2 added a couple of backticks

From the point of view of geometry, the crucial fact about $\rho$ is that the corresponding line bundle on the flag manifold is (upt to a sign) a (the) square-root of the canonical bundle (top exterior power of $T^*B T^*_B G/B \simeq bb_-$ is the sum of the negative roots). This is of course equivalent to Alain Valette's description in terms of the modular character of the Borel. In other words its sections in the real world are half-densities (things for which we can define the $L^2$ inner product).

It is a universal fact about passage from the classical world to the quantum world (in particular the geometric construction of representations) involves a shift by the square root of the canonical bundle. There are many ways to explain or motivate this. For example if we seek unitary representations we need to be able to define an $L^2$ inner product, which means considering not sections of the bundle we might have expected but sections times half-densities (again this is Alain's answer restated). From the point of view of rings of differential operators, the adjoint of a differential operator acting on functions (or on sections of a bundle $L$) is invariantly not another diffop (on $L$) but a differential operator acting on volume forms (or on sections of $L$ tensor the canonical bundle) --- so the self dual twist of differential operators is by half-forms, ie $\rho$-shifted. (Put another way, Serre duality is a reflection centered at half-forms!)

My favorite explanation is in Beilinson-Bernstein's Proof of Jantzen Conjectures and doesn't involve self-adjointness or unitarity: it's a consistency condition for deformation quantization of symbols (functions on the cotangent bundle): if you want this deformation quantization to be correctly normalized to order two (this is not the right question to go into that) you find you need to look at differential operators twisted by half-forms, not functions. On the flag variety this means a $\rho$-shift, and from the D-module POV on representation theory this is one fundamental place where that shift is forced on you, independent of thinking of inner products. This is in particular one way to see why it comes up in the Weyl character formula, through the geometric proof via Atiyah-Bott or via the BGG resolution, both of which involve the geometry of the flag variety.

1

From the point of view of geometry, the crucial fact about $\rho$ is that the corresponding line bundle on the flag manifold is (upt to a sign) a (the) square-root of the canonical bundle (top exterior power of $T^*B G/B \simeq b-$ is the sum of the negative roots). This is of course equivalent to Alain Valette's description in terms of the modular character of the Borel. In other words its sections in the real world are half-densities (things for which we can define the $L^2$ inner product).

It is a universal fact about passage from the classical world to the quantum world (in particular the geometric construction of representations) involves a shift by the square root of the canonical bundle. There are many ways to explain or motivate this. For example if we seek unitary representations we need to be able to define an $L^2$ inner product, which means considering not sections of the bundle we might have expected but sections times half-densities (again this is Alain's answer restated). From the point of view of rings of differential operators, the adjoint of a differential operator acting on functions (or on sections of a bundle $L$) is invariantly not another diffop (on $L$) but a differential operator acting on volume forms (or on sections of $L$ tensor the canonical bundle) --- so the self dual twist of differential operators is by half-forms, ie $\rho$-shifted. (Put another way, Serre duality is a reflection centered at half-forms!)

My favorite explanation is in Beilinson-Bernstein's Proof of Jantzen Conjectures and doesn't involve self-adjointness or unitarity: it's a consistency condition for deformation quantization of symbols (functions on the cotangent bundle): if you want this deformation quantization to be correctly normalized to order two (this is not the right question to go into that) you find you need to look at differential operators twisted by half-forms, not functions. On the flag variety this means a $\rho$-shift, and from the D-module POV on representation theory this is one fundamental place where that shift is forced on you, independent of thinking of inner products. This is in particular one way to see why it comes up in the Weyl character formula, through the geometric proof via Atiyah-Bott or via the BGG resolution, both of which involve the geometry of the flag variety.