For Which Pseudo Reflection Groups Are thep-adic Polynomial Invariants Again a Polynomial Algebra?

LetWbe a finite group acting on a latticeLover thep-adic integers p. The analysis of the ring of invariants of the associatedW-action on the algebra p[L] of polynomial functions onLis a classical question of invariant theory. Ifpis coprime to the order ofW, classical results show thatWis a pseudo reflection group, if and only if the ring of invariants is again polynomial. We analyze the situation for odd primes dividing the order ofWand, in particular, determine those pseudo reflection groups for which the ring of invariants p[L]Wis a polynomial algebra.