1$. \end{lemm} \begin{proof} First, say $\sum_{n\,\geqs\,1} d_n>1$. Let $V = \bigoplus_{n=-N}^{N}\C^{d_n}$ and let $X = V\aq \C^*$. The standard contracting homotopy of $V$, namely $H_t (v) = tv$, induces a contracting homotopy of $X$, so in particular $X$ has the homology of a point. Moreover, Lemma~\ref{hom-loc-model-lem} shows that the rational homology of $(\bigoplus_{n\,\neq\,0}\C^{d_n})\aq\C^*$ is finitely generated, and is not that of a sphere. By Lemma~\ref{homology-lem}, the homology of $X-\{[0]\}$ is not that of a sphere either. However, if $X$ had a Euclidean neighborhood $U \isom \bbR^k$ around $[0]$, we would~have \[ H_* (X, X - \{[0]\}) \isom H_* \left(\bbR^k, \bbR^k - \{0\}\right) \] by excision, and by comparing the long exact sequences of these pairs we find that $H_* (X - \{[0]\}) \isom H_* (\bbR^k - \{0\}) \isom H_* (S^{k-1})$. Hence $X$ has a topological singularity at $[0]$. Conversely, if $\sum_{n\,\geqs\,1} d_n=1$, then $(\C\oplus\C^{d_0}\oplus \C)\aq \C^*\cong \C^{d_0} \cross (\C \oplus \C)\aq\C^*$, where on the right $\C^*$ acts with weights $-n$ and $n$ on the left and right factors (respectively). The multiplication map $\C\cross \C\to \C$ induces a homeomorphism \[ (\C \oplus \C)\aq\C^* \srm{\isom} \C, \] so we find that $(\C\oplus\C^{d_0}\oplus \C)\aq \C^*$ is homeomorphic to $\C^{d_0+1}$ in this case. \end{proof} \subsection{Algebraic and Topological Singularities} Richardson showed the singular locus of $\X_r(G)$ is precisely the union of the bad locus with the reducible locus if $G$ is semisimple, $r\geqs 2$ and the Lie algebra of $G$ does not have any rank 1 simple factors~\cite{Ri}. \cite[Theorem~7.4]{FLR} extends Richardson's theorem to connected reductive groups $G$ such that the simple factors of the Lie algebra of its derived subgroup $DG$ have rank 2 or more (addressing~\cite[Conjectures~3.34 and~4.8]{FL2}). In this subsection, we generalize these results by showing that if $r\geqs 3$ and $G$ is connected and reductive (but allowing its derived subgroup to have local rank 1 factors), then the singular locus coincides with the union of the bad locus with the reducible locus. We also generalize results in~\cite{FL2}, by showing all algebraic singularities in $\X_r(G)$ are in fact topological (ugly); which shows that $\X_r(G)$, as a class of varieties, have properties in common with normal surfaces (\cite{Mumford}). \begin{rema}\label{rankexamples} \cite[Remark~3.33]{FL2} shows that $\XC{2}(\p\SL_2(\C))$ has smooth points which are reducible and singular points which are irreducible; so a condition on the rank of $G$ when $r=2$ is necessary. \cite[Examples~7.2 and~7.3]{FLR} show that there are Lie groups $H$ of arbitrarily large rank with the property that $\XC{2}(H)$ has smooth reducibles and singular irreducibles (and $H$ does not have to be a product with a rank 1 Lie group for this to happen; although $H$ does need to be a local product with a rank 1 Lie group). \end{rema} \begin{theo}\label{conj-thm} Let $G$ be a connected, reductive $\C$-group. Assume either $r\geqs 2$ and the Lie algebra of $DG$ has no rank 1 simple factors, or that $r\geqs 3$ with no additional conditions on $G$. Then \[ \XC{r}(G)^{sing}=\XC{r}(G)^{red}\cup\X_r(G)^{bad}, \] and all points in $\XC{r}(G)^{bad}$ are orbifold singularities. \end{theo} \begin{proof} We prove the theorem in the following steps: \begin{enumerate} \item\label{prooftheo5.10.1} $\XC{r}(G)^{red}\subset \XC{r}(G)^{sing}$, \item\label{prooftheo5.10.2} $\X_r(G)^{bad}\subset \XC{r}(G)^{sing}$, and all points in $\XC{r}(G)^{bad}$ are orbifold singularities, \item\label{prooftheo5.10.3} $\XC{r}(G)^{good}=\clX_r (G)$. \end{enumerate} If $r\geqs 2$ and the Lie algebra of $DG$ has only simple factors of rank 2 or more, then this is the content of Richardson~\cite{Ri} in the case with $G$ is semisimple, and~\cite[Theorem~7.4]{FLR} in the case when $G$ is connected and reductive. So we now assume $r\geqs 3$. The second part of~\eqref{prooftheo5.10.2} follows from the first part of~\eqref{prooftheo5.10.2} since $\X_r(G)^{irr}$ is always an orbifold and $\X_r(G)^{good} \subset \clX_r (G)$ is always a smooth manifold. This also shows that~\eqref{prooftheo5.10.3} follows from~\eqref{prooftheo5.10.1} and~\eqref{prooftheo5.10.2}. \eqref{prooftheo5.10.1}: Let $G$ be a connected, reductive $\C$-group, $DG$ the derived subgroup $[G,G]$, and $\widetilde{DG}$ the universal cover of $DG$. Then $\widetilde{DG}=\prod_i G_i$ is a finite product of simple Lie groups and~\cite[Proposition~2.9]{FL4} says $\X_r(\prod_i G_i)\cong \prod_i\X_r(G_i)$. Now, for $r\geqs 3$ and $G$ is simple, we have $\X_r(G)^{red}\subset \X_r(G)^{sing}$: for $\Rank(G)\geq 2$ this follows by~\cite[Theorem~7.4]{FLR}, and for $\Rank(G) = 1$ it follows by\cite[Theorem~3.21]{FL2} and~\cite[Corollary~7.9]{FLR}. Observe that $[\rho]\in\X_r(\prod_i G_i)\cong \prod_i\X_r(G_i)$ is reducible if and only if it is reducible in some $\X_r(G_i)$, and likewise it is singular if and only if it is singular in some $\X_r(G_i)$. Thus, if $r\geqs 3$, we have $\X_r(\widetilde{DG})^{red}\subset \X_r(\widetilde{DG})^{sing}$. However, \cite[Theorem~7.8]{FLR} proves that if $\X_r(\widetilde{DG})^{red}\subset \X_r(\widetilde{DG})^{sing}$, then $\X_r(G)^{red} \subset \X_r(G)^{sing}$, proving item~\eqref{prooftheo5.10.1}. \eqref{prooftheo5.10.2}: We will use the following lemma from~\cite{NR}: \begin{lemm}{\cite[Lemma~4.4]{NR}}\label{NR-lem} Let $f:X\to Y$ be a morphism from an $n$-dimensional complex manifold onto a normal $n$-dimensional variety $Y$. If the set $S$ of points at which $f$ is not locally injective is of codimension at least 2, then $f(S)$ is the singular set of $Y$. \end{lemm} Let $[\rho]\in \X_r(G)$ be bad. Then $\rho$ is polystable since $\rho$ is irreducible and all irreducible representations are stable. By Lemma~\ref{sing-lem}, there is a local model of $[\rho]\in\X_r(G)$ of the form $V\aq\Gamma:=H^1(\F_r;\fkg_{\Ad_\rho})\aq \stab(\rho)$, where $\Gamma:=\stab(\rho)$ is the finite stabilizer of $\rho$ in $PG$. We note that $V$ has the same dimension as $\X_r(G)$ since $\rho$ is irreducible, and since $\Gamma$ is finite the dimension of $V\aq\Gamma$ is also equal to $\X_r(G)$. Since $\Gamma$ is finite, every point in $V$ where $\pi_\Gamma:V\to V\aq \Gamma$ is not locally injective has a non-trivial stabilizer (in fact, the reverse implication also holds) and so Proposition~\ref{nopseudo} allows us to apply Lemma~\ref{NR-lem}. We thus conclude that $[0]$ is singular in $V\aq \Gamma$ and therefore $[\rho]$ is singular in $\X_r(G)$ by Lemma~\ref{sing-lem}. Thus, the first part of item~\eqref{prooftheo5.10.2} holds, and as noted above the theorem follows. \end{proof} \begin{rema} We note that item~\eqref{prooftheo5.10.1} in the above proof resolves the first part of~\cite[Conjecture~3.34]{FL2}. We note (again) that~\cite[Examples~7.2 and 7.3]{FLR} show when $r=2$, item~\eqref{prooftheo5.10.1} may be false if $DG$ locally has rank 1 factors, so item~\eqref{prooftheo5.10.1} in Theorem~\ref{conj-thm} is sharp and cannot be generally improved. \end{rema} As shown in~\cite{FL2}, if $G=\SL_n(\C)$ or $\GL_n(\C)$ then generally reducible representations are ugly. We now generalize this result. \begin{theo}\label{redugly-thm} All reducible representations are ugly if $r\geqs 3$ for any connected, reductive $\C$-groups $G$, or if $r\geqs 2$ and the Lie algebra of $DG$ has only simple factors of rank 2 or more. \end{theo} \begin{proof} First note that if a point $[\rho]\in\X_r(G)$ is not ugly, then there exists a Euclidean open set around $[\rho]$ that is homeomorphic to a Euclidean ball. Thus, all points in that Euclidean set are also not ugly. And so the collection of non-ugly points in $\X_r(G)$ is an open set; that is, being ugly is a closed condition in $\X_r(G)$. Since the conjugate of an ugly representation is still ugly (as it gives the same point in $\X_r(G)$), we conclude that the ugly locus in $\hom(\F_r,G)$ is also closed. This alone shows that all reducibles are ugly in $\X_r(\SL_n(\C))$ if $r\geqs 3$ and $n\geqs 2$ or if $n\geqs 3$ and $r\geqs 2$, since~\cite{FL2} shows that a Euclidean dense set of reducibles is ugly in these cases. We now generalize this to arbitrary $G$. Let $G$ be a connected, reductive $\C$-group. We say that $L$ is \emph{quasi-irreducible} if it is a Levi subgroup of a maximal parabolic subgroup of $G$. Let $\rho :\F_r \to G$ be a representation. We say that $\rho$ is \emph{quasi-irreducible} if its Zariski-closure is quasi-irreducible. (This definition can be viewed as a generalization of the reducible representations shown to be ugly in~\cite{FL2}; namely the representations in $\SL_n(\C)$ conjugate to a representation having two non-trivial irreducible blocks.) Up to conjugation we have only finitely many maximally non-irreducible subgroups in $G$ (at most the number of conjugacy classes of maximal parabolic subgroups of $G$; that is, the rank of $G$). We claim that quasi-irreducible representations are generic. Precisely, the subset of quasi-irreducible representations $\hom^{qi}(\F_r,G)$ is Euclidean dense in the (constructible) subset of polystable, reducible representations in $\hom (\F_r, G)$. This simply comes from the fact that completely reducible and non-irreducible representations $\rho:\F_r \to G$ have to be contained in a parabolic subgroup of $G$ and therefore a maximal parabolic subgroup. Since they are also completely reducible they factor through the inclusion of a maximally non-irreducible subgroup of $G$. Now the result follows from the general fact that Zariski-dense representations of a free group of rank at least 2 in a connected reductive group are Euclidean dense in the representation variety of that reductive group by Proposition~\ref{zd-prop}. Now, we prove that quasi-irreducible representations are ugly. First let us begin with the Zariski-closure of $\rho(\F_r)$, which we denote by $L$. Let us show that $Z_G(L)/Z(G)$ is isomorphic to $\C^*$. Fix a Cartan subgroup $H$ of $G$. Up to conjugation we may assume that $H$ is a subgroup of a Levi subgroup $L$ of the maximal parabolic subgroup $P$. Since $H$ contains the center of $G$, we may write its Lie algebra as $\fkz\oplus \fkh$, where $\fkz$ is the center of $\fkg$. Let $\fkl \subset \fkp \subset \fkg$ denote the Lie algebras of $L$ and $P$. Recall that we have the following decomposition: \[ \fkg=\fkz\oplus \fkh\oplus \bigoplus_{\alpha\,\in\,\Delta}\fkg_{\alpha}, \] where as usual, $\Delta$ is the set of generalized eigenvalues of $\fkh$ (these are linear forms on $\fkh$) and $\fkg_{\alpha}$ are the corresponding generalized eigenspaces of dimension~$1$. We fix a non-zero vector $X_{\alpha}$ in each. We recall that we may choose in $\Delta$ a set of simple roots $\alpha_1,\,\dots, \,\alpha_m$ so that any root is a positive sum of these $\alpha_i$ or a negative sum of these $\alpha_i$. Define $(H_1,\,\dots,\,H_m)$ to be the dual basis in $\fkh$ to $(\alpha_1,\,\dots,\,\alpha_m)$ in $\fkh^*$. For a fixed $\alpha_i$, we may construct: \[ \fkp_i:=\fkz\oplus\fkh\oplus \bigoplus_{\alpha\,\in\, \Delta^+}\fkg_{\alpha}\oplus \bigoplus_{\alpha\,\in\, \Delta^-\cap\,\left\langle \alpha_j\,\middle|\,j\,\neq\,i\right\rangle}\fkg_{\alpha}. \] Up to conjugation, the Lie algebras of maximal parabolic subgroups are of the form above. Thus we may assume that $\fkp=\fkp_i$. Finally, if we define \[ \fkl_i:=\fkz\oplus\fkh\oplus \bigoplus_{\alpha\,\in\, \Delta\,\cap\,\left\langle \alpha_j\middle|\,j\,\neq\,i\right\rangle}\fkg_{\alpha}, \] then $\fkl_i$ is conjugate to the Lie algebra of each Levi subgroup of $P$. Thus we may assume that $\fkl=\fkl_i$. Now note that $P=\Rad_U(P)\rtimes L$ and since both $P$ and its unipotent radical $\Rad_U(P)$ need to be connected, $L$ is connected as well. It follows that for $x\in G$, $x\in Z_G(L)$ if and only if $x\in Z_G(\fkl)$. Furthermore, $L$ contains the Cartan subgroup $H$, which is its own centralizer. It follows that if $x\in Z_G(L)$ then $x\in Z_G(H)=H$. Therefore, we may write any element $x\in Z_G(L)$ as $x=z\exp_G(\lambda_1H_1+\cdots+\lambda_m H_m)$ where $\lambda_1,\,\dots,\,\lambda_m\in \C$ and $z\in Z(G)$. Finally, any $x$ of this form will commute with $\fkh$ and for $\alpha=n_1\alpha_1+\,\cdots\,+n_m\alpha_m\in \Delta$, we have \[ \Ad(x)\cdot X_{\alpha}=\exp \left(\alpha\left(\lambda_1H_1+\cdots+\lambda_mH_m\right)\right)X_{\alpha}=\exp\left(n_1\lambda_1+\cdots+n_m\lambda_m\right)X_{\alpha}. \] Thus in the case $\alpha = \alpha_j$, we see that in order to have $\adl(x)\cdot X_{\alpha_j}=X_{\alpha_j}$, we need $\lambda_j\in 2\pi \sqrt{-1}\bbZ$. It follows that $x=z\exp_G(X) \in \fkh$ belongs to $Z_G(L)$ if and only if $X\equiv \lambda H_i$ in $\fkh/\langle 2\pi\sqrt{-1}H_j\mid 1\leq j\leq r\rangle_{\bbZ}$, where $\langle 2\pi\sqrt{-1}H_j\mid 1\leq j\leq r\rangle_{\bbZ}$ is the additive subgroup of $\fkh$ generated by the elements $2\pi\sqrt{-1}H_j$. It follows that $Z_G(L)$ is generated by $Z(G)$ and a $1$-parameter subgroup $t\mapsto \exp(tH_i)$, whence $Z_G(L)/Z(G)$ is isomorphic to $\C^*$. We now denote for $n\neq 0$, \begin{align*} \fku_n:=& \bigoplus_{\substack{\alpha\,\in\,\Delta\\ \alpha\,=\,n\alpha_i\,+\,\sum\limits_{j\,\neq\,i}n_j\alpha_j}}\fkg_{\alpha}. \\ \intertext{Then it is clear that} \fkg=& \fkl\oplus \bigoplus_{n\,\neq\,0}\fku_n. \end{align*} Furthermore, with respect to $\C^*\cong Z_G(L)/Z(G)$, $\C^*$ acts trivially on $\fkl$ and for $\lambda\in \C^*$ and $v\in \fku_n$, we have $\lambda\cdot v=\lambda^n v$. Now $\stab(\rho)\leq PG$ is exactly $Z_G(L)/Z(G)\cong\C^*$, so the infinitesimal action on $H^1(\F_r;\fkg_{\Ad_\rho})$ is given by the action (as above) on the corresponding coefficients of: \[ H^1\left(\F_r;\fkg_{\Ad_\rho}\right)=H^1\left(\F_r;\fkl_{\Ad_\rho}\right)\oplus \bigoplus_{n\,\neq\,0}H^1\left(\F_r;(\fku_n)_{\Ad_\rho}\right), \] that is, $\C^*$ acts trivially on $H^1(\F_r;\fkl_{\Ad_\rho})$ and acts on $H^1(\F_r;(\fku_n)_{\Ad_\rho})$ by $\lambda\cdot v=\lambda^nv$. By Lemma~\ref{sing-lem}, $[\rho]$ is ugly if and only if $[0]$ is a topological singularity in \[ H^1\left(\F_r;\fkg_{\Ad\rho}\right)\aq\stab(\rho), \] and we have established that, for some $N>0$, \[ H^1\left(\F_r;\fkg_{\Ad_\rho}\right)\aq \stab(\rho)\cong \left(\bigoplus_{n\,=\,-N}^{N}\C^{d_n}\right)\aq\C^*\cong \C^{d_0}\cross \left(\bigoplus_{n\,\neq\,0}\C^{d_{n}}\right)\aq\C^*, \] as in Lemma~\ref{weighted-lem} below (we include $H^1(\F_r;\fkl_{\Ad_\rho})$ as the factor $\C^{d_0}$). Now by direct computation we have: \begin{align*} \dim_\C H^1\left(\F_r;(\fku_n)_{\Ad_\rho}\right)&=\dim_\C Z^1\left(\F_r,(\fku_n)_{\Ad_\rho}\right)-\dim_\C B^1\left(\F_r,(\fku_n)_{\Ad_\rho}\right)\\ &=\dim_\C Z^1\left(\F_r,(\fku_n)_{\Ad_\rho}\right)-\dim_\C \fku_n+\dim_\C Z^0\left(\F_r,(\fku_n)_{\Ad_\rho}\right)\\ &=r\dim_\C \fku_n-\dim_\C \fku_n+\dim_\C \fku_n^{\rho(\F_r)}\\ &=(r-1)\dim_\C \fku_n, \end{align*} because the Zariski-closure of $\rho(\F_r)$ contains a Cartan subgroup of $G$ which has no non-zero fixed point on $\fku_n$. Since $\dim_\C \fku_n\geqs 1$, Lemma~\ref{weighted-lem} finishes the proof so long as $r\geqs 3$. Finally, if $r=2$, it suffices to prove that $\sum_{n\,\geq\,1}\dim_\C \fku_n\geqs 2$ provided that $\fkg$ has no simple factor of rank $1$. This is equivalent to the fact that $\fkg$ has at least two positive roots. One may easily check this for any simple complex Lie algebra $\fkg$ which is not $\fksl_2$ and therefore we may again apply Lemma~\ref{weighted-lem} to complete the proof. \end{proof} \begin{rema} The first paragraph in the proof of Theorem~\ref{redugly-thm}, that the ugly locus is closed, proves~\cite[Conjecture~4.8]{FL2} to be true since~\cite{FL2} shows that a Euclidean dense set of reducibles is ugly in the cases considered in~\cite[Conjecture~4.8]{FL2}. \end{rema} \begin{rema} We note that with respect to the actual application of Lemma~\ref{weighted-lem} to Theorem~\ref{redugly-thm}, that $d_1\geqs 1$ since there is always a simple root with eigenvalue $1$. Moreover, the situation when $\sum_{n\,\geqs\,1} d_n=1$ does in fact occur in \[ \hom\left(\F_2,\SL_2(\C)\right)\aq \SL_2(\C)\cong \C^3, \] see~\cite{FL2}. \end{rema} A priori it is not clear if bad representations are ugly or not. Such a representation has a local model that is a finite group quotient of affine space. By the Chevalley--Shephard--Todd Theorem~\cite{Che,ShTo}, the quotient is smooth if and only if the finite group is generated by pseudoreflections. But such a quotient, even if singular, can be topologically a manifold as the next example shows. \begin{exam}\label{starr-ex} Let $\Gamma$ be the binary icosahedral group; that is, the group of symmetries of the icosahedron (this group is of order 120, and is isomorphic to $\SL_2 (\bbF_5)$). The rotational symmetries of the icosahedron are naturally a subgroup of $\SO(3)$ and $\Gamma$ is the inverse image of this subgroup under the double covering $\SU(2) \to \SO(3)$. We call the inclusion \[ \Gamma\injects \SU(2) \injects \GL_2 (\C) \] the ``standard representation.'' Now consider the homomorphism $\alpha:\Gamma\to \GL_3(\C)$ equal to the direct sum of the standard representation of $\Gamma$ with a trivial representation. Then $\alpha$ defines an action on $\C^3$, and the quotient $\C^3/\Gamma$ is a normal, topological manifold that is algebraically singular~\cite[Jason Starr's example]{MOsing}. \end{exam} We now show that bad representations are ugly if $r\geqs 3$, which implies the situation of the above example does not occur for the varieties $\X_r(G)$. \begin{theo}\label{badugly-thm} Let $G$ be a connected, reductive $\C$-group, and assume either $r\geqs 3$, or $r\geqs 2$ and the Lie algebra of $DG$ has only simple factors of rank 2 or more. Let $[\rho]\in \X_r(G)$. If $\rho$ is bad, then $\rho$ is ugly. \end{theo} \begin{proof} We repeat the set-up from Lemma~\ref{sing-lem}. Let $[\rho]\in \X_r(G)$ be bad; we assume that $\rho$ is polystable~(in fact, stable). There is a local model of $[\rho]$ of the form $V\aq \Gamma$, where $V:=H^1(\F_r;\fkg_{\Ad_\rho})$ is a $\C$-vector space, $[0]$ corresponds to $[\rho]$, and $\Gamma:=\stab(\rho)$ is a finite subgroup of $PG$ acting linearly (and so fixes $0$). Since the ugly locus is closed and the bad locus is closed, it suffices to show that generic bad representations are ugly. The set of bad representations with abelian stabilizer is generic by Proposition~\ref{ab-gen}, so we will assume $\Gamma$ is abelian. Under the assumptions of the theorem, Proposition~\ref{nopseudo} now shows that the action of $\Gamma$ on $V$ does not include any pseudoreflections. We also note that if $\Gamma$ does not act effectively (faithfully), then $\Delta:=\cap_{v\,\in\,V}\stab_\Gamma(v)$ is a normal subgroup and $V\aq\hat{\Gamma}\cong V\aq \Gamma$ where $\hat{\Gamma}:=\Gamma/\Delta$. Since we have shown that bad representations are singular, we know $\Gamma$ cannot act trivially on $V$. So without loss of generality, we may assume $\Gamma$ is a non-trivial abelian group, acting effectively. Let $F \subset V$ denote the set of vectors fixed by some non-trivial element of $\Gamma$; this is a union of linear subspaces, invariant under the action of $\Gamma$. Since $\Gamma$ does not act by pseudoreflections, $F$ has codimension at least 2. We now prove that $V\aq \Gamma$ has a topological singularity at $[0]$. In what follows we replace $\aq$ with $/ $ since $\Gamma$ is finite and hence these quotients are equivalent. Say $\dim_\C (V) = n$. For a space $X$, let $X^+$ denote its one-point compactification. Then $V^+ \homeo S^{2n}$, and since $F$ is a union of linear subspaces, $F^+$ is a union of spheres (of dimension at most $2n-4$). We claim $(\star)\ F^+/\Gamma \isom (F/\Gamma)^+$ is locally contractible, and $(\star\, \star)$ has no integral homology in dimensions greater than $2n-3$. We prove $(\star)$ and $(\star\, \star)$ below, but first let us see how they are used. Since $\Gamma$ acts freely on the complement of $F$, the quotient map $V - F \to V/\Gamma - F/\Gamma$ is a covering map, and $V - F$ is simply connected because $F$ is a union of smooth submanifolds of $V$, each with real codimension greater than 2. So $\pi_1 (V/\Gamma - F/\Gamma) = \Gamma$, and since $\Gamma$ is abelian, $H_1 (V/\Gamma - F/\Gamma) = \Gamma$ as well. We have $(V/\Gamma)^+ - (F/\Gamma)^+ = V/\Gamma - F/\Gamma$. Assume $F \neq \{0\}$; the case $F = \{0\}$ is easier and treated at the end. If $V/\Gamma$ is a topological manifold around $[0]$, then by Proposition~\ref{sing-lem}, $V/\Gamma$ is homeomorphic to a Euclidean space; by dimension count, we must in fact have $V/\Gamma\homeo V$, and so $(V/\Gamma)^+ \homeo S^{2n}$. By $(\star)$, we may apply Alexander Duality, yielding: \[ H_1 (V/\Gamma - F/\Gamma) \cong H_1 \left((V/\Gamma)^+ - (F/\Gamma)^+\right) \cong H^{2n-2} \left((F/\Gamma)^+\right). \] This contradicts $(\star\, \star)$, since the left hand side is non-zero but the right hand side is zero. If $F = \{0\}$, then $\Gamma$ acts freely on $V - \{0\} \heq S^{2n-1}$ and so $\pi_1 (V/\Gamma - \{[0]\})\isom \Gamma$. This is also a contradiction, because $V/\Gamma - \{[0]\} \homeo \C^n - \{0\} \heq S^{2n-1}$. Therefore, we have shown that if $\rho$ is a bad representation such that $\stab(\rho)$ is abelian and acts on $V$ without pseudoreflections, then $\X_r (G)$ has a topological singularity at $[\rho]$. Since such representations form a generic subset of the bad locus, we have shown that bad representations are ugly (under the hypotheses of the theorem). Proof of $(\star)$: Local contractibility at all points other than $\infty$ is immediate since $F/\Gamma$ is an algebraic subset of $V/\Gamma$, hence triangulable (see~\cite{Hofmann}). Let $|-|$ be a $\Gamma$-invariant norm on $V$ (as in the proof of Lemma~\ref{sing-lem}). We claim that $U_N:=F^+ \cap \{v\in V : |v| > N\}$ admits a $\Gamma$-equivariant deformation retraction to $\infty\in F^+$, which then descends to a deformation retraction of $U_N/\Gamma$ to $[\infty]\in F^+/\Gamma$ (giving the desired contractible neighborhoods around $[\infty]$). The deformation retraction is just given by sending $v \mapsto (1/(1-t))v$ at time $t$. This is $\Gamma$-equivariant since $\Gamma$ acts linearly. Proof of $(\star\, \star)$: We study the homology of $(F/\Gamma)^+$ using the Mayer--Vietoris sequence for the open cover consisting of $F/\Gamma$ and $(F/\Gamma)^+ - \{[0]\}$. The latter set is contractible (as proven in the previous paragraph), so it will suffice to show that $F/\Gamma$ and \[ F/\Gamma \cap \left((F/\Gamma)^+ - \{[0]\}\right) = F/\Gamma - \{[0]\} \] each have no integral homology in dimensions greater than $2n-4$. By the Universal Coefficient Theorem, it is enough to check this in cohomology. Both spaces are locally contractible, so their integral cohomology agrees with their \v{C}ech cohomology, and thus it is enough to verify that these spaces have topological covering dimension at most $2n-4$ (see~\cite{gentop} for a discussion \v{C}ech cohomology and covering dimension). But $F$ is a simplicial complex of dimension at most $2n-4$, and finite quotients do not increase covering dimension (\cite[Proposition~9.2.16]{Pears}) so $F/\Gamma$ also has covering dimension at most $2n - 4$, as does its open subspace $F/\Gamma- \{[0]\}$. This completes the proof of $(\star\, \star)$. \end{proof} We note that the binary icosahedral group has trivial abelianization, so the arguments in the proof of Theorem~\ref{badugly-thm} do not apply to Example~\ref{starr-ex}. \begin{rema} In~\cite{FLR}, it is shown that $\pi_2(\X_r(G))=0$ if $DG$ has its Lie algebra isomorphic to a product of special linear groups. Given the results in Section~\ref{Schur} this shows that if $G$ is a CI group, then $\pi_2(\X_r(G))=0$. As in the proof of the above Theorem~\ref{badugly-thm}, let $[\rho]\in \X_r(G)$ be bad and consider a local model of $[\rho]$ of the form $V\aq \Gamma$, where $V$ is a $\C$-vector space, $[0]$ corresponds to $[\rho]$, and $\Gamma$ is a finite subgroup of $PG$ acting linearly. The Topological Form of Zariski's Main Theorem (\cite{Redbook}) implies any path-connected open neighborhood around $[0]$ has its smooth part $($the complement of the singular locus$)$ path-connected. Let $S$ be the singular locus in $V\aq \Gamma$ $($which has codimension at least 2 by normality$)$, and let $p:V\to V\aq \Gamma$ be the quotient map $($which has path-lifting by~\cite{LR}$)$. By Proposition~\ref{nopseudo} the set of points where $\Gamma$ does not act locally injective is codimension at least 2, and so by Lemma~\ref{NR-lem}, $p^{-1}(S)$ is this collection. Let $U$ be a path-connected open neighborhood of $[0]$. Since $\{0\}=p^{-1}([0])$ and $[0]\in U$, we conclude that $p^{-1}(U)$ is path-connected and so by codimension $W:=p^{-1}(U-S)$ is path-connected too. Since $p$ is a quotient map, $W$ is saturated and so $\Gamma$ acts on $W$. It acts freely since we have removed the points that map to singularities. Since $\Gamma$ is finite and so discrete, we have that $p:W\to U-S$ is a non-trivial covering map. Thus, $U-S$ cannot be simply connected. Thus, the proof in~\cite{FLR} that $\pi_2(\X_r(\SL_n(\C)))=0$ cannot generalize to other reductive $\C$-groups $G$ if $G$ is not CI since the proof in~\cite{FLR} required the existence of neighborhoods, in particular around bad representations, whose smooth locus was simply connected $($which we just showed is impossible if $G$ is not CI since in that case there will always be bad representations$)$. \end{rema} \section{Homotopy Groups of Good Representations} In this section we compute the homotopy groups of the good locus of $\X_r(G)$, in a range of dimensions tending to infinity with $r$, when $G$ is a connected, reductive $\C$-group and when $r\geqs 2$. As per our previous results, in most of these cases the smooth locus $\clX_r(G)$ is equal to the good locus $\X_r(G)^{good}=\hom(\F_r,G)^{good}/G$. \begin{rema} We exclude the $r=1$ case, since if $r=1$ and $G$ is non-abelian, $\X_1(G)\cong T/W\times_F Z$ where $T$ is a maximal torus in $DG$, $W$ is its Weyl group, $Z$ is the center in $G$, and $F=DG\cap Z$. In this case, $Z$ is a complex torus and $T/W$ is contractible. Moreover, the irreducible locus and hence the good locus is empty in this case. When $G$ is abelian, the topology is also understood. In this case, $\X_r(G)\cong (\C^*)^{rn}$ since $G\cong (\C^*)^n$, and all representations are good. The results in this section are trivially true when $G$ is abelian, and so in our proofs we only treat the non-abelian case. \end{rema} We begin by reviewing some lemmata in~\cite{BL,FLR} that we will find useful. \begin{lemm}[{\cite[Lemma~4.4]{FLR}}]\label{nullhomotopy} Let $G$ be a connected Lie group, and assume $r\geqs 2$. Then for each $\rho \in G^r$, the map $PG\to G^r$ given by $[g]\mapsto [g]\rho[g]^{-1}$ is nullhomotopic. \end{lemm} %\pagebreak \begin{lemm}[{\cite[Lemma~2.2]{BL}}]\label{lem22BL} Let $G$ be a connected, reductive $\C$-group, and assume $r\geqs 2$. Then \[ \hom\left(\F_r,G\right)^{good}\to \XC{r}(G)^{good} \] is a principal $PG$-bundle. \end{lemm} Lemma~\ref{nullhomotopy} is proven by considering a path $\rho_t$ from $\rho$ to the trivial representation. Lemma~\ref{lem22BL} follows from the fact that the conjugation action of $PG$ on the good locus is free and proper. Now, we put together the main theorem from a previous section (Theorem~\ref{codimbad}) with a generalization of~\cite[Theorem~2.9]{FLR}. \begin{lemm}\label{lem-gencodim} Let $G$ be a connected, reductive $\C$-group, and assume $r\geqs 2$. Then \[ \codim_\C\hom\left(\F_r,G\right)^{red}\geqs (r-1)\Rank(DG). \] \end{lemm} \begin{proof} This result is essentially contained in the proof of~\cite[Theorem~2.9]{FLR}; we briefly outline the computation. First, one finds (similar to Lemma~\ref{codim}) that \[ \codim_\C \left(\hom\left(\F_r,G\right)^{red} \right) \geqs (r-1)\left(\dim_\C \left(G/P_{max}\right)\right), \] where $P_{max}$ is a maximal dimensional proper parabolic subgroup of $G$. The Bruhat decomposition of the flag variety $G/P_{max} \isom DG/(P_{max}\cap DG)$ shows this quotient contains a maximal torus $T$ of $DG$, giving \[ \codim_\C \left(\hom(\F_r,G)^{red} \right) \geqs (r-1)\left(\dim_\C \left(G/P_{max}\right)\right)\geqs(r-1)\Rank(DG). \] \end{proof} Recall that by~\cite[Proposition~1.3]{JM}, the set $\hom(\F_r,G)^{good}$ is Zariski open in $\hom(\F_r,G)$, so its complement $\hom(\F_r, G)^{bad}\cup\hom(\F_r,G)^{red}$ is algebraic. We define \[ C_{pasbon}:=\codim_\R\left(\hom(\F_r, G)^{bad}\cup\hom(\F_r,G)^{red}\right). \] The authors emphasize that the codimension in the definition of $C_{pasbon}$ is considered over $\R$. Combining Theorem~\ref{codimbad} and Lemma~\ref{lem-gencodim} gives the following lower bound on $C_{pasbon}$. \begin{theo}\label{gencodim} Let \begin{align*} R_G &:=\min\left\{\Rank(G')\, \middle|\, G' \text{ is a simple factor of } \widetilde{DG}\right\}. \\ \intertext{Then} C_{pasbon} &\geqs 2\min\left\{2(r-1)R_G,(r-1)\Rank(DG)\right\}. \end{align*} In particular, $C_{pasbon}\geqs 2$ if $r\geqs2$ and $G$ is non-abelian, and $C_{pasbon}$\linebreak grows unboundedly in $r$ and in the minimum rank of a simple factor of $[\fkg,\fkg]$. \end{theo} We now turn our attention to homotopy groups, beginning with $\hom(\F_r,G)^{good}$. \begin{lemm}\label{homotopy-codim} Assume $r\geqs 2$. The inclusion map induces an isomorphism \[ \pi_k \left(\hom\left(\F_r, G\right)^{good}\right) \stackrel{\isom}{\maps} \pi_k \left(\hom\left(\F_r, G\right)\right)\isom \pi_k (G)^r \] for $k \leqs C_{pasbon} -2$, and is a surjection for $k=C_{pasbon} -1.$ \end{lemm} \begin{proof} Since $\hom(\F_r, G)^{bad}\cup\hom(\F_r,G)^{red}$ is Zariski closed in $\hom(\F_r,G)$, it is a finite union of locally closed submanifolds, and the codimension~$C_{pasbon}$ is a lower bound on the real codimension these submanifolds. Since $\hom(\F_r,G)\cong G^r$ is a smooth manifold, transversality shows that every map $S^k \to \hom(\F_r,G)$ with $k\leqs C_{pasbon}-1$ is homotopic to a map with image in the good locus, and for $k\leqs C_{pasbon}-2$ every homotopy between such maps can be deformed into the good locus.\footnote{Our use of transversality in this context is analogous to~\cite[Corollary~4.8]{Ramras}.} \end{proof} We now combine Lemmas~\ref{nullhomotopy} and~\ref{homotopy-codim}. \begin{lemm}\label{fiber-null} Let $r\geqs 2$. For any $\rho\in \hom(\F_r, G)^{good}$, the orbit-inclusion map $PG \to \hom(\F_r, G)^{good}$, $[g] \mapsto [g]\rho [g]^{-1}$, induces the zero map on homotopy groups in dimensions at most $C_{pasbon}-2$. \end{lemm} \begin{proof} By Lemma~\ref{nullhomotopy}, the composite map \[ PG \to \hom\left(\F_r, G\right)^{good} \to \hom\left(\F_r, G\right) \] is nullhomotopic, and by Lemma~\ref{homotopy-codim}, the second map in this composition is an isomorphism in the stated range. \end{proof} \begin{theo}\label{pi0pi1} Let $r\geqs 2$ and $G$ a connected, reductive $\C$-group. Then $\pi_0(\X_r(G)^{good})$ is trivial, and if $r\geqs 3$ or the rank of $DG$ is at least 2, then \begin{align*} \pi_1\left(\X_r(G)^{good}\right) &\cong \pi_1(G)^r \\ \intertext{and} \pi_2\left(\X_r(G)^{good}\right) &\cong \pi_1 (PG). \end{align*} \end{theo} \begin{proof} Since $\hom(\F_r,G)\cong G^r$ is irreducible, then so is $\hom(\F_r,G)^{good}$ as it is Zariski open. Thus, $\X_r(G)^{good}$ is irreducible too and hence connected. Since $\hom(\F_r,G)^{good}\to \X_r(G)^{good}$ is a principal $PG$-bundle, there is a long exact sequence in homotopy: \begin{align*} \cdots & \to \pi_2 (PG)\to \pi_2\left(\hom\left(\F_r,G\right)^{good}\right)\to \pi_2\left(\X_r(G)^{good}\right)\\ & \to \pi_1(PG)\to \pi_1\left(\hom\left(\F_r,G\right)^{good}\right) \to \pi_1\left(\X_r(G)^{good}\right)\to \pi_0(PG)=0. \end{align*} Lemma~\ref{fiber-null} and Lemma~\ref{homotopy-codim} then imply $\pi_1(\X_r(G)^{good})\cong \pi_1(G)^r$ as long as $1\linebreak\leqs C_{pasbon}-2$. Since $\pi_2 (G) = 0$, the same reasoning shows that $\pi_2(\X_r(G)^{good}) \isom \pi_1 (PG)$ when $1\leqs C_{pasbon}-2$. By Theorem~\ref{gencodim}, this bound holds if either $r\geqs 3$ or the rank of $DG$ is at least 2. \end{proof} \begin{rema} The fundamental group of $G$, always abelian, is the same as that of its maximal compact subgroup $K$ since $G$ deformation retracts to $K$. A standard result $($see~\cite{Hall}$)$ gives the fundamental group of $K$. Precisely, let $\fkk$ be the Lie algebra of $K$ and let $\fkt$ be a maximal commutative subalgebra of $\fkk$. Then $\pi_1(K)\cong \Gamma/\Lambda$, where $\Gamma$ is the kernel of the exponential mapping for $\fkt$ and $\Lambda$ is the lattice generated by the real co-roots. Thus, the fundamental group of $\X_r(G)^{good}$ is isomorphic to $(\Gamma/\Lambda)^r$ under the conditions of Theorem~\ref{pi0pi1}. \end{rema} With the first few homotopy groups computed for the good locus, we now turn our attention to the higher homotopy groups. By work in~\cite[Theorem~3.3]{FLR} it suffices to consider the case where $G$ is semisimple, since \[ \pi_k\left(\X_r(G)^{good}\right)\cong \pi_k\left(\X_r(DG)^{good}\right) \] for $k\geqs 2$. We also note that $\pi_k(PG)=\pi_k(G)$ for $k\geqs 2$ since $G\to PG$ is a fibration whose fiber (an algebraic torus cross a finite group) is $\pi_k$-trivial for $k\geqs 2$. This then implies $\pi_k(G)=\pi_k(DG)$ for $k\geqs 2$ since $PG=PDG$. \begin{theo}\label{splitting} Let $r\geqs 2$. Assume $1 \leqs k \leqs C_{pasbon}-2$. Then \[ \pi_k\left(\X_r(G)^{good}\right)\cong \pi_k(G)^r\times \pi_{k-1}(PG). \] \end{theo} \begin{proof} Since $\hom(\F_r, G)$ is an irreducible algebraic set, every non-empty Zariski open subset of $\hom(\F_r, G)$ is path connected, and it follows that $\XC{r}(G)^{good}$ is also path connected. By Lemma~\ref{lem22BL}, \[ PG\to \hom\left(\F_r, G\right)^{good}\to \XC{r}(G)^{good} \] is a $PG$-bundle (where the first map is the inclusion of an adjoint orbit). Hence we have an exact sequence \begin{equation}\label{es} \cdots \to \pi_1(PG)\to \pi_1(\hom(\F_r, G)^{good})\to \pi_1(\XC{r}(G)^{good})\to 0. \end{equation} When $2 \leqs k \leqs C_{pasbon}-2$, Lemma~\ref{fiber-null} implies the long exact sequence breaks into short exact sequences: \[ 0\to \pi_k\left(\hom\left(\F_r,G\right)^{good}\right)\to\pi_k\left(\X_r(G)^{good}\right)\to \pi_{k-1}(PG)\to 0. \] Lemma~\ref{homotopy-codim} implies $\pi_k(\hom(\F_r,G)^{good})\cong \pi_k(G)^r$ and so we can write the short exact sequences as: \begin{equation}\label{eq:SES} 0\to \pi_k(G)^r\to\pi_k\left(\X_r(G)^{good}\right)\to \pi_{k-1}(PG)\to 0. \end{equation} By Proposition~\ref{good-htpy} in the following Section~\ref{splitsection}, these short exact sequences split (non-canonically) if $1\leqs k\leqs C_{pasbon}-2$. \end{proof} \begin{coro}\label{pi34-cor} Assume $C_{pasbon}\geqs 6$ and $r\geqs 2$, and let $s$ be the number of simple factors of the Lie algebra of $DG$, and $t$ the number of those factors of type $A_1,B_1,$ or $C_n$ for $n\geqs 1$. Then $\pi_3(\X_r(G)^{good})\cong \Z^{sr}$ and $\pi_4(\X_r(G)^{good})\cong (\Z_2)^{rt}\times \Z^s.$ \end{coro} \begin{proof} The universal cover of $DG$ has the form $\prod_{i=1}^s G_i$ where each $G_i$ is simple and simply connected, and $s$ is the number of simple factors of the Lie algebra of $DG$. Since $\pi_k(G)\cong \pi_k(DG)$ for $k\geqs 2$, we conclude that $\pi_k(G)\cong \oplus_i\pi_k(G_i)$ for $k\geqs 2$. By~\cite{Bott56}, $\pi_3(G)=\Z$ if $G$ is simple. By~\cite{BottSam}, $\pi_4(G)=0$ or $\pi_4(G)\cong \Z_2$ if $G$ is simple; it is $\Z_2$ exactly when $G$ is of type $A_1$, $B_1$, or $C_n$ for $n\geqs 1$ (and 0 otherwise). Thus, because of the splitting in Theorem~\ref{splitting}, we have that \begin{align*} \pi_3\left(\X_r(G)^{good}\right) &\cong \Z^{sr}, \\ \intertext{and} \pi_4\left(\X_r(G)^{good}\right) &\cong (\Z_2)^{rt}\times \Z^s \end{align*} where $t$ is the number of simple factors of type $A_1,B_1,$ or $C_n$ for $n\geqs 1$ in the Lie algebra of $DG$. \end{proof} Note that by Theorem~\ref{gencodim}, if $r\geqs 4$, then $C_{pasbon}\geqs 6$. \begin{rema} Let $r\geqs 3$. Then the good locus is the smooth locus, and so the above theorem and corollary are true when replacing $\X_r(G)^{good}$ by the smooth locus $\clX_r(G)$. \end{rema} \begin{rema}\label{cor-per} By Theorems~\ref{gencodim} and~\ref{splitting}, if $G$ is simple then $\pi_k(\X_r(G)^{good})\cong \pi_k(G)^r\times \pi_{k-1}(PG)$ for $1\leqs k\leqs 2(r-1)\Rank(G)-2$. Given Bott periodicity for the classical groups $A_n$, $B_n$, $C_n$, and $D_n$~\cite{bott-per}, it follows that $\X_r(G)^{good}$ also exhibits periodic homotopy within appropriate stable ranges for the classical groups. \end{rema} See Example~\ref{ex-per} for a more precise formulation of Remark~\ref{cor-per}. \begin{exam} By similar reasoning as used in the proof of Corollary~\ref{pi34-cor}, if $r\geqs 2$ and $C_{pasbon}\geqs 7$ (which holds if $r\geqs 7$), we conclude that: \[ \pi_5\left(\X_r(G)^{good}\right)\cong \bigoplus_i^m\pi_5(G_i)^r\oplus (\Z_2)^t, \] where $t$ is the number of simple factors of type $A_1,B_1,$ or $C_n$ for $n\geqs 1$ in the universal cover of $DG$, itself a product of the simple simply connected $\C$-groups $G_1,\,\dots,\,G_m$. By results in~\cite{bott-per, BottSam,Mim-G2F4,MimTod, MimTod-Sym} we can calculate the fifth homotopy group of all simple $G$. In particular, if $G_i$ is exceptional, then $\pi_5(G_i)=0$. If $G_i$ is of type $A_n$, then $\pi_5(G_i)\cong \Z$ if $n\geqs 2$ and $\pi_5(G_i)\cong \Z_2$ if $n=1$. If $G_i$ is of type $B_n$, then $\pi_5(G_i)\cong 0$ if $n\geqs 3$, and $\pi_5(G_i)\cong \Z_2$ if $n=1,2$. If $G_i$ is of type $C_n$, then $\pi_5(G_i)\cong \Z_2$ for all $n\geqs 1$. If $G_i$ is of type $D_n$, then $\pi_5(G_i)\cong 0$ if $n=1$ or $n\geqs 4$, $\pi_5(G_i)\cong \Z$ if $n=3$, and $\pi_5(G_i)\cong (\Z_2)^2$ if $n=2$. So although it is not a clean formula, this completely describes the fifth homotopy groups of $\X_r(G)^{good}$. \end{exam} In short, if one knows the homotopy groups of $G$, then Theorem~\ref{splitting} allows one to compute the $k^{\rm th}$ homotopy groups of $\X_r(G)^{good}$ for sufficiently large $r$. As an example of this, we next list the $k^{\rm th}$ homotopy groups for $0\leqs k\leqs 15$ when $G$ is an exceptional Lie group. \begin{exam} We consider the complex adjoint type of each exceptional Lie group below. They are all simply connected except $E_6$ and $E_7$ with fundamental group $\Z_3$ and $\Z_2$ respectively. Since they are of adjoint type, $G=PG$ in each case. We assume that $r\geqs 2$ generally. However, if a cell is highlighted red in Table~\ref{splitting}, then we have assumed $r\geqs 3$, if it is highlighted orange then we have assumed $r\geqs 4$, if it is highlighted yellow then we have assumed $r\geqs 5$, and if it is highlighted green, then we have assumed $r\geqs 6$. A ``?'' in a cell of the table means that the homotopy groups needed for the computation are not known $($as far as we know$)$. Although for $E_6$, $E_7$, and $E_8$, for the cases where there is an ? and beyond, the $2$ and $3$-primary parts of the homotopy groups are known; see~\cite{Kachi, KaMi}. So one can obtain corresponding facts about the homotopy groups of $\X_r(G)^{good}$ in these cases. Our main sources of reference for the computations in Table~\ref{splitting}, aside from Theorem~\ref{splitting}, are~\cite{BottSam, Mim-G2F4}. \begin{table}[!ht] \caption{$\pi_k(\X_r(G)^{good})\cong \pi_k(G)^r\oplus\pi_{k-1}(PG)$}\label{exceptionalcases} \begin{tabular}{c||c|c|c|c|c} $k\setminus G$&$G_2$&$F_4$&$E_6$&$E_7$&$E_8$\\ \hline \hline $0$ & $0$& $0$&$0$ &$0$ &$0$ \\ \hline $1$ & $0$&$0$ &$(\Z_3)^r$ &$(\Z_2)^r$ & $0$\\ \hline $2$ &$0$ &$0$ &$\Z_3$ &$\Z_2$ & $0$\\ \hline $3$ &\cellcolor{red!25}$\Z^r$ & $\Z^r$& $\Z^r$& $\Z^r$&$\Z^r$ \\ \hline $4$ & \cellcolor{red!25}$\Z$& $\Z$& $\Z$& $\Z$&$\Z$ \\ \hline $5$ &\cellcolor{red!25}$0$ &$0$ & $0$&$0$ & $0$\\ \hline $6$ &\cellcolor{red!25}$(\Z_3)^r$ & $0$&$0$ &$0$ &$0$ \\ \hline $7$ &\cellcolor{orange!25}$\Z_3$ &\cellcolor{red!25} $0$& $0$& $0$&$0$ \\ \hline $8$ &\cellcolor{orange!25}$(\Z_2)^r$ &\cellcolor{red!25} $(\Z_2)^r$ &$0$ &$0$ &$0$ \\ \hline $9$ &\cellcolor{orange!25}$(\Z_6)^r\oplus\Z_2$ &\cellcolor{red!25}$(\Z_2)^{r+1}$ & $\Z^r$& $0$& $0$\\ \hline $10$ &\cellcolor{orange!25}$\Z_6$ &\cellcolor{red!25}$\Z_2$ & ?& $0$&$0$ \\ \hline $11$ &\cellcolor{yellow!25} $\Z^r\oplus (\Z_2)^r$&\cellcolor{red!25}$\Z^r\oplus (\Z_2)^r$ & ?&$\Z^r$ &$0$ \\ \hline $12$ &\cellcolor{yellow!25}$\Z\oplus \Z_2$ &\cellcolor{red!25}$\Z\oplus \Z_2$ & ?& ?& $0$\\ \hline $13$ &\cellcolor{yellow!25}$0$ &\cellcolor{red!25}$0$ & ?& ?& $0$\\ \hline $14$ &$(\Z_{168})^r\oplus(\Z_2)^r$\cellcolor{yellow!25} &\cellcolor{red!25}$(\Z_2)^r$ & ?& ?& $0$\\ \hline $15$ &\cellcolor{green!25} $(\Z_2)^{r+1}\oplus\Z_{168}$&\cellcolor{orange!25}$\Z^r\oplus \Z_2$ &? &? &\cellcolor{red!25} $\Z^r$\\ \hline \end{tabular} \end{table} While the last row of Table~\ref{exceptionalcases} stops at $k=15$, one can easily compute the homotopy groups up to $k=22$ for $G_2$ and $F_4$ using~\cite{Mim-G2F4}. For example, one finds that $\pi_{22}(\X_r(G_2)^{good})\cong \Z_{1386}\oplus\Z_8$ if $r\geqs 7$ and $\pi_{18}(\X_r(F_4)^{good})\cong \Z_{720}\oplus\Z_3$ if $r\geqs 4$. \end{exam} We now illustrate the periodicity that comes from Theorem~\ref{splitting} for the classical groups $A_n$, $B_n$, $C_n$, and $D_n$ (Remark~\ref{cor-per}). \begin{exam}\label{ex-per} For this example, we refer to~\cite{bott-per}. We assume $r\geqs 2$. First, if $k\leqs n-2$ then $k\leqs 2(r-1)\Rank\left(\SO_n(\C)\right)-2$ which then implies \[ k+8\leqs 2(r-1)\Rank\left(\SO_{n+8}(\C)\right)-2, \] since \[ \Rank\left(\SO_n(\C)\right)= \begin{cases} n/2,&\text{ if }n\text{ is even}\\ (n-1)/2,&\text{ if }n\text{ is odd.} \end{cases} \] Thus, if $2\leqs k\leqs n-2$, then \begin{align*} \pi_k\left(\X_r(\SO_n(\C))^{good}\right)&\cong \pi_k\left(\SO_n(\C)\right)^r\oplus \pi_{k-1}\left(\SO_n(\C)\right)\\&\cong \pi_{k+8}\left(\SO_{n+8}(\C)\right)^r\oplus \pi_{k+7}\left(\SO_{n+8}(\C)\right)\\ &\cong \pi_{k+8}\left(\X_r\left(\SO_{n+8}(\C)\right)^{good}\right). \end{align*} So in particular, $\pi_{k}(\X_r(\SO_n(\C))^{good})\cong \Z $ for all $k\equiv 7\mod 8$ and $n\equiv 9\mod 8$ so long as $k\geqs 7$ and $n\geqs 9$. Likewise, there is $\pi_k$-periodicity in the $A_n$ series for $k\leqs 2n+1$ $($shift in $n$ is $+1$ and shift in $k$ is $+2)$ and $C_n$ series for $k\leqs 4n+1$ (shift in $n$ is $+4$ and shift in $k$ is~$+8$). \end{exam} On the other hand, our result shows that the homotopy groups can vary consistently in $r$ once $r$ gets sufficiently large. \begin{exam} From~\cite{Mim-G2F4}, \[ \pi_{22}\left(\X_r\left(\SO_9(\C)\right)^{good}\right)\cong \left(\Z_{11!/32}\right)^r\oplus(\Z_8)^r\oplus(\Z_2)^{2r}\oplus \Z_{12}\;\text{ for all }\;r\geqs 4. \] There are many other examples along these lines. \end{exam} \section{Splitting short exact sequences}\label{splitsection} The goal of this section is to prove the following proposition. \begin{prop}\label{good-htpy} In dimensions~$1\leqs k \leqs C_{pasbon} - 2$, we have \[ \pi_k \left(\XC{r}(G)^{good}\right) \isom \pi_k (G)^r \cross \pi_{k-1} (PG). \] \end{prop} The following algebraic fact now implies that the short exact sequences~\eqref{eq:SES} admit (non-canonical) splittings. \begin{lemm} If $0\to A\srt{i} B\srt{q} C\to 0$ is a short exact sequence of finitely generated abelian groups, and there exists an isomorphism $B\srt{\isom} A\cross C$, then the sequence splits. \end{lemm} \begin{proof} This follows from the results in~\cite{Miyata}. \end{proof} We now prepare for the proof of Proposition~\ref{good-htpy}, which will be at the end of this section. Given a topological group $K$, we let $BK$ be its classifying space and $\pi\co EK \to BK$ denote a universal principal $K$-bundle; that is, a (right) principal $K$-bundle with $EK$ contractible. We note that there are at least two functorial constructions of $EK$: Milnor's infinite join construction (which works for all topological groups) and the standard simplicial bar construction (which works for all Lie groups). Either of these models will suffice for our purposes below. \begin{defi} Let $K$ be a topological group, and let $X$ be a $($left$)$ $K$-space. Then the \emph{homotopy orbit space} for the action of $K$ on $X$ is the space \[ X_{hK} := EK \cross_K X = (EK \cross X)/K, \] where $K$ acts by $(e, x)\cdot k = (ek, k^{-1} x)$. \end{defi} We record some standard facts regarding homotopy orbit spaces. There is a natural map $p_X \co X_{hK}\to BK$, induced by the projection~$EK\to BK$. This map is a fiber bundle with fiber $X$, locally trivial over each open set in $BK$ over which $EK$ is trivial. The next fact may be found, for instance, in Atiyah--Bott~\cite[Section~13]{AB}. \begin{lemm}\label{free} Let $X$ be a $K$-space such that the projection map $X\to X/K$ is a principal $K$-bundle. Then the map $X_{hK} \to X/K$, sending $[(e, x)]\in EK\cross_K X$ to $[x]\in X/K$, is a weak homotopy equivalence. \end{lemm} Recall that a map $f\co X\to Y$ is said to be $n$-connected if the induced map on homotopy groups is an isomorphism in degrees less than $n$ and is surjective in degree $n$. \begin{lemm}\label{n-ctd} If $X\to Y$ is an equivariant map of $K$-spaces, and $f$ is $n$-connected, then so is the map $f_{hK}\co X_{hK}\to Y_{hK}$ induced by $f$. \end{lemm} \begin{proof} The lemma follows by applying the Five Lemma to the diagram of long exact sequences in homotopy induced by the commutative diagram \[ \xymatrix{X\ar[r]\ar[d]^f &X_{hK} \ar[r]^{p_X} \ar[d]^{f_{hK}} & BK \ar[d]^=\\ Y \ar[r] &Y_{hK} \ar[r]^{p_Y} & BK. } \] \end{proof} With these lemmata complete, we now prove Proposition~\ref{good-htpy}. \begin{proof}[Proof of Proposition~\ref{good-htpy}] Consider the principal $PG$-bundle \[ PG \maps \hom\left(\F_r,G\right)^{good} \maps \XC{r}(G)^{good}. \] Lemma~\ref{free} shows that we have a weak equivalence \begin{equation}\label{eq:good1} \XC{r}(G)^{good} \heq \left(\hom\left(\F_r,G\right)^{good}\right)_{hPG}. \end{equation} By Lemma~\ref{homotopy-codim}, the inclusion \[ \hom\left(\F_r,G\right)^{good}\injects G^r \] is a $(C_{pasbon} - 1)$--connected map, so Lemma~\ref{n-ctd} implies the induced map \begin{equation}\label{eq:good2} \left(\hom\left(\F_r,G\right)^{good}\right)_{hPG} \maps (G^r)_{hPG} \end{equation} is $(C_{pasbon} - 1)$--connected as well. Since the identity element $e\in G^r$ is fixed under conjugation, the fibration \begin{equation}\label{eq:hPG} (G^r)_{hPG}\to BPG \end{equation} admits a splitting, defined by $[x]\mapsto [x, e] \in EPG\cross_{PG} G^r$; this splitting is continuous because the map $EPG \to EPG\cross_{PG} G^r$, $x\mapsto [x,e]$, is continuous, and $BG \isom EPG/PG$ (as holds for all principal bundles). It follows that the long exact sequence associated to~\eqref{eq:hPG} breaks up into \emph{split} short exact sequences of the form \begin{equation}\label{eq:hPG2} 0 \maps \pi_k (G^r) \maps \pi_k\left((G^r)_{hPG}\right) \maps \pi_k (BPG) \maps 0. \end{equation} For any Lie group $H$, we have $\pi_k (BH) \isom \pi_{k-1} (H)$ for each $k\geqs 1$, so the split short exact sequences~\eqref{eq:hPG2} yield \[ \pi_k \left((G^r)_{hPG}\right)\isom \pi_k (G)^r \cross \pi_{k-1} (PG) \] for $k\geqs 1$. We saw above that the map~\eqref{eq:good2} is $(C_{pasbon} - 1)$-connected, so this completes the proof of Proposition~\ref{good-htpy}. \end{proof} \section{A generalization of Schur's lemma}\label{Schur} In this section we will characterize connected, reductive $\C$-groups containing no bad subgroup. These are called CI-groups (see~\cite{Si4} for definition). We will also give a rather simple description for the bad locus of character varieties in simply connected semisimple $\C$-groups. If $G$ is semisimple, let $\Lambda_G$ denote the lattice generated by the roots in the dual of the Lie algebra of a fixed Cartan subgroup of $G$ (see~\cite[Chapter~23]{FH}). We begin with a lemma about the centralizers of BdS subgroups. We shall use the fact, which follows from the definition, that a BdS subgroup of $G$ is defined, up to conjugation, by a sub-root system of the root system of $G$ with identical rank. The lemma itself comes from Borel and de Siebenthal's article (done in the compact case but is essentially identical), see~\cite{BdS}. \begin{lemm}\label{ZBdS} Let $S$ be a BdS subgroup of a connected, reductive $\C$-group $G$. Then $Z_G(S)=Z(S)$ and furthermore, we have an isomorphism between \[ Z_G(S)/Z(G)\text{ and } \Lambda_G/\Lambda_S. \] \end{lemm} \begin{proof} Let $H$ be a Cartan subgroup of $S$ (it is then a Cartan subgroup in $G$). Since $H$ is a Cartan subgroup of $G$, $Z_G(H)=H$ whence $Z_G(S)\leqs Z_G(H)\leqs S$, therefore $Z_G(S)=Z(S)$. Fix $\alpha_1,\,\dots,\,\alpha_r$ a system of simple roots for $G$ and $\beta_1,\,\dots,\,\beta_r$ a system of simple roots for $S$. Let $\Gamma_G$ (respectively $\Gamma_S$) be the lattice of elements in $\fkh$ which are sent to integers via the functionals in $\Lambda_G$ (respectively $\Lambda_S$). Using the functoriality of the exponential map, one sees that an element $h=\exp(X)\in H$ will commute with all elements in $G$ (respectively $S$) if and only if $X$ belongs to $2\sqrt{-1}\pi \Gamma_G$ (respectively $2\sqrt{-1}\pi \Gamma_S$). Whence $Z_G(S)/Z(G)$ is isomorphic to $2\sqrt{-1}\pi \Gamma_S/2\sqrt{-1}\pi \Gamma_G$ which is isomorphic to $\Gamma_S/\Gamma_G$. Finally, since $\Lambda_G/\Lambda_S$ is finite, there is a perfect pairing between $\Lambda_G/\Lambda_S$ and $\Gamma_S/\Gamma_G$ induced by the perfect pairing $\fkh^*\times \fkh\to \C$. In particular, $\Gamma_S/\Gamma_G$ is isomorphic to~$\Lambda_G/\Lambda_S$. \end{proof} We immediately deduce the following corollary: \begin{coro} Any BdS subgroup in a connected, reductive $\C$-group is bad. \end{coro} \begin{proof} Let $G$ be a connected, reductive $\C$-group and $S$ be a BdS subgroup. Because of the preceding lemma, if we had $Z_G(S)=Z(G)$ then we would have $\Lambda_G=\Lambda_S$. This is impossible because this would imply that $G=S$. As a result, $Z_G(S)\neq Z(G)$. It is a routine verification to show that if $L$ is a Levi subgroup of a parabolic subgroup of $G$ then $\dim_\C Z_G(L)>\dim_\C Z(G)$. If $S$ were contained in a parabolic subgroup then it would be contained in one of its Levi subgroups since $S$ is reductive and we would therefore have $\dim Z_G(S)>\dim Z(G)$. The preceding lemma implies that $\dim Z_G(S)=\dim Z(G)$, and so $S$ is not contained in any parabolic subgroup of $G$. Therefore, $S$ is irreducible. \end{proof} It is easy to see that $\SL_n(\C)$ and $\GL_n(\C)$ are CI by Schur´s lemma (see~\cite[Lemma~3.5]{FL2}), and $\Orm_n(\C),\Sp_{2n}(\C), \p\SL_n(\C)$ are not CI (see~\cite[Proposition~3.32]{FL2}). In~\cite[Question 19]{Si4}, Sikora asks: Are $\GL_n(\C)$ and $\SL_n(\C)$ the only CI-groups? We now give a characterization of such groups, answering Sikora's question. \begin{theo}\label{CI-thm} A connected, reductive $\C$-group is a CI-group if and only if its derived subgroup is a product of special linear groups. \end{theo} \begin{proof} First, notice that if $\pi:G_1\to G_2$ is a finite covering of connected, reductive $\C$-groups and $S$ is a bad subgroup of $G_1$ then $\pi(S)$ is a bad subgroup of~$G_2$. Secondly, for any connected, reductive $\C$-group $G$ there is a finite cover $Z(G)\times [G,G]\to G$ sending $(z,s)$ to $zs$. If $S$ is a bad subgroup of $[G,G]$ then it is a bad subgroup of $Z(G)\times [G,G]$ whence it is a bad subgroup of $G$. On the other hand, if $S$ is a bad subgroup of $G$ then $\langle Z(G),S\rangle\cap [G,G]$ is a bad subgroup of $[G,G]$. So that $G$ contains a bad subgroup if and only if $[G,G]$ contains a bad subgroup. As a result, it suffices to show that among semisimple groups, the only CI-groups are the ones that are products of $\SL_n(\C)$ for potentially varying $n$. If $G$ is simply connected and semisimple then $G$ is isomorphic to a product of simple simply connected groups $G_1,\,\dots,\,G_m$. Thus, $G$ contains a bad subgroup if and only if there exists $i$ such that $G_i$ contains a bad subgroup. Because of Schur's lemma, simple groups isomorphic to $\SL_n(\C)$ do not contain bad subgroups and because of Table~\ref{BdS} any other simply connected simple group contains a BdS subgroup and thus a bad subgroup by the preceding corollary. So the only CI-groups among simply connected semisimple groups are products of special linear groups. Thus, if $G$ is semisimple and a CI-group, the first sentence of this proof implies that the universal cover of $G$ has to be a product of special linear groups. Furthermore, one can construct a bad subgroup in any non-trivial quotient of a special linear group (see Lemma~\ref{SLquo}). Therefore, $G$ is CI if and only if $DG$ is isomorphic to a product of special linear groups. \end{proof} \begin{lemm}\label{SLquo} Let $G$ be a product of special linear groups and $C$ be a non-trivial central subgroup of $G$, then $G/C$ is not a CI-group. \end{lemm} \begin{proof} Let $n>1$, let $\xi$ be a primitive $n^{\rm th}$ root of the unity and $d$ dividing $n$. We define \[ g_{n,d}=\lambda_{n,\,d} \begin{pmatrix}I_{\frac{n}{d}}&&&\\ &\xi^{\frac{n}{d}}I_{\frac{n}{d}}&&\\&&\ddots&\\&&&\xi^{\frac{n}{d}(d-1)}I_{\frac{n}{d}} \end{pmatrix}\quad\text{and}\quad M_{n,\,d}:=\lambda_{n,\,d} \begin{pmatrix}&&&I_{\frac{n}{d}}\\ I_{\frac{n}{d}}&&&\\&\ddots&&\\&&I_{\frac{n}{d}}& \end{pmatrix} \] where $\lambda_{n,\,d}$ is chosen so that $\det(g_{n,\,d})=\det(M_{n,\,d})=1$. It follows that $g_{n,\,d}$ and $M_{n,\,d}$ are in $\SL_n(\C)$ and satisfy $[g_{n,\,d},M_{n,\,d}]=\xi^{\frac{n}{d}}$. One sees that $M_{n,\,d}$ acts by conjugation on the subgroup $D_{n,\,d}$ of $\SL_n(\C)$ generated by unimodular matrices which are diagonal by blocks of size $n/d$.The group generated by $D_{n,\,d}$ and $M_{n,\,d}$ acts naturally on $\C^n$ and fixes no proper non-trivial subspace of $\C^n$. It follows that the group generated by $D_{n,\,d}$ and $M_{n,\,d}$ is irreducible. Let $G$ be $\SL_{n_1}(\C)\times\cdots\times \SL_{n_s}(\C)$ and $C$ a non-trivial central subgroup of $G$. We take $c\in C$ such that $c\neq 1_G$ and write \[ c=\left(\xi_1^{\frac{n_1}{d_1}}I_{n_1},\,\dots,\,\xi_s^{\frac{n_s}{d_s}}I_{n_s}\right) \] where $d_i$ divides $n_i$ and $\xi_i$ is a $n_i^{\rm th}$ root of unity. We denote $\pi :G\to G/C$ the natural projection. Let $g=(g_{n_1,\,d_1},\,\dots,\,g_{n_s,\,d_s})$, $M=(M_{n_1,\,d_1},\,\dots,\,M_{n_s,\,d_s})$ and $S$ be the group generated by $M$ and $D_{n_1,\,d_1}\times\cdots\times D_{n_s,\,d_s}$. Because the projection of $S$ for each factor of $G$ is irreducible, $S$ is itself irreducible and thus $\pi(S)$ is too. Furthermore $[g,M]=c$ by construction. Since $g$ commutes with $D_{n_1,\,d_1}\times\cdots\times D_{n_s,\,d_s}$, we deduce from this $Z_ {G/C}(\pi(S))$ contains $\pi(g)$. Since $g$ is not central in $G$, it follows that $G/C$ is not a CI-group. \end{proof} \begin{rema} In Section~\ref{GBU-sec}, it is shown that $\X_r(G)^{red}\subset \X_r(G)^{sing}$ if $r\geqs 3$, or \linebreak$r\geqs 2$ and the rank of the simple factors of the Lie algebra of $DG$ are at least 2. Conversely, if $r=2$ there are semisimple Lie groups $G$ of arbitrarily large rank so $\X_r(G)$ contains smooth reducibles; \cite[Example~7.2]{FLR}. These two facts together resolve the first part of~\cite[Conjecture~3.34]{FL2}. The second part of~\cite[Conjecture~3.34]{FL2} states that $\X_r(G)^{red}=\X_r(G)^{sing}$ if and only if $DG$ is isomorphic to a product of special linear groups. Given that we have shown in Section~\ref{sect2}, that bad representations are singular whenever $r\geqs 3$, or $r\geqs 2$ and the rank of the simple factors of the Lie algebra of $DG$ are at least 2, this conjecture is equivalent to statement that the only CI groups are those whose derived subgroup is a product of special linear groups. So the above theorem affirmatively resolves the second part of~\cite[Conjecture~3.34]{FL2} too. \end{rema} As a result of the above Theorem~\ref{CI-thm}, Schur's Lemma (elements commuting with an irreducible subgroup are central) is true in only one simple $\C$-group: the special linear group. The main reason for CI-groups $G$ are interesting is that the irreducible locus of the $G$-character variety of a free group/surface group is a manifold (see~\cite{FL2,Si4}). Next, we focus on the case when $G$ is simply connected. The first lemma is fundamental to our discussion. It is true in greater generality than we state (see~\cite[Chapter~4]{OnVi}). \begin{lemm} Let $G$ be a semisimple simply connected $\C$-group and $g$ a semisimple element in $G$. Then $Z_G(g)$ is connected. \end{lemm} \begin{proof} A proof is given in~\cite{Humphreys-conjugacy}, for example. \end{proof} The next corollary is a direct consequence of this lemma. \begin{coro}\label{bad-cor} Let $G$ be a semisimple simply connected $\C$-group and $\rho:\F_r\to G$ a bad representation. Then $\rho(\F_r)$ is contained in a BdS subgroup. \end{coro} \begin{proof} Let $g$ be an element commuting with $\rho(\F_r)$. Since $\rho$ is irreducible, $g$ is semisimple (by Proposition~\ref{commsemi}). From Lemma~\ref{semisimplecentralizers}, it follows that $\fkz_{\fkg}(g)$ is either contained in a parabolic subalgebra or is a BdS subalgebra. Because of the preceding lemma, $Z_G(g)$ needs to be connected. As a result, if $\fkz_{\fkg}(g)$ were contained in a parabolic subalgebra then $Z_G(g)$ would be contained in a parabolic subgroup which would contradict the irreducibility of $\rho$. It follows that $\fkz_{\fkg}(g)$ is a BdS subalgebra and therefore $Z_G(g)$ is a BdS subgroup. \end{proof} As a result, if we want to compute the bad locus of $G$-character varieties when $G$ is simply connected, it suffices to understand the irreducible characters (equivalence classes of irreducible representations) that factor through the inclusion of maximal BdS subgroups in $G$. We now illustrate this principle with the lowest rank exceptional Lie group. By definition (see for example~\cite{FH,B-S} or~\cite{Rac}), $G_2$ is the automorphism group of a non-commutative, non-associative complex algebra $\bbO_\C:=\bbO\otimes_\R \C$ of complex dimension~$8$ (the \emph{bi-octonians}), where $\bbO$ is the usual octonians. Since $G_2$ is simply connected, bad subgroups are contained in BdS subgroups by Corollary~\ref{bad-cor}. From Table~\ref{BdS}, we see that for $G_2$ there are only two types of BdS subgroups: type $A_2$ and $A_1\times A_1$. \begin{figure}[!ht] \begin{tikzpicture}[scale=0.68] \node[scale=0.65] at (0,-3.8) {Root system of $\fkg_2$}; \draw[ultra thick,->,color=black] (0,0)--({2.0011*cos(0)},{2.0011*sin(0)}); \draw[ultra thick,->] (0,0)--({2.0011*cos(60)},{2.0011*sin(60)}); \draw[ultra thick,->] (0,0)--({2.0011*cos(120)},{2.0011*sin(120)}); \draw[ultra thick,->] (0,0)--({2.0011*cos(180)},{2.0011*sin(180)}); \draw[ultra thick,->] (0,0)--({2.0011*cos(240)},{2.0011*sin(240)}); \draw[ultra thick,->] (0,0)--({2.0011*cos(300)},{2.0011*sin(300)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(30)},{2.0011*1.73205081*sin(30)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(90)},{2.0011*1.73205081*sin(90)}); \draw[ultra thick,->,color=black](0,0)--({2.0011*1.73205081*cos(150)},{2.0011*1.73205081*sin(150)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(30)},{-2.0011*1.73205081*sin(30)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(90)},{-2.0011*1.73205081*sin(90)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(150)},{-2.0011*1.73205081*sin(150)}); \end{tikzpicture} \hfill \begin{tikzpicture}[scale=0.68] \node[scale=0.65] at (0,-3.8) {Root system of $\fksl_3$ inside $\fkg_2$}; \draw[ultra thick,->,dotted] (0,0)--({2.0011*cos(0)},{2.0011*sin(0)}); \draw[ultra thick,->,dotted] (0,0)--({2.0011*cos(60)},{2.0011*sin(60)}); \draw[ultra thick,->,dotted] (0,0)--({2.0011*cos(120)},{2.0011*sin(120)}); \draw[ultra thick,->,dotted] (0,0)--({2.0011*cos(180)},{2.0011*sin(180)}); \draw[ultra thick,->,dotted] (0,0)--({2.0011*cos(240)},{2.0011*sin(240)}); \draw[ultra thick,->,dotted] (0,0)--({2.0011*cos(300)},{2.0011*sin(300)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(30)},{2.0011*1.73205081*sin(30)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(90)},{2.0011*1.73205081*sin(90)}); \draw[ultra thick,->,color=black](0,0)--({2.0011*1.73205081*cos(150)},{2.0011*1.73205081*sin(150)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(30)},{-2.0011*1.73205081*sin(30)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(90)},{-2.0011*1.73205081*sin(90)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(150)},{-2.0011*1.73205081*sin(150)}); \end{tikzpicture} \hfill \begin{tikzpicture}[scale=0.68] \node[scale=0.65] at (0,-3.8) {Root system of $\fkso_4$ inside $\fkg_2$}; \draw[ultra thick,->,color=black] (0,0)--({2.0011*cos(0)},{2.0011*sin(0)}); \draw[ultra thick,->,dotted] (0,0)--({2.0011*cos(60)},{2.0011*sin(60)}); \draw[ultra thick,->,dotted] (0,0)--({2.0011*cos(120)},{2.0011*sin(120)}); \draw[ultra thick,->] (0,0)--({2.0011*cos(180)},{2.0011*sin(180)}); \draw[ultra thick,->,dotted] (0,0)--({2.0011*cos(240)},{2.0011*sin(240)}); \draw[ultra thick,->,dotted] (0,0)--({2.0011*cos(300)},{2.0011*sin(300)}); \draw[ultra thick,->,dotted](0,0)--({2.0011*1.73205081*cos(30)},{2.0011*1.73205081*sin(30)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(90)},{2.0011*1.73205081*sin(90)}); \draw[ultra thick,->,color=black,dotted](0,0)--({2.0011*1.73205081*cos(150)},{2.0011*1.73205081*sin(150)}); \draw[ultra thick,->,dotted](0,0)--({2.0011*1.73205081*cos(30)},{-2.0011*1.73205081*sin(30)}); \draw[ultra thick,->](0,0)--({2.0011*1.73205081*cos(90)},{-2.0011*1.73205081*sin(90)}); \draw[ultra thick,->,dotted](0,0)--({2.0011*1.73205081*cos(150)},{-2.0011*1.73205081*sin(150)}); \end{tikzpicture} \caption{BdS Subalgebras of $\fkg_2$.}\label{g2root} \end{figure} In the second diagram in Figure~\ref{g2root}, the sub-root system is of index $3$ while in the third diagram, the sub-root system is of index $2$. It follows (see~\cite[Chapter~23 \S~2]{FH}) that the center of the BdS subgroup of type $A_2$ is of order $3$ and the center of the BdS subgroup of type $A_1\times A_1$ is of order $2$. Then, the two BdS subgroups are identified as $\SL_3(\C)$ and $\SO_4(\C)$. These two subgroups may be constructed using the minimal dimensional representation of $G_2$. The algebra $\bbO_\C=\bbO\otimes_\R \C$ contains a copy of $\C\otimes_{\bbR}\C$ and $\bbH\otimes_{\R}\C$ as subalgebras (where $\bbH$ is the quaternions). The subgroup $\SL_3(\C)$ can be identified as the subgroup of $G_2$ that point-wise fixes the sub-algebra $\C\otimes_{\bbR}\C$, while $\SO_4(\C)$ can be identified as the stabilizer of the sub-algebra $\bbH\otimes_{\R}\C$. As a result bad representations in $G_2$ correspond to irreducible representations stabilizing non-degenerate sub-algebras of $\bbO_\C$. Lastly, note that the map from $\X_r(S)$ to $\X_r(G)$ induced by the inclusion of $S$ into $G$ has no reason to be injective in general. For instance, one may check that $\X_r(\SL_3(\C))^{irr}$ to $\X_r(G_2)^{irr}$ is $2$-to-$1$ onto its image. This follows from the fact that $\SL_3(\C)$ is of index $2$ in its $G_2$-normalizer. The corresponding map for $\SO_4(\C)$ is more complicated. \vspace{26pt} \pagebreak \appendix \section{Maximal Parabolic and BdS Subalgebras} \label{appa} In this appendix, we compute the codimension of Lie subalgebras of simple Lie algebras referred to in the proof of Theorem~\ref{codimbad}. In the first table, we consider the codimension of a Levi subalgebra $\fkl$ in a maximal parabolic subalgebra of the corresponding simple Lie algebra. \begin{table}[!ht] \caption{Classification of Levi subalgebras in maximal parabolic subalgebras of simple Lie algebras.}\label{Levi} \begin{tabular}{|c|c|l|c|c|} \hline $\fkg$ & $\dim_{\C}\fkg$ & \multicolumn{1}{|c|}{$[\fkl_k,\fkl_k]$}& $\codim_{\C}(\fkg,\fkl_k)$ & $\min_{k}\codim_{\C}(\fkg,\fkl_k)$\\ \hline\hline $A_r$ & $r(r+2)$& $A_{k-1}+A_{r-k},\, 1\leqs k\leqs r$ & $2k(r+1-k)$ & $2r$ \\ \hline $B_r$& $r(2r+1)$ & $A_{k-1}+B_{r-k},\, 1\leqs k\leqs r$ & $k(4r+1-3k)$ & $2(2r-1)$ \\ \hline $C_r$ & $r(2r+1)$ & $A_{k-1}+C_{r-k},\, 1\leqs k\leqs r$ & $k(4r+1-3k)$ & $2(2r-1)$\\ \hline $D_r$ & $r(2r-1)$ & $\!A_{k-1}+D_{r-k}, 1\leqs k\leqs r-3$ & $k(4r-1-3k)$ & $r(r-1),\text{ if }r=3,4$\\ & & $A_{r-3}+A_1+A_1,\, k=r-2$& $r^2+3r-10$ & $4(r-1),\text{ if }r> 4$\\ & & $A_{r-1},\, k=r-1,r$ & $r^2-r$ &\\ \hline $G_2$& $14$& $A_1,\, k=1,2$ & $10$ & $10$\\ \hline $F_4$ & $52$ & $C_3,\, k=1$ & $30$ & $30$ \\ & & $A_1+A_2,\, k=2,3$ & $40$ & \\ & & $B_3,\, k=4$& $30$ & \\ \hline $E_6$& $78$ & $D_5,\, k=1,5$ & $32$ & $32$ \\ & & $A_1+A_4,\, k=2,4$ & $50$ & \\ & & $A_1+A_2+A_2,\, k=3$ & $58$ & \\ & & $A_5,\, k=6$ & $42$& \\ \hline $E_7$ & $133$ & $D_6,\ k=1$ & $66$ & $54$ \\ && $A_1+A_5,\, k=2$ & $94$ &\\ && $A_1+A_2+A_3,\, k=3$ & $106$ &\\ && $A_4+A_2,\, k=4$ & $100$ &\\ && $D_5+A_1,\, k=5$ & $84$ &\\ && $E_6,\, k=6$ & $54$ &\\ && $A_6,\, k=7$ & $84$ & \\ \hline $E_8$ & $248$ & $E_7,\, k=1$ & $114$ & $114$\\ & & $A_1+E_6,\, k=2$ & $166$ & \\ & & $A_2+D_5,\, k=3$ & $194$ & \\ & & $A_3+A_4,\, k=4$ & $208$ & \\ & & $A_4+A_2+A_1,\, k=5$ & $212$ & \\ & & $A_6+A_1,\, k=6$ & $196$ & \\ & & $D_7,\, k=7$ & $156$ & \\ & & $A_7, k=8$ & $184$ &\\ \hline \end{tabular} \end{table} To emphasize the ambient algebra, we write $\codim_{\C}(\fkg,\fks)$ for the codimension of $\fks$ in $\fkg$ in the above (and below) table. We recall, that once we choose a Cartan subalgebra $\fkh$ and a set of simple roots $\{\alpha_1,\,\dots,\,\alpha_r\}$ to go with it, conjugacy classes of maximal parabolic subalgebras in simple Lie algebras are in one-to-one correspondence with the set of simple roots (see~\cite[Chapter~IV, 14.17]{Borel} for instance). Now $\fkl_k$ refers to a Levi subalgebra in the maximal parabolic subalgebra $\fkp_k$ associated to the simple root $\alpha_k$ (this description of simple roots is as in~\cite[Chapter~22]{FH}). Using this correspondence, the root system (and thus the isomorphism class) of $[\fkl_k,\fkl_k]$ can easily be seen by removing the corresponding node on the Dynkin diagram. Finally, one uses the fact that for Levi subalgebras of maximal parabolic subalgebras, one has \[ \fkl_k=\left[\fkl_k,\fkl_k\right]\oplus\C. \] Using the fact that $\codim_{\C}(\fkg,\fkl_k)=2\codim_{\C}(\fkg,\fkp_k)$, one also has the minimal codimension of parabolic subalgebras. In the second table, we compute the codimension of BbS subalgebras relevant to Theorem~\ref{codimbad}. \begin{table}[!ht] \caption{Classification of maximal BdS subalgebras in simple Lie algebras.}\label{BdS} \begin{tabular}{|c|c|l|c|c|} \hline $\fkg$ & $\dim_{\C}\fkg$ & \multicolumn{1}{|c|}{$\fks$} & $\codim_{\C}(\fkg,\fks)$ &$\min_{\fks}\codim_{\C}(\fkg,\fks)$\\ \hline\hline $A_r$ & $r(r+2)$& \multicolumn{1}{|c|}{$\emptyset$}& $\emptyset$ & $\emptyset$ \\ \hline $B_r$ & $r(2r+1)$ & $D_k+B_{r-k},\ 2\leqs k\leqs r$ & $2k(2r-2k-1)$ & $2r$ \\ \hline $C_r$& $r(2r+1)$ & $C_k+C_{r-k},\ 1\leqs k\leqs r-1$ & $4k(r-k)$ & $4(r-1),\ r\geqs 2$ \\ \hline $D_r$ & $r(2r-1)$ & $D_k+D_{r-k},\ 2\leqs k \leqs r-2$ & $4k(r-k)$ & $8(r-2),\ r\geqs 3$\\ \hline $G_2$& $14$& $A_1+\widetilde{A_1},\, k=1$ & $8$ & $6$ \\ & & $A_2,\, k=2$ & $6$ & \\ \hline $F_4$& $52$ & $A_1+C_3,\, k=1$ & $28$ & $16$\\ & & $A_2+\widetilde{A_2},\, k=2$ & $36$ & \\ & & $A_3+\widetilde{A_1},\, k=3$ & $34$ & \\ & & $B_4,\, k=4$ & $16$ & \\ \hline $E_6$ & $78$ & $A_5+A_1,\, k=2$ & $40$ & $40$ \\ & & $A_2+A_2+A_2,\, k=3$ & $54$ & \\ \hline $E_7$ & $133$ & $D_6+A_1,\, k=1,6$ & $64$ & $64$ \\ & & $A_7,\, k=2$ & $90$ & \\ & & $A_5+A_2,\, k=3,5$ & $100$ & \\ & & $A_3+A_3+A_1,\, k=4$ & $70$ & \\ \hline $E_8$ & $248$ & $D_8,\, k=1$ & $112$ & $112$\\ & & $A_8,\, k=2$ & $162$ & \\ & & $A_1+A_7,\, k=3$ & $188$ & \\ & & $A_1+A_2+A_5,\, k=4$ & $200$ &\\ & & $A_4+A_4,\, k=5$ & $202$ &\\ & & $D_5+A_3,\, k=6$ & $182$ &\\ & & $A_2+E_6,\, k=7$ & $128$ & \\ & & $A_1+E_7,\, k=8$ & $168$&\\ \hline \end{tabular} \end{table} We recall from~\cite{Dyn} or~\cite{Tit} that one can associate to any simple root of $\fkg$ a BdS subalgebra of $\fkg$. Furthermore, all maximal BdS subalgebras (if any) can be chosen among these subalgebras (however, not all such BdS subalgebras are maximal, see~\cite{Tit}). In Table~\ref{BdS} we use the same enumeration as in Table~\ref{Levi}. Also, the symbol $\widetilde{X}$ used in this table denotes a non-conjugate copy of the group $X$. The isomorphism class of the corresponding BdS subalgebra can also be read off the Dynkin diagram. In practice, one needs to add the minimal root of the root system to the Dynkin diagram and delete the $k^{\rm th}$ node to get the Dynkin diagram of the BdS subalgebra. One can compute its dimension from this. \vspace{35pt} \bibliography{guerin} \end{document}