Commutation relations of differential operators and Whittaker functions on \(\text{Sp}_ 2(\mathbb{R})\)
The author considers the action of the dual pairs \(\text{Sp}_4 (\mathbb{R})\times (\text{SL}_2 (\mathbb{R})\times \text{SL}_2 (\mathbb{R}))\) and \(\text{Sp}_4 (\mathbb{R})\times \text{SL}_4(\mathbb{R})\) on the Weil representation; note that \(\text{SL}_2(\mathbb{R})\times \text{SL}_2 (\mathbb{R})\) is isogenous to \(\text{SO}(2,2)\) and \(\text{SL}_4 (\mathbb{R})\) is isogenous to \(\text{SO}(3,3)\). With the help of computer algebra the author calculates the actions of the corresponding universal enveloping algebras on the Weil representation. Using the global Weil representation he deduces a (known) formula for Whittaker functions on \(\text{Sp}_4 (\mathbb{R})\). Finally he determines the Mellin transform of this function.
commutation relations of differential operators
scientific article; zbMATH DE number 897686