Equidistribution of intersections with homogeneous subspaces

I will discuss the following general problem in homogeneous dynamics: Let $G$ be a semisimple Lie group, $K$ a compact subgroup of $G$ and $\Gamma$ a lattice in $G$. Let $O_n$ be an equidistributing sequence of locally homogeneous subspaces of $\Gamma \backslash G/K$. What is the distribution of the intersection of $O_n$ with a fixed analytic subspace $V$?

 

In a joint work with Salim Tayou, we show that, under the correct hypotheses, this intersection equidistributes towards a certain $G$-invariant form on $G/K$. Determining this form leads to an interesting question about the cohomology of compact homogeneous spaces. This result has interesting applications in arithmetics and in Hodge theory, which I will mention briefly if time permits.