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51) which represents a bounded Cp -analytic function Lµ : XS −→ Cp . 52) Indeed, the boundedness of the function Lµ is obvious since all characters x ∈ XS take values in Op and µ also is bounded. 51), namely that it depends analytically on the parameter x ∈ XS . However, we give below a pure algebraic proof of this fact which is based on a description of the Iwasawa algebra. This description will also imply that every bounded Cp -analytic function on XS is the Mellin transform of a certain measure µ.

For any commutative Q-algebra A the group Wm acts on the set (A× )m+1 , therefore any homomorphism of Q-algebras λ : L → A can be identified with some element (α0 , α1 , · · · , αm ) ∈ [(A× )m+1 ]Wm . 7. 4 Hecke operators Any double coset (g) = (Γ gΓ ) (g ∈ ∆ = ∆m q (N )) can be represented as a disjoint union of left cosets: t(g) Γ gi , (g) = i=1 therefore any element X ∈ L of the Hecke algebra L takes the form of a finite linear combination t(X) X= µi (Γ gi ), i=1 52 2 Siegel modular forms and the holomorphic projection operator with µi ∈ Q, gi ∈ ∆.

However, we give a more direct computation of these coefficients in terms of the corresponding measures. Let us consider the component aUp of the set Z× S where a ∈ (Z/pν Z)× × Z× q , q= and let µa (x) = µ(ax) be the corresponding measure on Up defined by restric∼ tion of µ to the subset aUp ⊂ Z× S . Consider the isomorphism Up = Zp given by: y = γ x (x ∈ Zp , y ∈ Up ), with some choice of the generator γ of Up (for example, we can take γ = 1+pν ). Let µa be the corresponding measure on Zp . 1 and [Mah]).

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