Is there a well-defined non-holomorphic index-raising operator $V_l$, taking index-$m$ Jacobi-Maass forms to index-$ml$ Jacobi-Maass forms (same weight), analogous to the holomorphic operator defined in Eichler-Zagier, p.41? I haven't found it in the literature yet.
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It may well be that no explicit expressions have been written down for the corresponding operator for Jacobi-Maass forms, but the analogous operators surely exist and behave reasonably, for the following reason(s). The Eichler-Zagier operator (and Maass-Shimura operators, in other contexts) are in-coordinates expressions for differential operators coming from the enveloping algebra of the Lie algebra, regardless of the weight/index and/or holomorphy/waveform features, which would refer to the (irreducible or not) representation in which we are "moving around". That is, the intrinsic prescription of the operators (as opposed to the formulaic/in-coordinates description) does not depend on the holomorphy-or-waveform feature, nor upon the index and/or weight. Yes, the expressions for the operators certainly depend, but this is in principle a secondary issue. Thus, one can be reassured that it is possible to obtain the desired operators, and be assured in advance that they have the expected/desired properties. |
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