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We propose a group theoretical method to study Isgur-Wise functions. Acurrent matrix element splits into a heavy quark matrix element and an overlapof the initial and final clouds, related to the IW functions, that contain thelong distance physics. The light cloud belongs to the Hilbert space of aunitary representation of the Lorentz group. Decomposing into irreduciblerepresentations one obtains the IW function as an integral formula,superposition of irreducible IW functions with positive measures, providingpositivity bounds on its derivatives. Our method is equivalent to the sum ruleapproach, but sheds another light on the physics and summarizes and gives allits possible constraints. We expose the general formalism, thoroughly applyingit to the case j = 0 for the light cloud, relevant to the semileptonic decayLambda_b -> Lambda_c + l + nu. In this case, the principal series of therepresentations contribute, and also the supplementary series. We recover thebound for the curvature of the j = 0 IW function xi_Lambda (w) that we didobtain 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 elementaryfunction. We give criteria to decide if any ansatz for the Isgur-Wise functionis compatible or not with the sum rules. We apply the method to some simplemodel forms proposed in the literature. Dealing with a Hilbert space, the sumrules are convergent, but this feature does not survive hard gluon radiativecorrections.

Isgur-Wise functions and unitary representations of the Lorentz group : the baryon case j = 0