Arithmetic differential equations on $GL_n$, II: arithmetic Lie–Cartan theory

Abstract

Motivated by the search of a concept of linearity in the theory of arithmetic differential equations (Buium in Arithmetic differential equations. Math. surveys and monographs, vol 118. American Mathematical Society, Providence, 2005), we introduce here an arithmetic analogue of Lie algebras, of Chern connections, and of Maurer–Cartan connections. Our arithmetic analogues of Chern connections are certain remarkable lifts of Frobenius on the p-adic completion of $GL_n$ which are uniquely determined by certain compatibilities with the “outer” involutions defined by symmetric (respectively, antisymmetric) matrices. The Christoffel symbols of our arithmetic Chern connections will involve a matrix analogue of the Legendre symbol. The analogues of Maurer–Cartan connections can then be viewed as families of “linear” flows attached to each of our Chern connections. We will also investigate the compatibility of lifts of Frobenius with the inner automorphisms of $GL_n$; in particular, we will prove the existence and uniqueness of certain arithmetic analogues of “isospectral flows” on the space of matrices.

Appendix: Comparison with classical differential equations

It is interesting to see what the theory in this paper corresponds to in the classical case of usual differential geometry. The easiest way to explain this is to place ourselves in the framework of differential algebra [15]; this framework approximates, in the algebraic setting, the situation classically encountered in the theory of Lie and Cartan. So, unlike in the body of the paper, we assume, in this “Appendix”, that R is a (commutative, unital) ${\mathbb {Q}}$-algebra equipped with a derivation $\delta$. For any scheme X over R, we denote by X(R) the set of its R-points. (The typical example we have in mind is that of the ring R of smooth or analytic, real or complex functions on ${\mathbb {R}}$ or ${\mathbb {C}}$; $\delta$ is then the usual derivative.)

1.1 Maurer–Cartan connection

The Kolchin logarithmic derivative

$$\begin{aligned} l\delta :GL_n(R)\rightarrow {\mathfrak {g}}{\mathfrak {l}}_n(R) \end{aligned}$$

(7.1)

is the map $l\delta (a)=\delta a \cdot a^{-1}$ where if $a=(a_{ij})$ then $\delta a:=(\delta a_{ij})$. This map is an algebraic incarnation of the Maurer–Cartan connection and our map 1.6 is an arithmetic analogue of it. (Note, however, that unlike the Kolchin logarithmic derivative 7.1, our map 1.6 is not intrinsically associated with $GL_n$ but also depends on an extra datum which is a lift of Frobenius on $GL_n$. Pinning down the lift of Frobenius, and hence 1.6, in terms of compatibilities with outer involutions defining the classical groups was the main purpose of Theorems 1.1 and 1.5.) If $G\subset GL_n$ is a smooth subgroup scheme defined by equations with coefficients in the ring of constants $R^{\delta }=\{c\in R;\delta c=0\}$, then one gets that $l\delta$ induces a map $l\delta :G(R)\rightarrow {\mathfrak {g}}(R)$ where ${\mathfrak {g}}=L(G)\subset {\mathfrak {g}}{\mathfrak {l}}_n$ is the Lie algebra of G. For any $\alpha \in {\mathfrak {g}}{\mathfrak {l}}_n(R)$, one can consider the linear differential equation $\delta u=\alpha u$ with unknown $u\in GL_n(R)$; this equation is, of course, equivalent to the equation $l\delta u=\alpha$ of which 1.8 is an arithmetic analogue.

1.2 Connections in principal bundles

Let us consider now $G=GL_n$ and the ring $\mathcal O(G)=R[x,\det (x)^{-1}]$. Consider the unique derivation $\delta _{G,0}:\mathcal O(G)\rightarrow \mathcal O(G)$ extending $\delta :R\rightarrow R$ such that $\delta _{G,0}x=0$. Also, for each $\alpha \in {\mathfrak {g}}{\mathfrak {l}}_n={\mathfrak {g}}{\mathfrak {l}}_n(R)$, consider the unique derivation $\delta _G:=\delta _{G,0}^{\alpha }:\mathcal O(G)\rightarrow \mathcal O(G)$ extending $\delta :R\rightarrow R$ such that

$$\begin{aligned} \delta _{G} x=\alpha x. \end{aligned}$$

(7.2)

This derivation can be viewed as an algebraic incarnation of a connection in a principal bundle.

1.3 Compatibility with outer involutions

Consider an involution $x\mapsto x^{\tau }$ on G, let ${{\mathcal {H}}}:\mathcal O(G)\rightarrow \mathcal O(G)$ be the R-algebra map induced by the map $G\rightarrow G$, $x\mapsto x^{-\tau }x$, consider, for $g\in G$, the R-algebra map ${{\mathcal {H}}}_g:\mathcal O(G)\rightarrow \mathcal O(G)$ induced by the map $G\rightarrow G$, $x\mapsto gx^{-\tau }x$, and let ${{\mathcal {B}}}_g: \mathcal O(G)\rightarrow \mathcal O(G)\otimes _R \mathcal O(G)$ be the R-algebra map defined by the map $G\times G\rightarrow G$, $(x_1,x_2)\mapsto gx_1^{-\tau }x_2$. Let us say that $\delta _{G}$ is ${{\mathcal {H}}}_g$-horizontal with respect to $\delta _{G,0}$ if the following diagram is commutative:

$$\begin{aligned} \begin{array}{ccc} \mathcal O(G) &{}\quad \mathop {\longleftarrow }\limits ^{\delta _{G}} &{}\quad \mathcal O(G)\\ {{\mathcal {H}}}_g \uparrow &{}\ &{}\uparrow {{\mathcal {H}}}_g\\ \mathcal O(G) &{}\quad \mathop {\longleftarrow }\limits ^{\delta _{G,0}} &{}\quad \mathcal O(G)\end{array} \end{aligned}$$

(7.3)

Similarly, let us say that $\delta _{G}$ is ${{\mathcal {B}}}_g$-symmetric with respect to $\delta _{G,0}$ if the following diagram is commutative:

$$\begin{aligned} \begin{array}{rcl} \mathcal O(G) &{}\quad \mathop {\longleftarrow }\limits ^{\delta _{G}\otimes 1+1\otimes \delta _{G,0}} &{}\quad \mathcal O(G)\otimes _R \mathcal O(G)\\ \delta _{G,0}\otimes 1+1\otimes \delta _{G} \uparrow &{} \ &{}\quad \uparrow {{\mathcal {B}}}_g\\ \mathcal O(G)\otimes _R \mathcal O(G) &{} \quad \mathop {\longleftarrow }\limits ^{{{\mathcal {B}}}_g} &{}\quad \mathcal O(G)\end{array} \end{aligned}$$

(7.4)

These diagrams can be viewed as differential algebraic analogues of the diagrams 1.20 and 1.21, respectively.

Theorem 1.1 should be viewed as an arithmetic analogue of the following facts. Let $x^{\tau }=q^{-1}(x^t)^{-1}q$ where $q\in G(R)=GL_n(R)$, $q^t=\pm q$. Hence ${{\mathcal {H}}}(x)=q^{-1}x^tqx$ and ${{\mathcal {H}}}_q(x)=x^tqx$. Then $\delta _{G}$ is ${{\mathcal {H}}}_q$-horizontal with respect to $\delta _{G,0}$ if and only if

$$\begin{aligned} \delta q+ \alpha ^tq+q\alpha =0. \end{aligned}$$

(7.5)

On the other hand, $\delta _{G}$ is ${{\mathcal {B}}}_q$-symmetric with respect to $\delta _{G,0}$ if and only if

$$\begin{aligned} \alpha ^tq-q\alpha =0. \end{aligned}$$

(7.6)

Note that the system consisting of Eqs. 7.5 and 7.6 (where q is viewed as given and $\alpha$ is viewed as unknown) has a unique solution $\alpha \in {\mathfrak {g}}{\mathfrak {l}}_n$ equal to

$$\begin{aligned} \alpha =-\frac{1}{2}q^{-1}\delta q. \end{aligned}$$

(7.7)

In other words, there is a unique $\alpha \in {\mathfrak {g}}{\mathfrak {l}}_n$ such that $\delta _G:=\delta _{G,0}^{\alpha }$ is ${{\mathcal {H}}}_q$-horizontal and ${{\mathcal {B}}}_q$-symmetric with respect to $\delta _{G,0}$, and this unique $\alpha$ is given by Eq. 7.7. Let us refer to $\delta _G$ as the Chern connection attached to q; this is an analogue of hermitian Chern connection on a hermitian bundle on a complex manifold, cf. [13], p. 73, and also of the Duistermaat connections in [11]. The analogy is not an entirely direct one, see the discussion in subsection 7.4 below. By the way, our formula 1.29 should be viewed as an arithmetic analogue of formula 7.7 above.

1.4 Chern connections versus hermitian Chern connections

Assume we have an involution on R, i.e., a ring automorphism $R\rightarrow R$, $a\mapsto \overline{a}$, whose square is the identity: $\overline{\overline{a}}=a$; we do not require that this automorphism be different from the identity. Also define the derivation $\overline{\delta }$ on R by the formula $\overline{\delta }(a):=\overline{\delta (\overline{a})}$; we further assume that $\delta$ and $\overline{\delta }$ commute on R. (The typical example we have in mind is that where R is the ring of complex valued smooth functions on a domain in ${\mathbb {C}}$ and $\delta =\frac{\partial }{\partial z}$ where z is a complex coordinate on ${\mathbb {C}}$.) Let

$$\begin{aligned} q^t=q=\left( \begin{array}{r@{\quad }r}q_1 &{} q_2\\ -q_2 &{} q_1\end{array}\right) \in GL^c_r(R)\subset GL_{2r}(R) \end{aligned}$$

and consider the attached matrix $q^c:=q_1+\sqrt{-1}\cdot q_2\in GL_r(R)$. Let $x={\left( \begin{array}{c@{\quad }c}a &{} b\\ c &{} d\end{array}\right) }$ be a $2r\times 2r$ matrix of variables and v, w be matrices of $r\times r$ variables. Consider now the natural isomorphism

$$\begin{aligned} GL_r^c\rightarrow GL_r\times GL_r \end{aligned}$$

(7.8)

given by the R-algebra map from

$$\begin{aligned} \mathcal O(GL_r\times GL_r)=R[v,\det (v)^{-1}]\otimes _R R[w,\det (w)^{-1}] \end{aligned}$$

to

$$\begin{aligned} \mathcal O(GL_r^c)=R[x,\det (x)^{-1}]/(a-d,b+c)=R\left[ a,b,\det \left( \begin{array}{r@{\quad }r}a &{} b\\ -b &{} a\end{array}\right) ^{-1}\right] \end{aligned}$$

(7.9)

sending v and w into the classes of $x^c:=a+\sqrt{-1}\cdot b$ and $(x^t)^c:=a-\sqrt{-1}\cdot b$, respectively. Denote by $\delta _{GL_r}$ the derivation on $R[v,\det (v)^{-1}]$ which is $\delta$ on R and satisfies $\delta _{GL_r} v=\alpha ^c v$, where $\alpha :=-\frac{1}{2}q^{-1}\delta q$, hence

$$\begin{aligned} \alpha ^c=-\frac{1}{2}(q^c)^{-1}(\delta q)^c=-\frac{1}{2} (q^c)^{-1}\delta (q^c). \end{aligned}$$

(7.10)

We may refer to $\delta _{GL_r}$ as the hermitian Chern connection on

$$\begin{aligned} GL_r=Spec\ R[v,\det (v)^{-1}] \end{aligned}$$

attached to the matrix $q^c$. On the other hand, let $\delta _{GL_{2r}}$ be the Chern connection attached to $q\in GL_{2r}(R)$. So if $\delta _{GL_{2r},0}$ is the derivation on $R[x,\det (x)^{-1}]$ that lifts $\delta$ on R and vanishes on x then $\delta _{GL_{2r}}$ is the unique derivation on $R[x,\det (x)^{-1}]$ that lifts $\delta$ on R and is ${{\mathcal {H}}}_q$-horizontal and ${{\mathcal {B}}}_q$-symmetric with respect $\delta _{GL_{2r},0}$. By 7.7, $\delta _{GL_{2r}} x=\alpha x$. Then the link between the hermitian Chern connection $\delta _{GL_r}$ attached to $q^c$ and the Chern connection $\delta _{GL_{2r}}$ attached to q is given as follows. The derivation $\delta _{GL_{2r}}$ induces a derivation, which we denote by $\delta _{GL^c_{r}}$, on the ring 7.9 (which we could refer to as the hermitian Chern connection on $GL_r^c$ attached to q); then we claim that the derivation $\delta _{GL^c_{r}}$ sends $R[v,\det (v)^{-1}]$ into itself and its restriction to $R[v,\det (v)^{-1}]$ equals $\delta _{GL_r}$. In other words, if $c:GL_r^c\rightarrow GL_r$ is the composition of the isomorphism 7.8 with the second projection and we still denote by c the induced algebra map between the ring of regular functions of the two groups, then we claim that the following diagram is commutative:

$$\begin{aligned} \begin{array}{rcl} \mathcal O(GL_r^c) &{}\quad \mathop {\longleftarrow }\limits ^{\delta _{GL_r^c}} &{}\quad \mathcal O(GL_r^c)\\ c \uparrow &{} \ &{}\quad \uparrow c\\ \mathcal O(GL_r) &{}\quad \mathop {\longleftarrow }\limits ^{\delta _{GL_r}} &{}\quad \mathcal O(GL_r)\end{array} \end{aligned}$$

(7.11)

To check the claim set $\alpha ={\left( \begin{array}{r@{\quad }r} \alpha _1 &{} \alpha _2\\ -\alpha _2&{}\alpha _1\end{array}\right) }$; then, since $\delta _{GL_{2r}}x=\alpha x$, we have

$$\begin{aligned} \delta _{GL_{2r}}v= & {} \delta _{GL_{2r}}(a+\sqrt{-1}\cdot b)\\= & {} \alpha _1 a-\alpha _2 b+\sqrt{-1}(\alpha _1 b+\alpha _2 a)\\= & {} \alpha ^c\cdot v, \end{aligned}$$

which proves our claim.

The existence of a derivation $\delta _{GL_r}$ making the diagram 7.11 commute is in stark contrast to assertion 3 of our Theorem 1.1. In other words, the hermitian Chern connection $\delta _{GL_r^c}$ above has a direct arithmetic analogue, $\phi _{GL_r^c}$, while the hermitian Chern connection $\delta _{GL_r}$ does not have a direct arithmetic analogue: the natural candidate $\phi _{GL_r}$ does not exist.

A fuller picture of the analogy with hermitian geometry is obtained as follows. Denote by $\overline{\delta }_{GL_r}$ the unique derivation on $R[v,\det (v)^{-1}]$ that sends v into 0 and equals the derivation $\overline{\delta }$ on R. Then $\delta _{GL_{2r,0}}$ and $\overline{\delta }_{GL_r}$ are related as follows. The derivation $\delta _{GL_{2r},0}$ induces a derivation, which we denote by $\delta _{GL^c_{r},0}$, on the ring 7.9; this induced derivation sends $R[v,\det (v)^{-1}]$ into itself, and if we denote by $\delta _{GL_r,0}:R[v,\det (v)^{-1}]\rightarrow R[v,\det (v)^{-1}]$ the restriction of $\delta _{GL^c_{r},0}$, we have the equality

$$\begin{aligned} \overline{\delta }_{GL_r}=\overline{(\ \ )}\circ \delta _{GL_r,0}\circ \overline{(\ \ )}, \end{aligned}$$

(7.12)

where $\overline{(\ \ )}$ is the automorphism of $R[v\det (v)^{-1}]$ that fixes v and lifts $a\mapsto \overline{a}$ on R. The pair of derivations $(\delta _{GL_r},\overline{\delta }_{GL_r})$ on $R[v,\det (v)^{-1}]$ is an analogue of the hermitian “Chern” connection on a hermitian vector bundle on a complex manifold [13], p. 73, cf. also the discussion in subsection 7.5.

From now on we assume, for simplicity that R is an algebraically closed field of characteristic zero; varieties over R will be identified with their sets of R-points.

1.5 Connections in vector bundles

We include, in what follows, a short digression on the “vector bundle” version of the above “principal bundle” formalism. Start with an n-dimensional vector space V over R and, again a derivation $\delta$ on R. By a connection on V, we understand here an additive group homomorphism $\nabla :V\rightarrow V$ such that

$$\begin{aligned} \nabla (a v)=(\delta a)v+a \nabla v \end{aligned}$$

for all $v\in V$, $a\in R$. Assume we are given a non-degenerate R-bilinear map

$$\begin{aligned} B:V\times V\rightarrow V \end{aligned}$$

which is either symmetric or antisymmetric (which we view as an analogue of a Riemannian metric or a 2-form, respectively). We say that $\nabla$ is B-horizontal if

$$\begin{aligned} \delta (B(u,v))=B(\nabla u,v)+B(u,\nabla v) \end{aligned}$$

(7.13)

for all $u,v\in V$. (If this is the case, we view $\nabla$ as an analogue of a connection that is compatible with a metric or a 2-form, respectively.) Moreover, given an R-linear linear endomorphism $\varLambda$ of V, we say that $\varLambda$ is B-symmetric if

$$\begin{aligned} B(\varLambda u, v) =B(u,\varLambda v) \end{aligned}$$

(7.14)

for all $u,v\in V$. It is easy to see that for any given non-degenerate bilinear map B and any connection $\nabla _0$ there is exactly one connection $\nabla$ such that $\nabla$ is B-horizontal and $\nabla -\nabla _0$ is B-symmetric. The uniqueness is clear. The existence is proved exactly in the same way one proves the existence of Chern connections. Indeed, let $V^*$ be the R-linear dual of V, equipped with the dual connection

$$\begin{aligned} \nabla _0^*:V^*\rightarrow V^* \end{aligned}$$

defined as the unique connection with the property that

$$\begin{aligned} \delta \langle u^*,v\rangle =\langle \nabla _0^* u^*,v\rangle +\langle u^*,\nabla _0 v\rangle , \end{aligned}$$

for $u^*\in V^*$ and $v\in V$; here $\langle \ ,\ \rangle :V^*\times V\rightarrow R$ is the duality pairing. Let $B^*:V\rightarrow V^*$ be the linear map defined by

$$\begin{aligned} \langle B^*(u),v\rangle :=B(u,v). \end{aligned}$$

Then define the connection $\nabla :V\rightarrow V$ by

$$\begin{aligned} \nabla :=\frac{1}{2}\nabla _0+\frac{1}{2}\cdot (B^*)^{-1}\circ \nabla _0^*\circ B^*. \end{aligned}$$

(7.15)

It is easy to see that $\nabla$ is B-horizontal and $\nabla -\nabla _0$ is B-symmetric.

The concepts of B-horizontality and B-symmetry relate to our concepts of ${{\mathcal {H}}}_q$-horizontality and ${{\mathcal {B}}}_q$-symmetry as follows. Start with a matrix $\alpha \in {\mathfrak {g}}{\mathfrak {l}}_n$ and consider the R-linear space $V=R^n$ whose elements we view as column vectors. Consider the connection

$$\begin{aligned} \nabla :V\rightarrow V,\ \ \ \nabla v=\delta v-\alpha v, \end{aligned}$$

for $v\in V$, where if $v=(v_i)$ has components $v_i$ then $\delta v:=(\delta v_i)$. Also for $q\in GL_n$ with $q^t=\pm q$, we consider the non-degenerate bilinear (symmetric, respectively, antisymmetric) map

$$\begin{aligned} B:V\times V\rightarrow R,\ \ \ B(u,v):=u^tqv. \end{aligned}$$

It is trivial then to see that $\nabla$ is B-horizontal if and only Eq. 7.5 holds and hence if and only if $\delta _G:=\delta _{G,0}^{\alpha }$ is ${{\mathcal {H}}}_q$-horizontal with respect to $\delta _{G,0}$. This makes the commutativity of the diagram 1.20 an analogue of the condition for a connection to be compatible with a metric or a 2-form, respectively. On the other hand, we may consider the connection

$$\begin{aligned} \nabla _0:V\rightarrow V,\ \ \ \nabla _0 v=\delta v; \end{aligned}$$

this is the unique connection that kills the standard basis of $R^n$. Then $\nabla -\nabla _0$ is B-symmetric if and only if the Eq. 7.6 holds, hence if and only if $\delta _G=\delta _{G,0}^{\alpha }$ is ${{\mathcal {B}}}_q$-symmetric with respect to $\delta _{G,0}$.

Note that, in our present context, there is no analogue of the concept of torsion in Riemannian geometry (because we only have one distinguished derivation $\delta$ on the base). In particular, our $\nabla$ is not an analogue of the Levi-Civita connection of a Riemannian metric.

Let us also record the hermitian paradigm. Assume the field R has the form $R_0\otimes _{{\mathbb Z}}{\mathbb Z}[\sqrt{-1}]$ where $R_0$ is some field and assume $a\mapsto \overline{a}$ is the $R_0$-automorphism of R sending $\sqrt{-1}\mapsto -\sqrt{-1}$. Let V is an n-dimensional vector space over R and let $\delta$ and $\overline{\delta }$ be two commuting derivations on R such that $\overline{\delta } (\overline{a})=\overline{\delta a}$ for all $a\in R$. Also let $H:V\times V\rightarrow R$ be a hermitian form (with respect to the involution on R). Define a $\delta$ -connection on V to be an additive operator $\nabla _{\delta }:V\rightarrow V$ such that $\nabla _{\delta }(a v)=\delta a \cdot v+ a\cdot \nabla _{\delta } v$ for $v\in V$ and $a\in R$; similarly define a $\overline{\delta }$ -connection on V to be an additive operator $\nabla _{\overline{\delta }}:V\rightarrow V$ such that $\nabla _{\overline{\delta }}(a v)=\overline{\delta } a \cdot v+ a\cdot \nabla _{\overline{\delta }} v$. Define a hermitian connection to be a pair $\nabla =(\nabla _{\delta },\nabla _{\overline{\delta }})$ consisting of a $\delta$-connection and a $\overline{\delta }$-connection on V. Say that $\nabla$ is compatible with H if one of the following two equivalent conditions is satisfied:

$$\begin{aligned} \delta H(v,w)= & {} H(\nabla _{\delta } v,w)+H(v,\nabla _{\overline{\delta }}w),\ \ v,w\in V\\ \overline{\delta } H(v,w)= & {} H(\nabla _{\overline{\delta }} v,w)+H(v,\nabla _{\delta }w),\ \ v,w\in V. \end{aligned}$$

Assume we are given a hermitian form H and a $\overline{\delta }$-connection $\nabla _0$ on V such that the kernel of $\nabla _0$ spans the R-vector space V. Then there exists a unique hermitian connection $\nabla =(\nabla _{\delta },\nabla _{\overline{\delta }})$ on V compatible with H and such that $\nabla _{\overline{\delta }}=\nabla _0$. Indeed, take an R-basis $(e_i)$ of V killed by $\nabla _0$ and consider the matrix $h=(h_{ij})$, where $h_{ij}=H(e_i,e_j)$. Then take $\nabla _{\overline{\delta }}=\nabla _0$ and define $\nabla _{\delta }$ by $\nabla _{\delta }e_i=\sum _j \overline{\beta }_{ji}e_j$, where the matrix $\beta =(\beta _{ij})$ is defined by $\beta =h^{-1}\delta h$. Cf. the computation of the hermitian “Chern” connection in [13], p. 73. This and Eq. 7.10 justify the terminology of hermitian Chern connection used in relation to Eq. 7.10.

1.6 Prime integrals

Going back to our general discussion of the involution $x^{\tau }=q^{-1}(x^t)^{-1}q$ on $G=GL_n$, let SO(q) be the identity component of the fixed group $G^+=\{x\in G;x^{\tau }=x\}$ and let ${\mathfrak s}{\mathfrak o}(q)$ be its Lie algebra. If $\alpha \in {\mathfrak s}{\mathfrak o}(q)$ and $\delta _G:=\delta _{G,0}^{\alpha }$, then, trivially,

$$\begin{aligned} \delta _{G}({{\mathcal {H}}}(x))=0; \end{aligned}$$

(7.16)

so ${{\mathcal {H}}}$ is a prime integral of the equation $\delta u=\alpha u$ in the sense that for any solution $u\in G$ of this equation we have $\delta ({{\mathcal {H}}}(u))=0$. There is a similar “classical” analogue of the $SL_n$ case of our Theorem 1.5. In particular, for ${{\mathcal {H}}}^*(x):=\det (x)$ and any $\alpha \in {\mathfrak s}{\mathfrak {l}}_n$, we have

$$\begin{aligned} \delta _{G}({{\mathcal {H}}}^*(x))=0, \end{aligned}$$

(7.17)

and hence ${{\mathcal {H}}}^*$ is a prime integral of the equation $\delta u=\alpha u$ in the sense that for any solution $u\in G$ of this equation we have $\delta (\det (u))=0$. The equality 7.17 follows from the general fact that for any ring equipped with a derivation $\delta$ and for any invertible matrix z with coefficients in that ring we have

$$\begin{aligned} \delta (\det (z))=\text {tr}(\delta z \cdot z^{-1})\det (z). \end{aligned}$$

(7.18)

The statement i) of Remark 1.6 is an arithmetic analogue of the statements 7.16 and 7.17. This suggests that the family of p-derivations $\delta _{G}^{\alpha }$ in 1.11 should be viewed as an arithmetic analogue of the “classical” family of derivations $\delta _{G,0}^{\alpha }$ in 7.2. The latter can be viewed as linear flows on G; this justifies viewing the p-derivations in 1.11 as arithmetic analogues of linear flows. The main difference between the “classical” and the arithmetic case is that the derivation $\delta _{G,0}$ in the classical case does not depend on $\tau$ (it simply maps x into 0), whereas the p-derivations $\delta _{G}=\delta _{G}^0$ in Theorem 1.5 depend (and they do so in a rather interesting way) on $\tau$.

1.7 Compatibility with inner involutions

If, instead of the above “outer” involutions, we consider “inner” involutions $x^{\tau }=q^{-1}xq$ with $q^2=1$, then the following hold. First note that ${{\mathcal {H}}}(x)=q^{-1}x^{-1}qx$ and the fixed group of $\tau$ is the centralizer of q, $C(q)=\{x\in G;xq=qx\}$. The Lie algebra of the latter is, of course, ${\mathfrak {c}}(q)=\{\alpha \in {\mathfrak {g}}{\mathfrak {l}}_n;\alpha q=q\alpha \}$. Now if $\alpha \in {\mathfrak {c}}(q)$ then, for $\delta _G:=\delta _{G,0}^{\alpha }$,

$$\begin{aligned} \delta _{G}({{\mathcal {H}}}(x))=-q^{-1}x^{-1}(\alpha x)x^{-1} qx +q^{-1}x^{-1}q(\alpha x)= 0 \end{aligned}$$

so ${{\mathcal {H}}}$ is a prime integral of the equation $\delta u=\alpha u$ in the sense that for any solution $u\in G$ of this equation we have $\delta ({{\mathcal {H}}}(u))=0$. This fact does not seem to have a direct analogue in the arithmetic setting, cf. assertion 4 in Theorem 1.8.

In seeking a classical fact of which assertion 3 in Theorem 1.8 and part (iii) in Remark 1.9 are arithmetic analogues, consider again an arbitrary derivation $\delta _{G}:\mathcal O(G)\rightarrow \mathcal O(G)$, $\delta _{G}x=\varDelta (x)$ and consider the diagram

$$\begin{aligned} \begin{array}{rcl} \mathcal O(G) &{}\quad \mathop {\longleftarrow }\limits ^{\delta _{G}} &{}\quad \mathcal O(G)\\ P \uparrow &{} \ &{}\quad \uparrow P\\ R[z] &{}\quad \mathop {\longleftarrow }\limits ^{\delta _0} &{}\quad R[z]\end{array} \end{aligned}$$

(7.19)

where $R[z]=R[z_1,\ldots ,z_n]$, $P(z_i)={\mathcal {P}}_i(x)$, $\det (s\cdot 1-x)=\sum _{i=0}^n(-1)^i{\mathcal {P}}_i(x)s^{n-i}$, $\delta _0z_i=0$. Then diagram 7.19 can be viewed as an analogue of the diagram 1.34. Assertion 3 in Theorem 1.8 is analogous to the easily checked fact that if

$$\begin{aligned} \varDelta (x)=\alpha x \end{aligned}$$

with $\alpha =\alpha (x)$ a diagonal matrix with entries in $\mathcal O(G)$ then 7.19 is commutative if and only if $\alpha =0$. Also part iii) in Remark 1.9 is an analogue of the fact that if

$$\begin{aligned} \varDelta (x)=[\alpha ,x]:=\alpha x-x\alpha \end{aligned}$$

where $\alpha =\alpha (x)$ is any matrix with coefficients in $\mathcal O(G)$ then the diagram 7.19 is commutative. Indeed, to check the latter is equivalent to checking that $\delta _{G}({\mathcal {P}}_i(x))=0$ for all i. Consider the ring $\mathcal O(G)[s,\det (s\cdot 1-x)^{-1}]$ and the unique derivation $\delta _{G}$ on this ring that extends $\delta _{G}:\mathcal O(G)\rightarrow \mathcal O(G)$ and sends $s\mapsto 0$. We want to show that $\delta _{G}(\det (s\cdot 1-x))$ vanishes. But

$$\begin{aligned} \delta _{G}(\det (s\cdot 1-x)) = \text {tr}(\delta _{G}(s\cdot 1-x)\cdot (s\cdot 1-x)^{-1})\det (s\cdot 1-x); \end{aligned}$$

so it is enough to show that the above trace vanishes. Now

$$\begin{aligned} \text {tr}(\delta _{G}(s\cdot 1-x)\cdot (s\cdot 1-x)^{-1})= & {} \text {tr}((x\alpha -\alpha x) \cdot (s\cdot 1-x)^{-1})\\= & {} \text {tr}(x\alpha (s\cdot 1-x)^{-1})-\text {tr}(\alpha x (s\cdot 1-x)^{-1})\\= & {} \text {tr}(\alpha (s\cdot 1-x)^{-1}x)-\text {tr}(\alpha x (s\cdot 1-x)^{-1})\\= & {} 0, \end{aligned}$$

because x and $(s\cdot 1-x)^{-1}$ commute, and our claim is proved. In particular, one gets that the polynomials ${\mathcal {P}}_i(x)$ are prime integrals of the equation

$$\begin{aligned} \delta u=[\alpha (u),u], \end{aligned}$$

(7.20)

equivalently of the equation

$$\begin{aligned} l\delta u=-(u\star \alpha (u))+\alpha (u), \end{aligned}$$

(7.21)

in the sense that for any solution u of this equation we have $\delta ({\mathcal {P}}_i(u))=0$. Here $\star$ is the adjoint action, $u\star v:=uvu^{-1}$. This is analogous to part iii) in Remark 1.9; especially Eq. 7.21 is analogous to Eq. 1.38. Cf. Remark 6.9 in the body of the paper. Equation 7.20 can be viewed as an “isospectral flow” on the space of matrices; so the lift of Frobenius $\phi _{G^{**}}$ in assertion 3 of Theorem 1.8 can be viewed as an arithmetic analogue of such an “isospectral flow”.