G,\theta)$ yields a closed embedding
$V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)\to
V^{\operatorname{st}}(G,\theta)$. It is $G_{\rho}$-invariant. We obtain a
morphism $i_{\rho}\colon
V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho}\to
V^{\operatorname{st}}(G,\theta)/G$. We will show the following in this
section.

######  Theorem 5.1.

The induced morphism $i_{\rho}\colon
V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho}\to
V^{\operatorname{st}}(G,\theta)/G$ is a closed immersion.

In a previous version of this paper, the above theorem was only shown under
the assumption that there were no $T$-invariants for a maximal torus $T$ of
$G$. Sergey Mozgovoy suggested to us an argument to get rid of this
assumption. It goes along the lines of [Moz23, Lemma 6.6] and uses the
machinery of étale slices. After we received his suggestion, we found a
related but different argument which allows us to show a finiteness statement
on the affine quotients, which might be of independent interest. This is
Proposition 5.4. From this proposition and two other lemmas, Theorem 5.1 will
follow.

We use the following criterion for closed immersions. It can be stated in
greater generality. We have adapted it to our situation. A proof of (a more
general version of) the criterion below is due to Osserman.

######  Proposition 5.2.

Let $f\colon X\to Y$ be a morphism of complex varieties. If

  1. (1)

$f$ is proper,

  2. (2)

$f$ is injective on closed points, and

  3. (3)

the induced map $d_{x}f\colon T_{x}X\to T_{f(x)}Y$ is injective for every
closed point $x\in X$,

then $f$ is a closed immersion.

We will prove these three properties for the morphism $i_{\rho}$. We need some
auxiliary results. The following is the main result of [Lun75].

######  Theorem 5.3 (Luna).

Let $X$ be an affine variety, $G$ a reductive algebraic group which acts on
$X$, and $H\subseteq G$ a closed reductive subgroup. Then $N_{G}(H)$ is
reductive, and $X^{H}/\\!\\!/N_{G}(H)\to X/\\!\\!/G$ is a finite morphism.

The proof uses the machinery of étale slices, which requires the ground field
to be algebraically closed of characteristic zero. Let us mention that a
version of this result exists in positive characteristic as well; see [BGM19,
Theorem 1.1]. We use Luna’s result to show the following.

######  Proposition 5.4.

The morphism $V_{\rho}/\\!\\!/G_{\rho}\to V/\\!\\!/G$ is finite.

###### Proof.

Let $X=V\times\mathbb{C}^{r}$, where $r$ is the rank of $\mathscr{T}$. On $X$
we consider the action of $\overline{G}=G\times\mathscr{T}$ via

$(g,t)*(v,w)=(gt.v,tw).$

Consider the reductive subgroup ${}_{\rho}:=\\{(\rho(t)^{-1},t)\mid
t\in\mathscr{T}\\}$ of $\overline{G}$. We see that
$N_{\overline{G}}({}_{\rho})=G_{\rho}\times\mathscr{T}$ and
$X^{{}_{\rho}}=V_{\rho}\times\\{0\\}$. The action of
$G_{\rho}\times\mathscr{T}$ on $V_{\rho}\times\\{0\\}$ is given by
$(g,t)*(v,0)=(gt.v,0)=(g\rho(t)v,0)$. This shows that

$\mathbb{C}\left[V_{\rho}\times\\{0\\}\right]^{G_{\rho}\times\mathscr{T}}\cong\mathbb{C}\left[V_{\rho}\right]^{G_{\rho}}.$

We obtain the following commutative diagram:

${{V_{\rho}/\\!\\!/G_{\rho}}}$${{V/\\!\\!/G}}$${{\left(V_{\rho}\times\\{0\\}\right)/\\!\\!/\left(G_{\rho}\times\mathscr{T}\right)}}$${{(V\times\mathbb{C}^{r})/\\!\\!/(G\times\mathscr{T})\hbox
to0.0pt{.\hss}}}$

The right-hand map comes from $V\to V\times\mathbb{C}^{r},\ v\mapsto(v,0)$.
The bottom map is a finite morphism by Luna’s result, so the top map is as
well. ∎

######  Lemma 5.5.

The morphism $i_{\rho}$ is proper.

###### Proof.

Consider the commutative diagram

${{V_{\rho}^{\operatorname{sst}}\left(G_{\rho},\theta\right)/\\!\\!/G_{\rho}}}$${{V^{\operatorname{sst}}(G,\theta)/\\!\\!/G}}$${{V_{\rho}/\\!\\!/G_{\rho}}}$${{V/\\!\\!/G\hbox
to0.0pt{.\hss}}}$

Both vertical morphisms are projective, in particular proper. The bottom map
is proper by Proposition 5.4. Therefore,
$\smash{V_{\rho}^{\operatorname{sst}}(G_{\rho},\theta)/\\!\\!/G_{\rho}}\to
V^{\operatorname{sst}}(G,\theta)/\\!\\!/G$ is proper. As, by Theorem 4.6,
$i_{\rho}$ arises as the base change of this morphism, $i_{\rho}$ is also
proper. ∎

######  Lemma 5.6.

The map $i_{\rho}$ is injective on closed points.

###### Proof.

Let $v_{1},v_{2}\in V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)$ be such
that $i_{\rho}(v_{1})=i_{\rho}(v_{2})$. Then there exists a $g\in G$ such that
$gv_{1}=v_{2}$. We need to show that $g\in G_{\rho}$, so we take
$t\in\mathscr{T}$ and show $g$ commutes with $\rho(t)$. We see that

$\rho(t)gv_{1}=\rho(t)v_{2}=t.v_{2}=t.gv_{1}=gt.v_{1}=g\rho(t)v_{1}.$

We have used that $v_{1}$ and $v_{2}$ belong to $V_{\rho}$. We deduce that the
commutator of $g$ and $\rho(t)$ lies in the stabilizer
$\operatorname{Stab}_{G}(v_{1})$. As $v_{1}$ lies in
$V^{\operatorname{st}}(G,\theta)$ by Theorem 4.6 and as stable points have
trivial stabilizers by Assumption 2.4, we obtain the desired commutativity. ∎

We now consider the tangent spaces. The tangent space of a $G$-principal
bundle quotient $Y=X/G$ at a point $y=\pi(x)$ may be identified as

$T_{y}Y\cong T_{x}X/\operatorname{im}(d_{e}\operatorname{act}_{G}(\underline{\
\ },x)),$

where $\operatorname{act}_{G}\colon G\times X\to X$ is the action map and
$d_{e}\operatorname{act}_{G}(\underline{\ \ },x)\colon\mathfrak{g}\to T_{x}X$
is the derivative at $e$ of the map $G\to X,\ g\mapsto gx$. Here,
$X=V^{\operatorname{st}}(G,\theta)$ is open in a vector space, so $T_{x}X\cong
V$. As the action is linear, $V$ becomes a $\mathfrak{g}$-module via the
derivative $d_{e}\sigma\colon\mathfrak{g}\to\mathfrak{gl}(V)$. We see that

$((d_{e}\sigma)(\xi))(v)=(d_{e}\operatorname{act}_{G}(\underline{\ \
},v))(\xi)$

for all $\xi\in\mathfrak{g}$ and $v\in V$. We write $\xi\cdot v$ for
$((d_{e}\sigma)(\xi))(v)$. For $g\in G$, $\xi\in\mathfrak{g}$, and $v\in V$,
$g(\xi\cdot v)=\operatorname{Ad}_{g}(\xi)\cdot gv$ holds; see for instance
[Kra84, Section II.2.3].

######  Lemma 5.7.

Let $v\in V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)$. Then the induced map

$d_{\pi_{\rho}(v)}i_{\rho}\colon
T_{\pi_{\rho}(v)}(V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho})\longrightarrow
T_{\pi(v)}(V^{\operatorname{st}}(G,\theta)/G)$

is injective.

###### Proof.

Abbreviate $Y:=V^{\operatorname{st}}(G,\theta)/G$ and
$Y_{\rho}:=V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho}$. Let
$x:=\pi_{\rho}(v)$ and $y:=\pi(v)=i_{\rho}(x)$. We have already argued that

$\displaystyle T_{y}Y$ $\displaystyle\cong
V/\operatorname{im}(d_{e}\operatorname{act}_{G}(\underline{\ \ },v)),$
$\displaystyle T_{x}Y_{\rho}$ $\displaystyle\cong
V_{\rho}/\operatorname{im}(d_{e}\operatorname{act}_{G_{\rho}}(\underline{\ \
},v)).$

Let $K$ be the kernel of $d_{x}i_{\rho}$. We have the following commutative
diagram with exact rows and columns:

${0}$${0}$${0}$${K}$${0}$${\mathfrak{g_{\rho}}}$${V_{\rho}}$${T_{x}Y_{\rho}}$${0}$${0}$${\mathfrak{g}}$${V}$${T_{y}Y}$${0}$${\mathfrak{g}/\mathfrak{g}_{\rho}}$${V/V_{\rho}}$${0}$${0\hbox
to0.0pt{.\hss}}$$\scriptstyle{d_{e}\operatorname{act}_{G_{\rho}}(\underline{\
\
},v)}$$\scriptstyle{d_{x}i_{\rho}}$$\scriptstyle{d_{e}\operatorname{act}_{G}(\underline{\
\ },v)}$

The snake lemma then tells us that there exists a linear map
$K\to\mathfrak{g}/\mathfrak{g}_{\rho}$ such that the sequence

$0\longrightarrow
K\longrightarrow\mathfrak{g}/\mathfrak{g}_{\rho}\longrightarrow V/V_{\rho}$

is exact. In order to show that $K=0$, it suffices to check that
$\mathfrak{g}/\mathfrak{g}_{\rho}\to V/V_{\rho}$ is injective.

The Lie algebra $\mathfrak{g}_{\rho}$ is given as follows. Recall that
$G_{\rho}$ is the centralizer of the image of $\rho$. Therefore,

$\mathfrak{g}_{\rho}=\\{\xi\in\mathfrak{g}\mid\operatorname{Ad}_{\rho(t)}(\xi)=\xi\text{
for all }t\in\mathscr{T}\\}.$

Now let $\xi\in\mathfrak{g}$, and suppose that $\xi\cdot v\in V_{\rho}$. We
show that this implies $\xi\in\mathfrak{g}_{\rho}$, which concludes the proof.
To this end, let $t\in\mathscr{T}$, and consider
$\operatorname{Ad}_{\rho(t)}(\xi)$. We apply this element to $\rho(t)v$ and
obtain

$\displaystyle\operatorname{Ad}_{\rho(t)}(\xi)\cdot\rho(t)v$
$\displaystyle=\rho(t)\cdot(\xi\cdot v)$ $\displaystyle=t.(\xi\cdot v)$
$\displaystyle=\xi\cdot(t.v)$ $\displaystyle=\xi\cdot(\rho(t)v)$

as both $v$ and $\xi\cdot v$ belong to $V_{\rho}$. This shows that
$\operatorname{Ad}_{\rho(t)}(\xi)-\xi$ lies in the Lie algebra of the
stabilizer $\operatorname{Stab}_{G}(\rho(t)v)$. By the stability of $\rho(t)v$
and Assumption 2.4, the stabilizer is trivial. Hence we may conclude that
$\operatorname{Ad}_{\rho(t)}(\xi)=\xi$, which finishes the proof. ∎

###### Proof of Theorem 5.1.

We have established the three requirements of Proposition 5.2. This proves
Theorem 5.1. ∎

## 6\. Analysis of the fixed point locus

In this section, we will prove the main result of the paper, a description of
the torus fixed point locus of the stable quotient. First, we are concerned
with the index set of the decomposition of the fixed point locus into
irreducible components.

######  Lemma 6.1.

Let $\rho_{1},\rho_{2}\colon\mathscr{T}\to G$ be morphisms such that
$F_{\rho_{1}}\cap F_{\rho_{2}}\neq\emptyset$. Then $\rho_{1}$ and $\rho_{2}$
are $G$-conjugate, and $F_{\rho_{1}}=F_{\rho_{2}}$.

###### Proof.

First of all, if $\rho_{1}$ and $\rho_{2}$ are $G$-conjugate, then it is
immediate that $F_{\rho_{1}}$ and $F_{\rho_{2}}$ agree.

Concerning the other assertion, let $y\in F_{\rho_{1}}\cap F_{\rho_{2}}$. Then
there exist $x_{i}\in V_{\rho_{i}}\cap V^{\operatorname{st}}(G,\theta)$ such
that $\pi(x_{i})=y$ for $i=1,2$. As $\pi$ is a geometric quotient, we find a
$g\in G$ such that $x_{2}=gx_{1}$. Then

$t.x_{2}=\rho_{2}(t)x_{2}=\rho_{2}(t)gx_{1}$

and on the other hand

$t.x_{2}=t.gx_{1}=gt.x_{1}=g\rho_{1}(t)x_{1}.$

This implies that $g^{-1}\rho_{2}(t)gx_{1}=\rho_{1}(t)x_{1}$, and the
stability of $x_{1}$ implies that $\rho_{1}(t)=g^{-1}\rho_{2}(t)g$ as the
$G$-action on the stable locus is free. ∎

We fix a maximal torus $T\subseteq G$. Let $W=N_{G}(T)/T$ be the corresponding
Weyl group. Now observe that for any $g\in G$, we get
$F_{\rho_{x}}=F_{\rho_{gx}}$ and $\rho_{gx}=g\rho_{x}g^{-1}$. The image of the
morphism $\rho_{x}$ is a torus; it is therefore contained in a maximal torus
of $G$. As far as the sets $F_{\rho_{x}}$ are concerned, we may hence without
loss of generality assume the image of $\rho_{x}$ to be contained in the fixed
maximal torus $T\subseteq G$—for if not, we may replace $x$ with $gx$ for a
suitable $g$ as all maximal tori are conjugate. By a slight abuse of notation,
we also denote the morphism $\mathscr{T}\to T$ by $\rho$.

######  Lemma 6.2.

Let $\rho_{1},\rho_{2}\colon\mathscr{T}\to T$ be morphisms. Assume that there
exists an $h\in G$ such that $\rho_{2}=h\rho_{1}h^{-1}$. Then there exists a
$g\in N_{G}(T)$ such that $\rho_{2}=g\rho_{1}g^{-1}$.

###### Proof.

Let $s_{1},s_{2}\in T$, and suppose they are conjugate. Then, by [Car93,
Proposition 3.7.1], there exists a $g\in N_{G}(T)$ such that
$s_{2}=gs_{1}g^{-1}$. The book [Car93] deals with finite groups of Lie type,
but the proposition holds true for every algebraic group which has a BN pair.

Hence for every $t\in\mathscr{T}$, there exists a $g_{t}\in N_{G}(T)$ such
that $\rho_{2}(t)=g_{t}\rho_{1}(t)g_{t}^{-1}$. The element $g_{t}$ is unique
up to $N_{G}(T)\cap C_{G}(\rho_{1}(t))$. Let $U=\\{t\in\mathscr{T}\mid
C_{G}(\rho_{1}(t))=C_{G}(\rho_{1})\\}$. This is an open dense subset of
$\mathscr{T}$. Recall that $C_{G}(\rho_{1})=G_{\rho_{1}}$. We obtain a map
$U\to N_{G}(T)/(N_{G}(T)\cap G_{\rho_{1}})$. This map is locally constant, and
the irreducibility of $U$ forces it to be constant, say of value $w$. We
choose a lift $g$ of $w$ to $N_{G}(T)$. Then $\rho_{2}(t)=g\rho_{1}(t)g^{-1}$
for all $t$ in the dense subset $U$. This proves the lemma. ∎

We now state our main result, the description of the components of the fixed
point locus of $V^{\operatorname{st}}(G,\theta)/G$ under the action of
$\mathscr{T}$.

######  Theorem 6.3.

The $\mathscr{T}\\!$-fixed point locus
$(V^{\operatorname{st}}(G,\theta)/G)^{\mathscr{T}}\\!$ decomposes into
connected components
$\smash{\bigsqcupop\displaylimits_{\rho}}\smash{F_{\rho}}$ indexed by a full
set of representatives of morphisms of tori $\rho\colon\mathscr{T}\to T$ up to
conjugation with $W$. The component $F_{\rho}$ equals $\pi(V_{\rho}\cap
V^{\operatorname{st}}(G,\theta))$, it is irreducible, and it is isomorphic to

$V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho}.$

###### Proof.

We have seen that every fixed point is contained in some subset $F_{\rho}$
which was defined as $\pi(V_{\rho}\cap V^{\operatorname{st}}(G,\theta))$. By
Lemma 6.1, different conjugacy classes yield disjoint $F_{\rho}$. It suffices
to consider morphisms $\rho\colon\mathscr{T}\to T$ up to Weyl group action by
Lemma 6.2 and the reasoning before the lemma. By Theorem 4.6,
$F_{\rho}=\pi(\smash{V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)})$, and by
Theorem 5.1, the morphism $i_{\rho}$ provides an isomorphism from
$\smash{V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho}}$ onto its
image, which is $F_{\rho}$, and which is closed. A variety of the form
$V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho}$ is obviously
irreducible. ∎

## 7\. A necessary condition for a non-empty fixed point component

The index set of the disjoint union in Theorem 6.3 is infinite; however, there
are just finitely many $\rho$ for which $F_{\rho}$ is non-empty. It seems hard
in this generality to give a necessary and sufficient condition for
$F_{\rho}\neq\emptyset$. Still, it is possible to find a necessary condition
which leaves us with just finitely many possibilities.

Recall the notation $N_{T}(V)$ for the set of weights of the action of $T$ on
$V$. We also want to introduce the symbol $\mathscr{N}_{T}(V)$ for the
$\mathbb{Q}$-linear subspace of $X^{*}(T){Q}$ which is spanned by $N_{T}(V)$.

######  Lemma 7.1.

If there exists a vector $v\in V$ with a finite stabilizer, then
$\mathscr{N}_{T}(V)=X^{*}(T){Q}$.

###### Proof.

Assume there are $\mu\in X^{*}(T)-(\mathscr{N}_{T}(V)\cap X^{*}(T))$. Let
$N_{T}(V)=\\{\mu_{1},\ldots,\mu_{r}\\}$. Then we find a one-parameter subgroup
$\lambda\colon\mathbb{C}^{\times}\to T$ such that
$\langle\mu,\lambda\rangle\neq 0$ while $\langle\mu_{i},\lambda\rangle=0$ for
$i=1,\ldots,r$. The subgroup $\lambda(\mathbb{C}^{\times})$ acts trivially on
$V$, and as $\lambda$ is non-trivial, this subgroup
$\lambda(\mathbb{C}^{\times})$ is infinite. ∎

In particular, we see that $\mathscr{N}_{T}(V)=X^{*}(T){Q}$ is necessary for
the existence of stable points for the $G$-action (with respect to any
stability parameter $\theta$).

######  Proposition 7.2.

  1. (1)

Let $\rho\colon\mathscr{T}\to T$ be a morphism. If $F_{\rho}\neq\emptyset$,
then $\mathscr{N}_{T}(V_{\rho})=X^{*}(T){Q}$.

  2. (2)

The set $\\{\rho\colon\mathscr{T}\to T\text{ morphism
}\mid\mathscr{N}_{T}(V_{\rho})=X^{*}(T){Q}\\}$ is finite.

###### Proof.

The first assertion follows from Lemma 7.1 and the characterization of
$F_{\rho}$ as $V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho}$. The
second claim follows from the identity $N_{T}(V_{\rho})=\\{\mu\in
X^{*}(T)\mid(\mu,\rho^{*}(\mu))\in N_{T\times\mathscr{T}}(V)\\}$ (see Remark
4.5) and the following lemma. ∎

######  Lemma 7.3.

Let $X$ and $Y$ be lattices. Let $E\subseteq X\times Y$ be a finite subset.
For a $\mathbb{Z}$-linear map $f\colon X\to Y$, consider $E_{f}:=\\{x\in
X\mid(x,f(x))\in E\\}$. Then the set

$\left\\{f\in\operatorname{Hom}{Z}(X,Y)\mid\langle
E_{f}\rangle{Q}=X{Q}\right\\}$

is finite.

###### Proof.

Let $F$ be the set in question. Let $E_{X}$ and $E_{Y}$ be the images of $E$
under the projections $X\times Y\to X$ and $X\times Y\to Y$. Assume that
$F\neq\emptyset$. Then $\langle E_{X}\rangle{Q}=X{Q}$.

We equip $X{R}$ and $Y{R}$ with norms, both denoted by $\lVert\cdot\rVert$.
For $f\in\operatorname{Hom}(X{R},Y{R})$, let $\left\lVert
f\right\rVert=\max_{\left\lVert x\right\rVert\leq 1}\left\lVert
f(x)\right\rVert$ be the operator norm associated with these two norms. Define

$R:=\max_{y\in E_{Y}}\left\lVert y\right\rVert.$

Let $B\subseteq E_{X}$ be a subset which is an $\mathbb{R}$-basis of $X{R}$.
Then we find a unique scalar product on $X{R}$ for which $B$ is an orthonormal
basis. Let $\left\lVert\cdot\right\rVert_{B}$ be the associated norm on
$X{R}$, and let $\left\lVert f\right\rVert_{B}=\max_{\left\lVert
x\right\rVert_{B}\leq 1}\left\lVert f(x)\right\rVert$ be the corresponding
operator norm on $\operatorname{Hom}(X{R},Y{R})$. As
$\operatorname{Hom}(X{R},Y{R})$ is finite-dimensional, this norm is equivalent
to the operator norm $\left\lVert\cdot\right\rVert$ from above. We find
$d_{B},D_{B}>0$ such that $d_{B}\left\lVert f\right\rVert\leq\left\lVert
f\right\rVert_{B}\leq D_{B}\left\lVert f\right\rVert$ for all
$f\in\operatorname{Hom}(X{R},Y{R})$. Let $d>0$ be the minimum of all $d_{B}$,
where $B$ ranges over all bases $B\subseteq E_{X}$.

Now let $f\in F$. Then $\langle E_{f}\rangle{R}=X{R}$ and $f(E_{f})\subseteq
E_{Y}$. Choose a basis $B=\\{x_{1},\ldots,x_{n}\\}\subseteq E_{f}$ of $X{R}$.
Let $x=\sumop\displaylimits\lambda_{i}x_{i}\in X$ be such that $\left\lVert
x\right\rVert_{B}\leq 1$. This implies
$\smash{\left\lVert\lambda\right\rVert_{2}^{2}}=\sumop\displaylimits\lvert\lambda_{i}\rvert^{2}\leq
1$, where $\lambda=(\lambda_{1},\ldots,\lambda_{n})$. It is well known that
$\left\lVert\lambda\right\rVert_{1}\leq\sqrt{n}\left\lVert\lambda\right\rVert_{2}$.
Now

$\displaystyle\left\lVert
f(x)\right\rVert\leq\underbrace{\sumop\displaylimits_{i=1}^{n}\lvert\lambda_{i}\rvert}_{\leq\sqrt{n}}\underbrace{\left\lVert
f(x_{i})\right\rVert}_{\leq R}\leq R\sqrt{n}$

and therefore $\left\lVert f\right\rVert_{B}\leq R\sqrt{n}$. This tells us
$\left\lVert f\right\rVert\leq\frac{1}{d_{B}}\left\lVert
f\right\rVert_{B}\leq\frac{R\sqrt{n}}{d}$. We have shown that the set
$\\{f\in\operatorname{Hom}(X{R},Y{R})\mid\langle E_{f}\rangle{R}=X{R}\\}$ is
bounded. The set $F$ is the set of lattice points of this set and is therefore
finite. ∎

## 8\. Application: Quiver moduli

Before we discuss the general theory of torus actions on quiver moduli, let us
look at a well-known example first.

######  Example 8.1.

Let $G=\operatorname{GL}_{m}(\mathbb{C})$, which acts on $V=M_{m\times
n}(\mathbb{C})$, the space of $m\times n$ matrices, by left multiplication.
Let $V_{0}=\mathbb{C}^{m}$ be the natural representation of
$\operatorname{GL}_{m}(\mathbb{C})$. Then $V\cong V_{0}\otimes\mathbb{C}^{n}$.

Assume that $m\leq n$. Let $\theta=\det\in X^{*}(G)$. The ring
$\mathbb{C}[V]^{G,\det}$ is generated by the maximal minors of an $m\times n$
matrix. This shows that a matrix $A$ is $\theta$-semi-stable if and only if
its rank is $m$. Every semi-stable point is therefore stable. The quotient
$V^{\operatorname{st}}(G,\theta)/G$ identifies with the Grassmannian
$\operatorname{Gr}_{n-m}(\mathbb{C}^{n})$ by assigning to a matrix $A$ of full
rank its kernel. Fix the maximal torus $T$ of diagonal matrices. The character
lattice is
$X^{*}(T)=\mathbb{Z}\varepsilon_{1}\oplus\ldots\oplus\mathbb{Z}\varepsilon_{m}$,
where $\varepsilon_{i}(\operatorname{diag}(s_{1},\ldots,s_{m}))=s_{i}$. Note
that the first fundamental weight is $\omega_{1}=\varepsilon_{1}$ and the
natural representation $V_{0}$ has highest weight $\omega_{1}$.

Let $\mathscr{T}=\mathbb{C}^{\times}$. We choose integers
$w_{1},\ldots,w_{n}$. We let $\mathscr{T}$ act on $V$ by right multiplication
with $\operatorname{diag}(t^{w_{1}},\ldots,t^{w_{n}})$. Let us assume that
$w_{1}\geq\ldots\geq w_{n}$, and let $0=p_{0}<p_{1}<\ldots<p_{k}=n$ be such
that

$w_{1}=\dots=w_{p_{1}}>w_{p_{1}+1}=\dots=w_{p_{2}}>\dots>w_{p_{k-1}+1}=\dots=w_{n}.$

Define $q_{j}=p_{j}-p_{j-1}$ for $j=1,\ldots,k$.

We now consider the fixed point components. Let
$\rho\colon\mathbb{C}^{\times}\to T$ be a one-parameter subgroup; it is given
by $\rho(t)=\operatorname{diag}(t^{r_{1}},\ldots,t^{r_{m}})$. As we consider
one-parameter subgroups up to conjugation by $W=S_{m}$, we may assume
$r_{1}\geq\ldots\geq r_{m}$. Let $0=s_{0}<s_{1}<\ldots<s_{l}=m$ be such that

$r_{1}=\dots=r_{s_{1}}>r_{s_{1}+1}=\dots=r_{s_{2}}>\dots>r_{s_{l-1}+1}=\dots=r_{m},$

and define $t_{i}=s_{i}-s_{i-1}$ for $i=1,\ldots,l$. We get
$G_{\rho}=\operatorname{GL}_{t_{1}}(\mathbb{C})\times\dots\times\operatorname{GL}_{t_{l}}(\mathbb{C})$,
viewed as block-diagonal matrices in $\operatorname{GL}_{m}(\mathbb{C})$. The
linear subspace $V_{\rho}\subseteq V$ by definition consists of all matrices
$A=(a_{ij})$ for which right multiplication with
$\operatorname{diag}(t^{w_{1}},\ldots,t^{w_{n}})$ agrees with left
multiplication with $\operatorname{diag}(t^{r_{1}},\ldots,t^{r_{m}})$. This
forces $a_{ij}=0$ whenever $r_{i}\neq w_{j}$. Let $I\subseteq\\{1,\ldots,l\\}$
be the subset of those $i$ for which there exists a $j\in\\{1,\ldots,k\\}$
such that

$r_{s_{i}}=w_{p_{j}}.$

Such an index $j$ is unique. Let $I=\\{i_{1}<\dots<i_{u}\\}$. The index
corresponding to $i_{\nu}$ will be denoted by $j_{\nu}$. We hence obtain an
increasing sequence $j_{1}<\dots<j_{u}$. As a $G_{\rho}$-representation, the
subspace $V_{\rho}$ is isomorphic to

$\bigoplusop\displaylimits_{\nu=1}^{u}M_{t_{i_{\nu}}\times
q_{j_{\nu}}}(\mathbb{C}),$

where $G_{\rho}$ acts on the $\nu^{\mathrm{th}}$ summand by left
multiplication with the $i_{\nu}^{\mathrm{th}}$ factor in $G_{\rho}$.

For $F_{\rho}\neq\emptyset$, it is necessary by Proposition 7.2 that
$\mathscr{N}_{T}(V_{\rho})=X^{*}(T){Q}$, which says that a matrix in
$V_{\rho}$ must have no zero rows. Thus $I=\\{1,\ldots,l\\}$. (This tallies
with the observation that an $m\times n$ matrix with rank $m$ cannot have zero
rows.) We get $i_{\nu}=\nu$ and $k\geq l$. A matrix $A\in V_{\rho}$ therefore
has the following block shape:

$\begin{pmatrix}\ldots&\boxed{t_{1}\times q_{j_{1}}}&\ldots&\\\
&&\ldots&\boxed{t_{2}\times q_{j_{2}}}&\ldots\\\ &&&&&\mathinner{\mkern
1.0mu\raise 7.0pt\vbox{\kern 7.0pt\hbox{$.$}}\mkern 2.0mu\raise
4.0pt\hbox{$.$}\mkern 2.0mu\raise 1.0pt\hbox{$.$}\mkern 1.0mu}\\\
&&&&&&\ldots&\boxed{t_{l}\times q_{j_{l}}}&\ldots\\\ \end{pmatrix}$

A matrix of this block shape can only have maximal rank if $q_{j_{i}}\geq
t_{i}$ for all $i=1,\ldots,l$. If this is satisfied, then the fixed point
component is, by Theorem 6.3, isomorphic to

$F_{\rho}\cong\prodop\displaylimits_{i=1}^{l}\operatorname{Gr}_{t_{i}}(\mathbb{C}^{q_{j_{i}}}).$

The one-parameter subgroups $\rho$ which occur are parametrized by the
following finite set of data:

  * •

a number $l\leq\min\\{m,k\\}$ and a sequence $0=s_{0}<s_{1}<\dots<s_{l}=m$,

  * •

a sequence $1\leq j_{1}<\dots<j_{l}\leq k$

such that $t_{i}:=s_{i}-s_{i-1}\leq q_{j_{i}}$ for every $i=1,\ldots,l$.

Torus actions on quiver moduli contain Example 8.1 as a special case. Let $Q$
be a quiver, i.e. an oriented graph. Formally, $Q$ consists of two sets
$Q_{0}$ and $Q_{1}$ and two maps $s,t\colon Q_{1}\to Q_{0}$. The set $Q_{0}$
is the set of vertices of $Q$, the set $Q_{1}$ is the set of arrows, and for
$a\in Q_{1}$, the vertices $s(a)$ and $t(a)$ are the source and target of $a$,
respectively. We assume that for any two vertices $i$ and $j$, there are just
finitely many $a\in Q_{1}$ such that $s(a)=i$ and $t(a)=j$. A representation
$M$ of $Q$ is a collection of complex vector spaces $(M_{i})_{i\in Q_{0}}$
together with $\mathbb{C}$-linear maps $(M_{a})_{a\in Q_{1}}$ such that
$M_{a}\colon M_{s(a)}\to M_{t(a)}$. If $\bigoplusop\displaylimits_{i\in
Q_{0}}M_{i}$ is finite-dimensional, we say that $M$ is finite-dimensional and
call $\operatorname{\underline{\mathrm{dim}}}M:=(\dim{C}M_{i})_{i\in
Q_{0}}\in\smash{\mathbb{Z}_{\geq 0}^{Q_{0}}}$ the dimension vector of $M$.
Given two representations $M$ and $N$ of $Q$, a homomorphism $f\colon M\to N$
is a collection $f_{i}\colon M_{i}\to N_{i}$ of linear maps such that
$N_{a}f_{s(a)}=f_{t(a)}M_{a}$ holds for all $a\in Q_{1}$. In this way we
obtain a $\mathbb{C}$-linear abelian category. We refer to [ASS06, Chapter II]
for more details.

Let $\Lambda(Q)=\\{\alpha=(\alpha_{i})_{i\in
Q_{0}}\in\mathbb{Z}^{Q_{0}}\mid\alpha_{i}=0\text{ for all but finitely many
}i\\}$, and let ${}_{+}(Q)=\Lambda(Q)\cap\smash{\mathbb{Z}_{\geq 0}^{Q_{0}}}$.
Fix a vector $\alpha\in{}_{+}(Q)$. Fix complex vector spaces $V_{i}$ of
dimension $\alpha_{i}$. Consider the (finite-dimensional) vector space

$V:=R(Q,\alpha):=\bigoplusop\displaylimits_{a\in
Q_{1}}\operatorname{Hom}(V_{s(a)},V_{t(a)}).$

An element of $V$ can be regarded as a representation of $Q$ of dimension
vector $\alpha$. The group $G_{\alpha}=\prodop\displaylimits_{i\in
Q_{0}}\operatorname{GL}(V_{i})$ acts on $V$ by change of basis. Two elements
of $V$ are isomorphic as representations if and only if they lie in the same
$G_{\alpha}$-orbit. The central subgroup
$\Delta:=\mathbb{C}^{\times}\cdot\operatorname{id}$ acts trivially on $V$.
Therefore, the action descends to an action of
$G:=PG_{\alpha}:=G_{\alpha}/\Delta$.

We will make a connection between weights of the action of a maximal torus of
$G$ and the support $\operatorname{supp}(\alpha)=\\{i\in
Q_{0}\mid\alpha_{i}\neq 0\\}$ of the dimension vector. To this end, fix a
basis of every vector space $V_{i}$. Let $T_{\alpha}$ be the maximal torus of
$G_{\alpha}$ of diagonal matrices with respect to our choices of bases. The
image of $T_{\alpha}$ under $G_{\alpha}\to PG_{\alpha}=G$ is a maximal torus
$T$ of $G$. For $i\in Q_{0}$ and $r\in\\{1,\ldots,\alpha_{i}\\}$, let
$y_{i,r}\in X^{*}(T_{\alpha})$ be the character which selects from
$t=(t_{i})_{i\in Q_{0}}$ the $r^{\mathrm{th}}$ diagonal entry of the matrix
$t_{i}$. Then $X^{*}(T_{\alpha})$ is the free abelian group generated by the
basis elements $y_{i,r}$, and

$X^{*}(T)=\left\\{\sumop\displaylimits_{i\in
Q_{0}}\sumop\displaylimits_{r=1}^{\alpha_{i}}a_{i,r}y_{i,r}\ \middle|\
\sumop\displaylimits_{i\in
Q_{0}}\sumop\displaylimits_{r=1}^{\alpha_{i}}a_{i,r}=0\right\\}\subseteq
X^{*}(T_{\alpha});$

it is generated as an abelian group by all differences $y_{j,s}-y_{i,r}$.
Recall that $N_{T}(V)$ is the set of weights of the $T$-action on $V$ and
$\mathscr{N}_{T}(V)$ is its $\mathbb{Q}$-linear span inside $X^{*}(T){Q}$. As
$G$ acts by conjugation, we have

$N_{T}(V)=\left\\{y_{t(a),s}-y_{s(a),r}\mid a\in Q_{1},\
r\in\\{1,\ldots,\alpha_{s(a)}\\},\ s\in\\{1,\ldots,\alpha_{t(a)}\\}\right\\}.$

######  Lemma 8.2.

The full subquiver supported on $\operatorname{supp}(\alpha)$ is connected if
and only if $\mathscr{N}_{T}(V)=X^{*}(T){Q}$.

###### Proof.

We may without loss of generality assume that
$\operatorname{supp}(\alpha)=Q_{0}$. To show the direction from left to right,
assume that $Q$ is connected. This means that for any two vertices $i$ and
$j$, there exists an unoriented path which connects them. More precisely,
there exist arrows $a_{1},\ldots,a_{k}$ and signs
$\varepsilon_{1},\ldots,\varepsilon_{k}\in\\{\pm 1\\}$ such that

$s(a_{1}^{\varepsilon_{1}})=i,\quad
t(a_{1}^{\varepsilon_{1}})=s(a_{2}^{\varepsilon_{2}}),\quad\ldots,\quad
t(a_{k-1}^{\varepsilon_{k-1}})=s(a_{k}^{\varepsilon_{k}}),\quad
t(a_{k}^{\varepsilon_{k}})=j.$

Here, $a^{1}:=a$ and $a^{-1}$ is a formal symbol for which we define
$s(a^{-1})=t(a)$ and $t(a^{-1})=s(a)$. Given a character $y_{j,s}-y_{i,r}$, we
choose an unoriented path connecting $i$ and $j$ as above and indices
$r_{1},\ldots,r_{k},s_{1},\ldots,s_{k}$ with
$r_{\nu}\in\\{1,\ldots,\smash{\alpha_{s(a_{\nu}^{\varepsilon_{\nu}})}}\\}$ and
$s_{\nu}\in\\{1,\ldots,\smash{\alpha_{t(a_{\nu}^{\varepsilon_{\nu}})}}\\}$
such that

$r_{1}=r,\quad s_{1}=r_{2},\quad\ldots,\quad s_{k-1}=r_{k},\quad s_{k}=s.$

This yields

$y_{j,s}-y_{i,r}=\sumop\displaylimits_{\nu=1}^{k}\left(y_{t(a_{\nu}^{\varepsilon_{\nu}}),s_{\nu}}-y_{s(a_{\nu}^{\varepsilon_{\nu}}),r_{\nu}}\right)\in\mathscr{N}_{T}(V).$

Suppose conversely that $Q$ is not connected. Then there exist two non-empty
disjoint subsets $C,D\subseteq Q_{0}$ such that $C\sqcup D=Q_{0}$ and such
that there are no arrows from one to the other. We let $\mathbb{C}^{\times}$
act on $V_{i}$ by weight $0$ if $i\in C$ and by weight $1$ if $i\in D$. This
provides us with a morphism $\varphi\colon\mathbb{C}^{\times}\to T$; let
$\varphi^{*}\colon X^{*}(T)\to\mathbb{Z}$ be the induced homomorphism on
characters. If $i\in C$ and $j\in D$, then

$\varphi^{*}(y_{j,s}-y_{i,r})=1$

for any $r\in\\{1,\ldots,\alpha_{i}\\}$ and $s\in\\{1,\ldots,\alpha_{j}\\}$.
But as all arrows in $Q$ start and end in either $C$ or $D$, we see that
$N_{T}(V)\subseteq\ker(\varphi^{*})$. Therefore,
$y_{j,s}-y_{i,r}\notin\mathscr{N}_{T}(V)$. ∎

Let $\theta=(\theta_{i})_{i\in Q_{0}}\in\mathbb{Z}^{Q_{0}}$ be such that
$\theta^{T}\cdot\alpha=\sumop\displaylimits_{i}\theta_{i}\alpha_{i}=0$. We may
interpret $\theta$ as a character of $G_{\alpha}$ by the assignment
$g\mapsto\smash{\prodop\displaylimits_{i\in
Q_{0}}}\smash{\det(g_{i})^{\theta_{i}}}$. The vanishing condition guarantees
that it descends to a character of $G$; by a slight abuse of notation, we also
denote this character by $\theta$. By [Kin94, Proposition 3.1] a
representation $M\in V$ is semi-stable with respect to $\theta$ if and only if
$\theta(U)\geq 0$ for all subrepresentations $U\subseteq M$; here $\theta(U)$
is by definition $\theta^{T}\cdot\operatorname{\underline{\mathrm{dim}}}U$.
Moreover, $M$ is $\theta$-stable if and only if $\theta(U)>0$ for all non-zero
proper subrepresentations $U$ of $M$. As the $\theta$-stable representations
are the simple objects in an appropriate abelian category (see [Rei03, Lemma
2.3]), Schur’s lemma tells us that the automorphism group of a $\theta$-stable
representation $M$ is $\mathbb{C}^{\times}$. Therefore, $G$ acts freely on the
stable locus $V^{\operatorname{st}}(G,\theta)$. Assumption 2.4 is hence
fulfilled. Set

$M^{\theta\operatorname{\\!-st}}(Q,\alpha):=V^{\operatorname{st}}(G,\theta)/G.$

Now assume that $Q_{0}$ (and hence also $Q_{1}$) is a finite set. Let
$\mathscr{T}=(\mathbb{C}^{\times})^{Q_{1}}$. Let a group element
$t=(t_{a})_{a\in Q_{1}}\in\mathscr{T}$ act on $M=(M_{a})_{a\in Q_{1}}\in V$ by

$t.M=(t_{a}M_{a})_{a\in Q_{1}}.$

It commutes with the $G$-action, so we obtain a $\mathscr{T}$-action on
$M^{\theta\operatorname{\\!-st}}(Q,\alpha)$. This action was studied by Weist
in [Wei13]. We recall his result and show how it follows from our description.

Let $\widehat{Q}$ be the (infinite) quiver given by
$\widehat{Q}_{i}=Q_{i}\times X^{*}(\mathscr{T})$ for $i=0,1$ and
$\widehat{s},\widehat{t}$ be defined by

$\widehat{s}(a,\chi)=(s(a),\chi),\quad\widehat{t}(a,\chi)=(t(a),\chi+x_{a}).$

Here, $x_{a}\colon\mathscr{T}\to\mathbb{C}^{\times}$ is the character given by
$x_{a}(t)=t_{a}$. A vector $\beta\in{}_{+}(\widehat{Q})$ is called a _cover_
of $\alpha$ if $\sumop\displaylimits_{\chi}\beta_{i,\chi}=\alpha_{i}$ for
every $i\in Q_{0}$. On ${}_{+}(Q)$ there is an action of the group
$X^{*}(\mathscr{T})$ given by $\xi.\beta=(\beta_{i,\chi+\xi})_{i,\chi}$. We
call two vectors translates of one another if they lie in the same orbit. We
define

$\widehat{\theta}=\left(\widehat{\theta}_{i,\chi}\right)_{(i,\chi)\in\widehat{Q}_{0}}\in\mathbb{Z}^{\widehat{Q}_{0}}$

by $\widehat{\theta}_{i,\chi}=\theta_{i}$. The following is [Wei13, Theorem
3.8].

######  Theorem 8.3 (Weist).

The fixed point locus
$M^{\theta\operatorname{\\!-st}}(Q,\alpha)^{\mathscr{T}}$ is the disjoint
union $\bigsqcupop\displaylimits_{\beta}F_{\beta}$ which ranges over all
$\beta\in{}_{+}(Q)$ which cover $\alpha$, up to translation. The component
$F_{\beta}$ is isomorphic to

$M^{\widehat{\theta}\operatorname{\\!-st}}(\widehat{Q},\beta).$

The index set of the disjoint union is infinite, but if $F_{\beta}$ is non-
empty, then the support of $\beta$ is connected. This reduces the
consideration to a finite subset.

Choose a basis of each of the vector spaces $V_{i}$. Let $T_{\alpha}\subseteq
G_{\alpha}$ be the maximal torus of tuples of invertible diagonal matrices.
Let $T=T_{\alpha}/\Delta$. It is a maximal torus of $G$. Let
$\rho\colon\mathscr{T}\to T$ be a morphism. Then

$V_{\rho}=\\{M\in V\mid t_{a}M_{a}=\rho(t)_{j}M_{a}\rho(t)_{i}^{-1}\text{ for
all }a\colon i\to j\\}.$

Via $\rho$, the torus $\mathscr{T}$ operates on $V_{i}$, inducing a weight
space decomposition $V_{i}=\smash{\sumop\displaylimits_{\chi}}V_{i,\chi}$. The
same argument as in [Wei13, Lemma 3.4] shows that $M$ lies in $V_{\rho}$ if
and only if we have $M_{a}(V_{i,\chi})\subseteq V_{j,\chi+\varepsilon_{a}}$.
This shows that $V_{\rho}=R(\smash{\widehat{Q}},\beta)$, where
$\beta_{i,\chi}=\dim V_{i,\chi}$. The group $G_{\beta}$ is identified with the
subgroup $\\{g\in G_{\alpha}\mid g_{i}(V_{i,\chi})\subseteq V_{i,\chi}\text{
for all }i,\chi\\}$ of $G_{\alpha}$. This is precisely the centralizer of
$\rho$: Let $g\in G_{\beta}$, $t\in\mathscr{T}$. To show that $g_{i}$ commutes
with $\rho(t)_{i}$, we apply it to $v\in V_{i}$. Write $v$ as
$v=\smash{\sumop\displaylimits_{\chi}v_{\chi}}$ with $v_{\chi}\in V_{i,\chi}$.
Then

$\displaystyle g_{i}\rho(t)_{i}v$
$\displaystyle=\sumop\displaylimits_{\chi}g_{i}\rho(t)_{i}v_{\chi}=\sumop\displaylimits_{\chi}\chi(t)g_{i}v_{\chi},$
$\displaystyle\rho(t)_{i}g_{i}v$
$\displaystyle=\sumop\displaylimits_{\chi}\rho(t)_{i}\underbrace{g_{i}v_{\chi}}_{\in
V_{i,\chi}}=\sumop\displaylimits_{\chi}\chi(t)g_{i}v_{\chi}.$

Conversely, suppose that $g\in G_{\alpha}$ commutes with $\rho$. Let $v\in
V_{i,\chi}$. We want to show that $g_{i}v\in V_{i,\chi}$. Let
$t\in\mathscr{T}$, and compute

$\rho(t)_{i}g_{i}v=g_{i}\rho(t)_{i}v=g_{i}\chi(t)v=\chi(t)g_{i}v.$

We have shown that
$V_{\rho}^{\operatorname{st}}(G_{\rho},\theta)/G_{\rho}\cong
M^{\widehat{\theta}\operatorname{\\!-st}}(\widehat{Q},\beta)$.

Lemma 8.2 shows that the support of $\beta$ is connected if and only if
$\mathscr{N}_{T}(V_{\rho})=X^{*}(T){Q}$ holds.

######  Example 8.4.

Let $Q$ be the quiver

${1}$${2,}$$\scriptstyle{a,b,c}$

the so-called $3$-Kronecker quiver. Fix the dimension vector $\alpha=(2,3)$.
Then $V=M_{3\times 2}(\mathbb{C})^{3}$, and $G$ is the quotient of
$G_{\alpha}=\operatorname{GL}_{2}(\mathbb{C})\times\operatorname{GL}_{3}(\mathbb{C})$
by the central one-parameter subgroup
$\Delta=\\{(t\operatorname{id},t\operatorname{id})\mid
t\in\mathbb{C}^{\times}\\}$. A representative $(g,h)$ of an element in $G$
acts on $(A,B,C)\in V$ by

$(g,h)\cdot(A,B,C)=(hAg^{-1},hBg^{-1},hCg^{-1}).$

Let $\theta\in X^{*}(G)$ be the character defined by
$\theta(g,h)=\det(g)^{-3}\det(h)^{2}$. An application of [Kin94, Proposition
3.1] yields that $\theta$-semi-stability and $\theta$-stability agree and that
a representation $(A,B,C)$ is $\theta$-stable if and only if $\dim\langle
Ax,Bx,Cx\rangle\geq 2$ for every $x\in\mathbb{C}^{2}\setminus\\{0\\}$. The
$\theta$-stable moduli space
$M^{\theta\operatorname{\\!-st}}(Q,\alpha)=V^{\operatorname{st}}(G,\theta)/G$
has dimension six.

Let $\mathscr{T}=(\mathbb{C}^{\times})^{3}$, which acts linearly on $V$ by

$(t_{a},t_{b},t_{c}).(A,B,C)=(t_{a}A,t_{b}B,t_{c}C).$

We will describe the fixed points under this torus action. We fix the maximal
torus $T_{\alpha}$ of $G_{\alpha}$ of invertible diagonal matrices and the
induced maximal torus $T=T_{\alpha}/\Delta$ of $G$. The Weyl group is
$W=S_{2}\times S_{3}$. By Theorem 6.3 the $\mathscr{T}$-fixed point locus of
the quotient $V^{\operatorname{st}}(G,\theta)/G$ is the disjoint union
$\smash{\bigsqcupop\displaylimits_{\rho}}F_{\rho}$, where
$F_{\rho}=\smash{V_{\rho}^{\operatorname{st}}}(G_{\rho},\theta)/G_{\rho}$; the
union ranges over all morphisms $\rho\colon\mathscr{T}\to T$ which satisfy
$\mathscr{N}_{T}(V_{\rho})=X^{*}(T){Q}$, up to $W$-conjugation. There are
three types of such morphisms:

  1. (1)

The morphism $\rho\colon\mathscr{T}\to T$ is defined by

$\rho(t_{a},t_{b},t_{c})=\left(\begin{pmatrix}1&\\\
&1\end{pmatrix},\begin{pmatrix}t_{a}&&\\\ &t_{b}&\\\
&&t_{c}\end{pmatrix}\right).$

For $V_{\rho}$ and $G_{\rho}$, we obtain

$V_{\rho}=\left\\{\left(\begin{pmatrix}*&*\\\ \phantom{*}&\phantom{*}\\\
\phantom{*}&\phantom{*}\end{pmatrix},\begin{pmatrix}\phantom{*}&\phantom{*}\\\
*&*\\\
\phantom{*}&\phantom{*}\end{pmatrix},\begin{pmatrix}\phantom{*}&\phantom{*}\\\
\phantom{*}&\phantom{*}\\\ *&*\end{pmatrix}\right)\right\\},\quad
G_{\rho}=\operatorname{GL}_{2}(\mathbb{C})\times(\mathbb{C}^{\times})^{3}.$

The fixed point component
$F_{\rho}=\smash{V_{\rho}^{\operatorname{st}}}(G_{\rho},\theta)/G_{\rho}$
consists of a single point.

  2. (2)

For $m_{1},\ldots,m_{4}\in\\{a,b,c\\}$ with $m_{1}\neq m_{2}\neq m_{3}\neq
m_{4}$, let $\rho\colon\mathscr{T}\to T$ be defined by

$\rho(t_{a},t_{b},t_{c})=\left(\begin{pmatrix}1&\\\
&t_{m_{2}}t_{m_{3}}^{-1}\end{pmatrix},\begin{pmatrix}t_{m_{1}}&&\\\
&t_{m_{2}}&\\\ &&t_{m_{2}}t_{m_{3}}^{-1}t_{m_{4}}\end{pmatrix}\right).$

The morphism associated with $(m_{1},\ldots,m_{4})$ is $W$-conjugate to the
morphism associated with $(m_{4},\ldots,m_{1})$. Up to this symmetry, there
are 12 such quadruples, namely

$abab,\ abac,\ abca,\ abcb,\ acab,\ acac,\ acbc,\ babc,\ bacb,\ bcac,\ bcbc,\
cabc.$

The associated morphisms are pairwise not $W$-conjugate. For all such $\rho$,
$G_{\rho}=T$ holds. The subspaces $V_{\rho}$ are all four-dimensional. For
instance for $m_{1}\ldots m_{4}=abac$, we get

$V_{\rho}=\left\\{\left(\begin{pmatrix}*&\phantom{*}\\\ \phantom{*}&*\\\
\phantom{*}&\phantom{*}\end{pmatrix},\begin{pmatrix}\phantom{*}&\phantom{*}\\\
*&\phantom{*}\\\
\phantom{*}&\phantom{*}\end{pmatrix},\begin{pmatrix}\phantom{*}&\phantom{*}\\\
\phantom{*}&\phantom{*}\\\ \phantom{*}&*\end{pmatrix}\right)\right\\}.$

In all of the 12 cases, the fixed point component $F_{\rho}$ consists of a
single point.

  3. (3)

For $m_{1},\ldots,m_{4}\in\\{a,b,c\\}$ such that
$\\{m_{1},m_{2},m_{3}\\}=\\{a,b,c\\}$ and $m_{4}\neq m_{3}$, let
$\rho\colon\mathscr{T}\to T$ be defined by

$\rho(t_{a},t_{b},t_{c})=\left(\begin{pmatrix}1&\\\
&t_{m_{3}}t_{m_{4}}^{-1}\end{pmatrix},\begin{pmatrix}t_{m_{1}}&&\\\
&t_{m_{2}}&\\\ &&t_{m_{3}}\end{pmatrix}\right).$

The morphisms obtained from $(m_{1},m_{2},m_{3},m_{4})$ and
$(m_{2},m_{1},m_{3},m_{4})$ are $W$-conjugate. Up to this symmetry we are left
with the following six quadruples (which lead to pairwise non-$W$-conjugate
morphisms):

$abca,\ abcb,\ acba,\ acbc,\ bcab,\ bcac.$

For all of those morphisms, $G_{\rho}=T$ holds. The subspaces $V_{\rho}$ are
four-dimensional. For example for $abcb$, it looks as follows:

$V_{\rho}=\left\\{\left(\begin{pmatrix}*&\phantom{*}\\\
\phantom{*}&\phantom{*}\\\
\phantom{*}&\phantom{*}\end{pmatrix},\begin{pmatrix}\phantom{*}&\phantom{*}\\\
*&\phantom{*}\\\
\phantom{*}&*\end{pmatrix},\begin{pmatrix}\phantom{*}&\phantom{*}\\\
\phantom{*}&\phantom{*}\\\ *&\phantom{*}\end{pmatrix}\right)\right\\}.$

However, although $\mathscr{N}_{T}(V_{\rho})=X^{*}(T){Q}$, the fixed point
component for this morphism $\rho$, and all others of the third type, is empty
because $V_{\rho}$ does not intersect $V^{\operatorname{st}}(G,\theta)$.
Indeed, for $abcb$ the span $\langle Ae_{2},Be_{2},Ce_{2}\rangle$ is one-
dimensional for every $(A,B,C)\in V_{\rho}$.

The above considerations show that $\mathscr{T}$ acts on
$V^{\operatorname{st}}(G,\theta)/G$ with 13 isolated fixed points.

## 9\. Application: Torus quotients

We are going to discuss quotients $V^{\operatorname{st}}(G,\theta)/G$ in the
case when $G$ is a torus. In this case, $V^{\operatorname{st}}(G,\theta)/G$ is
a toric variety. We will first describe the toric structure of
$V^{\operatorname{st}}(G,\theta)/G$. We believe this should be well known, but
we could not find it in the literature in the form we need it for our
purposes. As a reference for toric geometry we use Fulton’s book [Ful93].

Let $V$ be a vector space of dimension $m$ and $G=(\mathbb{C}^{\times})^{r}$ a
torus which we assume to act linearly on $V$. Let $a\colon
G\to\operatorname{GL}(V)$ be the induced morphism of algebraic groups.
Decompose $V$ into weight spaces $V=\smash{\bigoplusop\displaylimits_{\chi\in
X^{*}(G)}}V_{\chi}$. Let $\chi_{1},\ldots,\chi_{p}$ be the characters which
occur in this decomposition, and let $e_{s,1},\ldots,e_{s,m_{s}}$ be a basis
of $V_{\chi_{s}}$ for $s=1,\ldots,p$. Let $T$ be the maximal torus of
$\operatorname{GL}(V)$ which corresponds to the basis
$(e_{1,1},\ldots,e_{1,m_{1}},\ldots,e_{p,1},\ldots,e_{p,m_{p}})$ of $V$. Then
$a$ factors through a morphism of tori $G\to T$, which we also call $a$.
Concretely,

$a(g)=(\underbrace{\chi_{1}(g),\ldots,\chi_{1}(g)}_{m_{1}\text{
times}},\ldots,\underbrace{\chi_{p}(g),\ldots,\chi_{p}(g)}_{m_{p}\text{
times}}).$

Let $x_{s,k}\in X^{*}(T)$ be the respective coordinate function.

Let $\theta\in X^{*}(G)$. We describe the semi-stable and the stable locus
with respect to $\theta$. Let

$I=\\{(s,k)\mid s\in\\{1,\ldots,p\\},\ k\in\\{1,\ldots,m_{s}\\}\\}.$

Write a vector $v\in V$ as $\smash{\sumop\displaylimits_{(s,k)\in
I}}v_{s,k}e_{s,k}$. We define $\operatorname{supp}(v)=\\{(s,k)\in I\mid
v_{s,k}\neq 0\\}$. Let $\eta\colon\mathbb{C}^{\times}\to G$ be a one-parameter
subgroup. Then

$\eta(z)v=\sumop\displaylimits_{(s,k)\in
I}z^{\langle\chi_{s},\eta\rangle}v_{s,k}e_{s,k},$

so the limit $\lim_{z\to 0}\eta(z)v$ exists if and only if
$\langle\eta,\chi_{s}\rangle\geq 0$ for all $(s,k)\in\operatorname{supp}(v)$.
This is a condition which depends just on the support of $v$, so
$\theta$-semi-stability and $\theta$-stability do as well. Let us be more
precise. Let $\operatorname{pr}\colon I\to\\{1,\ldots,p\\}$ be the projection
to the first coordinate. For $S\subseteq I$, we define

$\tau_{S}:=\operatorname{cone}(\chi_{s})_{s\in\operatorname{pr}(S)}\subseteq
X^{*}(G){R}.$

It is a rational convex polyhedral cone. Its dual

$\displaystyle\tau_{S}^{\vee}$ $\displaystyle=\\{y\in X_{*}(G){R}\mid\langle
x,y\rangle\geq 0\text{ (all $x\in\tau_{S}$)}\\}$ $\displaystyle=\\{y\in
X_{*}(G){R}\mid\langle\chi_{s},y\rangle\geq 0\text{ (all
$s\in\operatorname{pr}(S)$)}\\}$

is again a rational convex polyhedral cone. Denote by $\theta^{\vee}$ the dual
of the ray through $\theta$, and let $\theta^{+}=\\{y\in
X_{*}(G)\mid\langle\theta,y\rangle>0\\}$. The Hilbert–Mumford criterion tells
us the following.

######  Lemma 9.1.

Let $v\in V$ and $S:=\operatorname{supp}(v)$.

  1. (1)

The vector $v$ is $\theta$-semi-stable if and only if
$\tau_{S}^{\vee}\subseteq\theta^{\vee}$.

  2. (2)

The vector $v$ is $\theta$-stable if and only if
$\tau_{S}^{\vee}\setminus\\{0\\}\subseteq\theta^{+}$.

We say a subset $S\subseteq I$ is $\theta$-_semi-stable_ if
$\tau_{S}^{\vee}\subseteq\theta^{\vee}$, and we say $S$ is $\theta$-_stable_
if $\tau_{S}^{\vee}\setminus\\{0\\}\subseteq\theta^{+}$. The notions of
(semi-)stability depend only on $\operatorname{pr}(S)$.

######  Lemma 9.2.

If $S\subseteq I$ is $\theta$-stable, then $(\chi_{s})_{(s,k)\in S}$ spans
$X^{*}(G){R}$.

###### Proof.

Assume that the $\mathbb{R}$-linear span $U$ of $(\chi_{s})_{(s,k)\in S}$ is a
proper subspace of $X^{*}(G){R}$. Then there exists a $y\in
X_{*}(G)\setminus\\{0\\}$ such that $\langle x,y\rangle=0$ for all $x\in U$.
Therefore, both $y$ and $-y$ lie in
$\tau_{S}^{\vee}\setminus\\{0\\}\subseteq\theta^{+}$, which implies
$\langle\theta,y\rangle>0$ and $\langle\theta,-y\rangle>0$, giving a
contradiction. ∎

We call $S$ _minimally_ $\theta$-_stable_ if $S$ is $\theta$-stable and if no
proper subset of $S$ is $\theta$-stable.

######  Lemma 9.3.

For a $\theta$-stable subset $S\subseteq I$, the following are equivalent:

  1. (1)

The subset $S$ is minimally $\theta$-stable.

  2. (2)

We have $\left|S\right|=r$.

  3. (3)

The set $(\chi_{s})_{(s,k)\in S}$ is a basis of $X^{*}(G){R}$.

###### Proof.

Let us assume $S$ is minimally stable, and let us show that
$(\chi_{s})_{(s,k)\in S}$ is a basis of $X^{*}(G){R}$. By Lemma 9.2, it is
enough to prove linear independence. Assume there are
$(s^{\prime},k^{\prime})\in S$ such that $\chi_{s^{\prime}}$ lies in the span
of the $\chi_{s}$ with $(s,k)\in
S^{\prime}:=S\setminus\\{(s^{\prime},k^{\prime})\\}$. Then
$\tau_{S^{\prime}}^{\vee}=\tau_{S}^{\vee}$, and therefore $S^{\prime}$ is also
stable. This contradicts the minimality of $S$.

Now let $(\chi_{s})_{(s,k)\in S}$ be a basis of $X^{*}(G){R}$. As $\dim
X(G){R}=\dim G=r$, it is obvious that $\left|S\right|=r$.

Finally, if $\left|S\right|=r$, then there can be no proper $\theta$-stable
subset $S^{\prime}$ of $S$ because the span of $(\chi_{s})_{(s,k)\in
S^{\prime}}$ is a proper subspace of $X^{*}(G){R}$. ∎

Lemma 9.1 shows in particular that $V^{\operatorname{sst}}(G,\theta)$ and
$V^{\operatorname{st}}(G,\theta)$ are $T$-invariant. The torus $T$ acts with a
dense open orbit, isomorphic to $T$, on $V^{\operatorname{st}}(G,\theta)$ and
$V^{\operatorname{sst}}(G,\theta)$, provided that these loci are non-empty. We
obtain an action of $T$ on the semi-stable and stable quotient.

We are going to assume in the following that $V^{\operatorname{st}}(G,\theta)$
is non-empty and that $G$ acts freely on the stable locus. Assumption 2.4 is
hence fulfilled.

Let $\mathscr{T}$ be the cokernel of $a\colon G\to T$. It is a torus of rank
$\dim V-\dim G$. We obtain a short exact sequence

$1\longrightarrow
G\xrightarrow[]{\;a\;}T\xrightarrow[]{\;\pi\;}\mathscr{T}\longrightarrow 1;$

note that Assumption 2.4 implies that $a$ is injective. The $T$-action on
$V^{\operatorname{sst}}(G,\theta)/\\!\\!/G$ and on
$V^{\operatorname{st}}(G,\theta)/G$ descends to an action of $\mathscr{T}$.
The torus $\mathscr{T}$ acts on both with a dense open orbit isomorphic to
$\mathscr{T}$. The quotients are therefore toric varieties. Let us describe
the toric fan of $V^{\operatorname{st}}(G,\theta)/G$. Let
$M:=X^{*}(\mathscr{T})$ and $N:=X_{*}(\mathscr{T})$. Consider the short exact
sequence of the lattices of one-parameter subgroups

$0\longrightarrow
X_{*}(G)\xrightarrow[]{\;a_{*}\;}X_{*}(T)\xrightarrow[]{\;\pi_{*}\;}N\longrightarrow
0.$

We have $X_{*}(T)=\bigoplusop\displaylimits_{(s,k)\in
I}\mathbb{Z}\varepsilon_{s,k}$, where
$\varepsilon_{s,k}\colon\mathbb{C}^{\times}\to T$ is defined by

$\varepsilon_{s,k}(z)e_{t,l}=\begin{cases}ze_{s,k}&\text{if }(s,k)=(t,l),\\\
e_{t,l}&\text{otherwise.}\end{cases}$

Note that the map $a_{*}\colon X_{*}(G)\to X_{*}(T)$ is given by
$a_{*}(\eta)=\smash{\sumop\displaylimits_{(s,k)\in
I}}\langle\chi_{s},\eta\rangle\varepsilon_{s,k}$; we write $X_{*}(T)$
additively here. For a subset $J\subseteq I$, we define
$\sigma_{J}:=\operatorname{cone}(\pi_{*}\varepsilon_{s,k})_{(s,k)\in J}$, a
rational convex polyhedral cone in $N{R}$. We define

$\displaystyle\Phi:={}^{\operatorname{st}}(\theta)$
$\displaystyle:=\\{J\subseteq
I\mid\tau_{J^{c}}^{\vee}\setminus\\{0\\}\subseteq\theta^{+}\\}=\\{J\subseteq
I\mid J^{c}\text{ is $\theta$-stable}\\},$
$\displaystyle\Delta:={}^{\operatorname{st}}(\theta)$
$\displaystyle:=\\{\sigma_{J}\mid J\in\Phi\\}.$

In the above equation, $J^{c}=I\setminus J$.

######  Lemma 9.4.

Let $J\in\Phi$.

  1. (1)

The elements $\pi_{*}\varepsilon_{s,k}$ with $(s,k)\in J$ are linearly
independent over $\mathbb{R}$. Therefore, $\sigma_{J}$ is a simplex of
dimension $\left|J\right|$.

  2. (2)

If $J^{\prime}\subseteq J$, then $J^{\prime}\in\Phi$ and $\sigma_{J^{\prime}}$
is a face of $\sigma_{J}$. Every face of $\sigma_{J}$ is of this form.

  3. (3)

For $J_{1},J_{2}\in\Phi$, we get
$\sigma_{J_{1}}\cap\sigma_{J_{2}}=\sigma_{J_{1}\cap J_{2}}$.

###### Proof.

(1) Assume there are $\lambda=\smash{\sumop\displaylimits_{(s,k)\in
J}}a_{s,k}\varepsilon_{s,k}\in X_{*}(T)$ with $\lambda\neq 0$ such that
$\pi_{*}\lambda=0$. Then there exists an $\eta\in X_{*}(G)\setminus\\{0\\}$
such that $\lambda=a_{*}\eta$. For all $(s,k)\notin J$, we have

$\langle\chi_{s},\eta\rangle=\langle a^{*}x_{s,k},\eta\rangle=\langle
x_{s,k},\lambda\rangle=0$

because $\lambda$ is supported on $J$. Therefore,
$\langle\chi_{s},\eta\rangle=0$ for all $s\in\operatorname{pr}(J^{c})$. In
particular, $\eta\in\tau_{J^{c}}^{\vee}$. As $J\in\Phi$, we obtain
$\langle\theta,\eta\rangle>0$. But by the same argument, we also obtain
$-\eta\in\tau_{J^{c}}^{\vee}$, which implies $\langle\theta,-\eta\rangle>0$,
giving a contradiction.

(2) The cone $\sigma_{J^{\prime}}$ is the intersection of $\sigma_{J}$ with
the supporting hyperplanes $(\pi_{*}\varepsilon_{s,k})^{\bot}$, where $(s,k)$
ranges over $J\setminus J^{\prime}$. Now let $\sigma$ be a face of
$\sigma_{J}$. Then there exists a $u\in\sigma_{J}^{\vee}$ such that
$\sigma=\sigma_{J}\cap u^{\bot}$. It is then easy to see that
$\sigma=\sigma_{J^{\prime}}$, where
$J^{\prime}=J\cap\operatorname{supp}(\pi^{*}u)$.

(3) Assume there are

$\lambda_{1}=\sumop\displaylimits_{(s,k)\in
J_{1}}a_{s,k}\varepsilon_{s,k},\quad\lambda_{2}=\sumop\displaylimits_{(s,k)\in
J_{2}}b_{s,k}\varepsilon_{s,k}$

with $a_{s,k},b_{s,k}\in\mathbb{R}_{\geq 0}$ such that
$\lambda_{1}\neq\lambda_{2}$ but $\pi_{*}\lambda_{1}=\pi_{*}\lambda_{2}$. Then
there exists an $\eta\in X_{*}(G)$ such that
$a_{*}\eta=\lambda_{1}-\lambda_{2}$. For all $(s,k)\in J_{2}^{c}$, we then
have

$\langle\chi_{s},\eta\rangle=\langle
x_{s,k},\lambda_{1}-\lambda_{2}\rangle=a_{s,k}-b_{s,k}=a_{s,k}\geq 0.$

As $J_{2}\in\Phi$, this implies $\langle\theta,\eta\rangle>0$. But the same
argument shows that we also have $\langle\theta,-\eta\rangle>0$. We therefore
again have a contradiction. ∎

######  Proposition 9.5.

The toric fan of $V^{\operatorname{st}}(G,\theta)/G$ is
${}^{\operatorname{st}}(\theta)$.

###### Proof.

For $J\in\Phi$, let us define
$\sigma_{J}^{+}=\smash{\operatorname{cone}(\varepsilon_{s,k})_{(s,k)\in J}}$.
The affine toric variety associated with the simplex $\smash{\sigma_{J}^{+}}$
is

$U_{\sigma_{J}^{+}}=\\{v\in V\mid J^{c}\subseteq\operatorname{supp}(v)\\}.$

For $v\in\smash{U_{\sigma_{J}^{+}}}$ with $S:=\operatorname{supp}(v)$, we get
$\tau_{S}^{\vee}\setminus\\{0\\}\subseteq\tau_{J^{c}}^{\vee}\setminus\\{0\\}\subseteq\theta^{+}$,
so $v$ is $\theta$-stable by Lemma 9.1. We define
${}^{+}=\\{\sigma_{J}^{+}\mid J\in\Phi\\}$. We have argued that the toric
variety $X_{{}^{+}}$ associated with the fan + is the stable locus
$V^{\operatorname{st}}(G,\theta)$. It is enough to show that there is a map
$\pi\colon X_{{}^{+}}\to X$ which is a categorical $G$-quotient. Consider the
map

$\pi_{J}\colon U_{\sigma_{J}^{+}}\longrightarrow U_{\sigma_{J}}$

induced by $\pi\colon T\to\mathscr{T}$. By definition and an easy
verification,

$U_{\sigma_{J}}=\operatorname{Spec}\mathbb{C}\left[M\cap\sigma_{J}^{\vee}\right]=\operatorname{Spec}\mathbb{C}\left[M\cap\left(\pi_{*}\sigma_{J}^{+}\right)^{\vee}\right]=\operatorname{Spec}\mathbb{C}\left[M\cap(\pi^{*})^{-1}\left(\left(\sigma_{J}^{+}\right)^{\vee}\right)\right].$

The ring $\mathbb{C}[M\cap(\pi^{*})^{-1}((\sigma_{J}^{+})^{\vee})]$ agrees
with $\mathbb{C}[X^{*}(T)\cap(\sigma_{J}^{+})^{\vee}]^{G}$, and therefore
$\pi_{J}$ is a universal categorical $G$-quotient. By Lemma 9.4(3), the maps
$\pi_{J}$ and $\pi_{J^{\prime}}$ agree on the intersection and therefore glue
to $\pi\colon X_{{}^{+}}\to X$ which, also by Lemma 9.4(3), is a categorical
$G$-quotient. ∎

We return to the action of $\mathscr{T}$. By general theory [Ful93, Section
3.1], the torus $\mathscr{T}$ acts on $V^{\operatorname{st}}(G,\theta)/G=X$
with finitely many orbits. The orbits are labeled by the faces of . In our
case, the orbit corresponding to $\sigma_{J}$ is

(#) $O_{\sigma_{J}}=\\{\pi(v)\mid v\in V\text{ such that
}\operatorname{supp}(v)=J^{c}\\}/G.$

The dimension of $O_{\sigma_{J}}$ is $\dim V-\dim
G-\dim\sigma_{J}=m-r-\left|J\right|$. The fixed points of the
$\mathscr{T}\\!$-action are therefore the orbits $O_{\sigma_{J}}$ associated
with those $J\in\Phi$ for which $\left|J\right|=m-r$. By Lemma 9.3, a set $J$
with cardinality $m-r$ is maximal in , and every maximal member of has
cardinality $m-r$.

We should like to apply our results to this setup and compare them with the
description of the fixed points coming from the toric fan. In order to do so,
we need $\mathscr{T}$ to act linearly on $V$. For this we choose, once and for
all, a section $c\colon\mathscr{T}\to T$ of $\pi\colon T\to\mathscr{T}$. Then
$\mathscr{T}$ acts on $V$ via $t.v=c(t)v$. This $\mathscr{T}\\!$-action
automatically commutes with the $G$-action, and the induced
$\mathscr{T}\\!$-action on $V^{\operatorname{st}}(G,\theta)/G$ agrees with the
$\mathscr{T}\\!$-action coming from the $T\\!$-action on $V$.

The group $G$ is already a torus. Let $\rho\colon\mathscr{T}\to G$ be a
morphism of algebraic groups. We get $G_{\rho}=G$ and

$\displaystyle V_{\rho}$ $\displaystyle=\\{v\in V\mid t.v=\rho(t)\cdot v\text{
(all $t\in\mathscr{T}$)}\\}$ $\displaystyle=\\{v\in V\mid
c(t)v=a\rho(t)v\text{ (all $t\in\mathscr{T}$)}\\}.$

For $(s,k)\in I$, let $c_{s,k}\colon\mathscr{T}\to\mathbb{C}^{\times}$ be such
that $c(t)=(c_{s,k}(t))_{(s,k)\in I}\in T=(\mathbb{C}^{\times})^{I}$. Then
$c(t)v=a\rho(t)v$ holds if and only if
$c_{s,k}(t)v_{s,k}=\chi_{s}(\rho(t))v_{s,k}$ for all $(s,k)\in I$. Define the
set $S_{\rho}\subseteq I$ as

$S_{\rho}=\\{(s,k)\in I\mid c_{s,k}(t)=\chi_{s}(\rho(t))\text{ (all
$t\in\mathscr{T}$)}\\}.$

We see that $V_{\rho}^{\operatorname{st}}(G,\theta)\neq\emptyset$ if and only
if $S_{\rho}$ is $\theta$-stable.

######  Proposition 9.6.

Let $\rho\colon\mathscr{T}\to G$ be a morphism of algebraic groups.

  1. (1)

If $S_{\rho}$ is $\theta$-stable, then $S_{\rho}$ is minimally
$\theta$-stable.

  2. (2)

If $\rho_{1},\rho_{2}\colon\mathscr{T}\to G$ are two morphisms of algebraic
groups such that $S_{\rho_{1}}=S_{\rho_{2}}$, then $\rho_{1}=\rho_{2}$.

  3. (3)

For every minimally $\theta$-stable subset $S\subseteq I$, there exists a
morphism $\rho\colon\mathscr{T}\to G$ such that $S_{\rho}=S$.

###### Proof.

(1) Let $S:=S_{\rho}$. By Lemma 9.3, it is enough to show that
$\left|S\right|\leq r$. Let $U$ be the $\mathbb{R}$-linear span of
$(x_{s,k})_{(s,k)\in S}$ inside $X^{*}(T){R}$. This is an
$\left|S\right|$-dimensional subspace. Consider the commutative diagram

${{X^{*}(\mathscr{T}){R}}}$${0}$${{X^{*}(\mathscr{T}){R}}}$${{X^{*}(T){R}}}$${{X^{*}(G){R}}}$${0}$$\scriptstyle{\pi^{*}}$$\scriptstyle{a^{*}}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{c^{*}}$$\scriptstyle{\rho^{*}}$

whose bottom row is exact. We know that $\dim U=\dim a^{*}(U)+\dim(U\cap\ker
a^{*})$. Let $u\in U\cap\ker a^{*}=U\cap\operatorname{im}\pi^{*}$; write it as
$u=\pi^{*}v$. Then $v=c^{*}\pi^{*}v=c^{*}u=\rho^{*}a^{*}u=0$ as $u\in\ker
a^{*}$. Therefore, $U\cap\ker a^{*}=0$ and thus

$\left|S\right|=\dim U=\dim a^{*}(U)\leq\dim X^{*}(G){R}=r.$

(2) Let $S:=S_{1}=S_{2}$. We have
$\chi_{s}(\rho_{1}(t))=c_{s,k}(t)=\chi_{s}(\rho_{2}(t))$ for all $(s,k)\in S$.
Let $\varepsilon_{1},\ldots,\varepsilon_{r}$ be a basis of $X_{*}(G)$. It is
enough to show that $\rho_{1}\varepsilon_{i}=\rho_{2}\varepsilon_{i}$ for all
$i=1,\ldots,r$. Consider $\eta_{i}\in X_{*}(G)$ defined by

$\eta_{i}(z)=\rho_{1}(\varepsilon_{i}(z))\rho_{2}(\varepsilon_{i}(z))^{-1}.$

Then $\langle\chi_{s},\eta_{i}\rangle=0$ for all $(s,k)\in S$. This implies
that both $\eta_{i}$ and $-\eta_{i}$ belong to $\tau_{S}^{\vee}$. If
$\eta_{i}$ is non-trivial, it follows that $\langle\theta,\eta_{i}\rangle>0$
and $\langle\theta,-\eta_{i}\rangle>0$, giving a contradiction. We have shown
that $\rho_{1}\varepsilon_{i}=\rho_{2}\varepsilon_{i}$.

(3) If $S$ is minimally $\theta$-stable, then $\smash{(\chi_{s})_{(s,k)\in
S}}$ is a basis of $X^{*}(G){R}$ and therefore also a basis of $X^{*}(G){Q}$.
Consider the homomorphism of abelian groups
$\mathbb{Z}^{S}=X^{*}((\mathbb{C}^{\times})^{S})\to X^{*}(G)$ which maps the
basis vector $x_{s,k}$ to $\chi_{s}$, i.e. the restriction of $a^{*}$. It
induces a homomorphism of bialgebras

$\mathbb{C}[\mathbb{Z}^{S}]=\mathbb{C}[(\mathbb{C}^{\times})^{S}]\longrightarrow\mathbb{C}[X^{*}(G)]=\mathbb{C}[G].$

This yields a morphism of algebraic groups, which on closed points is given as

$a_{S}\colon G\longrightarrow(\mathbb{C}^{\times})^{S},\quad
g\longmapsto(\chi_{s}(g))_{(s,k)\in S}.$

As $\mathbb{C}[(\mathbb{C}^{\times})^{S}]\to\mathbb{C}[G]$ is injective,
$a_{S}$ is dominant. Moreover, because $(\chi_{s})_{(s,k)\in S}$ generates
$X^{*}(G){Q}$, the morphism $a_{S}$ is finite. This implies that $a_{S}$ is
surjective. By our Assumption 2.4, $G$ acts freely on the non-empty set

$\\{v\in V^{\operatorname{st}}(G,\theta)\mid\operatorname{supp}(v)=S\\}.$

As a consequence, $a_{S}$ is also injective. In characteristic zero, a
bijective morphism of algebraic groups is an isomorphism. We define

$\rho:=a_{S}^{-1}\operatorname{pr}_{S}c,$

where $\operatorname{pr}_{S}\colon T\to(\mathbb{C}^{\times})^{S}$ is the
projection. Then

$a_{S}\rho=a_{S}a_{S}^{-1}\operatorname{pr}_{S}c=\operatorname{pr}_{S}c,$

which means $\chi_{s}(\rho(t))=c_{s,k}(t)$ for all $(s,k)\in S$. We have shown
that $S\subseteq S_{\rho}$. But $S$ is $\theta$-stable, so $S_{\rho}$ is as
well. By (1), it is then minimally $\theta$-stable, which forces $S=S_{\rho}$.
∎

Proposition 9.6 shows that the description of the fixed points using the toric
fan agrees with our description. More precisely, we have the following.

######  Corollary 9.7.

The assignment $\rho\mapsto S_{\rho}^{c}$ defines a bijection between the set

$\\{\rho\colon\mathscr{T}\to G\mid\rho\text{ morphism of algebraic groups such
that }V_{\rho}^{\operatorname{st}}(G,\theta)\neq\emptyset\\}$

and the set of maximal members of ${}^{\operatorname{st}}(\theta)$. Moreover,
for every $\rho\colon\mathscr{T}\to G$ for which
$V_{\rho}^{\operatorname{st}}(G,\theta)\neq\emptyset$, let $J:=S_{\rho}^{c}$.
Then the subsets $V_{\rho}^{\operatorname{st}}(G,\theta)/G$ and
$O_{\sigma_{J}}$ of $V^{\operatorname{st}}(G,\theta)/G$ agree and consist of a
single point.

###### Proof.

The bijectivity of $\rho\mapsto S_{\rho}^{c}$ was already established in
Proposition 9.6. The second statement follows from the description (# ‣ 9) of
$O_{\sigma_{J}}$ and from

$V_{\rho}^{\operatorname{st}}(G,\theta)=\\{v\in
V\mid\operatorname{supp}(v)=S_{\rho}\\}.\qed$

We finish this section, and the paper, with an example.

######  Example 9.8.

Fix a number $d\in\mathbb{Z}_{\geq 0}$. Consider the vector space
$V=\mathbb{C}^{4}$ with the action of $G=(\mathbb{C}^{\times})^{2}$ given by

$(g,h)\cdot(x_{0},x_{1},y,z)=(gx_{0},gx_{1},hy,g^{d}hz).$

This means, as a $G$-representation,
$V=\mathbb{C}(1,0)^{2}\oplus\mathbb{C}(0,1)\oplus\mathbb{C}(d,1)$. Let
$\theta\colon G\to\mathbb{C}^{\times}$ be the character given by
$\theta(g,h)=g^{d+1}h$. Using the Hilbert–Mumford criterion, we can show that

$V^{\operatorname{sst}}(G,\theta)=V^{\operatorname{st}}(G,\theta)=\\{v=(x_{0},x_{1},y,z)\mid(x_{0},x_{1})\neq(0,0)\neq(y,z)\\}.$

We see that the quotient $V^{\operatorname{st}}(G,\theta)/G$ is isomorphic to
$\mathbb{P}(\mathscr{O}_{\mathbb{P}^{1}}\oplus\mathscr{O}_{\mathbb{P}^{1}}(d))$,
the $d^{\mathrm{th}}$ Hirzebruch surface.

Observe that $G$ acts freely on the stable locus. So Assumption 2.4 is
fulfilled. We get $T=(\mathbb{C}^{\times})^{4}$, and the morphism $a\colon
G\to T$ is given by

$a(g,h)=(g,g,h,g^{d}h).$

The morphism $\pi\colon T\to\mathscr{T}=(\mathbb{C}^{\times})^{2}$ defined by
$\pi(z_{1},\ldots,z_{4})=(z_{1}z_{2}^{-1},z_{3}z_{2}^{d}z_{4}^{-1})$ is a
cokernel of $a$. Let $I=\\{1,\ldots,4\\}$ and
$\varepsilon_{1},\ldots,\varepsilon_{4}$ be the basis of $X_{*}(T)$ consisting
of the embeddings in the $i^{\mathrm{th}}$ diagonal entry. We read off

$\Phi=\\{J\subseteq\\{1,\ldots,4\\}\mid\\{1,2\\}\nsubseteq
J\nsupseteq\\{3,4\\}\\},$

and by Proposition 9.5, the toric fan is
$\Delta=\\{\operatorname{cone}(\pi^{*}\varepsilon_{i})_{i\in J}\mid
J\in\Phi\\}$. The maximal elements of are $\\{1,3\\}$, $\\{2,3\\}$,
$\\{1,4\\}$, and $\\{2,4\\}$. They correspond to the fixed points

$\pi(0,1,0,1),\quad\pi(1,0,0,1),\quad\pi(0,1,1,0),\quad\pi(1,0,1,0).$

A section of $\pi$ is given by $c\colon\mathscr{T}\to T$, which sends
$(t_{1},t_{2})$ to $(t_{1},1,t_{2},1)$. For $\rho\colon\mathscr{T}\to G$, we
obtain

$V_{\rho}=\\{v\in V\mid c(t)v=a\rho(t)v\text{ (all $t\in\mathscr{T}$)}\\}.$

Let $\rho(t)=(t_{1}^{m_{1}}t_{2}^{m_{2}},t_{1}^{n_{1}}t_{2}^{n_{2}})$. Then
for $v=(v_{1},\ldots,v_{4})$, the condition $c(t)v=a\rho(t)v$ is equivalent to
the four equations

$\displaystyle t_{1}v_{1}$ $\displaystyle=t_{1}^{m_{1}}t_{2}^{m_{2}}v_{1},$
$\displaystyle v_{2}$ $\displaystyle=t_{1}^{m_{1}}t_{2}^{m_{2}}v_{2},$
$\displaystyle t_{2}v_{3}$ $\displaystyle=t_{1}^{n_{1}}t_{2}^{n_{2}}v_{3},$
$\displaystyle v_{4}$
$\displaystyle=t_{1}^{dm_{1}+n_{1}}t_{2}^{dm_{2}+n_{2}}v_{4}.$

There are four possibilities for $\rho$ which admit a non-empty stable locus
$V_{\rho}^{\operatorname{st}}(G,\theta)$. They are
$\rho_{1}(t)=(t_{1},t_{2})$, $\rho_{2}(t)=(1,t_{2})$,
$\rho_{3}(t)=(t_{1},\smash{t_{1}^{-d}})$, and $\rho_{4}(t)=(1,1)$. They give
the fixed points

$\pi(1,0,1,0),\quad\pi(0,1,1,0),\quad\pi(1,0,0,1),\quad\pi(0,1,0,1).$

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