Inner product on $V_{-\rho}$? Prologue. To $M^n$ a compact real manifold with frame bundle $F$ (a principal $GL_n$ bundle), we associate a line bundle using the representation $M\mapsto \sqrt{|\det M|}$, the bundle of half-densities. Then the space of sections of this has a canonical inner product (multiply then integrate), invariant under all diffeomorphisms of $M$. If $M$ is oriented then one can use the representation $M\mapsto \sqrt{\det M}$, the "half-form" bundle.
Present day.
I was thinking about "parabolic category $\mathcal O$" and realized that the most parabolic is the one containing (only) the Verma module $L(-\rho)$ with highest weight $-\rho$. The corresponding Borel-Weil line bundle is the
half-form bundle, so its space of sections carries a canonical inner product... except there are no holomorphic ones. Or indeed any sheaf cohomology.

What special structures do the very special representation $L(-\rho)$ bear? Is there any shadow of the inner product that the finite-dimensional irrep would have, if it existed?

EDIT: "If it existed" was a lousy way to talk about that finite-dimensional irrep. As Ben points out, Borel-Weil-Bott makes clear that it does exist but is $0$.
 A: I'm not totally sure what will satisfy you on this account.  Of course you pointedly restricted yourself to the reals, but some complex manifolds do have holomorphic half-form bundles, and the flag manifold is one of them.  Of course, the half-form line bundle $\omega_X^{1/2}$ "should" have sections $V(-\rho)$ (actually, this should be the cohomology in all degrees) but in order to make sense of this you have to declare that $V(-\rho)=0$.  It's still true that multiplying two sections of this bundle gives a (now holomorphic) top form, but there just are no global sections.
One manifestation of this that is of interest to geometric representation theorists is that the ring and sheaf $D_{\omega^{1/2}}$ of holomorphic differential operators twisted in half-forms on any complex manifold (which make sense even if half-forms are not a well-defined line bundle) is self-opposite (the sheaf never is for other twists, sometimes the ring is various global reasons).  This makes it often the most natural twist to work with, and it plays a important role in D-module theory.
