Theta Lift Derivative & Automorphic Forms

• The derivative of a theta lift is an integral transform that measures the change in theta lifts, linking automorphic forms, harmonic weak Maass forms, and cycle cohomology.
• It employs regularization techniques and operator identities, such as the action of a lowering operator, to establish analytic continuation and adjointness properties.
• Arithmetic applications include explicit relations between period integrals and Hilbert–Eisenstein series, bridging automorphic representations with modular derivatives.

A theta lift is an integral transform (or correspondence) connecting automorphic forms and representations across dual reductive pairs, with both local and global variants. The study of its derivative captures deep aspects of cohomology, arithmetic, and representation theory. In particular, the derivative of a theta lift acts as a bridge between objects such as harmonic weak Maass forms, automorphic forms, and cycles on locally symmetric spaces. This article synthesizes the main constructions, regularization techniques, operator identities, and arithmetic applications of the derivative of a theta lift, organizing the discussion around both the real case (symmetric spaces for $\mathrm{SL}_N$) and the non-Archimedean theta correspondence.

1. Cohomological Theta Kernels and Classical Lifts

Let $V = \mathbb{Q}^N \oplus \mathbb{Q}^N$ with quadratic form $Q(v,w) = v \cdot w$ and $$X = \operatorname{GL}_N(\mathbb{R})^+ / \operatorname{SO}(N)$$, $$S = \operatorname{SL}_N(\mathbb{R})/\operatorname{SO}(N)$$. The Mathai–Quillen formalism produces Schwartz-valued differential forms:

• $$\varphi(z, v) \in \Omega^N(X) \otimes S(V_\mathbb{R})$$, closed and rapidly decaying,
• $$\alpha(z,v) \in \Omega^{N-1}(X) \otimes S(V_\mathbb{R})$$, its transgression.

These satisfy

$$g^*\varphi(z, v) = \varphi(z, \rho(g)^{-1}v), \quad d_X \alpha(z, \sqrt{y}v) = L_N\varphi(z, \sqrt{y}v),$$

where $L_N = -2i y^2 \partial \overline{\partial}_\tau$ is the lowering operator in the $\tau$-variable. A $\Gamma$-invariant finite adèle Schwartz function $\varphi \in S(V_{\mathbb{A}_f})$ allows construction of theta series

$$\Theta_\varphi(z, \tau, \varphi) = \sum_{v \in V} \varphi^0(z, \sqrt{y} v) \varphi(v) e(Q(v) \tau)$$

of weight $N$. Pushing forward over $u$ yields

$$E_\varphi(z_1, \tau, \varphi, s) = \int_0^\infty \Theta_\varphi(z, \tau, \varphi) u^{-2Ns} \, \frac{du}{u} \in \Omega^{N-1}(S)^\Gamma,$$

leading to the cohomological theta lift

$$E_\varphi(c, \tau, \varphi, s) = \int_c E_\varphi(z_1, \tau, \varphi, s),$$

for $(N-1)$-chains $c \subset S$, with holomorphic lift at $s=0$.

2. Regularization and Definition of the Regularized Theta Lift

For a harmonic weak Maass form $f \in H_{2-N}(\Gamma')$ (weight $2-N$, $\Gamma' \leq \mathrm{SL}_2(\mathbb{Z})$), the unregularized theta lift

$$\Phi_{\text{unreg}}(z_1, f, \varphi) = \int_{\Gamma'\backslash\mathbb{H}} E_\psi(z_1, \tau, \varphi) f(\tau) y^{-\rho} dx\,dy$$

(diverges at the cusps). Regularization involves truncating to a Siegel set $F_T \subset \Gamma'\backslash\mathbb{H}$. For $\rho = 0$ (critical shift), define

$$\Lambda(f; z_1, \varphi) = \operatorname{CT}_{s=0} \lim_{T \to \infty} \int_{F_T} E_\psi(z_1, \tau, \varphi) f(\tau) \, dx\,dy,$$

where $\operatorname{CT}_{s=0}$ takes the constant term of the Laurent expansion at $s=0$. Equivalently, subtracting the divergent term:

$$\Lambda(f; z_1, \varphi) = \lim_{T \to \infty} \left\{ \int_{F_T} E_\psi(z_1, \tau, \varphi) f(\tau) dx\,dy - \log T \sum_r w_r a_r^+(f, 0) e_\psi(z_1, \varphi_r) \right\},$$

with $w_r$ the cusp width, $\varphi_r$ a translated Schwartz function at the cusp $r$, and $a_r^+(f, 0)$, $e_\psi(z_1, \varphi_r)$ Fourier coefficients as above. For fixed $f$, $\Lambda(f; \cdot, \varphi)$ is smooth on $S_\Gamma$ away from finitely many special cycles $S_f(\varphi)$ (Branchereau, 28 Dec 2025).

3. Spectral Derivative and the Operator Identity

The lift $E_\varphi(z_1, \tau, \varphi, s)$ depends holomorphically on $s$, with a Taylor expansion

$$E_\varphi(z_1, \tau, \varphi, s) = E_\varphi(z_1, \tau, \varphi) + s\,E_\varphi'(z_1, \tau, \varphi) + O(s^2).$$

A key identity relates the derivative in $s$ to the action of the lowering operator $L_N$ and differential $d_S$. The transgression yields

$$d_S E_\psi(z_1, \tau, \varphi) = L_N E'_\varphi(z_1, \tau, \varphi),$$

with $E_\psi$ associated to the transgressed form $\psi(z,v) = \alpha(z,v) du/u$. This underpins the analytic properties and adjointness for the regularized lift and its derivative (Branchereau, 28 Dec 2025).

4. Adjointness Theorem for the Regularized Theta Lift Derivative

Given $c\in Z_{N-1}(S_\Gamma)$ a compactly supported cycle and the Petersson pairing $\langle F, g \rangle$ for $F \in M_N$, $g \in S_N$, the main adjointness statement reads:

$$2N(-1)^{N-1} \int_c \Lambda(f; z_1, \varphi) = \langle E_\varphi'(c, \tau, \varphi), g \rangle + \sum_r w_r \kappa_r(c, f, \varphi),$$

where the “error terms” $\kappa_r$ depend only on principal parts of $f$ at the cusps. When $f$ has vanishing constant terms at all cusps, the boundary terms vanish, yielding the pure adjointness relation:

$$\langle \partial_s \Phi(f; \cdot)|_{s=0}, \omega \rangle = \langle f, \Lambda(\omega) \rangle.$$

This identifies the derivative in $s$ of the cohomological theta lift with the adjoint of the regularized lift, highlighting a deep duality between automorphic objects and their associated cycles (see Proposition 4.14, Theorem 4.16 in (Branchereau, 28 Dec 2025)).

5. Arithmetic Applications: Periods and Hilbert–Eisenstein Series

For a totally real field $F/\mathbb{Q}$ of degree $N$, totally odd Hecke character $\chi$, and ideals $c \mid p$, the construction of Schwartz functions $\varphi_{\chi, c}$ and positive-norm-one cycles $c_\epsilon$ yields striking arithmetic identities. The cohomological theta lift

$E_\varphi(c_\epsilon, \tau, \varphi_{\chi, c}, s)$

recovers normalized Hilbert–Eisenstein series, whose $s$-derivative at $s=0$ gives a non-holomorphic modular form of weight $N$. The holomorphic projection has Fourier coefficients described by arithmetic divisor sums, and the constant term involves logarithms of algebraic numbers. The toric period of the regularized theta lift,

$$\int_{c_\epsilon} \Lambda(f; z_1, \varphi_{\chi, c}),$$

is explicitly related to both arithmetic data

$\log \alpha(f, \chi, c)$

and the Petersson pairing of the $s$-derivative of Hilbert–Eisenstein series with the shadow modular form $g(\tau)$:

$$\int_{c_\epsilon} \Lambda(f; z_1, \varphi_{\chi, c}) = \log \alpha(f, \chi, c) - \frac{1}{2N} \langle E'_{\chi, c}(\tau), g(\tau) \rangle - \frac{A_\chi}{2N} \left(a^+_f(0) + p \frac{\chi(c)}{N(c)} a^+_{f, 0}(0)\right),$$

where $A_\chi$ is an explicit transcendental constant and $\alpha(f, \chi, c)$ is algebraic (Branchereau, 28 Dec 2025).

6. Derivatives of Theta Lifts in Non-Archimedean Local Theta Correspondence

In the non-Archimedean setting, for a reductive dual pair $(G,H) = (G(W), H(V))$ over a local field $F$, the big theta lift $\Theta_{V,W}(\pi)$ (for an irreducible smooth representation $\pi$ of $G$) and the Bernstein–Zelevinsky derivative functor $D^{(k)}_\tau$ interact in a precise, functorial manner. For representations without critical exponent factors (lying in a large subcategory $R_V^\dagger(G)$), the main theorem asserts exactness:

$$D^{(k)}_{\tau^c} ( \Theta(\pi)) \simeq \Theta( D^{(k)}_{\tau} (\pi) ).$$

This compatibility is critical for understanding irreducibility properties of big theta lifts and for recursion on successive derivatives, tying representation-theoretic derivatives directly to the structure of the theta correspondence (Chen et al., 2023).

7. Summary Table: Core Constructions and Relations (Real Case)

Construction	Symbol / Equation	Context/Meaning
Cohomological theta kernel	$\varphi(z,v)$, $\alpha(z,v)$	Schwartz-valued forms on $X$
Theta series	$\Theta_\varphi(z, \tau, \varphi)$	Defines automorphic forms
Regularized theta lift	$\Lambda(f; z_1, \varphi)$	Regularized Maass–Maass lift
Spectral derivative	$\partial_s E_\varphi(z_1, \tau, \varphi, s)$	Appears in adjointness identities
Operator identity	$d_S E_\psi(z_1, \tau, \varphi) = L_N E'_\varphi$	Connects differential and lowering operator
Adjointness for periods	$2N(-1)^{N-1} \int_c \Lambda(f;z_1,\varphi) = \dots$	Relates regularized lift to $s$-derivative
Arithmetic toric periods	$$\int_{c_\epsilon} \Lambda(f; z_1, \varphi_{\chi,c})$$	Related to Hilbert–Eisenstein derivatives

This set of constructions and identities governs the theory of theta lift derivatives, linking analytic, geometric, and arithmetic aspects of automorphic forms, cohomology, and periods (Branchereau, 28 Dec 2025, Chen et al., 2023).