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This answer turned out to be much longer than what I'd intended to write, so let me summarize by saying that the frequent appearance of $\rho$ and the "dot" action in the representation theory of $\mathfrak g$ is very closely related to the geometry of the flag variety of $\mathfrak g$.]

This is actually a fairly deep question. Your suspicion that there may be multiple answers is correct, but there might be some surprising connections between seemingly unrelated answers. Let me give one possible thread of explanation.

In The underlying principle is that the appearance of $\rho$ and the "dot" action $w\cdot\lambda=w(\lambda+\rho)-\rho$ in representation theory , one is closely related to the geometry of the first appearances flag variety.

One of the first places one meets $\rho$ (and the dot action) is in the Weyl character formula, which states that if $V$ is an irrep of $\mathfrak g$ of highest weight $\lambda$ then$$\text{ch} V = \frac{\sum_{w\in W} (-1)^{l(w)} e^{w(\lambda+\rho)-\rho}}{\sum_{w \in W} (-1)^{l(w)} e^{w\rho - \rho}}.$$I won't bother explaining what all these symbols mean or where they live as all this is fairly standard. What I will point out is A theorem of Kostant shows that the right-hand side kind of looks like it could formula can be written as the ratio of two Lie algebra cohomology Euler characteristics. There is indeed a way to write it out as such. The key phrase here is "BGG resolution" but let me stick to something more basic (due to Bott and Kostant). If we let $\mathfrak g = \mathfrak n \oplus \mathfrak h \oplus \mathfrak n^-$ be the triangular decomposition of $\mathfrak g$From this perspective, then we have that$the appearance$\text{ch} V = w \frac{\chi(\mathfrak n, V)}{\chi(\mathfrak n, cdot \mathbb C)}, $$where lambda and \chi(\mathfrak n, V) denotes w\cdot0 in the Euler characteristic WCF is ultimately explained by the fact that these are the weights appearing in the weight space decomposition of the relevant Lie algebra cohomology modules, namely H^\ast(\mathfrak H^*(\mathfrak n, V)V^\lambda) and \mathbb C denotes the trivial module. To get the WCF from thisH^\ast(\mathfrak n, we use a theorem of Kostant which states thatV^0), where H^i(\mathfrak \mathfrak n , V) = \bigoplus_{w \in W, l(w)=i} bigoplus_{\alpha>0} \mathbb C_{w(\lambda+\rho)-\rho}.$$This is a decomposition of $H^i(\mathfrak n, V)$ as an mathfrak g_\alpha$and$\mathfrak h$-module. In view of this, we can explain V^\mu$ denotes the appearance irrep of $\rho$ and the "dot" action highest weight $w \cdot \lambda = w(\lambda+\rho)-\rho$\mu$. We can rephrase this in the WCF geometric terms by saying that invoking the weights $w\cdot\lambda$ are those that appear in $H^\ast(\mathfrak n, V)$. But this isn't really very satisfying, so let's work a bit harder. A result that is very closely related to "geometric analogue" of Kostant's theoremabout $H^\ast(\mathfrak n, V)$ is i.e. the Borel–Weil–Bott theorem; in fact, the BWB theorem is equivalent to Kostant's theoremin the sense that if you have one then you can prove the other. Roughly speaking, the BWB theorem describes the cohomology of certain line bundles over the flag variety Kostant's description of $\mathfrak g$. To give the actual statement we need to work in the group setting. So let $G$ be the simply connected complex semisimple Lie group with Lie algebra $\mathfrak g$ and let $B$ denote the Borel subgroup corresponding to the Borel subalgebra cohomology of $\mathfrak b^- n = \mathfrak h \oplus \mathfrak n^-$. Then the flag variety can be identified g /\mathfrak b^-$ with coefficients in an irrep translates into a representation-theoretic description of the space sheaf cohomology of certain line bundles $G/B$. Each L_\lambda$(integral) weight constructed using integral weights $\lambda$ produces a $G$-equivariant line bundle $L_\lambda := G \times_\lambda B$) over the flag variety $G/B$ and BWB tells us what G/B^-$ of $H^\ast(G/B,L_\lambda)$ is\mathfrak g$. There are two cases: • either there is no $w\in W$ such that $w\cdot \lambda$ is dominantConsequently, or • there is a unique $w\in W$ such that $w\cdot \lambda$ is dominant. • The second case is the one of interest to us: BWB tells us that dot action shows up in this case $H^i(G/B,L_\lambda)$ is zero except in degree $i=l(w)$description, in which case$H^{l(w)}(G/B,L_\lambda)$is the irreducible$G$-module of weight$w\cdot\lambda$. We're now very close to giving an explanation for the appearance of$\rho$and the dot actionthis time it's accompanied by a shift in degree. All we need is one last simple observation: the cohomology of line bundles over$G/B$must satisfy This in turn can be explained by Serre duality. In our setting, this is ; the assertion that$H^i(G/B,L_\lambda)$and$H^{n-i}(G/B, L_\lambda^\ast \otimes K)$are dual, where$n = \dim G/B$and$K$key fact is the that canonical bundle of$G/B$. It G/B^-$ turns out that $K$ itself is one of these $L_\lambda$'s: in fact, to be $K = L_{-2\rho}$.From here

So, a simple computation shows that in some sense, the appearance of $\rho$ and the dot action is in the WCF can be thought of as a manifestation of Serre duality.

There are also several other geometric explanations for the appearance of $\rho$ (e.g. the existence

[N.B. This is a condensed version of equivariant spin structures on the flag variety), but my answer is already long enough that I should stop here and maybe give another lengthy original answerlater.

Edit: But let me at least give an example of all this! Let $G = \operatorname{SL}_2\mathbb C$ and $B$ the subgroup of lower triangular matrices, so that $G/B = \mathbb P^1$. The set of weights here old version can be identified with $\mathbb Z$ and the line bundle $L_n$ corresponding to the weight $n\in\mathbb Z$ turns out to be $\mathcal O(n)$. Also, $\rho = 1$, so that the canonical bundle is $\mathcal O(-2)$. Serre duality then says that $H^0(\mathbb P^1, \mathcal O(n))$ and $H^1(\mathbb P^1, \mathcal O(n)^\ast \otimes O(-2)) = H^1(\mathbb P^1, \mathcal O(-n-2))$ are dual as representations of $G$, so they're in fact isomorphic. (For general $G$, if $V$ is an irrep of of highest weight $\lambda$ its dual $V^\ast$ is an irrep of highest weight $-w_0 \lambda$, where $w_0$ is the longest element of the Weyl group. For $\operatorname{SL}_2$, $w_0=-1$, so $V \cong V^\ast$.) Now compare this with what BWB says...

This example is actually very fundamental for the general case: found in the original proof of BWB, Serre duality applied to $\operatorname{SL}_2$-reps was at the core of the argumentedit history.]

 
 
 
 
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[This answer turned out to be much longer than what I'd intended to write, so let me summarize by saying that the frequent appearance of $\rho$ and the "dot" action in the representation theory of $\mathfrak g$ is very closely related to the geometry of the flag variety of $\mathfrak g$.]

This is actually a fairly deep question. Your suspicion that there may be multiple answers is correct, but there might be some surprising connections between seemingly unrelated answers. Let me give one possible thread of explanation.

In representation theory, one of the first appearances of $\rho$ is in the Weyl character formula, which states that if $V$ is an irrep of $\mathfrak g$ of highest weight $\lambda$ then $$\text{ch} V = \frac{\sum_{w\in W} (-1)^{l(w)} e^{w(\lambda+\rho)-\rho}}{\sum_{w \in W} (-1)^{l(w)} e^{w\rho - \rho}}.$$ I won't bother explaining what all these symbols mean or where they live as all this is fairly standard. What I will point out is that the right-hand side kind of looks like it could be the ratio of two Euler characteristics. There is indeed a way to write it out as such. The key phrase here is "BGG resolution" but let me stick to something more basic (due to Bott and Kostant). If we let $\mathfrak g = \mathfrak n \oplus \mathfrak h \oplus \mathfrak n^-$ be the triangular decomposition of $\mathfrak g$, then we have that $$\text{ch} V = \frac{\chi(\mathfrak n, V)}{\chi(\mathfrak n, \mathbb C)},$$ where $\chi(\mathfrak n, V)$ denotes the Euler characteristic of the Lie algebra cohomology $H^\ast(\mathfrak n, V)$ and $\mathbb C$ denotes the trivial module. To get the WCF from this, we use a theorem of Kostant which states that $$H^i(\mathfrak n, V) = \bigoplus_{w \in W, l(w)=i} \mathbb C_{w(\lambda+\rho)-\rho}.$$ This is a decomposition of $H^i(\mathfrak n, V)$ as an $\mathfrak h$-module. In view of this, we can explain the appearance of $\rho$ and the "dot" action $w \cdot \lambda = w(\lambda+\rho)-\rho$ in the WCF by saying that the weights $w\cdot\lambda$ are those that appear in $H^\ast(\mathfrak n, V)$. But this isn't really very satisfying, so let's work a bit harder.

A result that is very closely related to Kostant's theorem about $H^\ast(\mathfrak n, V)$ is the Borel–Weil–Bott theorem; in fact, the BWB theorem is equivalent to Kostant's theorem in the sense that if you have one then you can prove the other. Roughly speaking, the BWB theorem describes the cohomology of certain line bundles over the flag variety of $\mathfrak g$. To give the actual statement we need to work in the group setting. So let $G$ be the simply connected complex semisimple Lie group with Lie algebra $\mathfrak g$ and let $B$ denote the Borel subgroup corresponding to the Borel subalgebra $\mathfrak b^- = \mathfrak h \oplus \mathfrak n^-$. Then the flag variety can be identified with the space $G/B$. Each (integral) weight $\lambda$ produces a $G$-equivariant line bundle $L_\lambda := G \times_\lambda B$ over $G/B$ and BWB tells us what $H^\ast(G/B,L_\lambda)$ is. There are two cases:

1. either there is no $w\in W$ such that $w\cdot \lambda$ is dominant, or

2. there is a unique $w\in W$ such that $w\cdot \lambda$ is dominant.

The second case is the one of interest to us: BWB tells us that in this case $H^i(G/B,L_\lambda)$ is zero except in degree $i=l(w)$, in which case $H^{l(w)}(G/B,L_\lambda)$ is the irreducible $G$-module of weight $w\cdot\lambda$.

We're now very close to giving an explanation for the appearance of $\rho$ and the dot action. All we need is one last simple observation: the cohomology of line bundles over $G/B$ must satisfy Serre duality. In our setting, this is the assertion that $H^i(G/B,L_\lambda)$ and $H^{n-i}(G/B, L_\lambda^\ast \otimes K)$ are dual, where $n = \dim G/B$ and $K$ is the canonical bundle $G/B$. It turns out that $K$ itself is one of these $L_\lambda$'s: in fact, $K = L_{-2\rho}$. From here, a simple computation shows that the appearance of $\rho$ and the dot action is a manifestation of Serre duality.

There are also several other geometric explanations for the appearance of $\rho$ (e.g. the existence of equivariant spin structures on the flag variety), but my answer is already long enough that I should stop here and maybe give another answer later.

Edit: But let me at least give an example of all this! Let $G = \operatorname{SL}_2\mathbb C$ and $B$ the subgroup of lower triangular matrices, so that $G/B = \mathbb P^1$. The set of weights here can be identified with $\mathbb Z$ and the line bundle $L_n$ corresponding to the weight $n\in\mathbb Z$ turns out to be $\mathcal O(n)$. Also, $\rho = 1$, so that the canonical bundle is $\mathcal O(-2)$. Serre duality then says that $H^0(\mathbb P^1, \mathcal O(n))$ and $H^1(\mathbb P^1, \mathcal O(n)^\ast \otimes O(-2)) = H^1(\mathbb P^1, \mathcal O(-n-2))$ are dual as representations of $G$, so they're in fact isomorphic. (For general $G$, if $V$ is an irrep of of highest weight $\lambda$ its dual $V^\ast$ is an irrep of highest weight $-w_0 \lambda$, where $w_0$ is the longest element of the Weyl group. For $\operatorname{SL}_2$, $w_0=-1$, so $V \cong V^\ast$.V^\ast$.) Now compare this with what BWB says... This example is actually very fundamental for the general case: in the original proof of BWB, Serre duality applied to $\operatorname{SL}_2$-reps was at the core of the argument. 2 added 1145 characters in body; deleted 4 characters in body; added 1 characters in body; deleted 1 characters in body; edited body A result that is very closely related to Kostant's theorem about $H^\ast(\mathfrak n, V)$ is the Borel–Weil–Bott theorem; in fact, the BWB theorem is equivalent to Kostant's theorem in the sense that if you have one then you can prove the other. Roughly speaking, the BWB theorem describes the cohomology of certain line bundles over the flag variety $X$of$\mathfrak g$. To give the actual statement we need to work in the group setting. So let$G$be the simply connected complex semisimple Lie group with Lie algebra$\mathfrak g$and let$B$denote the Borel subgroup corresponding to the Borel subalgebra$\mathfrak b^- = \mathfrak h \oplus \mathfrak n^-$. Then the flag variety can be identified with the space$G/B$. Each (integral) weight$\lambda$produces a$G$-equivariant line bundle$L_\lambda := G \times_\lambda B$on  over $G/B$ and BWB tells us what $H^\ast(G/B,L_\lambda)$ is. There are two cases: We're now very close to given giving an explanation for the appearance of $\rho$ and the dot action. All we need is one last simple observation: the cohomology of line bundles over $G/B$ must satisfy Serre duality. In our setting, this is the assertion that $H^i(G/B,L_\lambda)$ and $H^{n-i}(G/B, L_\lambda^\ast \otimes K)$ are dual, where $n = \dim G/B$ and $K$ is the canonical bundle $G/B$. It turns out that $K$ itself is one of these $L_\lambda's$: L_\lambda$'s: in fact, $K = L_{-2\rho}$. From here, a simple computation shows that the appearance of $\rho$ and the dot action is a manifestation of Serre duality.

There are also several other geometric explanations for the appearance of $\rho$ (e.g. the existence of equivariant spin structures on the flag variety), but my answer is already long enough that I should stop here and maybe give another answer later.

Edit: But let me at least give an example of all this! Let $G = \operatorname{SL}_2\mathbb C$ and $B$ the subgroup of lower triangular matrices, so that $G/B = \mathbb P^1$. The set of weights here can be identified with $\mathbb Z$ and the line bundle $L_n$ corresponding to the weight $n\in\mathbb Z$ turns out to be $\mathcal O(n)$. Also, $\rho = 1$, so that the canonical bundle is $\mathcal O(-2)$. Serre duality then says that $H^0(\mathbb P^1, \mathcal O(n))$ and $H^1(\mathbb P^1, \mathcal O(n)^\ast \otimes O(-2)) = H^1(\mathbb P^1, \mathcal O(-n-2))$ are dual as representations of $G$, so they're in fact isomorphic. (For general $G$, if $V$ is an irrep of of highest weight $\lambda$ its dual$V^\ast$is an irrep of highest weight$-w_0 \lambda$, where$w_0$is the longest element of the Weyl group. For$\operatorname{SL}_2$,$w_0=-1$, so$V \cong V^\ast$.) Now compare this with what BWB says...

This example is actually very fundamental for the general case: in the original proof of BWB, Serre duality applied to $\operatorname{SL}_2$-reps was at the core of the argument.

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