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 $L(-\rho)$? 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 $0$.

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 $0$.

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 $L(-\rho)$? Is there any shadow of the inner product that the finite-dimension irrep would have, if it existed?

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 $L(-\rho)$? 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 $0$.