ΓΓ-invariant ideals in Iwasawa algebras

Let kGkG be the completed group algebra of a uniform pro-pp group GG with coefficients in a field kk of characteristic pp. We study right ideals II in kGkG that are invariant under the action of another uniform pro-pp group ΓΓ. We prove that if II is non-zero then an irreducible component of the characteristic support of kG/IkG/I must be contained in a certain finite union of rational linear subspaces of SpecgrkGSpecgrkG. The minimal codimension of these subspaces gives a lower bound on the homological height of II in terms of the action of a certain Lie algebra on G/GpG/Gp. If we take ΓΓ to be GG acting on itself by conjugation, then ΓΓ-invariant right ideals of kGkG are precisely the two-sided ideals of kGkG, and we obtain a non-trivial lower bound on the homological height of a possible non-zero two-sided ideal. For example, when GG is open in SLn(Zp)SLn(Zp) this lower bound equals 2n−22n−2. 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].