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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