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Irreducible cuspidal representations with prescribed local behavior

From: American Journal of Mathematics
Volume 133, Number 5, October 2011
pp. 1231-1258 | 10.1353/ajm.2011.0035



Let $G$ be a simple algebraic group defined over the global field $k$. In this paper, we use the simple trace formula to determine the sum of the multiplicities of the irreducible representations in the cuspidal spectrum of $G$, with specified local behavior at a finite set of places of $k$ and unramified elsewhere. This sum is expressed as the product of the values of modified Artin $L$-functions at negative integers.