Representations of \(\text{SL}(2,\mathbb{R})\) and symbolic calculi
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 \(G\).
scientific article; zbMATH DE number 604809
scientific article