Abstract
We classify HarishChandra modules generated by the pullback to the metaplectic group of harmonic weak Maaß forms with exponential growth allowed at the cusps. This extends work by SchulzePillot and parallels recent work by Bringmann–Kudla, who investigated the case of integral weights. We realize each of our cases via a regularized theta lift of an integral weight harmonic weak Maaß form. HarishChandra modules in both integral and halfintegral weight that occur need not be irreducible. Therefore, our display of the role that the theta lifting takes in this picture, we hope, contributes to an initial understanding of a theta correspondence for extensions of HarishChandra modules.
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Bringmann and Kudla [7] provided a classification of the HarishChandra modules generated by the pullback to \(\textrm{SL}_{2}(\mathbb {R})\) of harmonic weak Maaß forms of integral weight with exponential growth allowed at the cusps. They complemented their classification with explicit examples for all 9 possibilities, and thus showed that these arise from harmonic weak Maaß forms. The case of halfintegral weight was partially treated by SchulzePillot in earlier work [23]. He restricted himself to a subclass of functions satisfying a more restrictive growth condition at the cusps (see Remark 2.6). However, he did not explicitly realize the different modules that arise.
In this note we provide a classification of HarishChandra modules corresponding to the full class of harmonic weak Maaß forms of halfintegral weight. Moreover, we explicitly realize all cases as certain theta liftings of integral weight harmonic weak Maaß forms that occur in Bringmann and Kudla’s work. We therefore extend SchulzePillot’s work in two directions. Observe that it is equally possible to realize these cases by Poincaré series or via an abstract cohomological argument.
Our work also provides a representation theoretic perspective on local theta liftings of harmonic weak Maaß forms. Theta liftings are an explicit realization of the theta correspondence introduced by Howe [19] and are best understood for cusp forms. For example, they can be used to realize the correspondence of Shimura and Shintani in the classical setting of cusp forms [25, 26], whose representation theoretic counterpart appears in Waldspurger’s work [27]. To be able to lift forms with singularities at the cusps we consider lifts that are regularized using ideas of Harvey–Moore [18] and Borcherds [6].
Much less is known on the representation theoretic side. Kudla and Rallis analyzed invariant distributions [22], which arise from the Shintani lift of constants when viewed through the lens of the archimedean theta correspondence. They encountered reducible HarishChandra modules, as opposed to the irreducible ones that one finds when treating cusp forms. The HarishChandra modules that occurred in the work of Bringmann–Kudla and SchulzePillot and that occur in our work are generally reducible, too. In this sense, we give an initial sense of how the archimedean theta correspondence might function on reducible HarishChandra modules.
We illustrate our results by an example. HarishChandra modules in our setting can be visualized by their Ktype support and transitions, which reflect the behaviour of Maaß lowering and raising operators on harmonic weak Maaß forms, defined in (1.2) and the paragraph that follows it. We have the following two HarishChandra modules, one for \(\textrm{SL}_{2}(\mathbb {R})\) and the other one for \(\textrm{Mp}_{1}(\mathbb {R})\), whose visualization we explain in Sect. 2.
The first HarishChandra module corresponds to case III (b) in [7]. It can be realized by the Eisenstein series
The second one can be realized by taking the regularized Shintanilift of \(E_2^*\) (compare Sect. 3 for the definition). Specifically, its (twisted) Shintanilift was computed in [5]. We let \(\Delta \) be a negative fundamental discriminant. We have
where
with \(H(0) = \frac{1}{12}\) and \(H(D) = 0\) if \(D \ne 0\) is not a discriminant, is Zagier’s weight\(\frac{3}{2}\) Eisenstein series [29]. Here, \(\beta _{3/2}(s)= \int _1^\infty e^{st} t^{3/2}dt\).
Our work is organised as follows: We first review some necessary background on the metaplectic group and harmonic weak Maaß forms. In Sect. 2 we introduce the principal series and state the classification for the \(({\mathfrak {g}},K)\) modules corresponding to harmonic weak Maaß forms of halfintegral weight. Then we give a short overview on Millson and Shintani theta liftings of even integral weight harmonic weak Maaß forms and close with the explicit realization of all of the modules arising from our classification.
1 Preliminaries
1.1 The metaplectic group
We define the real metaplectic group as
equipped with the usual group law \((g, \omega ) (g',\omega ') = (g g', \tau \mapsto \omega (g' \tau ) \omega '(\tau ))\). Further, we write \(\pi _{\textrm{Mp}_{1}}\) for the projection from \(\textrm{Mp}_{1}(\mathbb {R})\) to \(\textrm{SL}_{2}(\mathbb {R})\) that sends \((g,\omega )\) to g. This turns \(\textrm{Mp}_{1}(\mathbb {R})\) into a connected double cover of \(\textrm{SL}_{2}(\mathbb {R})\).
We let K, M, and N be the preimages under \(\pi _{\textrm{Mp}_{1}}\) of \(\textrm{SO}_{2}(\mathbb {R})\), the subgroup of diagonal, and the subgroup of upper triangular unipotent matrices. We have a KMNdecomposition of \(\textrm{Mp}_{1}(\mathbb {R})\), and the subgroups K, M, and N are uniformized by
In the argument of k, we have \(\theta \in \mathbb {R}\) and \(\omega _{k(\theta )}:\, \mathbb {H}\rightarrow \mathbb {C}\) is uniquely defined by its value \(\omega _{k(\theta )}(i) = \exp (i \frac{1}{2} \theta )\). To specify the right hand side of m(a, s), we define the sign function \(\textrm{sgn}(i a):= i \textrm{sgn}(a)\) for \(a \in i \mathbb {R}\). Given \(a \in \mathbb {R}\), \(a > 0\), \(s \in \{\pm 1\}\), or \(a \in \mathbb {R}\), \(a < 0\), \(s \in \{\pm i\}\), we let \(sa^{\frac{1}{2}}\) be the square root of \(a^{1}\) with sign s. The argument of n is \(b \in \mathbb {R}\).
1.2 The Lie algebra of \(\textrm{Mp}_{1}(\mathbb {R})\)
Since \(\pi _{\textrm{Mp}_{1}}\) is a covering map of Lie groups, the (complexified) Lie algebras of \(\textrm{Mp}_{1}(\mathbb {R})\) and \(\textrm{SL}_{2}(\mathbb {R})\) are canonically isomorphic. We follow the notation in [7], and set
which is a basis for \(\mathfrak {g}=(\mathfrak {g}_0)_\mathbb {C}\cong \{ A \in \textrm{Mat}_{2}(\mathbb {C}) \,:\, \textrm{trace}(A) = 0 \}\), where we let \(\mathfrak {g}_0=\textrm{Lie}(\textrm{SL}_{2}(\mathbb {R}))\). We write \(\textrm{U}(\mathfrak {g})\) for the universal enveloping algebra of \(\mathfrak {g}\).
We have the commutator relations
The Casimir operator
is central as required. We have \(C = (H1)^2 + 4 X_+ X_  1\) and \(C = (H+1)^2 + 4 X_ X_+  1\), which is slightly more convenient for later purposes.
The action of \(X + i Y \in \mathfrak {g}\), \(X, Y \in \mathfrak {g}_0\), on smooth complex functions \(\tilde{f}:\, \textrm{Mp}_{1}(\mathbb {R}) \rightarrow \mathbb {C}\) is defined by
where we write \(\partial _{t=0}\) for the value at \(t = 0\) of the derivative with respect to t and \(\exp \) denotes the exponential map for the Lie group \(\textrm{Mp}_{1}(\mathbb {R})\).
1.3 Harmonic weak Maaß forms
The action of \(\textrm{SL}_{2}(\mathbb {R})\) on the Poincaré upper half plane \(\mathbb {H}\) extends to the metaplectic group via \(\pi _{\textrm{Mp}_{1}}\):
We define the slash action of weight \(k \in \frac{1}{2} \mathbb {Z}\) on functions \(f:\, \mathbb {H}\rightarrow \mathbb {C}\) by
We let \(\textrm{Mp}_{1}(\mathbb {Z})\) be the inverse image of \(\textrm{SL}_{2}(\mathbb {Z})\) under the covering map \(\textrm{Mp}_{1}(\mathbb {R})\rightarrow \textrm{SL}_{2}(\mathbb {R})\). An arithmetic type is a finite dimensional, complex representation \(\rho \) of a finite index subgroup \(\Gamma \subseteq \textrm{Mp}_{1}(\mathbb {Z})\). We write \(V(\rho )\) for the representation space of \(\rho \). Functions \(f:\, \mathbb {H}\rightarrow V(\rho )\) admit the following slash actions for \(k \in \frac{1}{2} \mathbb {Z}\):
The Weil representation is an arithmetic type that is most relevant in the context of theta lifts. Consider a finite quadratic module \(D = (M,q)\), i.e. a finite abelian group M together with a nonsingular quadratic form \(q: M \rightarrow \mathbb {Q}/\mathbb {Z}\), with corresponding bilinear form \(\langle \,\cdot ,\cdot \,\rangle _q\). We let \(e(x):=e^{2\pi i x}\). There is a unique representation \(\rho _D\) for which \(V(\rho _D)\) is the free module \(\mathbb {C}M\) with basis M and actions
where
Recall that \(\tau =u+iv\). We fix the normalization of the Maaß lowering and raising operators as
Then the weightk Laplace operator equals \(\Delta _k:=  \textrm{R}_{k2} \textrm{L}_k\).
A harmonic weak Maaß form of weight \(k \in \frac{1}{2} \mathbb {Z}\) and arithmetic type \(\rho \) for \(\Gamma \subseteq \textrm{Mp}_{1}(\mathbb {Z})\) is a smooth function \(f:\, \mathbb {H}\rightarrow V(\rho )\) with \(\Delta _k\, f = 0\) such that
and for some norm \(\Vert \,\cdot \,\Vert \) on \(V(\rho )\)
Note that we only have nonzero harmonic weak Maaß forms if \(\rho ((I,i)) = i^{2k}\).
We denote the space of such forms by \(\textrm{H}_k^{\textrm{mg}} (\Gamma ,\rho )\) as in [7]. Observe that while \(\textrm{mg}\) stands for moderate growth, the condition imposed in (1.3) differs from what is called the moderate growth condition in the context of modular forms.
If \((\rho ,V)\) is onedimensional, i.e., given by a character \(\chi :\Gamma \rightarrow \mathbb {C}^\times \), we write \(\textrm{H}_k^{\textrm{mg}} (\Gamma ,\chi )\), or even \(\textrm{H}_k^{\textrm{mg}} (\Gamma )\) if \(\chi \) is trivial. This subspace is referred to as the space of scalarvalued harmonic weak Maaß forms of weight k for \(\Gamma \).
Harmonic weak Maaß forms are related to classical spaces of modular forms by the \(\xi \)operator. Following Bruinier and Funke [12] we define it by
Proceeding as in [12] we see that
and that \(\xi _k\) is surjective. Here, \(\textrm{M}^!_{2k}\) denotes the subspace of weakly holomorphic modular forms (i.e., forms that are holomorphic on \(\mathbb {H}\) and have poles of finite order at the cusps). Moreover, \(\overline{\rho }\) is defined by \(\overline{\rho }(\gamma )v= \overline{\rho (\gamma )\overline{v}}\). Here we clearly need to assume that V is defined over \(\mathbb {R}\).
A natural subspace of \(\textrm{H}_k^{\textrm{mg}} (\Gamma ,\overline{\rho })\) consists of those functions that map to cusp forms under the \(\xi \)operator or alternatively for which there exists a polynomial \(P_f(\tau )\in V[q^{1}]\) such that
as \(v\rightarrow \infty \) for some \(\varepsilon >0\) (and similarly at the other cusps). We denote the subspace of these forms by \(\textrm{H}_k (\Gamma ,\rho )\). Its image under the \(\xi \)operator are cusp forms.
We now describe the Fourier expansion of such forms. A scalarvalued harmonic weak Maaß form of integral weight \(k\ne 1\) has a Fourier expansion of the form
at \(\infty \), where \(W_k(x)\) is the realvalued incomplete \(\Gamma \)function
with \(\Gamma (s,x)=\int _x^\infty e^{t} t^{s1} dt\). If \(k=1\), we have to replace the term \(v^{1k}\) in the nonholomorphic part by \(\log (v)\).
If \(f\in \textrm{H}_k(\Gamma )\), then we have \(c^_f(n)=0\) for all \(n\ge 0\), and more generally \(c^_{f _k \gamma }(n) = 0\) for all \(\gamma \in \textrm{Mp}_{1}(\mathbb {Z})\) and \(n \ge 0\).
We now let \(k \in \mathbb {Z}_{\le 0}\). A further differential operator which establishes relations between harmonic weak Maaß forms and classical modular forms is
By Bol’s identity we have \( D^{1k}= (4\pi )^{k1} \textrm{R}_k^{1k} \) and \(D^{1k}: \textrm{H}_k^{\textrm{mg}}(\Gamma ,\rho )\rightarrow \textrm{M}^!_{2k}(\Gamma ,\rho )\).
For scalarvalued forms we define the flipped space by
The spaces \(\textrm{H}_k(\Gamma )\) and \(\textrm{H}_k^{\#}(\Gamma )\) are “flipped” by the flipping operator
The flipping operator satisfies
It acts on the Fourier expansion (1.4) of a harmonic weak Maaß form \(f\in \textrm{H}^{\textrm{mg}}(\Gamma )\) by
2 \((\mathfrak {g},K)\)modules and harmonic weak Maaß forms
Recall that a \((\mathfrak {g},K)\)module is a simultaneous module for a Lie algebra \(\mathfrak {g}\) and a compact group K with \(\textrm{Lie}(K) \subseteq \mathfrak {g}\) and suitable compatibility conditions imposed. A HarishChandra module is an admissible \((\mathfrak {g},K)\)module, i.e., a \((\mathfrak {g},K)\)module with finite dimensional Kisotypical components. The latter are referred to as Ktypes. As before, we set \(\mathfrak {g}= \textrm{Lie}(\textrm{Mp}_{1}(\mathbb {R}))\), and \(K = \pi _{\textrm{Mp}_{1}}^{1}(\textrm{SO}_{2}(\mathbb {R}))\) has already been defined.
We visualize HarshiChandra modules by their Ktype support, i.e., the set of Kisotypical components that are nonzero, and the vanishing of Ktype transitions, i.e., those Ktypes on which X or Y in Sect. 1.2 act as zero. We label Ktypes by their eigenvalues under H. Note that in the setting of harmonic weak Maaß forms, Ktypes are at most onedimensional. It therefore suffice to indicate their nonvanishing. For instance, consider following diagram.
This diagram represents a HarishChandra module whose Ktypes of Heigenvalue in \(\frac{3}{2} + 2\mathbb {Z}_{\ge 0}\) vanish. The vertical line at \(\frac{3}{2}\) together with the arrow in positive direction indicate that any vector of Ktype \(\frac{3}{2}\) vanishes under \(X_\), that is, it points towards the Ktypes support of a subrepresentation of the HarishChandra module, which in the present case is zero.
2.1 Some principal series
We start by providing a sufficient supply of \((\mathfrak {g},K)\)modules by decomposing suitable degenerate principal series. This is analogue of Sect. 4 of [7] in the case of the metaplectic group \(\textrm{Mp}_{1}(\mathbb {R})\).
For halfintegers \(\epsilon \in \frac{1}{2} \mathbb {Z}/2 \mathbb {Z}\) and \(\nu \in \mathbb {C}\), we let \(\textrm{I}^{\textrm{sm}}(\epsilon , \nu )\) be the principal series representation of \(\textrm{Mp}_{1}(\mathbb {R})\) on the space of smooth functions with the property
To check that this space is not empty, we only need to see that the intersection of M and N is trivial.
We consider the associated \((\mathfrak {g},K)\)module \(\textrm{I}(\epsilon , \nu )\) of Kfinite functions in \(\textrm{I}^{\textrm{sm}}(\epsilon , \nu )\). Given an halfinteger j with \(j \in \epsilon + 2 \mathbb {Z}\), we let \(\phi _j \in \textrm{I}(\epsilon , \nu )\) be the unique function that satisfies
To see that this is welldefined it suffices to check that the defining property for \(\textrm{I}(\epsilon , \nu )\) holds on \(K \cap MN\). For \(\theta \in \pi \mathbb {Z}\), we have
This calculation also shows that no other \(\textrm{K}\)types but those corresponding to the functions \(\phi _j\), \(j \in \epsilon + 2 \mathbb {Z}\) can occur in \(\textrm{I}(\epsilon ,\nu )\).
The decomposition of these principle series was determined by Waldspurger [28], and conveniently reformulated by SchulzePillot [23]. We will encounter two families of irreducible representations, whose Ktypes are spanned by
They lie in the discrete series if \(\nu > 0\).
Proposition 2.1
(Lemma 6 of [23] and Proposition 6 of [28]) Let \(\epsilon \) and \(\nu \) be as in (2.1). Consider the case that \(\epsilon \in \frac{1}{2} + \mathbb {Z}\) and \(\nu \in \epsilon + 2 \mathbb {Z}\). Then we have \(\nu  1 \in \epsilon + 2 \mathbb {Z}\), and we have \(X_+\,\phi _{\nu 1} = 0\), which implies that \(\phi _{\nu 1}\) spans the maximal Ktype of a subrepresentation of \(I(\epsilon ,\nu )\). We have the short exact sequence
Consider the case that \(\epsilon \in \frac{1}{2} + \mathbb {Z}\) and \(\nu + 1 \in \epsilon + 2 \mathbb {Z}\). Then we have \(X_\,\phi _{\nu +1} = 0\), which implies that \(\phi _{\nu +1}\) spans the minimal Ktype of a subrepresentation of \(I(\epsilon ,\nu )\). We have the short exact sequence
Remark 2.2
The representations in Proposition 2.1 are “genuine” representations of \(\textrm{Mp}_{1}(\mathbb {R})\), that is, they do not arise via pullbacks along the projection from \(\textrm{Mp}_{1}(\mathbb {R})\) to \(\textrm{SL}_{2}(\mathbb {R})\). Note that they have two composition factors. This differs from the situation of integral \(\epsilon \), which yields nongenuine principle series with three composition factors except for the very special case of \(\nu = 0\).
2.2 HarishChandra modules associated to harmonic weak Maaß forms
We let \(\textrm{A}(\textrm{Mp}_{1}(\mathbb {R}))\) be the space of complexvalued, smooth functions on \(\textrm{Mp}_{1}(\mathbb {R})\) that are linear combinations of functions with the property that
We consider an arithmetic type \(\rho :\, \Gamma \rightarrow \textrm{GL}_{}(V(\rho ))\), \(\Gamma \subset \textrm{Mp}_{1}(\mathbb {Z})\). We let \(\textrm{A}(\textrm{Mp}_{1}(\mathbb {R}), V(\rho )):= \textrm{A}(\textrm{Mp}_{1}(\mathbb {R})) \otimes _\mathbb {C}V(\rho )\) be the space of smooth functions on \(\textrm{Mp}_{1}(\mathbb {R})\) that take values in \(V(\rho )\) and are linear combinations of functions with the same property (2.3). Finally, let \(\textrm{A}(\textrm{Mp}_{1}(\mathbb {R}), \rho ) \subseteq \textrm{A}(\textrm{Mp}_{1}(\mathbb {R}), V(\rho ))\) be the subspace of functions with the additional property that
We can associate HarishChandra modules, in fact submodules of \(\textrm{A}(\textrm{Mp}_{1}(\mathbb {R}), \rho )\) to harmonic weak Maaß forms of halfintegral weight and of type \(\rho \). We loosely follow the description of the integral weight case in [7].
In the rest of this subsection we fix \(f \in \textrm{H}^{\textrm{mg}}_k(\Gamma ,\rho )\), and set
Note that \(\tilde{f}\) depends on k, but it is customary to suppress this dependence from the notation. As long as f transforms like a modular form, the weight k can be recovered from the asymptotic expansion of \(f \circ \gamma \) for suitable \(\gamma \in \textrm{Mp}_{1}(\mathbb {Z})\).
We see that \(\tilde{f}\) is a function from \(\textrm{Mp}_{1}(\mathbb {R})\) to \(V(\rho )\), and verify that it satisfies (2.3) for \(j = k\). We calculate the action of K by right shifts to find that \(\tilde{f} \in \textrm{A}(\textrm{Mp}_{1}(\mathbb {R}), V(\rho ))\). A similar calculation also shows that \(H \tilde{f} = k \tilde{f}\). This merely reflects the fact that \(\tilde{f} = \tilde{f}_k\) was constructed via the weightk slash action in (2.5). In particular, it does not use any modular properties of f, let alone the fact that it is harmonic. Finally, inspecting the action of \(\textrm{Mp}_{1}(\mathbb {Z})\) by left shifts then shows that \(\tilde{f} \in \textrm{A}(\textrm{Mp}_{1}(\mathbb {R}), \rho )\).
Calculations for \(X_+\) and \(X_\) are significantly more involved. They yield the same results as in the integral weight case. The lowering and raising operators intertwine with the construction in (2.5) provided that the weight k is adjusted:
Only now we employ the fact that f is harmonic, i.e., that we have \(\textrm{R}_{k+2}\,\textrm{L}_k\,f = 0\). Recalling that \(C = (H1)^2 + 4 X_+ X_  1\), this corresponds via (2.6) to
The PoincaréBirkhoffWitt property of the generators \(H, X_\pm \) of \(\textrm{U}(\textrm{Lie}(\textrm{Mp}_{1}(\mathbb {R}))_\mathbb {C})\) in conjunction with (2.7) implies that \(\tilde{f}\) generates a \((\mathfrak {g},K)\)module \(\varpi (f,k) = \varpi _\infty (f,k) \subset \textrm{A}(\textrm{Mp}_{1}(\mathbb {R}), \rho )\), which is spanned by the functions
The commutation relations of H and \(X_\pm \) then imply that each Ktype in \(\varpi (f,k)\) occurs with multiplicity at most once. In particular, \(\varpi (f,k)\) is a HarishChandra module.
We finish with the eigenvalues of \(\tilde{f}\) under \(X_^r X_+^r\) and the eigenvalues of \(\tilde{f}_{k2}\) under \(X_+^r X_^r\). The next lemma will be helpful when identifying \(\varpi (f,k)\) in the context of our classification. It features the Pochhammer symbols
Lemma 2.3
Fix \(k \in \frac{1}{2} + \mathbb {Z}\) and let \(f:\, \mathbb {H}\rightarrow V\) be a smooth function with \(\Delta _k\, f = 0\) for some complex vector space V. Then for \(\tilde{f}\) defined in (2.5), we have
Proof
Since the Casimir element is central it acts by scalars on the module generated by \(\tilde{f}\). We have \(C \tilde{f}_{k + 2r} = ((k1)^2  1) \tilde{f}_{k + 2r}\) for all \(r \in \mathbb {Z}\). The action of H was determined before: \(H\, \tilde{f}_{k+2r} = (k + 2r) \tilde{f}_{k+2r}\).
We conclude that for \(r \ge 0\), we have
Similarly, if \(r \ge 0\), we have
This yields the recursions, valid for \(r > 0\),
The statement follows from these recursions by induction on r. \(\square \)
2.3 Classification
We next describe the HarishChandra modules \(\varpi (f,k)\) associated with harmonic weak Maaß forms in terms of the standard modules \(\varpi ^\pm (\pm \nu )\). The theory is much more stringent than in the case of integral weights.
Proposition 2.4
Let f be a weakly holomorphic modular form of weight \(k \in \frac{1}{2} + \mathbb {Z}\). Then \(\varpi (f,k)\) is isomorphic to \(\varpi ^+(k1)\).
Let f be a harmonic weak Maaß form of weight \(k \in \frac{1}{2} + \mathbb {Z}\) that is not weakly holomorphic. Then \(\varpi (f,k)\) fits into the nonsplit exact sequence
Remark 2.5
The existence of weakly holomorphic modular forms in all halfintegral weight cases is clear. The existence of harmonic weak Maaß forms in all weights follows along the lines of Bruinier–Funke [12], when removing the condition that the image under \(\xi _k\) has moderate growth.
Remark 2.6
SchulzePillot in Proposition 7 of [23] provided a classification of harmonic weak Maaß forms for which \(\xi _k\,f\) is a cusp form instead of a weakly holomorphic modular form as in our case.
Proof
One can appeal to a classification that identifies irreducible HarishChandra modules in terms of their eigenvalues under the Casimir element and their Ktypes support.
A more elementary approach was suggested by Bringmann–Kudla [7]. Since all Ktypes in \(\varpi (f,k)\) appear with multiplicity at most one, it suffices to compare the action of \(X_\pm ^r X_\mp ^r\) and \(X_\mp ^r X_\pm ^r\). We give the details in the case that f is not weakly holomorphic.
Observe that \(g:= \textrm{L}_k\,f\) is annihilated by \(\textrm{R}_{k2}\). Using the intertwining property (2.6) of \(\textrm{R}_{k2}\) and \(X_+\), we conclude that \(X_+\, \tilde{g}_{k2} = 0\). To show that the HarishChandramodule \(\varpi (g, k2)\) is isomorphic to \(\varpi ^(1k)\), it now suffices to calculate and compare the eigenvalues of \(\tilde{g} = \tilde{f}_{1}\) and of \(\phi _{k2}\) from Sect. 2.1 under \(X_+^r X_^r\) for all positive integers r. The former was given in (2.8) and the latter based on the discussion after (2.1) with \(\nu = k2\) and \(j = k2\).
The Ktype support of the quotient module \(\varpi (f,k) /\varpi (g,k2)\) corresponds to the \(k(\theta )\)eigenvalues \(e(i j \theta )\) for \(j \in k + 2 \mathbb {Z}_{\ge 0}\), which coincides with the Ktypes that appear in \(\varpi ^+(k1)\). Again because each Ktype occurs with multiplicity one, it suffices to compare the eigenvalues of \(\tilde{f}\) and \(\phi _k\) under \(X_^r X_+^r\). The former was also given in (2.8) and the latter based on the discussion after (2.1) with \(\nu = k2\) and \(j = k\). \(\square \)
2.4 Diagrams of Ktypes
Recall the visualization of HarishChandra modules introduced in Sect. 2. Let f be a harmonic weak Maaß form of weight k. If we have \(\textrm{L}_k\, f = 0\) and \(k \in \frac{1}{2} + 2\mathbb {Z}\) is greater than one, the associated HarishChandra module \(\varpi (f,k)\) yields the Ktype diagram
Observe that the Ktypes next to 0 in this diagram are labelled by \(\frac{3}{2}\) and \(\frac{1}{2}\). Integral weights do not support Ktypes in this diagram. If we have \(\textrm{L}_k\, f = 0\) and \(k \in \frac{3}{2} + 2\mathbb {Z}\) is greater than one, then the positively labelled Ktype next to zero is \(\frac{3}{2}\) and the negative one is \(\frac{1}{2}\). This yields the diagram
If \(\textrm{L}_k\, f \ne 0\), we again have two cases. One of the following two diagrams describes the Ktypes in \(\varpi (f,k)\):
The situation is very similar for k less than 1. Depending on \(\textrm{L}_k\,f\) and k, one of the following four diagrams arises from \(\varpi (f,k)\):
3 Theta lifts of harmonic weak Maaß forms
In this section we review results on the extension of the classical Shintani lifting to harmonic weak Maaß forms and the socalled Millson theta lifting of such forms obtained by the first named author in joint work with Markus Schwagenscheidt [4, 5]. These liftings are an explicit realization of the theta correspondence given as the integral of an input function transforming like a modular form of (even) weight k against a certain theta kernel function. We illustrate this procedure in a bit more detail. If f transforms of weight k one is led to consider the following integral (where the integration is carried over a suitable fundamental domain \({\mathcal {F}}\))
Here, \(\Theta (\varphi , \tau , z)\) is an integration kernel of weight k in the variable z. In the variable \(\tau \) the complex conjugate \(\overline{\Theta (\varphi , \tau , z)}\) is of halfintegral weight \(\ell \). Moreover, \(\varphi \) is a suitable Schwartz function, Provided the integral converges, it transforms like an automorphic form of weight \(\ell \).
For holomorphic modular forms such liftings have been investigated in the framework of the Shimura–Shintanicorrespondence [20, 21, 25, 26]. Ideas of Harvey and Moore [18] and Borcherds [6] led to the theory of regularized theta liftings allowing for inputs that are not holomorphic at the cusps. The lifts we consider in this work serve as generating series of traces of CM values and (regularized) geodesic cycle integrals of the input function.
We will consider twisted versions of the Shintani and Millson lift. These are obtained via twisting the theta kernel with a certain genus character (see [2] for a description of this procedure). This enables us to state our results for the full modular group. In particular, we let \(\Delta \in {\mathbb {Z}}\) be a fundamental discriminant.
We denote the Millson theta lift by \(\Lambda ^{\textrm{M}}\) and the Shintani theta lift by \(\Lambda ^{\textrm{Sh}}\).
Remark 3.1
We state the results in the following subsections for theta lifts of forms for the full modular group to halfintegral weight forms for the group \(\Gamma _0(4)\). Note that their results hold in greater generality (see [4, 5]).
3.1 The Shintani lifting
In the past years the classical Shintani theta lift of holomorphic forms has been generalized to weakly holomorphic modular forms by Bringmann, Guerzhoy and Kane [8, 9] and to differentials of the third kind by Bruinier, Funke, Imamoglu and Li [14]. In [5] the first author considered the Shintani lift of harmonic weak Maaß forms together with Schwagenscheidt.
Their results can be summarized as follows. We do not state the explicit Fourier expansion (since we do not need it in the course of this paper). The Fourier coefficients of the holomorphic part are given by the regularized traces of geodesic cycle integrals of the integral weight form.
Theorem 3.2
(Proposition 5.2 and Theorem 6.1 in [5]) Consider \(k\in \mathbb {Z}_{\ge 0}\) such that \((1)^{k+1}\Delta >0\). The regularized Shintani theta lift \(\Lambda ^{\textrm{Sh}}_{\Delta }(G,\tau )\) of a harmonic Maaß form \(G \in H_{2k+2}\) exists and defines a harmonic Maaß form in \(H_{3/2+k}\). If \(G \in M_{2k+2}^{!}\) is a weakly holomorphic modular form then \(\Lambda ^{\textrm{Sh}}_{\Delta }(G,\tau ) \in M_{3/2+k}\) is a holomorphic modular form, and if in addition \(a_{G}^{+}(0) = 0\) then \(\Lambda ^{\textrm{Sh}}_{\Delta }(G,\tau ) \in S_{3/2+k}\) is a cusp form.
3.2 The Millson theta lifting
In [30] Zagier considered traces of the values of the modular invariant j(z) at quadratic irrationalities. He showed that these traces are the Fourier coefficients of modular forms of halfintegral weight (both of weight 1/2 and 3/2).
Using the framework of [16] Bruinier and Funke [11] showed that such modularity results in weight 3/2 for generating series of traces of modular functions can be obtained via the KudlaMillson theta lift. Their work was generalized in various directions: to twisted traces in [2], to higher weight in [10] and [1]. In [3] and [4] a different theta lift, the socalled Millson theta lift, was considered which then fully recovered Zagier’s results.
We now briefly review the results of [4]. We remark that the Fourier coefficients of the holomorphic part are given by the traces of CM values of a suitable derivative of the input.
Theorem 3.3
(Theorem 1.1 in [4] and Proposition 5.5 in [5]) Let \(k \in \mathbb {Z}_{\ge 0}\) such that \((1)^{k}\Delta < 0\) and let \(F \in H_{2k}^{\textrm{mg}}\).

1.
Let \(k\ne 0\). The Millson theta lift \(\Lambda ^{\textrm{M}}_{\Delta }(F,\tau )\in H_{1/2k}^{\textrm{mg}}\) is a harmonic weak Maaß form of weight \({1/2k}\) for \(\Gamma _{0}(4)\) satisfying the Kohnen plus space condition. Further, if F is weakly holomorphic, then so is \(\Lambda ^{\textrm{M}}_{\Delta }(F,\tau )\).

2.
Let \(k=0\) and let F be such that the constant term of its nonholomorphic part vanishes. Then the Millson theta lift \(\Lambda ^{\textrm{M}}_{\Delta }(F,\tau )\in H^{\textrm{mg}}_{1/2}\) is a harmonic weak Maaß form of weight 1/2 for \(\Gamma _{0}(4)\) satisfying the Kohnen plus space condition.
Remark 3.4
The Millson and Shintani lifting are related via a differential equation satisfied by the two theta kernels:
4 Examples
In this section we give explicit examples for each of the cases occurring in Sect. 2.4. These are given as the Millson theta lifts of forms of weight \(2k\le 0\) and Shintani theta lifts of forms of weight \(2k\ge 2\). We denote the integral weights by \(2k\in 2\mathbb {Z}\) and the halfintegral weights by \(\ell \in \frac{1}{2}\mathbb {Z}{\setminus } \mathbb {Z}\).
Remark 4.1
To realize the cases associated to halfintegral weight it suffices to consider lifts of the scalarvalued examples that occur in [7]. Nonetheless, it would be interesting to extend the theory of theta liftings to symmetric power types along the lines of Funke and Millson’s work [17].
4.1 Weight \(\ell \le \frac{1}{2}\)
We first let \(\ell \le \frac{1}{2}\) and need to provide functions \(f\in H_{\ell =1/2k}^{\text {mg}}\) that satisfy \(\textrm{L}_\ell f=0\) and \(\textrm{L}_\ell f\ne 0\).
4.1.1 Weight \(\ell \le \frac{1}{2}\) and \(\textrm{L}_\ell f=0\)
First note that the Millson lift of the constant function, that gives an example for case I (a) in [7] (i.e., it satisfies \(\textrm{L}_0 f=0\) and \(\textrm{R}_0 f=0\)), is a weakly holomorphic modular form of weight \(\ell =1/2\) as can easily be deduced from Proposition 3.4.8 in [24].
If \(\ell \le 1/2\), we can take the Millson lift of a weakly holomorphic modular form of weight \(2k<0\) (compare case I (b) in [7]). We see from Theorem 3.3 that the Millson lift of a weakly holomorphic modular form \(F\in M^!_{2k}\) is again weakly holomorphic of weight \(\ell =1/2k\) if \(2k<0\).
4.1.2 Weight \(\ell \le \frac{1}{2}\) and \(\textrm{L}_\ell f\ne 0\)
If \(k=0\), the lift lies in the space of harmonic weak Maaß forms \(H_{1/2}^{\textrm{mg}}\). This already realizes the case of weight \(\ell =1/2\) when we require \(\textrm{L}_\ell f\ne 0\). For \(\ell \le 1/2\) we consider a function f satisfying \(\textrm{L}_{2k} f\ne 0\) and \(\textrm{R}_{2k}^{1+2k} f=0\) (corresponding to case I (c) in [7]). We lift the realization of [7]: We let \(F\in M_{2k}^!\setminus \{0\}\) and take \(G:={\mathcal {F}}_{2k}F\). Note that if
then, compare (1.5), we have
From Theorem 3.3 we easily deduce that the lift of G is in \(H^{\textrm{mg}}_{1/2k}\).
Remark 4.2
For the sake of completeness we explain the Millson lift of the remaining cases that Bringmann and Kudla consider. Their case IV (d) \(\textrm{L}_{2k} f\ne 0\) and \(\textrm{R}_{2k}^{1+2k} f\ne 0\) is realized by letting \(F\in M_{2k}^!\setminus \{0\}\) and taking \(G:=F+{\mathcal {F}}_{2k}F\). Lifting this we obviously obtain a combination of the previous two cases.
Moreover, we note that the lift of the Eisenstein series is again an Eisenstein series of weight \(1/2k\). This can be shown by standard arguments (for example using work of Crawford and Funke [15]).
Remark 4.3
We remark that the weight \(k=0\) (\(\ell =1/2\)) case can also be realized by the Siegel lift investigated in [13].
4.2 Weight \(\ell \ge \frac{3}{2}\)
4.2.1 Weight \(\ell \ge \frac{3}{2}\) and \(\textrm{L}_\ell f=0\)
Considering the Shintani lift of cusp forms of integral weight \(2k\ge 2\) we see that these give us examples of harmonic weak Maaß forms of weight \(\ell =3/2+k\ge 3/2\) satisfying \(\textrm{L}_\ell f=0\). Cusp forms of integral weight correspond to case III (a) in the classification of Bringmann and Kudla.
4.2.2 Weight \(\ell \ge \frac{3}{2}\) and \(\textrm{L}_\ell f\ne 0\)
We can realize the case of \(\textrm{L}_\ell f\ne 0\) by taking the Shintani lift of the weight 2 Eisenstein series (case III (b) in [7]) and sesquiharmonic Poincaré series (case III (c) in [7]).
To give an example for a function f with \(\textrm{L}_{3/2} f\ne 0\), we consider the lift of the weight 2 Eisenstein series
It was computed in [5]. We have
where
with \(H(0) = \frac{1}{12}\) and \(H(D) = 0\) if \(D \ne 0\) is not a discriminant, is Zagier’s weight 3/2 Eisenstein series (see [29]). Moreover, \(\beta _{3/2}(s)\) is defined as in the introduction.
The third case in [7] is characterized by \(\textrm{L}_{2k} F \ne 0\) and \(\textrm{L}_{2k}^{2k} F\ne 0\). An example is constructed via certain sesquiharmonic Poincaré series that are in fact harmonic (this relies on the vanishing of the dual space of cusp forms). We do not explicitly compute the lift of such series but note that according to Theorem 3.2 the lift is a harmonic weak Maaß form of moderate growth of weight \(\ell =3/2+k\). In analogy with the visualization presented in the introduction, this case yields
Remark 4.4
We remark that the weight 3/2 case can also be realized as the KudlaMillson lift of a harmonic weak Maaß form of weight 0 as in [11].
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References
Alfes, C.: Formulas for the coefficients of halfintegral weight harmonic Maaßforms. Math. Z. 277(3–4), 769–795 (2014)
Alfes, C., Ehlen, S.: Twisted traces of CM values of weak Maass forms. J. Number Theory 133(6), 1827–1845 (2013)
Alfes, C., Griffin, M., Ono, K., Rolen, L.: Weierstrass mock modular forms and elliptic curves. Res. Number Theory 1(1), 1–31 (2015)
AlfesNeumann, C., Schwagenscheidt, M.: On a theta lift related to the Shintani lift. Adv. Math. 328, 858–889 (2018)
AlfesNeumann, C., Schwagenscheidt, M.: Shintani theta lifts of harmonic Maass forms. Trans. Am. Math. Soc. 374(4), 2297–2339 (2019)
Borcherds, R.E.: Automorphic forms with singularities on Grassmannians. Invent. Math. 132(3), 491–562 (1998)
Bringmann, K., Kudla, S.: A classification of harmonic Maass forms. Math. Ann. 370(3–4), 1729–1758 (2018)
Bringmann, K., Guerzhoy, P., Kane, B.: Shintani lifts and fractional derivatives for harmonic weak Maass forms. Adv. Math. 255, 641–671 (2014)
Bringmann, K., Guerzhoy, P., Kane, B.: On cycle integrals of weakly holomorphic modular forms. Math. Proc. Camb. Philos. Soc. 158(3), 439–449 (2015)
Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125(1), 198–219 (2004)
Bruinier, J.H., Funke, J.: Traces of CM values of modular functions. J. Reine Angew. Math. 594, 1–33 (2006)
Bruinier, J.H., Ono, K.: Algebraic formulas for the coefficients of halfintegral weight harmonic weak Maass forms. Adv. Math. 246, 45–90 (2013)
Bruinier, J. H., Funke, J., Imamoḡlu, Ö.: Regularized theta liftings and periods of modular functions. J. Reine Angew. Math. 703, 43–93 (2015)
Bruinier, J. H., Funke, J., Imamoḡlu, Ö., Li, Y.: Modularity of generating series of winding numbers. Res. Math. Sci. 5(2), 23 (2018)
Crawford, J., Funke, J.: The Shimura–Shintani correspondence via singular theta lifts and currents (2021). arXiv:2112.11379 [math.NT]
Funke, J. P.: Rational quadratic divisors and automorphic forms. PhD Thesis, University of Maryland, College Park; roQuest LLC, Ann Arbor (1999)
Funke, J., Millson, J.: Spectacle cycles with coefficients and modular forms of halfintegral weight. In: Arithmetic Geometry and Automorphic Forms, vol. 19. Adv. Lect. Math. (ALM), pp. 91–154. International Press, Somerville (2011)
Harvey, J.A., Moore, G.: Algebras, BPS states, and strings. Nucl. Phys. B 463(2–3), 315–368 (1996)
Howe, R.: \(\Theta \)series and invariant theory. Automorphic forms, representations and Lfunctions. In: Proc. Symp. Pure Math. Am. Math. Soc., Corvallis 1977, vol. 33, 1, pp. 275–285 (1979)
Kohnen, W.: Modular forms of halfintegral weight on \(\Gamma _0\) (4). Math. Ann. 248(3), 249–266 (1980)
Kohnen, W.: Newforms of halfintegral weight. J. Reine Angew. Math. 333, 32–72 (1982)
Kudla, S.S., Rallis, S.: Degenerate principal series and invariant distributions. Isr. J. Math. 69(1), 25–45 (1990)
SchulzePillot, R.: Weak Maas forms and (g, K)modules. Ramanujan J. 26(3), 437–445 (2011)
Schwagenscheidt, M.: Regularized Theta Lifts of Harmonic Maass Forms. TU Darmstadt Dissertation (2018)
Shimura, G.: On modular forms of half integral weight. Ann. Math. 2(97), 440–481 (1973)
Shintani, T.: On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J. 58, 83–126 (1975)
Waldspurger, J.L.: Sur les coefficients de Fourier des formes modulaires de poids demientier. J. Math. Pures Appl. 9(60), 375–484 (1981)
Waldspurger, J.L.: Correspondance de Shimura. Progr. Math. 12, 357–369 (1981)
Zagier, D.: Nombres de classes et formes modulaires de poids 3/2. C. R. Acad. Sci. Paris Ser. AB 281(21), 883–886 (1975)
Zagier, D.: Traces of Singular Moduli. Motives, Polylogarithms and Hodge Theory, Part I (Irvine, CA, 1998), vol. 3, pp. 211–244. Int. Press Lect. Ser. International Press, Somerville (2002)
Acknowledgements
The authors thank Steve Kudla for sharing his ideas and Igor Burban, Jens Funke and Markus Schwagenscheidt for interesting discussions and comments. The second author thanks the Institut MittagLeffler, where parts of this work was conducted during the program on Moduli and Algebraic Cycles.
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C. AlfesNeumann was supported by the Daimler and Benz Foundation, the Klaus Tschira Boost Fund, and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—SFBTRR 358/1 2023—491392403. M. Raum was partially supported by Vetenskapsrådet Grant 201504139 and 201903551.
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AlfesNeumann, C., Raum, M. A classification of harmonic weak Maaß forms of halfintegral weight. Res. number theory 9, 48 (2023). https://doi.org/10.1007/s40993023004559
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DOI: https://doi.org/10.1007/s40993023004559