The technique of faithfully flat descent, and, in the case of vector spaces, Galois descent has been used quite a bit in Algebraic Geometry. However, the question of whether, say, a given $k$-vector space $V$ arises from some $L$-vector space $W$ seems like it could be asked in a wide variety of settings. I'm wondering, in particular, if anyone has seen descent in modular representation theory.



Applications of Descent?
The question is a little too vague to have a really satisfactory answer, but obviously descent
is omnipresent in (modular) representation theory. 


For example, the basic result that two representations of a group $G$ on some $k$-vector spaces are isomorphic iff they become isomorphic on some extension $L$ of $k$ is an instance of faithfully flat descent.


Or to take a less trivial example, let $R$ be a d.v.r with residual field $k$ algebraically closed (or just separably closed, or finite, or...) and of field of fraction $K$. Let $\rho$ be a representation of a group $G$ on some extension $L$ of $K$, whose characteristic polynomials of element of $g$ are in $R[x]$. (In characteristic zero, you can replace
that condition by "character takes values in $R$").
A natural question in this context is if you can "descend" $\rho$ to a representation over $R$. For example, the answer is yes when there exists a representation of $G$ over $k$
which is absolutely irreducible and whose characteristic polynomial agree with the reduction of those of $\rho$. The proof uses a lot of faithfully flat 
descent via the theory of Azumaya algebras.


Edit: to answer Martin's comment, I precise how the faithful flatness plays a role in the first example. 


Lemma: Let $G$ be a group. Let $k$ be a commutative noetherian ring, $k'$ a flat $k$-algebra. Let $M$, $N$, be $k[G]$-module, finitely generated as $k$-modules. let $M'=M \otimes\_k k'$ and $N'=N\otimes\_k k'$.
Then $Hom\_{k[G]}(M,N) \otimes\_k k' = Hom\_{k'[G]}(M',N')$. Moreover if $k'$ is a faithfully flat $k$-algebra, $Hom\_{k[G]}(M,N)$ is non-zero if and only if $Hom\_{k'[G]}(M',N')$ is non-zero.


Pf: Let $g\_1,\dots,g\_n$ be a finite family of elements of $g$ generating the image of $k[G]$ in $End\_{k}(N)$.
Let $t\_i$ be the linear endomorphism of $H:=Hom\_{k}(M,N)$ given by $\phi \mapsto \phi g\_i - g\_i \phi$. Consider the map $H \rightarrow H^n$ given by the $t\_i$. The kernel of this map
is $Hom\_{k[G]}(M,N)$. the kernel of the same map tensorized by $k'$ is $Hom\_{k'[G]}(M',N')$.
Hence the first result by flatness. The second is clear by faithful flatness.


So for example, if $k$, $k'$ are field, $M$, $N$ simple $k[G]$-modules, one retrieve
the classical assertion that $M \simeq N$ off $M' \simeq N'$. But I hope I have shown that 
the essence of this assertion is faithfully flat descent.


