An explicit construction of non-tempered cusp forms on $O(1,8n+1)$

Abstract

We explicitly construct non-holomorphic cusp forms on the orthogonal group of signature $(1,8n+1)$ for an arbitrary natural number n as liftings from Maass cusp forms of level one. In our previous works [31] and [24] the fundamental tool to show the automorphy of the lifting was the converse theorem by Maass. In this paper, we use the Fourier expansion of the theta lifts by Borcherds [4] instead. We also study cuspidal representations generated by such cusp forms and show that they are irreducible and that all of their non-archimedean local components are non-tempered while the archimedean component is tempered, if the Maass cusp forms are Hecke eigenforms. Our non-archimedean local theory relates Sugano’s local theory [39] to non-tempered automorphic forms or representations of a general orthogonal group in a transparent manner.

Résumé

Cet article décrit une construction explicite de formes paraboliques non-holomorphes sur le groupe orthogonal de signature $(1, 8n + 1)$, pour un entier n arbitraire, à partir de relèvements de formes de Maass paraboliques de niveau un. Cela fait suite aux travaux [31] et [24], où l’outil fondamental pour démontrer l’automorphie du relèvement est le “converse theorem” de Maass. Cet article utilise les séries de Fourier associées aux relèvements theta de Borcherds [4]. Nous étudions aussi les représentations cuspidales engendrées par de telles formes paraboliques et montrons qu’elles sont irréductibles et que toutes leurs composantes locales non-archimédiennes sont non-tempérées, alors que la composante archimédienne est tempérée, lorsque la forme de Maass de départ est vecteur propre pour les opérateurs de Hecke. Notre théorie locale non-archimédenne relie de maniére plus transparente la théorie locale de Sugano [39] aux formes et aux représentations automorphes non-tempérées d’un groupe orthogonal général.

Appendix: cuspidal representations generated by Oda–Rallis-Schiffmann lifts

We have used Sugano’s non-archimedean local theory in [39, Section 7] to study the Hecke theory of the cusp forms given by our lifting and the cuspidal representations generated by them. His local theory is originally motivated by studying non-archimedean local aspect of the lifting theory of Oda [28] and Rallis-Schiffmann [35]. We can therefore expect that the results in Sects. 4 and 5 naturally hold also for the lifting by Oda and Rallis-Schiffmann. In this appendix, still restricting ourselves to “the case of even unimodular lattices”, we carry out the argument similar to Sects. 4 and 5 to deduce similar results for the case of Oda–Rallis-Schiffmann lifting.

1.1 Basic notation

Let $({{\mathbb {Z}}}^{8n},S)$ be an even unimodular lattice with a positive definite symmetric matrix S and put $Q_1:= \begin{pmatrix} &{} &{} 1\\ &{} -S &{} \\ 1 &{} &{} \end{pmatrix}$. We then let $Q_2:= \begin{pmatrix} &{} &{} 1\\ &{} Q_1 &{} \\ 1 &{} &{} \end{pmatrix}$ and let ${{\mathcal {G}}}=O(Q_2)$ (respectively ${{\mathcal {H}}}=O(Q_1)$) be the orthogonal group over ${{\mathbb {Q}}}$ defined by $Q_2$ (respectively $Q_1$). We introduce several algebraic subgroups of ${{\mathcal {G}}}$. We first introduce the maximal parabolic subgroup ${{\mathcal {P}}}$ with a Levi decomposition ${{\mathcal {P}}}={{\mathcal {N}}}_1\rtimes {{\mathcal {L}}}_1$, where ${{\mathcal {N}}}_1$ and ${{\mathcal {L}}}_1$ are defined by the groups of ${{\mathbb {Q}}}$-rational points as follows:

$$\begin{aligned} {{\mathcal {N}}}_{1}({{\mathbb {Q}}})= & {} \left\{ \left. {n_{Q_1}(x)}=\begin{pmatrix} 1 &{} -{}^txQ_1 &{} -\frac{1}{2}{}^txQ_1x\\ &{} 1_{8n+2} &{} x\\ &{} &{} 1 \end{pmatrix}~\right| ~x\in {{\mathbb {Q}}}^{8n+2}\right\} ,\\ {{\mathcal {L}}}_{1}({{\mathbb {Q}}})= & {} \left\{ \left. \begin{pmatrix} a &{} &{} \\ &{} h &{} \\ &{} &{} a^{-1} \end{pmatrix}~\right| ~a\in {{\mathbb {Q}}}^{\times },~h\in O(Q_1)({{\mathbb {Q}}})\right\} . \end{aligned}$$

For $w\in {{\mathbb {Q}}}^{8n}$ let ${n_0(w):= \begin{pmatrix} 1 &{} {}^twS &{} \frac{1}{2}{}^twSw\\ &{} 1_{8n} &{} w\\ &{} &{} 1 \end{pmatrix}}$ and ${n_{1}(w):= \begin{pmatrix} 1 &{} &{} \\ &{} n_0(w) &{} \\ &{} &{} 1 \end{pmatrix}}$. We then introduce the maximal unipotent subgroup ${{\mathcal {N}}}$ of ${{\mathcal {G}}}$ defined by its group of ${{\mathbb {Q}}}$-rational points

$$\begin{aligned} {{\mathcal {N}}}({{\mathbb {Q}}}):=\{n(x,w)\mid x\in {{\mathbb {Q}}}^{8n+2},~w\in {{\mathbb {Q}}}^{8n}\}, \end{aligned}$$

where $n(x,w):=n_{Q_1}(x)n_{1}(w)$.

Let $G_{\infty }$ be the real Lie group ${{\mathcal {G}}}({{\mathbb {R}}})$. To describe an Iwasawa decomposition of $G_{\infty }$ we introduce

$$\begin{aligned} A_{\infty }:=\left\{ \left. \begin{pmatrix} a_1 &{} &{} &{} &{} \\ &{} a_2 &{} &{} &{} \\ &{} &{} 1_{8n} &{} &{} \\ &{} &{} &{} a_2^{-1} &{} \\ &{} &{} &{} &{} a_1^{-1} \end{pmatrix}~\right| ~a_1,~a_2\in {{\mathbb {R}}}^{\times }_+\right\} \end{aligned}$$

and a maximal compact subgroup

$$\begin{aligned} K_{\infty }:=\{k\in G_{\infty }\mid {}^tkRk=R\} \end{aligned}$$

of $G_{\infty }$, where ${R= \begin{pmatrix} 1_2 &{} &{} \\ &{} S &{} \\ &{} &{} 1_2 \end{pmatrix}}$ is the majorant of $Q_2$. We then have an Iwasawa decomposition of $G_{\infty }$ as follows:

$$\begin{aligned} G_{\infty }={{\mathcal {N}}}({{\mathbb {R}}})A_{\infty }K_{\infty }. \end{aligned}$$

We next introduce the symmetric domain of type IV, which is identified with the quotient $G_{\infty }/K_{\infty }$. We follow [25, Sect. 1.4] to describe it. Let $B_{Q_1}$ be the bilinear form on $V\times V$ defined by $Q_1$ with $V={{\mathbb {R}}}^{8n+2}$ and let $(V,\tau )$ be the Euclidean Jordan algebra equipped with the trace form

$$\begin{aligned} \tau :V\times V\ni (x,y)\mapsto \tau (x,y)=2B_{Q_1}(x\circ y,e), \end{aligned}$$

where

$$\begin{aligned} x\circ y:=B_{Q_1}(x,e)y+B_{Q_1}(y,e)x-B_{Q_1}(x,y)e,\quad (x,y\in V) \end{aligned}$$

with ${}^te=(\frac{1}{\sqrt{2}},0,\cdots ,0,\frac{1}{\sqrt{2}})$. This Euclidean Jordan algebra has the determinant $\Delta$ given by

$$\begin{aligned} \Delta (v):=\frac{1}{2}B_{Q_1}(v,v)\quad (v\in V). \end{aligned}$$

Let us introduce the symmetric cone $\Omega :=\{v\in V\mid B_{Q_1}(v,v)>0,~B_{Q_1}(v,e)>0\}$ of V. Then the symmetric domain of type IV corresponding to $G_{\infty }$ is realized as ${{\mathcal {D}}}:=V+\sqrt{-1}\Omega$. The Lie group $G_{\infty }$ acts on ${{\mathcal {D}}}$ by the linear fractional transformation, for which we use the notation $g\cdot z$ for $(g,z)\in G_{\infty }\times {{\mathcal {D}}}$. Let $J(g,z)\in {{\mathbb {C}}}$ be the automorphy factor for $(g,z)\in G_{\infty }\times {{\mathcal {D}}}$. For the definition of $g\cdot z$ and J(g, z) see [10, Sect. 1]. We can identify $G_{\infty }/K_{\infty }$ with ${{\mathcal {D}}}$ by $G_{\infty }\ni g\mapsto g\cdot (\sqrt{-1}e)\in {{\mathcal {D}}}$.

1.2 Review on Oda–Rallis-Schiffmann lifting

By $S_{\kappa }(SL_2({{\mathbb {Z}}}))$ we denote the space of holomorphic cusp forms on the complex upper half plane ${{\mathfrak {h}}}$ of weight $\kappa$ with respect to $SL_2({{\mathbb {Z}}})$. To review the Oda–Rallis-Schiffmann lift from these holomorphic cusp forms we introduce the archimedean Whittaker function $W_{\lambda ,\kappa }$ on $G_{\infty }$ with $\lambda \in \Omega$ and a positive integer $\kappa$ by

$$\begin{aligned}&W_{\lambda ,\kappa }(n(x,w)ak)\\&\qquad :=J(k_{\infty },\sqrt{-1}e)^{-\kappa }\Delta (\mathrm{Im}({n_{1}(w)}a\cdot \sqrt{-1}e))^{\frac{\kappa }{2}}\\&\quad \qquad \exp (2\pi \sqrt{-1}\tau (\lambda ,x+\sqrt{-1}\mathrm{Im}({n_{1}(w)}a\cdot \sqrt{-1}e)) \end{aligned}$$

for $(x,w,a,k)\in {{\mathbb {R}}}^{8n+2}\times {{\mathbb {R}}}^{8n}\times A_{\infty }\times K_{\infty }$, where $\mathrm{Im}(z)$ denotes the imaginary part of $z\in {{\mathcal {D}}}$.

Let $f\in S_{\kappa -4n+2}(SL_2({{\mathbb {Z}}}))$ be given by the q-expansion $f(\tau )=\sum _{m\ge 1}c(m)q^m$ (thus $\kappa$ has to be even and $\kappa -4n+2\ge 12$). We put $|\lambda |_{Q_1}:=\sqrt{\frac{1}{2}{}^t\lambda Q_1\lambda }=\sqrt{\Delta (\lambda )}$ for $\lambda \in V$. We introduce a smooth function $F_f$ on $G_{\infty }$ by

$$\begin{aligned} F_f(g_{\infty })=\sum _{\lambda \in {{\mathbb {Z}}}^{8n+2}\cap \Omega }C_{\lambda }W_{\lambda ,\kappa }(g_{\infty }), \end{aligned}$$

where

$$\begin{aligned} C_{\lambda }=\sum _{d|d_{\lambda }}d^{\kappa -1}c(\frac{|\lambda |_{Q_1}^2}{d^2}) \end{aligned}$$

with the greatest common divisor $d_{\lambda }$ of the entries of $\lambda$. For the maximal lattice ${{\mathbb {Z}}}^{\oplus 2}\oplus {{\mathbb {Z}}}^{8n}\oplus {{\mathbb {Z}}}^{\oplus 2}$ with respect to $Q_2$ we introduce an arithmetic subgroup

$$\begin{aligned} \Gamma _S:=\{\gamma \in {{\mathcal {G}}}({{\mathbb {Q}}})\mid \gamma ({{\mathbb {Z}}}^{\oplus 2}\oplus {{\mathbb {Z}}}^{8n}\oplus {{\mathbb {Z}}}^{\oplus 2})={{\mathbb {Z}}}^{\oplus 2}\oplus {{\mathbb {Z}}}^{8n}\oplus {{\mathbb {Z}}}^{\oplus 2}\}. \end{aligned}$$

We now state the following theorem by Oda [28, Corollary to Theorem 5] and Rallis-Schiffmann [35, Theorem 5.1].

Theorem 6.1

(Oda, Rallis-Schiffmann) For $\kappa >8n+4$ the smooth function $F_f$ is a holomorphic cusp form of weight $\kappa$ with respect to $\Gamma _S$, lifted from the domain ${{\mathcal {D}}}$ to the group $G_{\infty }$.

1.3 Cuspidal representations generated by Oda–Rallis-Schiffmann lifts

1.4 (1) Adelization of $F_f$

To consider the cuspidal representation generated by $F_f$ we adelize $F_f$. We carry out it following the argument similar to Sect. 3.3.

Let $K_f:=\prod _{p<\infty }K_p$ with $K_p:=\{g\in {{\mathcal {G}}}({{\mathbb {Q}}}_p)\mid g{{\mathbb {Z}}}_p^{8n+4}={{\mathbb {Z}}}_p^{8n+4}\}$. We remark that the strong approximation theorem of ${{\mathcal {G}}}({{\mathbb {A}}})$ with respect to the maximal compact subgroup $K_f$ holds, from which we deduce that the set of $\Gamma _S$-cusps is in bijection with ${{\mathcal {H}}}({{\mathbb {Q}}})\backslash {{\mathcal {H}}}({{\mathbb {A}}})/{{\mathcal {H}}}({{\mathbb {R}}})U_f$ with $U_f:=\prod _{p<\infty }U_p~(U_p:=\{h\in {{\mathcal {H}}}({{\mathbb {Q}}}_p)\mid h{{\mathbb {Z}}}_p^{8n+2}={{\mathbb {Z}}}_p^{8n+2}\})$. This is nothing but Lemma  2.1 for the case of ${{\mathcal {G}}}=O(Q_2)$.

For $h=(h_p)_{p\le \infty }\in {{\mathcal {H}}}({{\mathbb {A}}}_f)$ we put $L_h:=(\prod _{p<\infty }h_p{{\mathbb {Z}}}_p^{8n+2}\times {{\mathbb {R}}}^{8n+2})\cap {{\mathbb {Q}}}^{8n+2}$ and write $h=au^{-1}$ with $(a,u)\in GL_{8n+2}({{\mathbb {Q}}})\times (\prod _{p<\infty }SL_{8n+2}({{\mathbb {Z}}}_p)\times SL_{8n+2}({{\mathbb {R}}}))$. For $\lambda \in L_h\setminus \{0\}$ we denote by $d_{\lambda }$ the greatest common divisor of the entries of $a^{-1}\lambda \in {{\mathbb {Z}}}^{8n+2}$, which is checked to be well-defined by the same argument as the proof of Lemma 3.2.

We introduce a function $A_{\lambda }$ indexed by $\lambda \in {{\mathbb {Q}}}^{8n+2}\setminus \{0\}$ as follows:

where $\Lambda$ denotes the standard additive character of ${{\mathbb {A}}}/{{\mathbb {Q}}}$. This $A_{\lambda }$ is verified to be well-defined function on ${{\mathcal {G}}}({{\mathbb {A}}}_f)$ similarly as in the proof of Lemma 3.2. With this $A_{\lambda }$ we adelize $F_f$ by

$$\begin{aligned} F_f(g)=\sum _{\lambda \in {{\mathbb {Q}}}^{8n+2}\setminus \{0\}}A_{\lambda }(g_f)W_{\lambda ,\kappa }(g_{\infty }) \end{aligned}$$

for $g=g_fg_{\infty }\in {{\mathcal {G}}}({{\mathbb {A}}})$ with $(g_f,g_{\infty })\in {{\mathcal {G}}}(A_f)\times G_{\infty }$. By the definition of the adelized $F_f$, $F_f$ is right $K_f$-invariant. By the standard argument in terms of the strong approximation theorem the left $\Gamma _S$-invariance of the non-adelic $F_f$ then implies the left ${{\mathcal {G}}}({{\mathbb {Q}}})$-invariance of the adelic $F_f$. The adelized $F_f$ is a cusp form on ${{\mathcal {G}}}({{\mathbb {A}}})$.

1.5 (2) Cuspidal representation generated by $F_f$

To determine explicitly the cuspidal representation of ${{\mathcal {G}}}({{\mathbb {A}}})$ generated by $F_f$ we first provide an explicit formula for Hecke eigenvalues of the adelized $F_f$. We can apply the non-archimedean local theory in Sect. 4.1 to our situation that $m=4n+2,~q=p,~F={{\mathbb {Q}}}_p,$ and $\partial =n_0=0$. The p-adic group ${{\mathcal {G}}}({{\mathbb {Q}}}_p)$ is viewed as $G_{4n+2}$ in the notation of Sect.  4.1. We need the lemma as follows:

Lemma 6.2

As a function on ${{\mathcal {G}}}({{\mathbb {Q}}}_p)(\simeq G_{4n+2})$, $A_{\lambda }(g)\in {{{\mathcal {W}}}}^{{{\mathcal {M}}}}_{\lambda }$, where we regard $g\in {{\mathcal {G}}}({{\mathbb {Q}}}_p)$ as an element in ${{\mathcal {G}}}({{\mathbb {A}}})$ in the usual manner.

We then state the theorem on the Hecke eigenvalues of $F_f$.

Theorem 6.3

Suppose that f is a Hecke eigenform and let $\lambda _p$ be the Hecke eigenvalue of f at $p<\infty$.

• (1) $F_f$ is a Hecke eigenform.

• (2) Let $\mu _i$ be the Hecke eigenvalue of the Hecke operator for $C_{4n+2}^{(i)}$ with $1\le i\le 4n+2$. We have

  $$\begin{aligned} \mu _i= {\left\{ \begin{array}{ll} p^{4n+1}(p^{-(\kappa -4n-1)}\lambda _p^2+p^{4n}+\cdots +p+p^{-1}+\cdots +p^{-4n})&{}(i=1)\\ |R_{4n+1}^{(i-1)}|\left( \mu _1-\displaystyle \frac{p^{i-1}-1}{p^i-1}f_{4n+2,1}\right) &{}(2\le i\le 4n+2) \end{array}\right. }. \end{aligned}$$

Proof

We give just an overview of the proof since it is quite similar to the case of the lifting from the Maass cusp forms. The only difference is the recurrence relation for the Fourier coefficients of the holomorphic cusp form f as follows:

Lemma 6.4

Let $f(\tau )=\sum _{n\ge 1}c(m)q^m\in S_{\kappa -(4n-2)}(SL_2({{\mathbb {Z}}}))$ be a Hecke eigenform. We have

$$\begin{aligned} c(p^2m)&=(\lambda _p^2-p^{\kappa -(4n-1)})c(m)- {{\left\{ \begin{array}{ll} p^{\kappa -(4n-1)}\lambda _pc(m/p)&{}(p|m)\\ 0&{}(p\not \mid m) \end{array}\right. }},\\ c(p^2m)&=(\lambda _p^2-2p^{\kappa -(4n-1)})c(m)-p^{2(\kappa -(4n-1))}c(m/p^2), \end{aligned}$$

where we assume $p^2|m$ for the second formula.

This follows from the well known recurrence relation of the Fourier coefficients (cf. [37, Chapitre VII, Sect. 5.3, Corollaire 2]).

With this lemma and [39, Theorem 7.4] for ${{\mathcal {W}}}_{\lambda }^{{{\mathcal {F}}}}$ on $G_{4n+2}$ (similar to Proposition 4.9), we get the explicit formula for $\mu _1$ by the proof similar to that of Proposition 4.13. The formula for $\mu _i$ with $i\ge 2$ is then an immediate consequence from Proposition  4.6. $\square$

1.6 Cuspidal representation generated by $F_f$

We now state the theorem quite similar to Theorem  5.6.

Theorem 6.5

Let $\pi _{F_f}$ be the cuspidal representation generated by $F_f$ and suppose that f is a Hecke eigenform.

• (1) The representation $\pi _{F_f}$ is irreducible and thus has the decomposition into the restricted tensor product $\otimes '_{v\le \infty }\pi _v$ of irreducible admissible representations $\pi _v$.

• (2) For $v=p<\infty$, $\pi _p$ is the spherical constituent of the unramified principal series representation of $G_p$ with the Satake parameter

  $$\begin{aligned} {\text {diag}}\left( \left( {\frac{\lambda '_p+\sqrt{{\lambda '}_p^2-4}}{2}}\right) ^2,p^{4n},\cdots ,p,1,1,p^{-1},\cdots ,p^{-4n},\left( {\frac{\lambda '_p+\sqrt{{\lambda '}_p^2-4}}{2}}\right) ^{-2}\right) , \end{aligned}$$

  where $\lambda '_p:=p^{-\frac{\kappa -(4n-1)}{2}}\lambda _p$.

• (3) For every finite prime $p<\infty$, $\pi _p$ is non-tempered while $\pi _{\infty }$ is tempered.

Proof

Also for this theorem we sketch the proof since it is similar to that of Theorem 5.6. Let ${{\mathfrak {g}}}_{\infty }$ be the Lie algebra of $G_{\infty }$. The right translations of $F_f$ by $G_{\infty }$ generate the anti-holomorphic discrete series representation $\pi _{\kappa }$ with minimal $K_{\infty }$-type given by

$$\begin{aligned} K_{\infty }\ni k\mapsto J(k,\sqrt{-1}e)^{-\kappa } \end{aligned}$$

as a $({{\mathfrak {g}}}_{\infty },K_{\infty })$-module, and it is irreducible. This is known as a classical fact due to Rallis-Schiffmann (for instance see [34, Sect. 6]). Since $F_f$ is a Hecke eigenform under the assumption we see the irreducibility of $\pi _F$ by [26, Theorem 3.1]. This is nothing but the first assertion. Due to Theorem  6.3 the rest of the assertions are settled by the proof similar to parts 2 and 3 of Theorem  5.6. For the third assertion we remark that the discrete series representations of semi-simple real Lie groups are a well-known class of tempered representations. $\square$

As we deduce Corollary 5.7 from Theorem  5.6 we have the following as an immediate consequence from Theorem 6.5.

Corollary 6.6

For any prime p the local p-factor $L_p(\pi _{F_f},{\text {St}},s)$ of the standard L-function for $\pi _{F_f}$ (or $F_f$) is written as

$$\begin{aligned} L_p(\pi _{F_f},{\text {St}},s)=L_p({\text {sym}}^2(f),s)\prod _{j=0}^{8n}\zeta _p(s+j-4n), \end{aligned}$$

where $L_p({\text {sym}}^2(f),s)$ is the p-factor of the symmetric square L-function for f.

Remark 6.7

This result is essentially obtained in [39, Theorem 8.1], which expresses the standard L-functions of the Oda–Rallis-Schiffmann lifts in the Jacobi form formulation in terms of L-functions of Jacobi forms. Sugano has remarked that $L_p({\text {sym}}^2(f),s)$ is a local factor of the L-function of some Jacobi form.