  We propose a group theoretical method to study Isgur-Wise functions. A
current matrix element splits into a heavy quark matrix element and an overlap
of the initial and final clouds, related to the IW functions, that contain the
long distance physics. The light cloud belongs to the Hilbert space of a
unitary representation of the Lorentz group. Decomposing into irreducible
representations one obtains the IW function as an integral formula,
superposition of irreducible IW functions with positive measures, providing
positivity bounds on its derivatives. Our method is equivalent to the sum rule
approach, but sheds another light on the physics and summarizes and gives all
its possible constraints. We expose the general formalism, thoroughly applying
it to the case j = 0 for the light cloud, relevant to the semileptonic decay
Lambda_b -> Lambda_c + l + nu. In this case, the principal series of the
representations contribute, and also the supplementary series. We recover the
bound for the curvature of the j = 0 IW function xi_Lambda (w) that we did
obtain from the sum rule method, and we get new bounds for higher derivatives.
We demonstrate also that if the lower bound for the curvature is saturated,
then xi_Lambda (w) is completely determined, given by an explicit elementary
function. We give criteria to decide if any ansatz for the Isgur-Wise function
is compatible or not with the sum rules. We apply the method to some simple
model forms proposed in the literature. Dealing with a Hilbert space, the sum
rules are convergent, but this feature does not survive hard gluon radiative
corrections.
