Remarks on metaplectic representations of \(SL(2)\)
type: Item
instance of: scholarly article
title: Remarks on metaplectic representations of \(SL(2)\)
zbMATH Open document ID: 0786.22022
author: Hiroyuki Yoshida
published in: Journal of the Mathematical Society of Japan
publication date: 1993-01-17T00:00:00Z
review text: In specializing to the case of \(G=SL(2,k)\), the well-known Weil construction gives a projective representation of \(G\) such that \[ \bigl(\pi \Bigl(\begin{pmatrix} 1 &b\\ 0 &1\end{pmatrix}\Bigr) F\bigr)(x)= \psi(bx^ 2)F(x), \qquad (\pi(w)F)(x)= \gamma F^*(x), \] where \(F\in L^ 2(k)\), \(w=({0 \atop {-1}} {1\atop 0})\), \(\psi\) is a non-trivial additive character of \(k\), \(F^*\) is the Fourier image of \(F\) with respect to \(\psi\) and \(\gamma\) is a constant, depending on \(F\). When \(k\neq\mathbb{C}\) and lifted to the 2-fold covering group of \(G\), this projective representation \(\pi\) becomes a linear representation. The main problem is to construct analogous representations for \(n\)-fold covering. This problem was solved by Kubota for \(k=\mathbb{C}\) and by Yamazaki for \(k=\mathbb{R}\) and \(n\) even. In the paper under review, the author proposes a conceptually simpler and unified treatment of these representations including the case \(k=\mathbb{R}\) and \(n\) is odd.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 20G25
zbMATH DE Number: 98453
zbMATH Keywords: metaplectic representation
zbMATH Keywords: principal series
zbMATH Keywords: \(n\)-fold covering
reviewed by: Do Ngoc Diep
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04430351
OpenAlex ID: W2012048411
DOI: 10.2969/JMSJ/04430351
description: scientific article; zbMATH DE number 98453