Let $V\_{k}$ denote the complex representation of $\mathrm{GL}(2)$ given by $\mathrm{Sym}^k(V)$, where $V$ is the defining 2-dimensional representation. Assume that $k$ is even. I would like to compute the cohomology group
$$ H^1(\mathrm{GL}(2,\mathbf Z),V\_{k}). $$ 
By the Lyndon-Hochschild-Serre spectral sequence, this is the same as the $\mathbf Z/2$-invariants in $H^1(\mathrm{SL}(2,\mathbf Z),V\_k)$. The latter cohomology group is a direct sum of three terms, each of which corresponds to a space of modular forms of weight $k+2$: one to holomorphic cusp forms, one to antiholomorphic cusp forms, and one to Eisenstein series. I think that the $\mathbf Z/2$-invariants consist exactly of the holomorphic cusp forms, but I can't seem to figure out why. Can someone give an argument and/or a reference that explains this, preferably in geometric terms?




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Thanks to Joël's great answer I think I understand what's going on. Let me try to make really explicit where my confusion came from. If you write the Eichler-Shimura isomorphism
$$ H^1\_{\mathrm{cusp}}(\mathrm{SL}(2,\mathbf Z),V\_k) \cong S\_{k+2} \oplus \overline S\_{k+2}$$
then it's *not* true as I thought (and which led to nonsensical answers) that the subspace $S\_{k+2}$ is $\mathbf Z/2$-invariant and $\overline S\_{k+2}$ is anti-invariant. Instead the $\mathbf Z/2$-action interchanges these two subspaces according to the involution $$ f(z) \mapsto -f(-\overline z),$$
which maps holomorphic modular forms to antiholomorphic ones and vice versa. (I guess depending on how you've defined the Eichler-Shimura isomorphism there might be a sign difference here.) Still, there is of course a natural way to identify the invariant subspace with the space of holomorphic cusp forms.



Holomorphic cusp forms and cohomology of GL(2,Z)
Let me try an answer. Instead of working with $H^1(SL(2,{\bf Z}),V\_k)$, I'll work with a space which is naturally isomorphic to it, but more concrete, the space of modular symbols $Symb(V\_k)$, defined as $Hom(\Delta^0,V\_k)^\Gamma$. Here, $\Delta^0$ is the abelian group of divisors of degree $0$ on the projective rational line $\mathbb P^1(\bf Q)$, and $\Gamma = SL\_2(\bf Z)$ for shortness. The isomorphism between those two spaces is standard, follows from the long exact sequence of cohomology attached to the pair (Poincaré half-plane with cusps $P^1(\bf Q)$ attached, $P^1(\bf Q)$), and commute with the action of Hecke operator on both spaces, in particular, what matters to us, with the action of the "Hecke operator at $\infty$" $\iota$ defined by the matrix $\left( \begin{matrix} 1 & 0 \\ 0 & -1\end{matrix}\right)$ (a generator of $GL\_2({\bf Z})/SL\_2({\bf Z})$). Also I see $V\_k$ as the dual of the space of polynomial in one variable z of degree $\leq k$, with the natural homographic action of $\Gamma$. That's equivalent to your definition as a $Sym^k$.


Now we need to describe explicitly the relation you mention between modular forms and cohomology/modular symbols. Let us begin by cusp forms. In your post, the $\Gamma$-module $V\_k$ is a complex vector space, but of course it is naturally defined over $\bf Z$, and I will call $V\_k(\bf R)$ the version over $\bf R$. Let $S\_{k+2}=S\_{k+2}(\Gamma,\bf C)$ be the space of cusp forms. Then there is an injective $\bf R$-linear map
$$S\_{k+2} \rightarrow Symb(V\_k({\bf R})),\ \ f \mapsto \phi\_f,$$
where $\phi\_f$ is the modular symbol defined by $$\phi\_f(D)=(P \mapsto \Re( \int\_D f(z) P(z) dz)).$$
Here, for $D=[a] - [b]$, $\int\_D$ means the integral along any geodesic in the Poincaré half plane relying the cusp $a$ to the cusp $b$, for more general $D \in \Delta^0$, $\int\_D$ is defined by linearity, and we see the right hand side is a linear form on the space of polynomials $P$ of degree $\leq k$ with real coefficients, hence an element of $V\_k(\bf R)$ as it should be. 


Now if $f = \sum a\_n q^n \in S\_{k+2}$, let $f'=\sum \bar a\_n q^n \in S\_{k+2}$.
The $\bf R$-linear involution $f \mapsto f'$ is of course the complex conjugation on $S\_{k+2}$ associated with its canonical real structure given by the forms defined over $\bf R$ (= with real coefficients). Note that $\overline{f(z)} = \sum \bar a\_n \overline{e^{2 i \pi n z}} = \sum \bar a\_n e^{-2 i \pi n \bar z}=f'(-\bar z)$. If $D=[\infty]-[a] \in \Delta^0$, $a \in \bf Q$, one has $\iota(D)=[\infty]-[-a]$ one has:
$$\Re \left( \int\_D f(z) P(z) dz \right) = \Re \left( \int\_0^\infty f(a+iy) P(a+iy) i dy \right) \\ = \Re \left( \overline{ \int\_0^\infty f(a+iy) P(a+iy) i dy } \right) 
\\ = - \Re \left( \int\_0^\infty f'(- a+iy) P(a-iy) i dy \right) 
\\ = -\Re \int\_{\iota D} f'(z) P(-z) dz, $$
hence $$\phi\_{f'} = - \iota (\phi(f)),$$
that is the $\iota$ involution on modular symbols corresponds to minus the complex conjugation on modular cusp forms.


Now we complexify this real linear map, getting an injective map
$$S\_{k+2}(\Gamma,{\bf C}) \otimes\_{\bf R} {\bf C} \hookrightarrow Symb(V\_k({\bf C}),$$
where the left hand side can be identified with two copies of $S\_{k+2}(\Gamma,{\bf C})$ and clearly from the above and easy linear algebra, one copy is identified with the $+1$-eigenspace, an other with $-1$ eigenspace for $\iota$ on the RHS.
So this justifies your claim that the subspace invariant by $\iota$ in $Symb(V\_k)$ contains one copy of the holomorphic modular cusp forms.


There is also the question of Eisenstein series. Working in level 1 as you do,
there is only one of them for each k, and the question if whether the modular symbol corresponding to it is of eigenvalue $+1$ or $-1$ for $\iota$. This implies unfortunately that to have it right we need to be very careful in choosing a consistent system of conventions, which is a little bit too hard for me. 
If I interpret correctly my own computations with Samit Dasgupta however in our paper THE p-ADIC L-FUNCTIONS OF EVIL EISENSTEIN SERIES (Prop. 2.6), to appear in compositio, I think this sign is +1, which would mean that the Eisenstein series
is also in the invariant subspace of $Symb(V\_k)$, in other words also appears in $H^1(GL\_2({\bf Z}),V\_k)$. but maybe I'm wrong. What makes you think the latter is constituted of only *cuspidal* nodular forms?


