Groups, Geometry, and Dynamics


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Volume 11, Issue 1, 2017, pp. 57–74
DOI: 10.4171/GGD/387

Published online: 2017-04-20

Faithful representations of Chevalley groups over quotient rings of non-Archimedean local fields

Mohammad Bardestani[1], Camelia Karimianpour[2], Keivan Mallahi-Karai and Hadi Salmasian[3]

(1) University of Ottawa, Canada
(2) University of Ottawa, Canada
(3) University of Ottawa, Canada

Let $F$ be a non-Archimedean local field with the ring of integers $\mathcal{O}$ and the prime ideal $\mathfrak{p}$ and let $G=\mathbf G_{\mathrm {ad}} (\mathcal O/\mathfrak p^n)$ be the adjoint Chevalley group. Let $m_{\sf f} (G)$ denote the smallest possible dimension of a faithful representation of $G$. Using the Stone–von Neumann theorem, we determine a lower bound for $m_{\sf f} (G)$ which is asymptotically the same as the results of Landazuri, Seitz and Zalesskii for split Chevalley groups over $\mathbb F_q$. Our result yields a conceptual explanation of the exponents that appear in the aforementioned results.

Keywords: Chevalley groups, faithful representation, Heisenberg subgroups, local fields, Stone–von Neumann theorem

Bardestani Mohammad, Karimianpour Camelia, Mallahi-Karai Keivan, Salmasian Hadi: Faithful representations of Chevalley groups over quotient rings of non-Archimedean local fields. Groups Geom. Dyn. 11 (2017), 57-74. doi: 10.4171/GGD/387