# On certain representations of semi-simple algebraic groups

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## Abstract

Suppose that is satisfied. (The subscript

*k*is an algebraic number field,*G*a connected semi-simple algebraic group, and*ρ*a rational representation of*G*in a vector space*X*defined over*k*. The triple (*G,X*,ρ) or simply ρ is called*admissible*over*k*if the Weil criterion [9, p. 20] for the convergence of the integral over*G*_{ A }/*G*_{ k }of the generalized theta-series$$\sum\limits_{\xi \in X_k } {\Phi (\rho (g) \cdot \xi )} $$

*A*denotes the adelization functor relative to*k*and ϕ is an arbitrary Schwartz-Bruhat function on*X*_{ A }.) It is called*absolutely admissible*if it is admissible over any finite algebraic extension of*k*. We shall discuss a complete classification of all absolutely admissible representations. (If*G*is absolutely simple, there are essentially 6 infinite sequences of, and 26 isolated, absolutely admissible triples.) Then, in the general case, we shall discuss the existence and basic properties of what we call the*principal subset X′*of*X*. Although the adjoint representation of*G*can never be absolutely admissible, we can say that*X′*is analogous to the set of regular elements of the Lie algebra of*G*via its adjoint representation [cf. 4]. The significance of*X′*is that it permits us to formulate a conjectural Siegel formula in a precise, explicit form. (The restriction to*X′*corresponds in the classical Siegel case to considering only those terms which contribute to cusp forms.) We shall discuss a weak solution of this conjecture in the general case. Finally, if there exists only one*G*-invariant, we shall discuss its solution as well as the analytic continuation and the functional equation of the corresponding zeta-function.## Preview

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© Springer-Verlag 1971