# Random matrix models of disordered bosons

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Unlike the fermionic side where the underlying Lie groups of the classical ensembles are compact, the symmetric groups for ensembles of disordered bosons are typically noncompact. In that case the basic random matrix models consist of matrices in the Lie algebra g = sp_{n} (R). Assuming dynamical stability, their eigenvalues are required to be purely imaginary. In the lecture we will sketch our recent work with K. Schaffert (J. Phys. A: Math. Theor. 44(2011) 335207) where a method is proposed for constructing ensembles (E; P) of G-invariant sets E of such matrices with probability measures P. These arise as moment map direct images from phase spaces X which play an important role in complex geometry and representation theory. In the toy-model case of n = 1, where X is the complex bidisk and P is the direct image of the uniform measure, an explicit description of the spectral measure is given.