  Let X^\mu={X_t^\mu;t>=0}, \mu>0, be the n-dimensional hyperbolic Brownian
motion with drift, that is a diffusion on the real hyperbolic space H^n having
the Laplace-Beltrami operator with drift as its generator. We prove the
reflection principle for X^\mu, which enables us to study the process X^\mu
killed when exiting the hyperbolic half-space, that is the set D={x\in H^n:
x_1>0}. We provide formulae, uniform estimates and describe asymptotic behavior
of the Green function and the Poisson kernel of D for the process X^\mu.
Finally, we derive formula for the lambda-Poisson kernel of the set D.
