Equivariant first order differential operators on boundaries of symmetric spaces
type: Item
instance of: scholarly article
title: Equivariant first order differential operators on boundaries of symmetric spaces
zbMATH Open document ID: 0957.22018
published in: Inventiones Mathematicae
publication date: 2001-03-25T00:00:00Z
review text: Let \(G\) be a real semisimple Lie group with finite centre and MAN a minimal parabolic subgroup of \(G\). Representations of the principal series are induced representations \(T_{\delta, \mu}= \text{Ind}^G_{ \text{MAN}}(\delta \otimes\mu \otimes 1)\), where \(\delta\) is an irreducible representation of \(M\) on a vector space \(V\), \(a\mapsto a^\mu\) is a character of \(A\). The representation \(T_{\delta,\mu}\) acts on the space of functions \(f\in C^\infty (G,V)\) satisfying the condition \(f(gma)= \delta(m^{-1}) a^{-\mu} f(g)\). In the paper under review all those intertwining operators between representations of the principal series are determined which can be written as differential operators of the first order. Let us formulate the result: Let \(\theta\) be a Cartan involution of \(G\) such that its fixed point subgroup is a maximal compact subgroup \(K\) with \(G=KAN\) and \(M\) the centralizer of \(A\) in \(K\). Let \(a\) be the Lie algebra of \(A\). Let \(\lambda\) be an indivisible restricted root and \(g_\lambda\) the corresponding root subspace of the Lie algebra \(g\) of \(G\). Take an orthonormal basis \(Y_j\) in \(g_{-\lambda}\) (with respect to the scalar product defined via the Killing form \(B\) of \(g)\), then \(\theta Y_j\) is an orthonormal basis in \(g_\lambda\). Let \(\sigma_\lambda \) be the restriction to \(M\) of the adjoint representation of \(G\) on \(g^C_\lambda\). Then \(\delta\otimes \sigma_\lambda\) decomposes multiplicity free into irreducible representations \(\delta_l\). Let \(\text{pr}_{\delta_l}\) denote the corresponding projection. Define \(H_\lambda\in a\) by \(B(H,H_\lambda)= \lambda (H)\), \(H\in a\). Under an irreducible representation \(\pi\) a Casimir element of Lie \(M\) goes to the multiplication by a number \(c(\pi)\). At last, take a basis \(e_k\) in \(V\), let \(e^*_k\) denote the dual basis. Now, if \(\mu\) satisfies the condition \(2\mu(H_\lambda)= c(\delta)+ c(\sigma_\lambda)-c (\delta_l)\), then an operator \(…
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A77
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1443408
zbMATH Keywords: real semisimple Lie group
zbMATH Keywords: minimal parabolic subgroup
zbMATH Keywords: principal series
zbMATH Keywords: intertwining operators
zbMATH Keywords: differential operators
zbMATH Keywords: restricted root
author: Adam Korányi
author: Hans Martin Reimann
reviewed by: V. F. Molchanov
MaRDI profile type: Publication
full work available at URL: s002229900030
OpenAlex ID: W1993236879
DOI: 10.1007/S002229900030
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description: scientific article; zbMATH DE number 1443408