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We pursue the group theoretical method to study Isgur-Wise functions. Weapply the general formalism, formerly applied to the baryon case j^P = 0^+ (for\Lambda_b -> \Lambda_c \ell \nu), to mesons with j^P = 1/2^-, i.e.$\overline{B} -> D(D^{(*)})\ell\nu. In this case, more involved from theangular momentum point of view, only the principal series of unitaryrepresentations of the Lorentz group contribute. We obtain an integralrepresentation for the IW function xi(w) with a positive measure, recover thebounds for the slope and the curvature of xi(w) obtained from theBjorken-Uraltsev sum rule method, and get new bounds for higher derivatives. Wedemonstrate also that if the lower bound for the slope is saturated, themeasure is a delta-function, and xi(w) is given by an explicit elementaryfunction. Inverting the integral formula, we obtain the measure in terms of theIW function, allowing to formulate criteria to decide if a given ansatz for theIsgur-Wise function is compatible or not with the sum rule constraints.Moreover, we have obtained an upper bound on the IW function valid for anyvalue of w. We compare these theoretical constraints to a number of forms for\xi(w) proposed in the literature. The "dipole" function \xi(w) =(2/(w+1))^(2c) satisfies all constraints for c \geq 3/4, while the QCD Sum Ruleresult including condensates does not satisfy them. Special care is devoted tothe Bakamjian-Thomas relativistic quark model in the heavy quark limit and tothe description of the Lorentz group representation that underlies this model.Consistently, the IW function satisfies all Lorentz group criteria for anyexplicit form of the meson Hamiltonian at rest.

Isgur-Wise functions and unitary representations of the Lorentz group : the meson case j = 1/2