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So, assume that $G=\underline{G}({\mathbb R})$, where $\underline{G}$ is an algebraic group (scheme) over reals. Let $A$ be an Artin local ${\mathbb R}$-ring with the valuation $\nu: A\to {\mathbb R}$. Then the set of $A$-points $G_A:=\underline{G}(A)$ is a certain nilponent extension of the Lie group $G$ with the quotient $\nu_G: G_A\to G$ induced by $\nu$. Then, instead of analyzing the scheme $Hom(\pi, \underline{G})$ at $\rho$, you consider the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)\cong Hom_{\rho}(\pi, \underline{G})(A)$, consisting or representations $\tilde\rho: \pi\to G_A$ which project to $\rho$ under $\nu_G$. The point is that the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)$ "knows everything" (and even more!) about the singularity of $Hom(\pi, G)$. For instance, to recover the (Zariski) tangent space $T_\rho Hom(\pi, G)$, you just take $A$ to be the "dual numbers", which is the quotient ${\mathbb R}[t]/(t^2)$. Then $$ T_\rho Hom(\pi, G)\cong Hom_{\rho}(\pi, G_A)$$ for this choice of $A$.

So, assume that $G=\underline{G}({\mathbb R})$, where $\underline{G}$ is an algebraic group (scheme) over reals. Let $A$ be an Artin local ${\mathbb R}$-ring with the projection to the residue field $\nu: A\to {\mathbb R}$. Then the set of $A$-points $G_A:=\underline{G}(A)$ is a certain nilponent extension of the Lie group $G$ with the quotient $\nu_G: G_A\to G$ induced by $\nu$. Then, instead of analyzing the scheme $Hom(\pi, \underline{G})$ at $\rho$, you consider the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)\cong Hom_{\rho}(\pi, \underline{G})(A)$, consisting or representations $\tilde\rho: \pi\to G_A$ which project to $\rho$ under $\nu_G$. The point is that the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)$ "knows everything" (and even more!) about the singularity of $Hom(\pi, G)$. For instance, to recover the (Zariski) tangent space $T_\rho Hom(\pi, G)$, you just take $A$ to be the "dual numbers", which is the quotient ${\mathbb R}[t]/(t^2)$. Then $$ T_\rho Hom(\pi, G)\cong Hom_{\rho}(\pi, G_A)$$ for this choice of $A$.

By the way, here is where algebraic viewpoint is definitely superior to the Lie theoretic. Suppose that you are interested in understanding local structure of the analytic variety $Hom(\pi, G)$, where $G$ is a real Lie group, i.e., what singularity it has at a representation $\rho: \pi\to G$. In general it is a rather difficult problem as singularities could be "arbitrarily complicated." So, assume that $G=\underline{G}({\mathbb R})$, where $\underline{G}$ is an algebraic group (scheme) over reals. Let $A$ be an Artin local ${\mathbb R}$-ring with the valuation $\nu: A\to {\mathbb R}$. Then the set of $A$-points $G_A:=\underline{G}(A)$ is a certain nilponent extension of the Lie group $G$ with the quotient $\nu_G: G_A\to G$ induced by $\nu$. Then, instead of analyzing the scheme $Hom(\pi, \underline{G})$ at $\rho$, you consider the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)\cong Hom_{\rho}(\pi, \underline{G})(A)$, consisting or representations $\tilde\rho: \pi\to G_A$ which project to $\rho$ under $\nu_G$. The point is that the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)$ "knows everything" (and even more!) about the singularity of $Hom(\pi, G)$. For instance, to recover the (Zariski) tangent space $T_\rho Hom(\pi, G)$, you just take $A$ to be the "dual numbers", which is the quotient ${\mathbb R}[t]/(t^2)$. Then $$ T_\rho Hom(\pi, G)\cong Hom_{\rho}(\pi, G_A)$$ for this choice of $A$.

By the way, here is where algebraic viewpoint is definitely superior to the Lie theoretic. Suppose that you are interested in understanding local structure of the analytic variety $Hom(\pi, G)$, where $G$ is a real Lie group, i.e., what singularity it has at a representation $\rho: \pi\to G$. In general it is a rather difficult problem as singularities could be "arbitrarily complicated." See M.Kapovich, J.Millson, "On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex-algebraic varieties", Math. Publications of IHES, vol. 88 (1999), p. 5-95, one of the main results there is that singularities of character varieties could be arbitrary (defined over ${\mathbb Q}$).

So, assume that $G=\underline{G}({\mathbb R})$, where $\underline{G}$ is an algebraic group (scheme) over reals. Let $A$ be an Artin local ${\mathbb R}$-ring with the valuation $\nu: A\to {\mathbb R}$. Then the set of $A$-points $G_A:=\underline{G}(A)$ is a certain nilponent extension of the Lie group $G$ with the quotient $\nu_G: G_A\to G$ induced by $\nu$. Then, instead of analyzing the scheme $Hom(\pi, \underline{G})$ at $\rho$, you consider the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)\cong Hom_{\rho}(\pi, \underline{G})(A)$, consisting or representations $\tilde\rho: \pi\to G_A$ which project to $\rho$ under $\nu_G$. The point is that the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)$ "knows everything" (and even more!) about the singularity of $Hom(\pi, G)$. For instance, to recover the (Zariski) tangent space $T_\rho Hom(\pi, G)$, you just take $A$ to be the "dual numbers", which is the quotient ${\mathbb R}[t]/(t^2)$. Then $$ T_\rho Hom(\pi, G)\cong Hom_{\rho}(\pi, G_A)$$ for this choice of $A$.

By the way, here is where algebraic viewpoint is definitely superior to the Lie theoretic. Suppose that you are interested in understanding local structure of the analytic variety $Hom(\pi, G)$, where $G$ is a real Lie group, i.e., what singularity it has at a representation $\rho: \pi\to G$. In general it is a rather difficult problem as singularities could be "arbitrarily complicated." So, assume that $G=\underline{G}({\mathbb R})$, where $\underline{G}$ is an algebraic group (scheme) over reals. Let $A$ be an Artin local ${\mathbb R}$-ring with the valuation $\nu: A\to {\mathbb R}$. Then the set of $A$-points $G_A:=\underline{G}(A)$ is a certain nilponent extension of the Lie group $G$ with the quotient $\nu_G: G_A\to G$ induced by $\nu$. Then, instead of analyzing the scheme $Hom(\pi, \underline{G})$ at $\rho$, you consider the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)\cong Hom_{\rho}(\pi, \underline{G})(A)$, consisting or representations $\tilde\rho: \pi\to G_A$ which project to $\rho$ under $\nu_G$. The point is that the collection of real-algebraic sets $Hom_{\rho}(\pi, G_A)$ "knows everything" (and even more!) about the singularity of $Hom(\pi, G)$. For instance, to recover the (Zariski) tangent space $T_\rho Hom(\pi, G)$, you just take $A$ to be the "dual numbers", which is the quotient ${\mathbb R}[t]/(t^2)$. Then $$ T_\rho Hom(\pi, G)\cong Hom_{\rho}(\pi, G_A)$$ for this choice of $A$.

This staff is explained in the paper W.Goldman, J.Millson, The Deformation Theory of Representations of Fundamental Groups of Compact Kahler Manifolds, Publ. Math. I.H.E.S.; 67 (1988).