Representaciones de Weil del Grupo SL* (2,F,q [x] /xm

Abstract

Let A be a unitary ring with involution *. The groups ,9.L*(2, A) were defined by J. Pa.ntoja and J. Soto Andrade in [4]. These groups are a nonconmutative version of the special linear groups SL(2,F), defined over a commutative base field -F'- In particular, if A is ihe full matrix ring M(2, F) endowed with the trasposition as involution, ther SL*(2, A) coincides with the s¡mplectic group ,Sp(2n, ,F') in 2n rariables. In this thesis we study the truncated polinomial rings A- : Erl"l/ @-)) with k a ñnite field of odd cha.racteritic p, for any positive integer rn endovved with the uatu¡al involution given s é -:8. These rings, reminiscent of algebras de rn-jets, appear v¡hen rn is a power ofp, with a difierent involution as polinomial models for the non- semi-simple modular group algebras k[C*] over a base field of characteristic p, whose involution is given by inversion in the group. We obtain a full cha¡acterization of all the involutions in the ring ,4*, for any rn and the fact that there is only one isomorphism type of non trivial involutio¡¡s in á-. Moreover we obtain a representation ol the SL"(2, A*) in terms of the so called "Bruhat generatof' and a complete set of relations a.rnong themThe order of the goup SL*(z, A-) is also calculated. Finally we construct a remarkable linear complex representation of the group SL*(2,4*), for odd m, which generalizes the classical co¡structionof the Weil representation of the grcup SL(z,F), defi¡red over a finite base field -F of odd characteristic, associated to a non degenerate quadratic space over }.. A first decomposition of the Weil representation is indicated .