Abstract
In this paper we start the study of configurations of flags in closed orbits of real forms using mainly tools of GIT. As an application, using cross ratio coordinates for generic configurations, we identify boundary unipotent representations of the fundamental group of the figure eight knot complement into real forms of \({{\,\mathrm{PGL}\,}}(4,{\mathbb {C}})\).
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Notes
The quotient \(Y/{\mathfrak {S}}_r\) is described by the ring of invariants of binary quantics: it is the ring of polynomials in the variables \(a_0, \ldots , a_r\) that are invariant under the action of \({{\,\mathrm{SL}\,}}_2({\mathbb {C}})\) on \({\mathbb {C}}[a_0, \ldots , a_r]\) defined by \((g\cdot f)(x,y) = f(g^{-1}(x,y))\) where \(g \in {{\,\mathrm{SL}\,}}_2({\mathbb {C}})\) and \(f = \sum _i a_i x^{r-i} y^i\).
Indeed, semi-stability only depends on the orbit under \(G({\mathbb {C}})\).
To a quaternion \(q = a + bi + cj +dk \in {\mathbb {H}}\) one can associate the \(2 \times 2\) matrix with complex entries
$$\begin{aligned} m(q) := \begin{pmatrix} a + bi &{}\quad c + di \\ -c +di &{}\quad a - bi \end{pmatrix}. \end{aligned}$$This induces an isomorphism of non-commutative \({\mathbb {C}}\)-algebras \(m :H \otimes _{\mathbb {R}}{\mathbb {C}}\rightarrow {{\,\mathrm{End}\,}}({\mathbb {C}}^2)\). A \(2 \times 2\) matrix with quaternionic coefficients,
$$\begin{aligned} \begin{pmatrix} q_1 &{}\quad q_2 \\ q_3 &{}\quad q_4\end{pmatrix} \in {{\,\mathrm{GL}\,}}_2({\mathbb {H}}) \end{aligned}$$belongs to \({{\,\mathrm{SL}\,}}_2({\mathbb {H}})\) if and only if the associated \(4 \times 4\) matrix with complex coefficients,
$$\begin{aligned} \begin{pmatrix} m(q_1) &{}\quad m(q_2) \\ m(q_3) &{}\quad m(q_4) \end{pmatrix}, \end{aligned}$$has determinant 1. If \(q_1 \ne 0\) this is equivalent to saying that the norm of \(q_1 q_4 - q_1 q_3 q_1^{-1} q_2\) is 1.
That is, for every \(x \in U({\mathbb {C}})\), the closure of the orbit \(G({\mathbb {C}}) \cdot x\) is contained in \(U({\mathbb {C}})\).
As already mentioned in the Introduction, a competing symbol in the literature is \(X /\!\!/ G\).
That is, for every \(x \in U({\mathbb {R}})\) the closure of the orbit \(G({\mathbb {R}}) \cdot x\) is contained in \(U({\mathbb {R}})\).
Under this correspondence \(L = \iota ^*{\mathcal {O}}(1)\) and \(V = {{\,\mathrm{H}\,}}^0(X, L)^\vee \).
Again in the literature it can be denoted \(X /\!\!/ G\) or \(X^{\text {ss}}/\!\!/ G\).
This follows from the fact that the representations of G are completely reducible.
One could argue by remarking that \(\frac{1}{d} A\) is a doubly stochastic matrix. The Birkhoff–von Neumann theorem therefore states that \(\frac{1}{d} A\) is a convex combination of permutation matrices. It follows that A can be written as a sum
$$\begin{aligned} A = \sum _{\sigma \in {\mathfrak {S}}_r} \lambda _\sigma A_\sigma , \end{aligned}$$where for a permutation \(\sigma \in {\mathfrak {S}}_r\), \(A_\sigma \) is the associated permutation matrix and \(\lambda _\sigma \) is a non-negative real number. Nonetheless it is not clear that the coefficients \(\lambda _\sigma \) can be taken as integers (the decomposition is not unique).
Given \(g \in P_1 \cap P_2\) and \(h \in {{\,\mathrm{rad}\,}}^u(P_1)\) one has to find the unique element \(h' \in {{\,\mathrm{rad}\,}}^u(P_1)\) such that \( gh \cdot W_2 = h' \cdot W_2\). Since g belongs to stabilizer of \(W_2\) by hypothesis, \( gh \cdot W_2 = ghg^{-1} \cdot W_2\). On the other hand \(ghg^{-1}\) belongs to \({{\,\mathrm{rad}\,}}^u(P_1)\) because the unipotent radical is a normal subgroup of \(P_1\). Thus \(h ' = ghg^{-1}\).
This can be seen as follows. Write q as a matrix
$$\begin{aligned} A= \begin{pmatrix} a &{}\quad - {\bar{b}} \\ b &{}\quad {\bar{a}}\end{pmatrix}, \end{aligned}$$with \(a, b \in {\mathbb {C}}\). The matrix A commutes with its adjoint, thus there is an orthonormal basis of \({\mathbb {C}}^2\) made of eigenvectors. In other words there exists \(U \in {{\,\mathrm{U}\,}}(2)\) and \(\lambda \in {\mathbb {C}}\) such that
$$\begin{aligned} U A U^{-1} = \begin{pmatrix} \lambda &{}\quad 0 \\ 0 &{}\quad {\bar{\lambda }}\end{pmatrix}. \end{aligned}$$This is essentially the proof of (2) in the preceding Proposition.
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We thank Claudio Gorodski, Antonin Guilloux, Greg Kuperberg and Maxime Wolff for several discussions.
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The third author is partially supported by INDAM-GNSAGA, by the 2014 SIR Grant Analytic Aspects in Complex and Hypercomplex Geometry and by Finanziamento Premiale FOE 2014 Splines for accUrate NumeRics: adaptIve models for Simulation Environments of the Italian Ministry of Education (MIUR). Part of this project has been developed while she was an INdAM-COFUND fellow at IMJ-PRG Paris.
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Falbel, E., Maculan, M. & Sarfatti, G. Configurations of flags in orbits of real forms. Geom Dedicata 207, 95–156 (2020). https://doi.org/10.1007/s10711-019-00489-3
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DOI: https://doi.org/10.1007/s10711-019-00489-3