Jacquet–Rallis Relative Trace Formula

• Jacquet–Rallis RTF is a framework that relates automorphic period integrals on unitary groups to special values of L-functions.
• It equates geometric expansions from weighted orbital integrals with spectral sums of automorphic representations, aiding trace formula comparisons.
• The approach underpins key conjectures like Gan–Gross–Prasad and establishes test function transfers through the fundamental lemma.

The Jacquet–Rallis relative trace formula (RTF) is a trace formula framework specifically designed to relate automorphic period integrals on unitary groups (and related linear groups) to special values of $L$-functions, with applications to the theory of automorphic forms, representation theory, and the arithmetic Langlands program. This formalism plays a central role in the paper of periods such as Bessel and Fourier–Jacobi periods, the comparison of trace formulas across different groups, and ultimately in the proof of deep conjectures such as Gan–Gross–Prasad for unitary groups.

1. Conceptual Framework and Generalization

At its core, the Jacquet–Rallis RTF expresses an identity between a geometric expansion (sum over weighted orbital integrals) and a spectral expansion (sum over periods or relative characters of automorphic representations). The framework originated from the paper of period integrals for pairs of unitary groups $\mathrm{U}_n$ and $\mathrm{U}_{n-1}$, where the period (now called the "Bessel period") is given by a certain integral of automorphic forms against a nontrivial character of a nonreductive subgroup.

The key generalization presented in (Liu, 2010) extends the formula to arbitrary pairs $\mathrm{U}_n \times \mathrm{U}_m$ for $0 \leq m \leq n$. The type of period—Bessel or Fourier–Jacobi—is determined by the parity of $n-m$: if $n-m$ is odd, the period is Bessel; if $n-m$ is even, it becomes Fourier–Jacobi. When $m=0$, the construction recovers the Flicker–Kloosterman period, and for $0 < m < n-1$, the formula yields a hybrid between these cases.

Let $V$ and $W$ be Hermitian spaces over an extension $k'/k$, with $\dim V = n$ and $\dim W = m$. $U(W)$ embeds as a subgroup of $U(V)$ (e.g., the stabilizer of an orthogonal complement of a subspace), and periods are then integrals over a nonreductive subgroup $H'$ of $U(V)\times U(W)$: $$B_{\psi}\left(P_T,P_o\right)=\int_{H'(k')\backslash H'(\mathbb{A}')}\,P_T(h')\,P_o(h')\,\overline{\psi}(h')\,dh',$$ where $P_T$ and $P_o$ are automorphic forms, and $\psi$ is a generic character (determining the Bessel or Fourier–Jacobi model), with $H' = U' \times U(W)$ and $U'$ a suitable unipotent subgroup.

2. Geometric Side: Orbital Integrals and Transfer

The geometric side of the Jacquet–Rallis RTF is built from sums over orbits that capture the "relative" nature of the formula, reflecting the non-invariance under the full group but invariance relative to a subgroup such as $U(W)$. The distribution is constructed via a kernel function (on the group or Lie algebra): $$K_f(h_1,h_2) = \sum_{\gamma\in U(V)(k')}\,f(h_1^{-1}\gamma h_2)$$ leading to a global distribution: $$J(f) = \int_{[H']\times[H']} K_f(E(h),K(h))\,\psi(h_1h_2^{-1})\,dh_1\,dh_2.$$ This is unfolded and, by use of the Poisson summation formula, decomposed into a sum over orbits on $U(V)$. A crucial innovation is a bijection,

$N:\,[S_n(k')_{\mathrm{reg}]/H(k')\longrightarrow \bigsqcup_{W\subset V}\,[U_{\mathrm{reg}}(k')]/H'(k'),$

linking regular orbits in the symmetric space $S_n$ (corresponding to the linear group, often after base change) with those in the unitary setting.

One of the central conjectures (see Conjecture 4.12 in (Liu, 2010)) is the "smooth matching" of test functions: For every smooth, compactly supported $F$ on $S_n(k')$, there exists a collection $\{f^B\}$ on the unitary side such that their weighted orbital integrals match up to a transfer factor $t(S)$: $$O(F,S)=t(S)\,O(f^{B},B),\quad t(S)=(-1)^{\operatorname{val}\prod_{i=1}^{r} t_i(S)}.$$ This transfer is bidirectional: given test data on one side, one can construct matching functions on the other with coincident (normalized) orbital integrals.

3. Spectral Side: Periods and Relative Characters

On the spectral side, the trace formula organizes the sum over automorphic representations in terms of their "distinguished" periods, with each period corresponding to a distinguished model (Bessel or Fourier–Jacobi). Generic period integrals are related to automorphic $L$-functions via the Gan–Gross–Prasad conjectures and their variants.

For Bessel periods, the integral detects representations whose $L$-packet contains a Vogan packet member with a Bessel model. For Fourier–Jacobi periods, the setting incorporates an extra theta (Weil) factor, such as

$$FJ_{\psi}(P_T,P_o;\phi) = \int_{H'(k')\backslash H'(\mathbb{A}')}\,P_T(h')\,P_o(h')\,\Theta_{\phi}(h')\,dh'$$

where $\Theta_\phi$ is a theta series attached to a Bruhat–Schwartz function on an appropriate vector space.

The trace formula then equates the geometric and spectral contributions, with the spectral side naturally allied to central $L$-values via the theory of distinguished representations and base change.

4. The Fundamental Lemma and Matching of Test Functions

A major technical component is the proof of the Jacquet–Rallis fundamental lemma (JRFL): an identity between "unit elements" (characteristic functions of hyperspecial maximal compact subgroups or lattices) in the spherical Hecke algebras, comparing their local (normalized) orbital integrals. For $U_n \times U_n$ in positive characteristic $p > \max(n,2)$, (Liu, 2010) establishes: $$O^{\mathrm{GL}}(1_{S_n(o')}) = t(S)\,O^{\mathrm{U}}(1_{U_n(o')}),$$ where $O^{\mathrm{GL}}$ and $O^{\mathrm{U}}$ denote local weighted orbital integrals for corresponding regular orbits, and $t(S)$ is the relevant transfer factor. The proof reduces to combinatorial lattice-counting, employing techniques from Yun’s work on the Lie algebra case and connecting to Flicker’s earlier studies.

The importance of the fundamental lemma is twofold:

• It certifies the viability of function transfer in the comparison of trace formulas.
• It guarantees the "stability" of the RTF: i.e., the correct matching of orbital integrals at almost all local places, foundational for deducing global period–$L$-value identities.

5. Lie Algebra (Infinitesimal) Variants and Regularization

Further developments generalize the entire framework to infinitesimal versions, replacing group-theoretic kernels and orbits by their Lie algebraic analogues ((Zydor, 2013) for unitary groups, (Zydor, 2013) for linear groups).

In these infinitesimal trace formulas:

• One works with Schwartz–Bruhat functions on the Lie algebras and employs Arthur-style truncated kernels to ensure absolute convergence.
• The geometric side becomes a sum over conjugacy classes in the Lie algebra, with regular semisimple classes contributing via relative orbital integrals, and relatively regular semisimple classes captured through explicit zeta function regularizations of orbital integrals.
• The spectral side is controlled by invariance of distributions under Fourier transforms defined on subspaces of the Lie algebra.
• The invariance and Haar measure dependence of distributions is carefully analyzed, yielding canonical (up to measure choices) constructions.

For relatively regular semisimple classes, formulas of the type

$$J_{\mathfrak{o}}(f) = \sum_{I \subset I_0} \zeta_I(f)(0)$$

appear, where $\zeta_I(f)(\lambda)$ are zeta functions integrals over tori, admitting meromorphic continuation and regularized by taking the constant term.

6. Applications to Periods, L-functions, and Arithmetic

The Jacquet–Rallis RTF forms the analytic foundation for various instances of the Gan–Gross–Prasad conjectures. By relating the nonvanishing of periods (e.g., Bessel or Fourier–Jacobi) to central critical values of Rankin–Selberg or Asai $L$-functions, the formula can be used to deduce arithmetic information about automorphic forms, such as distinction properties, functorial transfer, and base change.

The machinery has also inspired and informed the development of general simple relative trace formulas (e.g., (Getz et al., 2014)) that encompass a broader class of period integrals, confirm relative Weyl laws, and support relative endoscopy comparison frameworks crucial for a deeper understanding of automorphic periods and cohomological cycles on Shimura varieties (Getz et al., 2017).

7. Technical Innovations and Broader Significance

Technical innovations originating in the Jacquet–Rallis RTF framework include:

• The construction of matching/transfer for regular and relatively regular semisimple orbital integrals between linear and unitary settings.
• Regularization using Arthur’s truncation and the introduction of test function transfer with explicit transfer factors.
• The geometric re-interpretation of orbital integrals via moduli spaces and fibers (as pursued, for example, by using Hitchin fibrations and counting methods in the proof of relative fundamental lemmas).

By establishing a method for relating geometric (orbital integrals) and spectral (periods of automorphic forms) data, the Jacquet–Rallis approach has become a paradigmatic tool for period–$L$-value comparison, base change, and distinction problems across various contexts, including the context of cohomological cycles and arithmetic invariant theory.

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The Jacquet–Rallis relative trace formula synthesizes techniques from harmonic analysis, algebraic geometry, and representation theory, operating as a central method for relating automorphic periods on unitary groups and related linear groups to special values of $L$-functions—thus providing both deep theoretical insights and the technical machinery for ongoing advances in the theory of automorphic forms and their arithmetic applications.