Harmonic Maaß-Jacobi forms of degree 1 with higher rank indices (Q2828366)

This paper defines and studies harmonic Maaß forms of degree \(1\) and rank greater than or equal to \(1\). The case when the rank is equal to \(1\) had been studied previously by a number of authors (see the introduction for a review of important contributions), and as pointed out by the authors, this class of functions includes the \(\mu\)-function considered by \textit{S. Zwegers} [Mock theta functions. Utrecht: Universiteit Utrecht (PhD Thesis) (2002)] to study mock theta functions. Interesting examples of functions that fall into the new higher rank functions studied in this paper include those involved in the multivariable generalization of the \(\mu\)-function.NEWLINENEWLINETo help explain their definition of a (higher rank) Maaß-Jacobi form of weight \(k\) and index \(L\) (where \(k\) is an integer and \(L\) is the Gram matrix with respect to a fixed basis of a positive definite integral even lattice of rank \(N \in \mathbb{N}\)), the authors first motivate and define a covariant operator \(\mathcal{C}^{k,L}\). Indeed, one of the defining characteristics of a Maaß-Jacobi form is that it is an eigenfunction of the operator \(\mathcal{C}^{k,L}\). The definition of \(\mathcal{C}^{k,L}\) is given in Section 2, where additional results pertaining to this operator are described. In particular, the authors also explain that \(\mathcal{C}^{k,L}\) is the Casimir operator (up to scalars) of the centrally extended rank \(N\) real Jacobi group \(\tilde{G}_N^J\) and lies in the center of the space of slash-invariant operators (see Theorem 2.4). Additionally, an expression of this operator in terms of the raising and lowering operators is given (see Proposition 2.7). Finally, the spaces of covariant operators of degree one, two, and three are described in terms of raising and lowering operators and the algebra of slash-invariant operators is shown to be generated by the degree three space (see Propositions 2.6 and 2.8). Theorem 2.4 and Propositions 2.6--2.8 are all proved in Section 5 of the paper.NEWLINENEWLINEThis paper also considers the generalization of the notion of H-harmonicity to the groups \(\tilde{G}_N^J\), as well as the notion of semi-holomorphic harmonic Maaß-Jacobi forms. The definitions and study of these two notions are given in Sections 3 and 4, respectively. An interesting result here is that for \(N\geq 2\), H-harmonicity coincides with semi-holomorphicity (see Theorem 3.4). Additionally, the authors show that the Maaß-Jacobi \(\theta\)-decomposition for a harmonic Maaß-Jacobi form gives a bijection between vector-valued weak harmonic Maaß-Jacobi forms and harmonic semi-holomorphic Maaß-Jacobi forms (Theorem 4.8). Also contained in Section 4 (in particular, Subsection 4.1), is a study of Maaß-Poincaré series for forms with lattice indices that may be larger than 1.NEWLINENEWLINEThe paper concludes with Section 5, which, as mentioned above, provides the detailed proofs of the main results described in Section 2. Also contained in this section, however, are descriptions of the necessary details used by the authors pertaining to Lie algebras (Subsection 5.1) and the algebraic study of the general theory of invariant differential operators (Subsection 5.3).