Polystability of Stokes representations and differential Galois groups

We continue our investigations of the nonabelian moduli spaces in 2d gauge theory (the theory of connections on curves). This article is essentially an appendix to the paper [13] that completed the construction of the wild character varieties of smooth curves as affine algebraic Poisson varieties (completing the sequence [5, 6, 8]). This is a purely algebro-geometric approach, complementary to the earlier analytic approaches [3, 1]. Here we will give an intrinsic characterisation of the points of the wild character varieties, generalising existing results in the tame case, and characterise the stable points (generalising a result of [8] in the untwisted wild case). For more background and applications see the reviews in [7, 9, 11] (the first large class of examples of wild character varieties is due to Birkhoff [2] and the simplest case underlies $U_{q}(\mathfrak{g})$, [7] §4).

First we will recall the basic statements in the tame case. Let $G$ be a connected complex reductive group, such as ${\mathop{\rm GL}}_{n}(\mathbb{C})$, and let $\Sigma^{\circ}$ be a smooth complex algebraic curve. Thus $\Sigma^{\circ}=\Sigma\setminus\alpha$ for some smooth compact complex algebraic curve $\Sigma$ (i.e. a compact Riemann surface), and a finite subset $\alpha\subset\Sigma$. Given a basepoint $b\in\Sigma^{\circ}$ one can consider the representation variety

which is a complex affine variety equipped with an action of $G$, conjugating representations. A point $\rho\in\mathcal{R}$ is a $G$-representation, i.e. a group homomorphism $\rho:\pi_{1}(\Sigma^{\circ},b)\to G$. In turn the $G$-character variety, or Betti moduli space

is the (affine) geometric invariant theory quotient of $\mathcal{R}$ by $G$. By definition this means that the points of the variety $\mathcal{M}_{\text{\rm B}}$ are the closed $G$ orbits in $\mathcal{R}$. The representations $\rho$ whose $G$ orbits are closed are called the polystable representations. A basic question is thus to characterise the polystable representations intrinsically. The answer (due to Richardson, building on earlier work in the general linear case), is as follows.

Let $\mathop{\rm Gal}(\rho)=A(\rho)\subset G$ be the Zariski closure (adhérence) of the image of $\rho$. A theorem of Schlesinger implies that if $\rho$ is the monodromy representation of an algebraic connection $(\nabla,E)$ on an algebraic principal $G$-bundle $E\to\Sigma^{\circ}$, with regular singularities at $\alpha$, then $\mathop{\rm Gal}(\rho)$ is the differential Galois group of $(\nabla,E)$. Recall that a complex affine algebraic group is a linearly reductive group if its identity component is reductive (i.e. has trivial unipotent radical). The basic characterisation is then:

A representation $\rho\in\mathcal{R}$ whose Galois group $\mathop{\rm Gal}(\rho)$ is linearly reductive is often called a semisimple representation¹¹1Indeed if $V\cong\mathbb{C}^{n}$ and $G={\mathop{\rm GL}}(V)\cong{\mathop{\rm GL}}_{n}(\mathbb{C})$ then $\rho:\pi_{1}(\Sigma^{\circ},b)\to G$ is a semisimple representation if and only if $V$ is a semisimple $\pi_{1}(\Sigma^{\circ},b)$-module.. Thus the theorem says that $\rho\in\mathcal{R}$ is polystable if and only if $\rho$ is semisimple, and so the points of $\mathcal{M}_{\text{\rm B}}$ are the isomorphism classes of semisimple representations. Note that for $G={\mathop{\rm GL}}_{n}(\mathbb{C})$ this statement is older (Artin/Procesi) and a full account appears in the book of Lubotzky–Magid [23].

This paper is concerned with the extension of this result to the case of Stokes representations, generalising the fundamental group representations, and the characterisation of the points of the wild character varieties, generalising the (tame) character varieties appearing above.

In brief any algebraic connection $(\nabla,E)$ on an algebraic principal $G$-bundle $E\to\Sigma^{\circ}$, has an invariant, its irregular class, at each point $a\in\alpha$. A connection is regular singular if and only if its irregular class is trivial. Our aim is to give the generalisation of Richardson’s results when the irregular classes are arbitrary. Due to work of many people it is known how to describe algebraic connections completely topologically, so we can work algebraically on the Betti/Stokes side (see the review of this story in [11]). The basic notions from the tame case are generalised as follows:

Once these generalisations are understood then the story proceeds similarly to the tame case (defining a wild representation variety $\mathcal{R}$ parameterising framed Stokes local systems, and then acting by a reductive group to forget the framings). A key novelty in the wild setting is that there is a breaking of structure group (“fission”) near the marked points so the group acting involves a (reductive) subgroup of $G$. This is intimately related to extra generators in Ramis’ description of the differential Galois group [25], generalising Schlesinger’s density theorem. However, as we will recall, the basic feature of the tame case remains, that $\mathcal{R}$ can be written explicitly in terms of a product of simpler pieces (doubles $\mathbb{D}$ or fission spaces $\mathcal{A}$), one for each marked point or handle on $\Sigma$:

if $\Sigma$ has genus $g$ and $m$ marked points.

As in [26] we use the terminology that an affine algebraic group over $\mathbb{C}$ is reductive if it is connected and has trivial unipotent radical. It is linearly reductive if its identity component is reductive.

Let $G$ be a linearly reductive group over $\mathbb{C}$. Recall that a point $x$ of an affine $G$-variety is said to be polystable if the orbit $G\cdot x$ is closed. It is said to be stable if it is polystable and the kernel of the action has finite index in its stabiliser $G_{x}$.

Let $n$ be a positive integer. For ${\bf x}=(x_{i})_{i=1}^{n}\in G^{n}$, let $A({\bf x})\subset G$ be the Zariski closure of the subgroup generated by $x_{1},x_{2},\dots,x_{n}$. In [26], Richardson examined the simultaneous conjugation action of $G$ on the product $G^{n}$ and obtained the following results:

In this section we will establish twisted versions of these results, in Thm. 7 and Thm. 12 (2) respectively. Two other characterisations of polystability will also be established, in Thm. 12 (1) and Cor. 18. The subsequent section (§3) will give a further generalisation.

Assume that $G$ is reductive and choose $\phi_{1},\phi_{2},\dots,\phi_{n}\in\operatorname{\mathop{\rm Aut}}(G)$. Let $\Gamma$ be the subgroup of $\operatorname{\mathop{\rm Aut}}(G)$ generated by the inner automorphism group $\operatorname{Inn}(G)$ and $\phi_{1},\phi_{2},\dots,\phi_{n}$. We assume that the quotient $\overline{\Gamma}:=\Gamma/\operatorname{Inn}(G)$ is finite and regard $\Gamma$ as an algebraic group with identity component $\operatorname{Inn}(G)\cong G/Z(G)$. For any $\phi\in\Gamma$ let $G(\phi)$ denote the $G$-bitorsor/“twisted group” $G\times\{\phi\}\subset G\ltimes\Gamma$, as in [13] §2 (as explained there, the monodromy of a $\mathcal{G}$-local system lies in such a space, and $G(\phi)$ embeds in the group of set-theoretic automorphisms of a fibre). Put

on which $G=G(\text{\rm Id})$ acts by the simultaneous conjugation. For ${\bf x}=(x_{i})_{i=1}^{n}\in X$, let $A({\bf x})$ be the Zariski closure (in the algebraic group $G\ltimes\Gamma$) of the subgroup generated by $x_{1},x_{2},\dots,x_{n}$. Let $\overline{A}({\bf x})\subset\Gamma$ be the image of $A({\bf x})$ under the homomorphism

where ${\mathop{\rm Ad}}(g)=g(\,\cdot\,)g^{-1}$. Note that $\overline{A}({\bf x})$ is contained in $\operatorname{\mathop{\rm Aut}}(G)$ and hence naturally acts on $G$. Thus in turn, via ${\mathop{\rm Ad}}$, the group $A({\bf x})$ naturally acts on $G$.

To see that Thm. 7 generalises Thm. 5, first note that:

Thus in particular it is now clear that Thm.  7 generalises Thm. 5.

Put $\widetilde{G}=G\ltimes\Gamma^{\prime}$. It is a linearly reductive group with identity component $G$ and $X$ is a closed subvariety of $\widetilde{G}^{n}$. By Theorem 5, a point ${\bf x}\in\widetilde{G}^{n}$ is polystable with respect to the simultaneous $\widetilde{G}$-conjugation if and only if $A({\bf x})\subset\widetilde{G}$ is linearly reductive. By Prop. 9 this happens if and only if $\overline{A}({\bf x})$ is linearly reductive. Together with Lemma 10 this implies the assertion. $\square$

Note that in general it really is necessary to work with $\overline{A}({\bf x})$ rather than ${A}({\bf x})$:

We prepare three lemmas.

(1) We should show that a subgroup $H\subset\widetilde{G}$ is linearly reductive if and only if any $H$-invariant parabolic in $G$ has an $H$-invariant Levi subgroup. First suppose that $H\subset\widetilde{G}$ is linearly reductive and $P\subset G$ is an $H$-invariant parabolic subgroup. Since $H$ is linearly reductive and contained in $N_{\widetilde{G}}(P)$, it is contained in some Levi subgroup $\widetilde{L}$ of $N_{\widetilde{G}}(P)$. Then $L:=\widetilde{L}\cap G$ is a Levi subgroup of $N_{\widetilde{G}}(P)\cap G=N_{G}(P)=P$ and normalised by $H$ (so it is $H$-invariant). Conversely, suppose any $H$-invariant parabolic in $G$ has an $H$-invariant Levi subgroup. By [26, Prop. 2.6], there exists a one-parameter subgroup $\lambda$ of $G$ such that $H\subset P_{\widetilde{G}}(\lambda)$ and $R_{u}(H)\subset U_{\widetilde{G}}(\lambda)$, where

and $R_{u}(H)$ is the unipotent radical of $H$. Put $P=P_{G}(\lambda)=P_{\widetilde{G}}(\lambda)\cap G$, which is a parabolic subgroup of $G$. Since $P$ is normalised by $P_{\widetilde{G}}(\lambda)$, it is normalised by $H$. Hence $P$ has an $H$-invariant Levi subgroup $L$ by assumption. We have $H\subset N_{\widetilde{G}}(L)$ and hence $R_{u}(H)\subset N_{G}(L)$ (recall that the unipotent radical is connected). Note that $R_{u}(H)$ is also contained in $U_{\widetilde{G}}(\lambda)\cap G=R_{u}(P)$, while any non-trivial element of $R_{u}(P)$ does not normalise the Levi subgroup $L$ by Lemma 13. Thus $R_{u}(H)$ is trivial, i.e. $H$ is linearly reductive.

(2) Suppose that ${\bf x}=(x_{i})_{i=1}^{n}\in X$ is stable and let $P$ be a ${A}({\bf x})$-invariant parabolic subgroup of $G$. By Theorem 7, $\overline{A}({\bf x})$ (and hence $A({\bf x})$) is linearly reductive. Since $A({\bf x})$ normalises $P$, there exists a Levi subgroup $\widetilde{L}$ of $N_{\widetilde{G}}(P)$ containing $A({\bf x})$. By [26, Prop. 2.4], there exists a one-parameter subgroup $\lambda$ of $G$ such that $N_{\widetilde{G}}(P)=P_{\widetilde{G}}(\lambda)$ and $\widetilde{L}=C_{\widetilde{G}}(\operatorname{Im}\lambda)$. Since $A({\bf x})\subset C_{\widetilde{G}}(\operatorname{Im}\lambda)$ each $x_{i}$ commutes with $\operatorname{Im}\lambda$ and hence $\operatorname{Im}\lambda\subset G_{\bf x}$. The stability now implies that $\operatorname{Im}\lambda$ is contained in the kernel $Z(G)^{\Gamma}$ and hence $P=G$. Conversely, suppose that ${\bf x}\in X$ is not stable. Then if the orbit $G\cdot{\bf x}$ is not closed we argue as follows: By the Hilbert–Mumford criterion, there exists a one-parameter subgroup $\lambda$ of $G$ and an element ${\bf y}=(y_{i})_{i=1}^{n}\in X$ such that $\lim_{t\to 0}\lambda(t)\cdot{\bf x}={\bf y}$ and $G\cdot{\bf y}$ is closed. We show that the parabolic subgroup $P:=P_{G}(\lambda)=P_{\widetilde{G}}(\lambda)\cap G$ is ${A}({\bf x})$-invariant and proper. For any $g\in P$ and $i=1,2,\dots,n$, the limit of

as $t\to 0$ exists. Hence $P$ is ${A}({\bf x})$-invariant. If $P$ is not proper, $\operatorname{Im}\lambda$ is contained in $Z(G)\cap G_{\bf y}=Z(G)^{\Gamma}$. Thus $\lambda(t)\cdot{\bf x}={\bf x}$ ($t\in\mathbb{C}^{*}$) and hence ${\bf x}={\bf y}$, which contradicts the assumption that $G\cdot{\bf x}$ is not closed. Hence $P$ is proper. Finally suppose $G\cdot{\bf x}$ is closed, but of the wrong dimension. Then the stabiliser $G_{\bf x}$ is linearly reductive ([26] 1.3.3) and the quotient $G_{\bf x}/Z(G)^{\Gamma}$ has non-trivial identity component. Hence there exists a one-parameter subgroup $\lambda$ of $G_{\bf x}$ such that $\operatorname{Im}\lambda\not\subset Z(G)^{\Gamma}$. Since each $x_{i}$ commutes with $\lambda$, the parabolic subgroup $P:=P_{G}(\lambda)$ is ${A}({\bf x})$-invariant. It is proper since Lemma 15 implies $\operatorname{Im}\lambda\not\subset Z(G)$. $\square$

Let us rephrase/abstract the first part of this in a way that will be useful later. Let $G$ be a reductive group and $\Lambda$ an algebraic group acting on $G$ by (algebraic) group automorphisms. Suppose that the action of $\Lambda$ is effective and the identity component $\Lambda^{0}$ acts by inner automorphisms. Then we may regard $\Lambda$ as a subgroup of $\operatorname{\mathop{\rm Aut}}(G)$ and its image in $\mathop{\rm Out}(G)$ is finite.

In the same setting there is a further characterisation:

Conversely, suppose that there exists a torus $S\subset G$ such that $L:=C_{G}(S)$ is $\Lambda$-invariant and has no proper $\Lambda$-invariant parabolic subgroups. Let $P\subset G$ be a $\Lambda$-invariant parabolic subgroup. Then the intersection $P\cap L$ is a $\Lambda$-invariant parabolic subgroup of $L$ and hence $P\cap L=L$, i.e. $L\subset P$. Since $L$ is reductive, there exists a Levi subgroup $M\subset P$ containing $L$. For any $\psi\in\Lambda$ the image $\psi(M)$ of $M$ is also a Levi subgroup of $P$ containing $L$. Since $L$ contains a maximal torus of $P$ and any maximal torus of $P$ is contained in a unique Levi subgroup, we have $\psi(M)=M$. Hence $M$ is $\Lambda$-invariant, which together with Prop. 16 shows that $\Lambda$ is linearly reductive. $\square$

Applying this to the set-up of the present section (with $\Lambda=\overline{A}({\bf x})$) yields:

Note that 3) and 4) are trivially equivalent since centralisers of tori in $G$ are exactly the Levi subgroups of parabolics.

The results of the previous section will now be generalised, in a form more directly useful in the context of Stokes local systems. Return to the set-up of Thm. 7 (with $n\geq 1$), but now further choose an integer $m\geq 1$ and tori $\mathbb{T}_{1},\ldots,\mathbb{T}_{m}\subset G$. Write ${\bf T}=\mathbb{T}_{1}\times\cdots\times\mathbb{T}_{m}\subset G^{m}$, let $H_{i}=C_{G}(\mathbb{T}_{i})$ and ${\bf H}=H_{1}\times\cdots\times H_{m}=C_{G^{m}}({\bf T})\subset G^{m}$. We allow some of the $\mathbb{T}_{i}$ to be a point, in which case $H_{i}=G$. In this section we will study the stability and polystability for the action of ${\bf H}$ on

given by

where ${\bf h}=(h_{1},\ldots,h_{m}),{\bf C}=(C_{2},\ldots,C_{m}),{\bf M}=(M_{1},\ldots,M_{n}),C_{i}\in G,M_{i}\in G(\phi_{i}).$

Thus if $m=1$ and $\mathbb{T}_{1}=1$ we recover the situation of Thm. 7. The case $m=1$ and $\mathbb{T}_{1}$ arbitrary but each $\phi_{i}=1$ was studied by Richardson in [26] Thm. 13.2,14.1 (taking $S=\mathbb{T}_{1}$ acting on $G$ by conjugation). More generally, in effect, [8] Cor. 9.6 studied the notion of stability in the case with $m$ arbitrary and each $\phi_{i}=1$, making the link to the differential Galois group of irregular connections (whence the $\mathbb{T}_{i}$ are the Ramis/exponential tori).

For ${\bf x}=({\bf C},{\bf M})\in X$ let $A({\bf x})\subset G\ltimes\Gamma$ be the Zariski closure of the subgroup generated by

Let $\overline{A}({\bf x})\subset\Gamma$ be the image of $A({\bf x})$ in $\Gamma\subset\operatorname{\mathop{\rm Aut}}(G)$, as before. Recall that a subset of $G$ is $A({\bf x})$-invariant if it is preserved by $\overline{A}({\bf x})\subset\operatorname{\mathop{\rm Aut}}(G)$.

As in the last section, one can rephrase polystability in several different ways:

Now choose ${\bf t}=(t_{1},\ldots,t_{m})\in{\bf T}$ so that $t_{i}$ generates a Zariski dense subgroup of $\mathbb{T}_{i}$ for each $i$. In particular $H_{i}=C_{G}(t_{i})$. Consider the simultaneous conjugation action of $G$ on $Y:=G^{m}\times\prod_{1}^{N}G(\phi_{i})$, and the $G$-equivariant embedding

Thus $\widetilde{X}/{\bf H}$ is identified with a closed subvariety of $Y$ and $\widetilde{X}$ is a $G$-equivariant principal ${\bf H}$-bundle over $\widetilde{X}/{\bf H}$. It follows that the $G\times{\bf H}$ orbit of $\widetilde{\bf x}$ is closed in $\widetilde{X}$ if and only if the $G$ orbit of $\pi(\widetilde{\bf x})$ is closed in $Y$.

Hence part (1) of the theorem follows from Thm. 7 (applied to $Y$). To deal with stability we need to consider the stabilisers and the kernels of the actions.