Nilpotent parts on a joint cyclic subspace give finite logarithmic dilation factors

Let $V$ be a complex vector space, let $A,B:V\to V$ be commuting linear maps, and let $v\in V$ satisfy $A^{r}v=0$ and $B^{s}v=0$ for positive integers $r,s$. Let $h,\bar{h}\in \mathbb{C}$. Let $U_z,U_{\zeta }\subset {\mathbb{C}}^{\times }$ be simply connected open sets, with chosen holomorphic logarithms ${\mathrm{\ell }}_z$ and ${\mathrm{\ell }}_{\zeta }$.

The span

$W_v={\mathrm{span}}_{\mathbb{C}}\{A^i{B}^{j}v:0\le i<r,0\le j<s\}$

is finite dimensional and invariant under $A$ and $B$. On $W_v$, define

$D_v(z,\zeta )=\exp \text{\hspace{0.17em}⁣}({\mathrm{\ell }}_z(z)(hI+A))\exp \text{\hspace{0.17em}⁣}({\mathrm{\ell }}_{\zeta }(\zeta )(\bar{h}I+B))\mathrm{.}$

Then

$D_v(z,\zeta )v=e^{h{\mathrm{\ell }}_z(z)+\bar{h}{\mathrm{\ell }}_{\zeta }(\zeta )}{\sum }_{i=0}^{r-1}{\sum }_{j=0}^{s-1}\frac{{\mathrm{\ell }}_z(z{)}^i{\mathrm{\ell }}_{\zeta }(\zeta {)}^j}{i!j!}{A}^i{B}^{j}v\mathrm{.}$

For every linear functional $\lambda :W_v\to \mathbb{C}$, the normalized matrix coefficient $e^{-h{\mathrm{\ell }}_z-\bar{h}{\mathrm{\ell }}_{\zeta }}\lambda (D_{v}v)$ is the finite polynomial

${\sum }_{i=0}^{r-1}{\sum }_{j=0}^{s-1}\frac{\lambda (A^i{B}^{j}v)}{i!j!}{\mathrm{\ell }}_z^i{\mathrm{\ell }}_{\zeta }^j\mathrm{.}$

Thus the nilpotent generator parts $A{\mathrm{\mid }}_{W_v},B{\mathrm{\mid }}_{W_v}$ supply possible logarithmic terms; a chosen pairing may annihilate their positive-degree coefficients. Under a Euclidean loop about the origin, ${\mathrm{\ell }}_z$ changes by $2\pi i$ and ${\mathrm{\ell }}_{\zeta }$ by $-2\pi i$, so the continued dilation is

$e^{2\pi i(hI+A)}{e}^{-2\pi i(\bar{h}I+B)}{D}_v\mathrm{.}$

Single-valuedness therefore requires a separate condition on the paired fields and matrix coefficients.