  To each simple Lie algebra g and an element w of the corresponding Weyl group
De Concini, Kac and Procesi associated a subalgebra U^w_- of the quantized
universal enveloping algebra U_q(g), which is a deformation of the universal
enveloping algebra U(n_- \cap w(n_+)) and a quantization of the coordinate ring
of the Schubert cell corresponding to w. The torus invariant prime ideals of
these algebras were classified by M\'eriaux and Cauchon [25], and the author
[30]. These ideals were also explicitly described in [30]. They index the the
Goodearl-Letzter strata of the stratification of the spectra of U^w_- into
tori. In this paper we derive a formula for the dimensions of these strata and
the transcendence degree of the field of rational Casimirs on any open
Richardson variety with respect to the standard Poisson structure [15].
