Differential operators and theta series

Abstract:

Let $f$ be a modular form on a congruence subgroup of ${\text {SL}}(2,\mathbb {Z})$—not necessarily holomorphic, but an eigenfunction of the weight $k$ Casimir operator. Maass introduced differential operators (coming from the complexified universal enveloping algebra) which raise and lower by $2$ the weight of such a form and shift the eigenvalue. Here we introduce differential operators on hyperbolic $3$ space analogous to the Maass operators. These change by $2$ the weight of a modular form for an imaginary quadratic field. Theorem. The Maass operators and the hyperbolic space operators are intertwined by the imaginary quadratic Doi-Naganuma (base change) lifting. That is, the following diagram is commutative: \[ \begin {array}{*{20}{c}} & F & {\underset {{{\text {operators}}}}{\overset {{{\text {hyperbolic space}}}}{\leftrightarrow }}} & {\tilde F} & \\ {{\text {Lift}}} & \updownarrow & & \updownarrow & {{\text {Lift}}.} \\ & f & {\underset {{{\text {operators}}}}{\overset {{{\text {Maass}}}}{\leftrightarrow }}} & {\tilde f} & \\ \end {array} \] Using similar techniques for the dual pair $({\text {SL}}(2,\mathbb {R}),\;{\text {SO}}(2,1))$, we give a simple proof that the Shimura correspondences preserve holomorphicity (for weight $\geqslant 5/2$) and an explanation for this property directly in terms of the theta series (Weil representation) integral kernel. We also establish similar results for the real quadratic Doi-Naganuma lifting.