Representation theory: terminology

Question

I am learning about representation theory. One of the things which continually trips me up is the (abuse of?) notation $V$ for a representation. Normally, one writes $(\rho, V)$ for a representation, where $\rho$ is a linear map from the group $G$ to $GL(V)$, ie invertible linear maps on $V$. However, sometimes people just say that $V$ is a representation. I then don't know what the corresponding $\rho$ is.

The following example illustrates this, taken from Harmonic Analysis on Finite Groups by Ceccherini-Silberstein, Scarabotti and Tolli. (Overall I really like this book!) Section 4.6, The Canonical Decomposition of $L(X)$ via Spherical Functions, starts as follows (slightly paraphrased).

Let $(G, K)$ be a Gelfand pair. Let $N+1$ be the number of orbits of $K$ on $X := G/K$. We know that $L(X)$ decomposes into $N+1$ distinct irreducible subrepresentations and that there exist distinct spherical functions, say $\phi_0 \equiv 1$, $\phi_1$, ..., $\phi_N$.

Denote by $V_n := \langle \lambda(g) \phi_n \mid g \in G \rangle$ (where $\lambda$ is the left regular representation) the subspace of $L(X)$ spanned by the $G$-translates of $\phi_n$, for $n = 0, 1, ..., N$.

...

The representation $V_n$ is called the spherical representation associated with the spherical function $\phi_n$. In particular, $V_0$ is the trivial representation.

I do not understand fully how $V_n$ is a representation. There must be some underlying linear map. If it isn't stated, is it just assumed to be $\lambda$, the left regular representation?

Answer 1

This is a relatively standard abuse of notation, which appears in other areas as well. Some examples:

1. "Let $X$ be a topological space" rather than "Let $(X, \tau)$ be a topological space"
2. "Let $V$ be a vector space" rather than "Let $(V, 0, +, -, \cdot)$ be a vector space".
3. "Let $R$ be a ring" rather than "Let $(R, 0, +, -, \cdot)$ be a ring"

(I welcome additional examples, as well as cases where it can be misleading)