On a theorem of Schmid
Let \(G\) be a semisimple and simply connected algebraic group over an algebraically closed field \(k\) of characteristic \(0\). Let \(P=LU\) be a parabolic subgroup of \(G\), and denote by \(U^{-}\) the unipotent radical of a parabolic opposite to \(P\). There is a natural \(L\)-equivariant map \(\varphi\) from the \(U\)-invariants of the coordinate ring of \(G/L\) to the coordinate ring \(k[U^{-}]\). The main result of the paper is a theorem stating precisely when \(\varphi\) is an isomorphism of \(k\)-algebras. One case is when \(U\) is commutative. In this situation, the Levi subgroup \(L\) is the group of fixed points of an involution of \(G\), and the authors show that decomposing \(k[U^{-}]\) into \(L\)-modules is then equivalent to decomposing \(k[G/L]\) into \(G\)-modules; i.e., the summands are parametrized by the same set of weights. This relates a theorem of Schmid (describing the \(L\)-decomposition of \(k[U]\)) to a theorem of Helgason on the spherical representations of \(G\). If \(U\) is not commutative, the authors prove that \(\varphi\) is an isomorphism if and only if \(G\) is of type \(F_4\) and \(L\) of type \(C_3\), or if \(G\) is of type \(B_n\) and \(L\) of type \(A_{n-1}\).
scientific article; zbMATH DE number 5292754