Evens norm, transfers and characteristic classes for extraspecial p-groups

Abstract

Let P be the extraspecial p-group of order p^{2n+1}, of p-rank n+1, and of exponent p if p>2. Let Z be the center of P and let kappa_{n,r} be the characteristic classes of degree 2^n - 2^r (resp. 2(p^n-p^r)) for p=2 (resp. p>2), 0 <= r <= n-1, of a degree p^n faithful irreducible representation of P. It is known that, modulo nilradical, the iotath powers of the kappa_{n,r}'s belong to T=Im(inf: H^*(P/Z,F_p)/sqrt{0} --> H^*(P,F_p)/sqrt{0}), with iota= 1 if p=2, iota= p if p>2. We obtain formulae in H^*(P,F_p)/sqrt{0} relating the kappa_{n,r}^iota terms to the ones of fewer variables. For p>2 and for a given sequence r_0,...,r_{n-1} of non-negative integers, we also prove that, modulo-nilradical, the element prod_{r_i}kappa^{r_i}_{n,i} belongs to T if and only if either r_0 >= 2, or all the r_i are multiple of p. This gives the determination of the subring of invariants of the symplectic group Sp_{2n}(F_p) in T.