Adjoint Orbits of Matrix Groups over Finite Quotients of Compact Discrete Valuation Rings and Representation Zeta Functions
This paper gives methods to describe the adjoint orbits of $\mathbf{G}(\mathfrak{o}_r)$ on $\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$ where $\mathfrak{o}_r=\mathfrak{o}/\mathfrak{p}^r$ ($r\in\mathbb{N}$) is a finite quotient of the localization $\mathfrak{o}$ of the ring of integers of a number fiel...
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Sprache: | eng |
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Zusammenfassung: | This paper gives methods to describe the adjoint orbits of
$\mathbf{G}(\mathfrak{o}_r)$ on $\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$
where $\mathfrak{o}_r=\mathfrak{o}/\mathfrak{p}^r$ ($r\in\mathbb{N}$) is a
finite quotient of the localization $\mathfrak{o}$ of the ring of integers of a
number field at a prime ideal $\mathfrak{p}$ and $\mathbf{G}$ is a closed
$\mathbb{Z}$-subgroup scheme of $\mathrm{GL}_{n}$ for an $n\in\mathbb{N}$ and
such that the Lie ring $\mathrm{Lie}(\mathbf{G})(\mathfrak{o})$ is quadratic..
The main result is a classification of the adjoint orbits in
$\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_{r+1})$ whose reduction
$\bmod\,\mathfrak{p}^{r}$ contains
$a\in\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$ in terms of the reduction
$\bmod\mathfrak{p}$ of the stabilizer of $a$ for the
$\mathbf{G}(\mathfrak{o}_r)$-adjoint action. As an application, this result is
then used to compute the representation zeta function of the principal
congruence subgroups of $\mathrm{SL}_{3}(\mathfrak{o})$. |
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DOI: | 10.48550/arxiv.1608.05725 |