Fix a prime p. In [H06a], we have shown that the geometric automorphism group of the irreducible component of the mod p Shimura variety of PEL type (of level away from p) associated to a reductive group G of unitary and symplectic type is almost identical to G1(A (p∞) ) modulo global center (cf. Lemma 1.1). Here G1 is the derived subgroup of G. In this paper, we give a (basically) characteristic p proof of the irreducibility of the Igusa tower over a reduction modulo p of the Shimura variety by showing that the stabilizer of an irreducible component of the tower is as large as possible under the PEL data. This is a characteristic p-version of the proof given in [PAF] Section 8.4 where we used mixed-characteristic results to show the maximality of the stabilizer. Here is a general axiomatic approach to prove the irreducibility of an étale covering π: Ig → S of an irreducible variety S over an algebraically closed residue field F. Suppose the following two axioms: (A1) A group G = M1×G1 acts on Ig and S compatibly so that M1 ⊂ Aut(Ig/S), G1 ⊂ Aut(S) and G1 acts trivially on π0(Ig)
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