  In quantum gravity the unitary evolution does not follow from the
Wheeler-DeWitt dynamics equation as it follows from the Schr\"odinger equation
in non-relativistic quantum mechanics. Therefore we can define a spin-foam
model based on SL(2,C) spinor finite non-unitary representations. The recently
discovered holomorphic Peter-Weyl theorem \cite{Huebschmann} made it possible
to decompose the delta function of a non-compact Lorentz group into the
convergent sum of the matrix coefficients. We calculate the vertex amplitude
with the help of that theorem and obtain a simple expression for our model. The
$SL(2,C)$ Hilbert space is defined from $SU(2)$ Hilbert space by
Huebschmann-Kirillov transform \cite{Huebschmann}. A new transform is simpler
than the well known Hall transform as it does not contain a heat kernel
convolution. We do not set Barbero-Immirzi constant $\gamma$ a priori, instead
we obtain it as a solution of the diagonal and off-diagonal simplicity
constraints being $\gamma = \frac{-in}{(|n| + 2p)}$ where $p$ is a non-negative
half-integer. When $p=0$ the solution corresponds to the Ashtekar's self-dual
connections. We point out that the Barbero-Immirzi becomes real when one
chooses a unitary representation. It is complex when the representation is
non-unitary principal series or non-unitary spinor representation.
