First one: In his paper he says $GL_2(A^{f})$could act on the direct limit of the first cohomology groups with compact support $H^1_c(M_n^{an},Sym^{k}(R^1f_{*}(Q)))$ with $n\rightarrow\infty$. Does he mean $GL_2(A^f)$ act on each finite level? If so, how does it exactly act on it, since the elements in $GL_2(A^f)$could have denominators. This confuses me for a long time.
Second one: In this paper he says the Hecke algebra is the algebra of integral measures on the discrete space $GL_2(Q_p)/GL_2(Z_p)$ invariant on the left by $GL_2(Z_p)$. This is the first time I see one interprets Hecke algebra as measures. Is this the measure in the sense of real analysis? Could any one suggest some reference on it? Many thanks.
Usual you have the action of $G(A)$ is equivalent to convolution by smooth, compactly supported functions on $G(A)$. These function factor $G(A) // K$ for some open compact $K$. I would bet that if $n$ is an ideal, you will get that the action factors through $K = \{ \gamma = (\gamma_p)_p = 1 \bmod n \}$ (the full modular group). Since it do not know what any $H^1( \dots)$ means precisely, this is just a comment. – Marc Palm Mar 25 '12 at 11:42