Representations of \(\text{SL}(2,\mathbb{R})\) and symbolic calculi
type: Item
instance of: scholarly article
title: Representations of \(\text{SL}(2,\mathbb{R})\) and symbolic calculi
zbMATH Open document ID: 0840.43018
DOI: 10.1007/BF01206295
published in: Integral Equations and Operator Theory
publication date: 1994-07-12T00:00:00Z
review text: The Weyl calculus of pseudodifferential operators on \(\mathbb{R}^n\) has been generalized to symbolic calculi of operators acting on the Hilbert spaces of representations taken from the holomorphic discrete series of various groups. If one considers instead the representation \(\pi_{+,i \lambda}\), taken from the principal series of representations of \(G = SL(2, \mathbb{R})\), and defined by \[ \left( \pi_{+, i \lambda} \left( {a \;b \choose c\;d} \right) u \right) (t) = |- ct + a |^{- 1 - i \lambda} u \left( {dt - b \over -ct + a} \right), \quad u \in L^2 (\mathbb{R}), \] it is the standard (convolution first)-antistandard calculus which can be generalized to this situation. Realizing the homogeneous space \(G/MA\) as \(\{(s,t) \in (\mathbb{R} \cup \{\infty\}) \times (\mathbb{R} \cup \{\infty\}) : s \neq t\}\), one can define two maps \(Op_0\) and \(Op_1\) from functions on \(G/MA\) to operators on \(L^2 (\mathbb{R})\) by the formulas \[ \begin{aligned} \bigl( Op_0 (f)u \bigr) (s) & = c_{- \lambda} \int f(s,t) |s - t |^{- 1 - i \lambda} (\theta u) (t) dt, \\ \bigl( \theta Op_1(f)u \bigr) (s) & = c_\lambda \int f(t,s) |s - t |^{-1 + i \lambda} u(t) dt, \end{aligned} \] where \(\theta\), the convolution by some distribution on \(\mathbb{R}\) coinciding with \(c_\lambda |s |^{- 1 + i \lambda}\) out of the origin, is the intertwining operator from \(\pi_{+, i \lambda}\) to \(\pi_{+, - i \lambda}\). We give the spectral decomposition of the operator \(F_\lambda\) that makes the identity \(Op_1 (F_\lambda f) = Op_0 (f)\) valid for every \(f \). Realizing, on the other hand, \(G/MA\) as a coadjoint orbit (a one-sheeted hyperboloid) in \({\mathfrak g}^*\), we show that an extension of \(Op_0\) (or \(Op_1)\) provides a symbolic calculus in the enveloping algebra of the representation, in which symbols are just the polynomial functions on the orbit. Similar results hold in connection with the discrete series of representations of \(G\).
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E99
Mathematics Subject Classification ID: 35S99
zbMATH DE Number: 604809
zbMATH Keywords: Weyl calculus
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: symbolic calculi
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: representations
zbMATH Keywords: principal series
zbMATH Keywords: homogeneous space
zbMATH Keywords: convolution
zbMATH Keywords: intertwining operator
zbMATH Keywords: coadjoint orbit
zbMATH Keywords: enveloping algebra
zbMATH Keywords: polynomial functions
zbMATH Keywords: discrete series
author: André Unterberger
author: Julianne Unterberger
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01206295
OpenAlex ID: W4229927415
description: scientific article; zbMATH DE number 604809