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Beyond endoscopy. (English) Zbl 1078.11033
Hida, Haruzo (ed.) et al., Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, May 14–17, 2002. Baltimore, MD: Johns Hopkins University Press (ISBN 0-8018-7860-8/hbk). 611-697 (2004).
The author speculates on two themes:
(I) If $$H$$ and $$G$$ are two reductive groups over the global field $$F$$ and the group $$G$$ is quasi-split then to each homomorphism $\phi:\,\,^LH\rightarrow\,\,^LG$ there is associated a transfer of automorphic representations of $$H$$ to automorphic representations of $$G$$.
(II) If $$\pi$$ is an automorphic representation of $$G$$ it should exist an algebraic subgroup $$^\lambda H_\pi$$ of $$^LG$$ such that if $$\rho$$ is a representation of $$^LG$$ then the multiplicity $$m_H(\rho)$$ of the trivial representation of $$^\lambda H_\pi$$ in the restriction of $$\rho$$ to $$^\lambda H_\pi$$ is the order $$m_\pi(\rho)$$ of the pole of the $$L$$-function $$L(s,\pi,\rho)$$ at $$s=1$$.
For the entire collection see [Zbl 1051.11005].

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
##### Keywords:
automorphic representation; L-function
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