On quasi-modular forms, almost holomorphic modular forms, and the vector-valued modular forms of Shimura

Shimura’s vector-valued modular forms, weight changing operators, and Laplacians

“…The reference [1] obtained some results for modular and quasi-modular forms on a larger class of Fuchsian groups. The prequel [22] to the current paper then showed how quasi-modular forms are related to the vectorvalued modular forms defined in [18] (and previously, in a different language, in [8]), that involve symmetric powers of the standard representation, and established some properties of these vector-valued modular forms. These objects complement other generalizations of modular forms, such as (scalar-valued and vector-valued) modular forms of arbitrary weight (appearing in [11,15], and many others), mock modular forms (first uncovered by [25], then expanded by [5] and others, including the development in [4] of the theory of the closely related harmonic weak Maass forms), or modular forms of higher order (see [6] for the initial definition, and [7] for a classification result).…”

“…We determine in this paper all the eigenspaces of the two Laplacians from the previous paragraph. We remark that the analysis of the eigenspaces in depth d becomes more difficult when the weight is an integer between d + 1 and 2d, a case that was also shown in [22] to be more delicate (e.g., this is the case where the dimension formulae from that reference depend on whether the Fuchsian group has cusps or not). Interestingly, this investigation leads to a special variant of the sesqui-harmonic modular forms defined, for example, in [2].…”

“…In this section we describe the weight changing operators on the various spaces of modular and quasi-modular forms considered in [22].…”

“…Explicitly, the action of γ = a b c d takes τ ∈ H to γ τ = aτ +b cτ +d , with the factor of automorphy j(γ , τ ) = cτ + d. We shall also use the notation j γ (τ ) for j(γ , τ ), so that the lower left entry of γ is just the derivative j γ of j γ (independently of τ ). We shall work, as in [22], with arbitrary Fuchsian groups ≤ SL 2 (R), namely discrete subgroups such that the quotient \H has finite volume with respect to the SL 2 (R)-invariant measure dxdy y 2 . Let k ∈ Z be a weight, and let ρ be a representation of a Fuchsian subgroup of SL 2 (R) on some finite-dimensional complex vector space V ρ .…”

“…In fact, all of our definitions and results will hold in a general context: k may be in 1 2 Z if is a subgroup of the metaplectic double cover of SL 2 (R), or alternatively k can be an arbitrary real (or even complex) number if we take ρ to be a (possibly vector-valued) multiplier system of weight k. For the definitions of the few notions required for these generalizations see, e.g., Sect. 1.1 of [22]. We do remark that an equivalent approach to multiplier systems in general weights can be obtained by considering subgroups of the universal covering group SL 2 (R) (a realization of which can be obtained by replacing the metaplectic data j(γ , τ ) attached to some γ ∈ SL 2 (R) by a choice of log j(γ , τ )), and taking representations of such groups.…”