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Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification

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  • Let $ \mathit{\boldsymbol{\mathrm{G}}} $ be a semisimple linear algebraic group defined over rational numbers, $ \mathrm{K} $ be a maximal compact subgroup of its real points and $ \Gamma $ be an arithmetic lattice. One can associate a probability measure $ \mu_{ \mathrm{H}} $ on $ \Gamma \backslash \mathrm{G} $ for each subgroup $ \mathit{\boldsymbol{\mathrm{H}}} $ of $ \mathit{\boldsymbol{\mathrm{G}}} $ defined over $ \mathbb{Q} $ with no non-trivial rational characters. As G acts on $ \Gamma \backslash \mathrm{G} $ from the right, we can push forward this measure by elements from $ \mathrm{G} $. By pushing down these measures to $ \Gamma \backslash \mathrm{G}/ \mathrm{K} $, we call them homogeneous. It is a natural question to ask what are the possible weak-$ * $ limits of homogeneous measures. In the non-divergent case this has been answered by Eskin–Mozes–Shah. In the divergent case Daw–Gorodnik–Ullmo prove a refined version in some non-trivial compactifications of $ \Gamma \backslash \mathrm{G}/ \mathrm{K} $ for $ \mathit{\boldsymbol{\mathrm{H}}} $ generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general $ \mathit{\boldsymbol{\mathrm{H}}} $ with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on $ {\text{SL}}_n $ proved by geometry of numbers and a theorem of Kleinbock–Margulis; (2) relations between partial Borel–Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory. 193 words.

    Mathematics Subject Classification: Primary: 37A17; Secondary: 22E46.


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