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On certain representations of semi-simple algebraic groups

  • Jun-ichi Igusa
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 185)

Abstract

Suppose that 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
$$\sum\limits_{\xi \in X_k } {\Phi (\rho (g) \cdot \xi )} $$
is satisfied. (The subscript 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.

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Copyright information

© Springer-Verlag 1971

Authors and Affiliations

  • Jun-ichi Igusa
    • 1
  1. 1.The Johns Hopkins UniversityBaltimore

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