, The Siegel operator ? g+1 : M k (? g+1 ) ? M k (? g ) is bijective
, This proves the first assertion. For the second, it is obvious that ? g is injective for g = 2k, hence for all g ? 2k as well by the relation ? g+1 ? ? g+1 = ? g . The surjectivity of ? g follows from the second equality of (5.5.2) for g > 2k. The surjectivity of ? 2k+1 , and the surjectivity of ?, The first equality in (5.5.2) means that ? g+1 is injective
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