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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.

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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
    
Measure = functions on discrete spaces. –  Marc Palm Mar 25 '12 at 12:27

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