Computing Lannes T functor (Q1262575)

Lannes' T functor, associated to a finite-dimensional \({\mathbb{F}}_ p\) vector space V, has the defining property \[ Hom(M,N\otimes H^*(BV))\cong Hom(T_ V(M),N), \] where M, N are either in the category \({\mathcal U}\) of unstable modules over the mod p Steenrod algebra, \({\mathcal A}_ p\), or \({\mathcal K}\), the category of unstable algebras over \({\mathcal A}_ p\). Here \(H^*\) means \(H*(-;{\mathbb{F}}_ p)\), for p a fixed odd prime, and \(BV=B{\mathbb{F}}_ p\times...\times B{\mathbb{F}}_ p\). For X a p-complete space, \(T_ V(H*X)\) is seen to be closely related to H*(map(BV,X)). Let \(T=T_{{\mathbb{F}}_ p}\), that is, \(T_ V\) for V the one-dimensional vector space. The author computes T(P(r)) for \(P(r)={\mathbb{F}}_ p[x]\) and deg x\(=2p^ r:\) \[ T(P(r))\cong \{{\mathbb{F}}_ p[\alpha]/(\alpha^ p-\alpha)\}\otimes P(r), \] where deg \(\alpha\) \(=0\). Extending the results to \(P(r_ 1,...,r_ n)=P(r_ 1)\otimes...\otimes P(r_ n)\) follows from the fact that T commutes with tensor products. To a subgroup G of \(Gl_ n{\mathbb{F}}_ p\) one associates the \({\mathcal A}_ p\)- subalgebra of invariants \(P^ G=P(0,...,0)^ G=H*(BT^ n)^ G\) for \(T^ n\) the n-torus. The author computes \(T_ V(P^ G)\) and relates the results to theorems of Adams-Wilkerson on embedding of unstable algebras in \(H*(BT^ n)\), and to results of Dwyer-Miller-Wilkerson on realizability of algebras of invariant as cohomology rings.