Let \(E|F\) be a quadratic extension of local fields of characteristic zero, and let \(W \subset V\) be a pair of Hermitian spaces such that \(\dim W^\perp\) is odd and \(U(W^\perp)\) is quasisplit. Set \(G = U(V) \times U(W)\). In the Gan-Gross-Prasad conjecture, one attaches to these data a subgroup \(H\) of \(G\) together with a continuous unitary character \(\xi\) of \(H(F)\); when \(\dim W^\perp = 1\), we have \(\xi = \mathrm{triv}\) and \(H\) is just the diagonal image of \(U(W)\). One’s aim is to understand the multiplicity \[ m(\pi) = \dim \operatorname{Hom}_{H(F)}(\pi^\infty, \xi) \] for all irreducible tempered representation \(\pi\) of \(G(F)\), where \(\pi^\infty\) means the space of smooth vectors in \(\pi\). The multiplicity-one theorem guarantees that \(m(\pi) \leq 1\). Assume that \(G\) and \(H\) are both quasisplit. The local Gan-Gross-Prasad conjecture (Theorem 1) asserts that for all tempered L-parameter \(\varphi\) of \(G\), there exists exactly one representation \(\pi\) in the disjoint union of tempered L-packets \(\Pi^{G_\alpha}(\varphi)\) – known as Vogan’s packet – such that \(m(\pi) = 1\), where \(\alpha\) ranges over \(H^1(F, H)\) and \(H_\alpha\), \(G_\alpha\) are the corresponding pure inner forms of \(H\), \(G\). This is the main aim of this important and hard-core work. Note that the \(p\)-adic case has been settled by Waldspurger; the novelty here is the Archimedean case \(F = \mathbb{R}\). Despite what the subtitle may suggest, the proof given here applies to all \(F\), and the real case is usually the hardest.
Let us give a sketch of the strategy. The starting point is the same as Waldspurger’s, namely one considers the right regular representation \(R\) of \(G(F)\) on \(L^2(H(F) \backslash G(F), \xi)\) and studies the trace by integrating the kernel \[ J(f) := \mathrm{Trace}\, R(f) = \int_{H(F) \backslash G(F)} K(f, x) \,\mathrm{d}x. \] Theorem 3 asserts its absolute convergence when \(f\) lies in the space \(\mathcal{C}_{\mathrm{scusp}}(G(F))\) of Schwartz-Harish-Chandra functions that are strongly cuspidal, in the sense that \(\int_{U(F)} f(mu)\,\mathrm{d}u = 0\) for all proper parabolic \(P = MU\) and all \(m \in M(F)\). As in Arthur’s local trace formula, \(J(f)\) admits geometric and spectral expansions. The spectral expansion (Theorem 4) reads \[ J(f) = \int_{\mathcal{X}(G)} D(\pi) \hat{\theta}_f(\pi) m(\pi) \,\mathrm{d}\pi \] where \(\mathcal{X}(G)\) is a space of tempered virtual characters defined a la Arthur, \(D(\pi)\) are certain determinant factors, and \(\hat{\theta}_f(\pi)\) are made from weighted characters. The properties of weighted characters simplify in this case as \(f\) is strongly cuspidal. In turn, the proof of Theorem 4 is essentially based on Theorem 5 asserting that all elements of \(\operatorname{Hom}_{H(F)}(\pi^\infty, \xi)\) arise from averaging matrix coefficients along \(H(F)\). For \(p\)-adic \(F\), this is originally done by Sakellaridis-Venkatesh, reflecting the fact that the Gan-Gross-Prasad spaces are strongly tempered.
The geometric expansion (Theorem 6) reads \[ J(f) = \lim_{s \to 0+} \int_{\Gamma(G, H)} c_f(x) D^G(x)^{\frac{1}{2}} \Delta(x)^{s - \frac{1}{2}} \,\mathrm{d}x \] where \(D^G\) is the Weyl discriminant, \(\Delta\) is again some determinant factor, \(\Gamma(G, H)\) is a measure space of semisimple classes in \(G(F)\) containing \(\{1\}\) as an atom, and \(c_f(x)\) is extracted from the weighted orbital integrals of \(f\).
One can show that the aforementioned results combine to yield a geometric formula for multiplicities (Theorem 2): \[ m(\pi) = \lim_{s \to 0+} \int_{\Gamma(G, H)} c_\pi(x) D^G(x)^{\frac{1}{2}} \Delta(x)^{s - \frac{1}{2}} \,\mathrm{d}x \] where \(c_\pi(x)\) is essentially the leading term of the local expansion of \(\theta_\pi\) (a so-called “quasi-character”) around \(x\). The main Theorem 1 then follows by summing over \(\alpha \in H^1(F, H)\) and using the transfer of stable tempered characters across pure inner forms, which leaves only the term indexed by \(\{1\} \in \Gamma(G, H)\). The remaining coefficient \(c_{\varphi, 1}(1) := \sum_{\pi \in \Pi^G(\varphi)} c_\pi\) can then be identified with the number of generic representations within the tempered L-packet, which equals \(1\).
In this work, the spectral expansion (Theorem 4) is proved first, and the multiplicity formula (Theorem 2) and geometric expansion (Theorem 6) are proved together in the inductive argument. The reason is that one has to know \(J\) is supported on the elliptic locus in proving Theorem 6, which is shown by applying the other ingredients to smaller triples \((G, H, \xi)\).
The appendix contains the required estimates, as well as some backgrounds on functional analysis that are indispensable when dealing with real groups and various integrals.