Remarks on metaplectic representations of \(SL(2)\)
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.
scientific article; zbMATH DE number 98453