Well, no one's explicitly talked about the relevance of spin structures to this story yet, so here's a sketch of the story as I understand it. For references see, for example, the nLab. I'll be blithely ignoring the difference between algebraic and topological K-theory, and $B$ denotes a positive Borel.

Roughly speaking, the Borel-Weil-Bott story is about constructing representations of $G$ by inducing them from $1$-dimensional representations of $B$. A geometric avatar of a finite-dimensional representation of $B$ is a $G$-equivariant vector bundle on $G/B$; in fact, this is an equivalence of categories, and so there is a natural isomorphism

$$K_G(G/B) \cong K_B(\text{pt}) \cong R(B)$$

from the equivariant K-theory $K_G(G/B)$ to the Grothendieck group of finite-dimensional representations of $B$. Similarly, there is a natural isomorphism

$$\text{Pic}_G(G/B) \ni L_{\lambda} \leftrightarrow \lambda \in \Lambda$$

from the group of $G$-equivariant line bundles on $G/B$ to the group of $1$-dimensional representations of $B$, which in turn can be identified with the weight lattice $\Lambda$.

In this language, induction can be interpreted as an attempt to construct a pushforward

$$K_G(G/B) \to K_G(\text{pt}) \cong R(G)$$

in equivariant K-theory from $G/B$ to a point; these kinds of pushforwards are one language for talking about geometric quantization. In algebraic geometry, the pushforward in K-theory to a point is given by taking sheaf cohomology, and Borel-Weil-Bott tells us exactly what happens when we push forward equivariant line bundles this way.

We might ask how we can take pushforwards in K-theory purely topologically, though. Recall, for example, that we can take the pushforward in cohomology of a map between compact oriented manifolds using Poincare duality. The analogous statement for real resp. complex K-theory is that we can take the pushforward of a map between compact oriented manifolds equipped with spin resp. complex spin structure. In both cases, the pushforward to a point is given by taking the index of a suitable Dirac operator. I believe all of this continues to be true equivariantly.

The happy fact about the algebro-geometric setting is that almost complex structures canonically induce complex spin structures; the corresponding Dirac operator is built from the Dolbeault operator, and with suitable hypotheses pushforward to a point is given by taking sheaf cohomology computed as Dolbeault cohomology.

What is the relevance of the Weyl vector to this story? Any complex spin structure has associated to it a canonical complex line bundle $\omega$. If we start with an almost complex structure, then $\omega$ is the canonical bundle. Then a choice of spin structure compatible with a complex spin structure is equivalent to a choice of square root $\sqrt{\omega}$. In our case, under the identification of $G$-equivariant vector bundles on $G/B$ with representations of $B$, we have

$$T(G/B) \mapsto \mathfrak{g}/\mathfrak{b}$$

where the latter has the adjoint action of $B$. This breaks up into a direct sum of $1$-dimensional weight spaces, one for each negative root, and hence under the identification of $G$-equivariant line bundles on $G/B$ with the weight lattice $\Lambda$, we have

$$\omega \mapsto 2 \rho.$$

Since $\Lambda$ is torsion-free, the isomorphism class of the square root $\sqrt{\omega}$ is unique, and as an element of $\Lambda$ it is precisely $\rho$.

So $\rho$ is special in this story because it represents the unique square root of the canonical bundle. From this perspective, the dot action

$$w \cdot \lambda = w (\rho + \lambda) - \rho$$

naturally arises as follows. Complex spin structures are canonically a torsor over complex line bundles; if $L$ is a complex line bundle, the action on complex spin structures modifies the canonical bundle by

$$\omega \mapsto \omega \otimes L^{\otimes 2}.$$

I again believe this continues to be true equivariantly, and so $G$-equivariant complex spin structures on $G/B$ are canonically a torsor over the weight lattice $\Lambda$. So we can noncanonically identify the two via

$$\lambda \mapsto \omega \otimes L_{\lambda}^{\otimes 2} \mapsto 2 \rho + 2 \lambda$$

where $\omega \otimes L_{\lambda}^{\otimes 2}$ denotes a complex spin structure, not just the corresponding canonical line bundle (and in particular it can be different from the complex spin structure given by $\omega$ even if $L_{\lambda}^{\otimes 2}$ is trivial, although that doesn't happen here). If $\lambda \in \Lambda$ is a weight, the pushforward of $L_{\lambda}$ with respect to $\omega$ can be identified with the pushforward of the trivial line bundle with respect to $\omega \otimes L_{\lambda}^{\otimes 2}$.

Now pick a maximal compact $K$ and identify $G/B$ with $K/T$, where $T = K \cap B$ is a maximal torus in $K$. Then the Weyl group $W = N_K(T)/T$ naturally acts on $K/T$; in fact it is precisely the $K$-equivariant automorphism group of $K/T$. This induces an action on $K$-equivariant line bundles, identified with characters of $T$, identified with the weight lattice $\Lambda$, which is the usual non-dot action.

The Weyl group *also* acts on $K$-equivariant complex spin structures on $K/T$, and this action is compatible with the torsor structure above, as well as with the map sending a complex spin structure to its canonical bundle. It is *not* compatible with the noncanonical identification between $K$-equivariant complex spin structures and $\Lambda$ above: instead, writing $2 \rho + 2 \lambda$ for the complex spin structure corresponding to $\omega \otimes L_{\lambda}^{\otimes 2}$, the Weyl group action is

$$w(2 \rho + 2 \lambda) = 2 \rho + 2 \left( w(\rho + \lambda) - \rho \right)$$

and using the noncanonical identification again we precisely recover the dot action! So, to summarize:

Geometrically, while the usual action of $W$ on $\Lambda$ is the natural action of $W$ on $K$-equivariant complex line bundles on $K/T$, the dot action is the natural action on $K$-equivariant complex spin structures on $K/T$; the Weyl vector $\rho$ appears when relating these because it corresponds to a distinguished $K$-equivariant complex spin structure associated to a choice of positive roots. It is the pushforward in K-theory with respect to such spin structures which lets us construct representations of $K$ from representations of $T$.

really greatexample of why MO is good. $\endgroup$