Theta correspondence and associated cycles

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We developed certain machinery for studying associated cycles of unitary representations of semisimple Lie groups by means of theta correspondence.To be more explicit, let (G_1 , G_2) be an irreducible, reductive dual pair of type I, which is in the stable range with G_2 as the smaller member. Take an irreducible representation π_2 of G_2 (or of its metaplectic double cover), and lift it to the representation π_1 of G_1 (or its metaplectic cover), called the theta lift ofπ_2. We study the cases where π_2 is a finite-dimensional unitary representation, or a representation in the holomorphic discrete series. Even if π_2 is the trivial representation, π_1 is an infinite dimensional unitary representation, which is supposed to be a unipotent representation, one of important objects in representation theory. In the following, we shall list up our results, where π_2 is a finite-dimensional unitary representation, or a representation in the holomorphic discrete series.We get the K-type formula for π_1.We describe the correspondence of associated varieties of (π_1, π_2) explicitly. Moreover, we succeed to get the multiplicities in their associated cycles and get a simple correspondence (still conjectural in general) between them.An associated variety can often be irreducible, and is the closure of a single nilpotent orbit. Using the results on the theta correspondence of representations, we investigate the corre spondence of nilpotent orbits. The main results are, a description of the structure of the function ring, an integral formula of degree of nilpotent orbits.