TITLE: Is there a 'nice' interpretation of virtual representations?
QUESTION [27 upvotes]: Let $G$ be a compact group and let $R(G)$ be the representation ring of $G$. Additively, $R(G)$ is generated by the irreducible representations of $G$. Usually one only deals with those representations which are nonnegative integer combinations of the irreducible representations. However, often one has formulas which apply to an arbitrary element of $R(G)$, and so includes virtual representations which (for this question at least) are the elements of $R(G)$ whose decomposition into irreducible components includes negative coefficients.

Question: Is there any 'natural' interpretation of virtual representations? In particular, aside from the obvious interpretation as the elements of the formal completion of the semiring of ordinary representations to a ring, is there a natural way to view these objects? Especially helpful would be pictures/ideas others use to assign meaning to virtual representations (if any).

Motivation: Often in decomposing various formulas involving characters, virtual representations arise in one way or another. For example, in Lie groups, the notion of a highest weight representation $\rho_\omega$ can be extended to arbitrary weights $t$ of $G$ via:
$\rho_{w(t)} = (-1)^w\rho_t$
In particular, if $t$ is not a dominant weight of $G$ then there is a unique $w$ in the Weyl Group of $G$ such that $w(t)$ is a dominant weight so that $\rho_t$ is defined for arbitrary weights $t$; note that the Weyl Dimension Formula agrees with this extension. If the length of $w$ is odd, then $\rho_t$ will have negative dimension from the dimension formula, hence is virtual.
Another simple example is that when considering the action of the Adams operation $\psi^k$ on $R(G)$, one has that $\psi^k(\rho)$ is in general a virtual representation.
It happens that from time to time I come across other instances of virtual representations appearing in equations I am considering, and I always work with them ignoring whether they have a physical interpretation or not, but at the same time it would be more satisfying if I could understand the equations as manifestations of some deeper structure.
Edit: Per Qiaochu's comment, yes, virtual representations can be fit into the framework of super-representations. If it is the case that virtual representations are often viewed as super-representations, then perhaps someone could elaborate on why super-representations are so natural and how one works around the dimension mismatches between virtual representations and super-representations, i.e. $dim(\rho_1\ominus\rho_2) = dim(\rho_1)-dim(\rho_2)$ but the dimension of the corresponding super-representation is $dim(\rho_1)+dim(\rho_2)$.

REPLY [5 votes]: Andre's answer above does a perfect job of explaining some of the abstract theory of "virtual" representations (of an arbitrary group).  I'd like to give one answer to the special case you bring up, when $G$ is a compact group.  Or, even better, I'll work with the case when $G = \mathfrak g$ is a (finite-dimensional) semisimple Lie algebra over $\mathbb C$.
Then one place that the signs you talk about come up is in the Weyl character formula.  Pick a triangular decomposition $\mathfrak g = \mathfrak n^- \oplus \mathfrak h \oplus \mathfrak n^+$.  For any weight $\lambda$, let $V(\lambda)$ denote the Verma module with heighest weight $\lambda$ (recall that $V(\lambda) = {\rm U}\mathfrak g \hspace{1ex}\otimes_{\rm U\mathfrak b}\hspace{1ex} \mathbb C_\lambda$, where $\mathfrak b =  \mathfrak h \oplus \mathfrak n^+$ is the standard Borel, and $\mathbb C_\lambda$ is the one-dimensional $\mathfrak b$-module on which $\mathfrak h$ acts by $\lambda$ and $\mathfrak n^+$ acts by $0$).  Let $L(\lambda)$ denote the quotient of $V(\lambda)$ by its maximal proper ideal; it is the unique irreducible $\mathfrak g$-module with heighest weight $\lambda$.  Recall that $\mathfrak h$ acts semisimply on any sufficiently nice $\mathfrak g$-module (including $M(\lambda)$ and its quotients and tensorands and summands; "sufficiently nice" is codified, for the present purposes, by "in the category $\mathcal O$"), and that the K-group of the semisimple $\mathfrak h$-modules is precisely the group ring of $\mathfrak h^*$ (the space of weights).  The character $\operatorname{ch}(M)$ of a $\mathfrak g$-module $M$ is its image after first forgetting from the $\mathfrak g$-action to just a $\mathfrak h$-action, and then looking at the element it represents in $\mathbb Z[e^{\mathfrak h^*}]$ (or rather in some completion; for category $\mathcal O$, I want to take the adic completion for the ideal generated by $e^{-\lambda}$ for simple roots $\lambda$).
The Weyl Character Formula asserts that, for $\lambda$ a dominant integral weight:
$$ \operatorname{ch}(L(\lambda)) = \sum_{w\in \mathfrak W} \operatorname{sign}(w) \; \operatorname{ch}(V(w(\lambda+\rho)-\rho)), $$
where $\rho$ is half the sum of the positive roots, $\mathfrak W$ is the Weyl group, and $\operatorname{sign}: \mathfrak W \to \{\pm 1\}$ is $w \mapsto \det_{\mathfrak h}w = (-1)^{\operatorname{length}(w)} $.
If there were not the signs there, you might hope that this formula came from some statement in the representation theory of $\mathfrak g$.  Recall that taking K-groups of a category turns exact sequences into addition equations.  Then a proof of the Weyl Character Formula can follow this outline:

Write down a matrix $b_{\lambda\mu}$ for the coefficient of $\operatorname{ch}(V(\mu))$ in $\operatorname{ch}(L(\lambda))$.
Then the inverse matrix $(b^{-1})_{\mu\lambda}$ expresses the coefficient of $\operatorname{ch}(L(\lambda))$ in an expansion of $\operatorname{ch}(V(\mu))$.
Recognize that each $V(\mu)$ is an extension of some $L(\lambda)$s — this is actually an equation in the representation theory of $\mathfrak g$ — and by studying how $V(\mu)$ is built out of $L(\lambda)$s, understand enough of the structure of $(b^{-1})_{\mu\lambda}$ to conclude the theorem.

The central point is that the inverse matrix $b^{-1}$ has only nonnegative integer entries, and so has a chance of descending from some extension problem in the representation theory.
So, what about the Weyl character formula itself?  Since it has negative coefficients, it cannot express that $L(\lambda)$ is some extension of Verma modules.  Instead, it descends from the fact that $L(\lambda)$ has a resolution in Verma modules, the so-called BGG resolution:
$$ 0 \to M(w_0(\lambda+\rho)-\rho) \to \cdots \to \bigoplus_{w\in W \text{ s.t. }\operatorname{length}(w) = k} M(w(\lambda + \rho)-\rho) \to $$
$$ \cdots \to \bigoplus_{s \text{ a simple reflection}} M(s(\lambda + \rho) - \rho) \to M(\lambda) \to 0 $$
has homology only in the last spot, where the homolgy is $L(\lambda)$.  (I think the differentials are just the obvious inclusions of Verma modules.  My reference for all of this is my collection of lecture notes from UC Berkeley's Lie Theory course (chapter 6.1), but it doesn't go into detail.)
So this might be the "nice interpretation" you're looking for.  In Andre's language, the point is to realize virtual modules as objects in the derived category of modules; i.e. chain complexes.  Then the chain complex above is isomorphic (in the derived category) to its homology $L(\lambda)$, by the projection $M(\lambda) \to L(\lambda)$, and the Weyl Character Formula is the decategorification of this isomorphism.  In general, the philosophy is that any naturally-occurring alternating sum of dimensions ought to come from a chain complex.  Conversely, any sum of dimensions out to come from an extension.