LANGLANDS CLASSIFICATIONS OF REDUCTIVE GROUPS

<正> Let G be a reductive algebraic group defined over R, and G(R) be its subgroup of realpoints. R. Langlands gave a parametrization of the irreducible admissible representations ofG(R) in terms of the L-group (see [1]). J. Adams and D. Vogan presented a substantiallymodified formulation of the Langlands classification by using a different definition of theL-group. This offers some benefit for the applications of the Langlands classification. Inthis paper,we discuss the connection between the Langlands classification and Adams-Vogan’smodified formulation. Especially, for the case of G(R) = Sp(n,R), we give a clear descrip-tion of the correspondence between the parametrization of the Langlands classification byusing discrete series representation and Adams-Vogan’s modified formulation.

Let G be a reductive algebraic group defined over R, and G(R) be its subgroup of realpoints. R. Langlands gave a parametrization of the irreducible admissible representations ofG(R) in terms of the L-group (see [1]). J. Adams and D. Vogan presented a substantiallymodified formulation of the Langlands classification by using a different definition of theL-group. This offers some benefit for the applications of the Langlands classification. Inthis paper,we discuss the connection between the Langlands classification and Adams-Vogan’smodified formulation. Especially, for the case of G(R) = Sp(n,R), we give a clear descrip-tion of the correspondence between the parametrization of the Langlands classification byusing discrete series representation and Adams-Vogan’s modified formulation.