Let me attempt a partial answer, trying to estimate the information contained in these characteristic classes. Since the explanations below are a
bit lengthy, I begin with the short version: I think that these
characteristic classes could have representation-theoretic
applications, but they likely do not provide more information than what is
also available by other means.
Setting - characteristic classes for singular varieties
Let us first examine the setting, and figure out how we could
understand the $L$-class. As explained in (S. Yokura: On
Cappell-Shaneson's homology $L$-classes of singular algebraic
varieties. Trans AMS. 347 (1995), 1005--1012), the $L$-class of
Cappell-Shaneson is the unique transformation $L_\ast:\Omega(-)\to
H_\bullet(-,\mathbb{Q})$ of covariant functors which agrees with
Hirzebruch's $L$-class when $X$ is smooth. Here, $\Omega(X)$ is
an algebraic group of cobordism classes of self-dual complexes on
$X$. The Goresky-MacPherson $L$-class of the question is obtained by applying
the Cappell-Shaneson $L$-class transformation to the class of the
intersection complex of $X$.
Now at this point, we can ask several slightly more precise questions:
for $X=G/B$ or a Schubert variety in there, what is $\Omega(X)$, what
is the map $L_\ast(X):\Omega(X)\to H_\bullet(X,\mathbb{Q})$ and what/how can we describe the class of the intersection complex in $\Omega(X)$?
I do not exactly know which of these questions you wanted to have answered, and I actually would not know the answer to any of them.
Instead of answering the specific question, let me
look at a slightly different, more general questions. In fact, the Cappell-Shaneson $L$-class
transformation is only a special case of a motivic Hirzebruch
transformation, cf. J.-P. Brasselet, J. Schürmann, S. Yokura:
Hirzebruch classes and motivic Chern classes for singular
spaces. J. Topol. Anal. 2 (2010) 1--55. The motivic Hirzebruch class
transformation is a map
$$
T_{y\ast}:K_0(\operatorname{Var}/X)\to
H_\bullet(X)\otimes\mathbb{Q}[y].
$$
This can be specialized to all sorts of characteristic classes,
Chern-Schwartz-MacPherson classes, $L$-classes, Todd classes, etc.
Now the new, more general question is: for $X=G/B$ or a Schubert
variety in there, can we define a class in
$K_0(\operatorname{Var}/X)$ - a universal motivic characteristic class - which specializes to the intersection
complex? How can we describe/compute it? And what does it mean?
Grothendieck group of varieties over $G/B$
Before that, it may be helpful to say a word about
$K_0(\operatorname{Var}/X)$. This is the Grothendieck ring of
quasiprojective varieties over $X$, for the application to the $L$-class
we may better look at compact complex analytic spaces. The relations
that need to be divided out come from the blow-up square,
cf. F. Bittner. The universal Euler characteristic for varieties of
characteristic zero. Compos. Math. 140 (2004), 1011--1032.
Universal characteristic classes
Now, for $X=G/B$, what class in $K_0(\operatorname{Var}/X)$ could best
specialize to the intersection complexes of Schubert varieties? Claim
1: all my bets are on
the classes of Bott-Samelson resolutions of Schubert
varieties. Claim 2: in fact, my guess
is that the abelian subgroup of $K_0(\operatorname{Var}/X)$ generated
by Bott-Samelson varieties is isomorphic to the split $K_0$ of the
category of pure weight 0 stratified mixed Tate motives on $X$ (resp. the
category of Soergel modules).
So, let's sum up. With a reasonable leap of faith, the information of
the class of the Bott-Samelson resolution in
$K_0(\operatorname{Var}/X)$ would seem to be closely related to the
information that is contained in the Kazhdan-Lusztig basis in the
Hecke algebra. (Ok, I'll try to make this more precise later, but let
it stand as it is for now. Making this precise would probably mean
doing some of these things equivariantly - talking about equivariant characteristic classes, Soergel bimodules categorifying the Hecke algebra and the relation between equivariant cohomology and the nil-Hecke algebra... too much for now) Now that we have a vague idea about the universal characteristic class, all that (any
specialization of) the motivic Hirzebruch transformation does from
this point on only forgets some of this information. This is why I said in the
very beginning that it is very likely that there are
representation-theoretic applications for these characteristic
classes. However, the most complete characteristic class - in
$K_0(\operatorname{Var}/X)$ - seems to be pretty close to what has
been used all along in Kazhdan-Lusztig theory.
Computations in the literature
Slightly different homology characteristic classes
(Chern-Schwartz-MacPherson classes) for Schubert varieties in
Grassmannians have been worked out in the following papers:
P. Aluffi and L.C. Mihalcea. Chern classes of Schubert cells and
varieties. J. Algebraic Geom. 18 (2009), 63--100.
Benjamin Jones: Singular Chern classes of Schubert varieties via
small resolutions. IMRN 2010, 1371--1416.
The first paper does the computation via Bott-Samelson resolution (as
explained above), and you can see from the computations with Young
tableaux that this is pretty close to applications in geometric
representation theory.
I hope this is useful for you.
