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 ; José Pantoja

Symmetry, integrability and geometry: methods and applications, Tome 11 (2015)

Résumé

We construct a complex linear Weil representation $\rho$ of the generalized special linear group $G=\mathrm{SL}_*^{1}(2,A_n)$ ($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 $\rho$.

Détail

Keywords: Weil representation; generalized special linear group.

Bibliographie

[1] Aubert A.-M., Przebinda T., “A reverse engineering approach to the Weil representation”, Cent. Eur. J. Math., 12 (2014), 1500–1585 | DOI | MR | Zbl

[2] Cliff G., McNeilly D., Szechtman F., “Weil representations of symplectic groups over rings”, J. London Math. Soc., 62 (2000), 423–436 | DOI | MR | Zbl

[3] Dutta K., Prasad A., “Combinatorics of finite abelian groups and Weil representations”, Pacific J. Math., 275 (2015), 295–324, arXiv: 1010.3528 | DOI | MR | Zbl

[4] Gérardin P., “Weil representations associated to finite fields”, J. Algebra, 46 (1977), 54–101 | DOI | MR

[5] Gow R., “Even unimodular lattices associated with the Weil representation of the finite symplectic group”, J. Algebra, 122 (1989), 510–519 | DOI | MR | Zbl

[6] Gutiérrez Frez L., “A generalized Weil representation for ${\rm SL}_*(2,A_m)$, where $A_m={\mathbb F}_q[x]/\langle x^m\rangle$”, J. Algebra, 322 (2009), 42–53 | DOI | MR | Zbl

[7] Gutiérrez Frez L., Pantoja J., Soto-Andrade J., “On generalized Weil representations over involutive rings”, New Developments in Lie Theory and its Applications, Contemp. Math., 544, Amer. Math. Soc., Providence, RI, 2011, 109–121, arXiv: 1009.0877 | DOI | MR | Zbl

[8] Herman A., Szechtman F., “The Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring”, J. Algebra, 392 (2013), 158–184, arXiv: 1306.3997 | DOI | MR | Zbl

[9] Howe R., “Remarks on classical invariant theory”, Trans. Amer. Math. Soc., 313 (1989), 539–570 | DOI | MR | Zbl

[10] Knus M.-A., Merkurjev A., Rost M., Tignol J.-P., The book of involutions, American Mathematical Society Colloquium Publications, 44, Amer. Math. Soc., Providence, RI, 1998 | MR | Zbl

[11] Pantoja J., “A presentation of the group ${\rm Sl}_*(2,A)$, $A$ a simple Artinian ring with involution”, Manuscripta Math., 121 (2006), 97–104 | DOI | MR | Zbl

[12] Pantoja J., Soto-Andrade J., “A Bruhat decomposition of the group ${\rm SL}_*(2,A)$”, J. Algebra, 262 (2003), 401–412 | DOI | MR | Zbl

[13] Pantoja J., Soto-Andrade J., “Bruhat presentations for $*$-classical groups”, Comm. Algebra, 37 (2009), 4170–4191 | DOI | MR | Zbl

[14] Prasad A., “On character values and decomposition of the Weil representation associated to a finite abelian group”, J. Anal., 17 (2009), 73–85, arXiv: 0903.1486 | MR | Zbl

[15] Soto-Andrade J., Représentations de certains groupes symplectiques finis, Mémoires de la Société Mathématique de France, 55–56, Société Mathématique de France, Paris, 1978

[16] Vera Gajardo A., “A generalized Weil representation for the finite split orthogonal group ${\rm O}_q(2n,2n)$, $q$ odd $>3$”, J. Lie Theory, 25 (2015), 257–270, arXiv: 1311.1174 | MR | Zbl

[17] Weil A., “Sur certains groupes d'opérateurs unitaires”, Acta Math., 111 (1964), 143–211 | DOI | MR | Zbl