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equidistribution de mesures algébriques

Published online by Cambridge University Press:  01 September 2005

laurent clozel
Affiliation:
laboratoire de mathématiques, université paris-sud, bâtiment 425, 91405 orsay cedex, francelaurent.clozel@math.u-psud.fr
emmanuel ullmo
Affiliation:
laboratoire de mathématiques, université paris-sud, bâtiment 425, 91405 orsay cedex, franceemmanuel.ullmo@math.u-psud.fr
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Abstract

let g be an algebraic group, $\gamma$ an arithmetic lattice of g and $x=\gamma\backslash g$. if h is an algebraic subgroup of g such that $h\cap \gamma$ is a lattice of h, then $\gamma\backslash \gamma h\subset x$ is endowed with a canonical h-invariant probability measure $\mu_h$. using ratner's theory, we give general examples where $\mu_{h_n}$ converges weakly to $\mu_g$ if hn is a strict sequence of algebraic subgroups of g. if $\gamma$ is a congruence subgroup of g, we define another probability measure $\mu_h^a$ on x by using the adelic description of the quotient. we conjecture that $\mu_{h_n}^a$ always converges weakly to $\mu_g$ if hn is a strict sequence. using automorphic forms and l-functions, we describe the case $g=\textit{sl}(2,f)$ for a number field f and a sequence of tori hn. the relation with similar problems on shimura varieties is explained.

Type
Research Article
Copyright
foundation compositio mathematica 2005

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