In his monograph Arthur (The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publications, American Mathematical Society, Providence, 2013) characterizes the L-packets of quasisplit symplectic groups and orthogonal groups. By extending his work, we characterize the L-packets for the corresponding similitude groups with desired properties. In particular, we show these packets satisfy the conjectural endoscopic character identities.

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The harmonic Bergman kernel$Q$_{Ω}for a simply, connected planar domain Ω can be expanded in terms of powers of the Friedrichs operator$F$_{Ω}║$F$_{Ω}║<1 in operator norm. Suppose that Ω is the image of a univalent analytic function ø in the unit disk with ø' ($z$)=1+ψ ($z$) where ψ(0)=0. We show that if the function ψ belongs to a space$D$_{$s$}($D$),$s$>0, of Dirichlet type, then provided that ║ψ║∞<1 the series for$Q$_{Ω}also converges pointwise in $$\bar \Omega \times \bar \Omega \backslash \Delta (\partial \Omega )$$ , and the rate of convergence can be estimated. The proof uses the eigenfunctions of the Friedrichs operator as well as a formula due to Lenard on projections in Hilbert spaces. As an application, we show that for every$s$>0 there exists a constant$C$_{$s$}>0 such that if ║ψ║_{$D$}_{s}($D$)≤$C$_{$s$}, then the biharmonic Green function for Ω=ø ($D$) is positive.

We construct a generalization of twistor spaces of hypercomplex manifolds and hyper-Kahler manifolds M, by generalizing the twistor P1 to a more general complex manifold Q. The resulting manifold X is complex if and only if Q admits a holomorphic map to P1. We make branched double covers of these manifolds. Some class of these branched double covers can give rise to non-Kahler Calabi-Yau manifolds. We show that these manifolds X and their branched double covers are non-Kahler. In the cases that Q is a balanced manifold, the resulting manifold X and its special branched double cover have balanced Hermitian metrics.