  Using the heavy quark effective theory framework put forward by Grinstein and
Pirjol we work out predictions for B -> K* l+ l-, l = (e, mu), decays for a
softly recoiling K*, i.e., for large dilepton masses sqrt{q^2} of the order of
the b-quark mass m_b. We work to lowest order in Lambda/Q, where Q = (m_b,
sqrt{q^2}) and include the next-to-leading order corrections from the charm
quark mass m_c and the strong coupling at O(m_c^2/Q^2, alpha_s). The leading
Lambda/m_b corrections are parametrically suppressed. The improved Isgur-Wise
form factor relations correlate the B -> K* l+ l- transversity amplitudes,
which simplifies the description of the various decay observables and provides
opportunities for the extraction of the electroweak short distance couplings.
We propose new angular observables which have very small hadronic
uncertainties. We exploit existing data on B -> K* l+ l- distributions and show
that the low recoil region provides powerful additional information to the
large recoil one. We find disjoint best-fit solutions, which include the
Standard Model, but also beyond-the-Standard Model ones. This ambiguity can be
accessed with future precision measurements.
