Hi all.


If $G$ is a finite group and $\varrho : G \to \text{GL}(V), \eta : G \to \text{GL}(W)$
are finite dimensional representations, $V\_0$ is a $G$-invariant subspace of $V$ 
and $f : V\_0 \to W$ is a homomorphism of representations, i.e. a homomorphism of vector spaces satisfying $f(\varrho(g)x) = \eta(g) f(x)$,


Question: is there any theory on the question whether or not this homomorphism can be continued to whole $V$?


Phrasing differently: How 'many' values of a homomorphism of representations can one prescribe?


Clearly, there is the theory of induced representations but unfortunately, in my case, it is not true that $V = \oplus\_{g \in G} ~ \varrho(g)V\_0$, neither is the sum direct nor does equality hold.


Providing more details:
In my case, $G$ is $\text{SL}\_2(\mathbb{Z}/N\mathbb{Z})$,
say $\text{SL}\_2(\mathbb{F}\_p)$ for the beginning.
$V$ is a certain group ring $\mathbb{C}[D]$ where $D$ is a discriminant form, i.e. $D=L'/L$ for some lattice $L$ that satisfies some properties (non degenerate, even, even signature, determinant is a $p$-power for the beginning, ... [i.e. all simplifications that seem senseful/necessary]), $\varrho$ is the Weil representation and $W$ is a certain subspace of $X = \mathbb{C}[D] \oplus \mathbb{C}[D'] \oplus ...$ i.e. a finite direct sum of the original group ring with some other group rings of other discriminant forms. The representation on $X$ is the direct sum of the Weil representations of the single discriminant forms.


Thanks in advance,


Fabian Werner



Continuation of homomorphisms of representations...
You don't explicitly say your representation is complex but I think your example shows that this is the case you're interested in. If so, then $V\_0$ has a $G$-invariant complement $V\_1$ by Maschke's Theorem <http://en.wikipedia.org/wiki/Maschke%27s_theorem>, and the unique linear extension of $f$ whose kernel contains $V\_1$ will be a homomorphism of representations $V\to W$ extending $f$. 


