### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 076, 15 pages      arXiv:1506.08071      https://doi.org/10.3842/SIGMA.2015.076

### Weil Representation of a Generalized Linear Group over a Ring of Truncated Polynomials over a Finite Field Endowed with a Second Class Involution

Luis Gutiérrez Frez a and José Pantoja b
a) Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Campus Isla Teja SN, Edificio Pugín, Valdivia, Chile
b) Instituto de Matemáticas, Pontificia Universidad Catolica de Valparaíso, Blanco Viel 596, Co. Barón, Valparaíso, Chile

Received July 03, 2015, in final form September 14, 2015; Published online September 26, 2015

Abstract
We construct a complex linear Weil representation $\rho$ of the generalized special linear group $G={\rm SL}_*^{1}(2,A_n)$ ($A_n=K[x]/\langle x^n\rangle$, $K$ the quadratic extension of the finite field $k$ of $q$ elements, $q$ odd), where $A_n$ is endowed with a second class involution. After the construction of a specific data, the representation is defined on the generators of a Bruhat presentation of $G$, via linear operators satisfying the relations of the presentation. The structure of a unitary group $U$ associated to $G$ is described. Using this group we obtain a first decomposition of $\rho$.

Key words: Weil representation; generalized special linear group.

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