On a theorem of Schmid
type: Item
instance of: scholarly article
title: On a theorem of Schmid
zbMATH Open document ID: 1150.22004
DOI: 10.4171/RLM/512
author: Francesco Esposito
author: Andrea Maffei
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2008-06-25T00:00:00Z
review text: 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 \(A_{n-1}\).
reviewed by: Annegret Paul
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 5292754
zbMATH Keywords: Semisimple algebraic groups
zbMATH Keywords: Homogeneous spaces
zbMATH Keywords: Symmetric spaces
zbMATH Keywords: Regular functions
MaRDI profile type: Publication
cites work: Projective normality of complete symmetric varieties
cites work: A Pieri rule for Hermitian symmetric pairs. II
cites work: A duality for symmetric spaces with applications to group representations
cites work: Projective normality of the wonderful compactification of semisimple adjoint groups
cites work: Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen
full work available at URL: https://doi.org/10.4171/rlm/512
OpenAlex ID: W2027732348
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description: scientific article; zbMATH DE number 5292754