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Is there any discussion of topology of space of matrices $ ABA^{-1}B^{-1} = E $ with $E$ diagonal matrix, $A,B \in \mathrm{GL}(n,\mathbb{C})$?

E.g. is this a variety of just a scheme? How many components? I'm trying to understand this space in relation to integrable systems

I know it's related to flat connections on the once-punctured torus.

REMARK: The question is really this abstract. I am looking at representations of the fundamental group by $n \times n$ matrices: $\pi_1(\mathbb{T}^2 - \{\cdot\})\to\operatorname{GL}(n, \mathbb{C})$. This is a system of $n^2$ polynomial equations in $2n^2$ unknowns. There should be some relation to Fenchel-Nielsen coordinates for the once-puctured torus.

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    $\begingroup$ @John, would you reformulate the definition of your space in a more formal and pedantic way? $\endgroup$
    – Włodzimierz Holsztyński
    Commented Jun 22, 2013 at 5:36
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    $\begingroup$ @John: You should revise the question. A meaningful question would be to fix the conjugacy class $C^E$ of a diagonal matrix $E$ and consider the "relative character variety", given by the equation $[A,B]\in C^E$. If this is what you are after, then google "parabolic Higgs bundles + character varieties". There is ample literature on this topic. I do not think you should worry about the scheme structure here; if I remember correctly, singularities are quadratic and the scheme is reduced (I am probably missing few exceptional cases for $E$ here, namely, among scalar matrices). $\endgroup$
    – Misha
    Commented Jun 22, 2013 at 11:25
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    $\begingroup$ arxiv.org/abs/math/0306275 studies the scheme $AB-BA=\mbox{diagonal}$. It is proved to be a reduced complete intersection, one of whose components is $AB-BA = 0$. Some intriguing combinatorial observations are made about the degrees of the components. $\endgroup$
    – David E Speyer
    Commented Jun 22, 2013 at 11:42
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    $\begingroup$ @Misha Your suggested question is a good one, but I don't understand what is wrong with studying the scheme formed by setting the off diagonal entries of $ABA^{-1} B^{-1}$ to $0$ (for $A$ and $B$ in $GL_n$.) $\endgroup$
    – David E Speyer
    Commented Jun 22, 2013 at 11:44
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    $\begingroup$ @David: Nothing wrong with the scheme-theoretic viewpoint per se (and, in general, character schemes of course, should be treated as such), but in this particular case, the scheme is reduced and essentially smooth. Thus, as far as I know, one does not gain much by looking at the scheme structure. On the other hand, if one looks at the complex variety, then one gains real-analytic isomorphism to the moduli space of parabolic Higgs bundles, which (to the best of my knowledge) is the most efficient tool for analyzing topology of this variety (which is what the OP is clearly after). $\endgroup$
    – Misha
    Commented Jun 27, 2013 at 4:20

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