An injectivity result for Hermitian forms over local orders

Let Lambda be a ring endowed with an involution a bar right arrow (a) over tilde. We say that two units a and b of Lambda fixed under the involution are congruent if there exists an element u is an element of Lambda(x) such that a = ub (u) over tilde. We denote by H(Lambda) the set of congruence classes. In this paper we consider the case where Lambda is an order with involution in a semisimple algebra A over a local field and study the question of whether the natural map H(Lambda) --> H(A) induced by inclusion is injective. We give sufficient conditions on the order Lambda for this map to be injective and give applications to hermitian forms over group rings.