Let be the completed group algebra of a uniform pro- group with coefficients in a field of characteristic . We study right ideals in that are invariant under the action of another uniform pro- group . We prove that if is non-zero then an irreducible component of the characteristic support of must be contained in a certain finite union of rational linear subspaces of . The minimal codimension of these subspaces gives a lower bound on the homological height of in terms of the action of a certain Lie algebra on . If we take to be acting on itself by conjugation, then -invariant right ideals of are precisely the two-sided ideals of , and we obtain a non-trivial lower bound on the homological height of a possible non-zero two-sided ideal. For example, when is open in this lower bound equals . This gives a significant improvement of the results of [K. Ardakov, F. Wei, J.J. Zhang, Reflexive ideals in Iwasawa algebras, Adv. Math. 218 (2008) 865–901].