Invariants of unipotent radicals (Q581481)

Let G be a reductive group over an algebraically closed field k of arbitrary characteristic. Let P, Q be parabolic subgroups of G with unipotent radicals \(U_ P, U_ Q\) and let \(L_ P=P/U_ P\), \(L_ Q=Q/U_ Q\). The coordinate ring k[G] is a \(G\times G\)-module. The main result is the exitence of a certain \(L_ P\times L_ Q\)-module filtration for the ring of invariants \(k[G]^{U_ P\times U_ Q}\). This gives a new proof of the theorem of Grosshans which states that \(k[G]^{U_ P}\) is finitely generated.