Plancherel formula for \(\mathrm{GL}_n(F)\setminus \mathrm{GL}_n(E)\) and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups. (English) Zbl 1482.22018
Let \(E|F\) be a quadratic extension of local fields of characteristic zero, and let \(Y_n\) be the homogeneous space \(\mathrm{GL}_n(F) \backslash \mathrm{GL}_n(E)\). The main goal of the long paper under review is to obtain an explicit decomposition of \(L^2(Y_n)\) as a unitary representation of \(\mathrm{GL}_n(E)\). Such a decomposition is called a Plancherel formula in harmonic analysis.
Fix an additive character \(\psi'\) of \(F\) to define Haar measures and an invariant measure on \(Y_n\). Denote by \(\mathrm{Temp}(\mathrm{GL}_n(E))\) the set of equivalence classes of tempered irreducible representations of \(\mathrm{GL}_n(E)\), and denote by \(\mathrm{Temp}(U(n))/\mathrm{st}\) the set of tempered \(L\)-parameters (i.e.tempered \(L\)-packets) of the quasisplit unitary group \(U(n)\) of rank \(n\) relative to \(E|F\). One endows the space \(\mathrm{Temp}(U(n))/\mathrm{st}\) with a measure that basically comes from unramified twists of parabolic inducing data. For each \(\sigma\) in \(\mathrm{Temp}(U(n))/\mathrm{st}\), let \(\mathrm{BC}_n(\sigma)\) denote its stable (resp.unstable) base change to \(\mathrm{GL}_n(E)\) if \(n\) is odd (resp.even).
In its abstract and weak form (Theorem 1), the decomposition reads \[ L^2(Y_n) \simeq \int^{\oplus}_{\mathrm{Temp}(U(n))/\mathrm{st}} \mathcal{H}_\sigma d\sigma \] where \(\sigma \mapsto \mathcal{H}_\sigma\) is a measurable field of unitary irreducible representations of \(\mathrm{GL}_n(E)\); in fact, \(\mathcal{H}_\sigma \simeq \mathrm{BC}_n(\sigma)\). This is in accordance with a general prediction of Sakellaridis and Venkatesh.
In order to obtain the explicit Plancherel formula, one follows the idea of Bernstein to consider the Schwartz spaces \(\mathcal{S}(\cdot)\), which have been defined in general, and seek an explicit spectral decomposition of the \(L^2\)-inner product \((\varphi_1, \varphi_2)_{Y_n}\) for \(\varphi_1, \varphi_2 \in \mathcal{S}(Y_n)\). The answer is given as follows. Fix an additive character \(\psi\) of \(E\) with \(\psi|_F = 1\), giving rise to the standard Whittaker datum \((N_n(E), \psi_n)\) for \(\mathrm{GL}_n(E)\). For all \(\pi \in \mathrm{Temp}(\mathrm{GL}_n(E))\), denote its Whittaker model as \(\mathcal{W}(\pi, \psi_n)\) and define an Hermitian form \((\cdot, \cdot)_{Y_n, \pi} \geq 0\) on \(\mathcal{S}(\mathrm{GL}_n(E))\) by \[ (f_1, f_2)_{Y_n, \pi} = \sum_{W} \beta(\pi(f_1) W) \overline{\beta(\pi(f_2)W)} \] where \(W\) ranges over an orthonormal basis of \(\mathcal{W}(\pi, \psi_n)\), with the scalar product induced from the Kirillov model (i.e. from \(L^2(N_n(E) \backslash P_n(E))\)), and \(\beta(W) = \int_{N_n(F) \backslash P_n(F)} W\), which turns out to converge. One shows that \((\cdot, \cdot)_{Y_n, \mathrm{BC}_n(\sigma)}\) factors through \(\mathcal{S}(Y_n)\). The explicit Plancherel formula (Theorem 2) asserts that \[ (\varphi_1, \varphi_2)_{Y_n} = \int_{\mathrm{Temp}(U(n))/\mathrm{st} } (\varphi_1, \varphi_2)_{Y_n, \mathrm{BC}_n(\sigma)} \cdot \frac{|\gamma^*(0, \sigma \mathrm{Ad}, \psi')|}{|\mathcal{S}_\sigma|} d\sigma. \] Here \(\mathcal{S}_\sigma\) is the usual component group in local Langlands correspondence, and \(\gamma^*(0, \cdots)\) is the value at \(s=0\) of \(\zeta_F(s)^{n_\sigma} \gamma(s, \cdots)\) where \(n_\sigma\) is the order of vanishing of \(\gamma(s, \cdots)\).
The overall strategy goes as follows. We may suppose that \(f_i \mapsto \varphi_i\) where \(f_i \in \mathcal{S}(\mathrm{GL}_n(E))\). Then \((\varphi_1, \varphi_2)_{Y_n} = \int_{\mathrm{GL}_n(F)} \overline{f_2} \star f_1^\vee\). We are thus led to the spectral decomposition of the form \(\int_{\mathrm{GL}_n(F)}\) on \(\mathcal{S}(\mathrm{GL}_n(E))\). The subsequent manipulations are arguably similar to the global ones in the factorization of Flicker–Rallis periods: a local unfolding leads to an integral \[ \int_{N_n(F) \backslash P_n(F)} dp \int_{N_n(F) \backslash \mathrm{GL}_n(F)} dh W_f(p, h) \] where \(W_f\) is a Whittaker function associated with \(f\), defined precisely in Section 2.14. Due to convergence issues, one cannot simply use the available spectral decomposition for \(W_f\). The workaround is to express the inner integral as the residue at \(s=0\) of some local Rankin-Selberg zeta integral for a test function \(\phi \in \mathcal{S}(F^n)\), perform the spectral decomposition in \(\operatorname{Re}(s) > 0\), and then use a local functional equation in terms of Asai \(\gamma\)-factors to reduce to a residue computabke via the available explicit formulas for Plancherel measures for \(\mathrm{GL}_n\).
The author gives three main applications. The first (Theorem 3) describes the Planchrel density for unitary groups in terms of adjoint \(\gamma\)-factors. The case \(F=\mathbb{R}\) is known by Harish-Chandra, whilst the special case of discrete series reduces to the formal degree conjecture of Hiraga-Ichino-Ikeda.
The second application (Theorem 4) establishes the matching of relative characters in the local unitary Gan-Gross-Prasad conjecture (codimension-one case). Compared to the earlier works, a remarkable feature is that the arguments here work uniformly for all \(F\). This confirms a conjecture of Wei Zhang.
The final application (Theorem 5) is that the Ichino-Ikeda conjecture in the local unitary Gan-Gross-Prasad case (codimension-one) holds when the representation \(\pi\) of \(G(\mathbb{A})\) is everywhere tempered and \(\mathrm{BC}(\pi)\) is supercuspidal at some place. These proofs are intertwined in a comparison of local Jacquet-Rallis trace formulas and their limits as \(\gamma \to 1\), and this is where the explicit Plancherel formula is applied.
Fix an additive character \(\psi'\) of \(F\) to define Haar measures and an invariant measure on \(Y_n\). Denote by \(\mathrm{Temp}(\mathrm{GL}_n(E))\) the set of equivalence classes of tempered irreducible representations of \(\mathrm{GL}_n(E)\), and denote by \(\mathrm{Temp}(U(n))/\mathrm{st}\) the set of tempered \(L\)-parameters (i.e.tempered \(L\)-packets) of the quasisplit unitary group \(U(n)\) of rank \(n\) relative to \(E|F\). One endows the space \(\mathrm{Temp}(U(n))/\mathrm{st}\) with a measure that basically comes from unramified twists of parabolic inducing data. For each \(\sigma\) in \(\mathrm{Temp}(U(n))/\mathrm{st}\), let \(\mathrm{BC}_n(\sigma)\) denote its stable (resp.unstable) base change to \(\mathrm{GL}_n(E)\) if \(n\) is odd (resp.even).
In its abstract and weak form (Theorem 1), the decomposition reads \[ L^2(Y_n) \simeq \int^{\oplus}_{\mathrm{Temp}(U(n))/\mathrm{st}} \mathcal{H}_\sigma d\sigma \] where \(\sigma \mapsto \mathcal{H}_\sigma\) is a measurable field of unitary irreducible representations of \(\mathrm{GL}_n(E)\); in fact, \(\mathcal{H}_\sigma \simeq \mathrm{BC}_n(\sigma)\). This is in accordance with a general prediction of Sakellaridis and Venkatesh.
In order to obtain the explicit Plancherel formula, one follows the idea of Bernstein to consider the Schwartz spaces \(\mathcal{S}(\cdot)\), which have been defined in general, and seek an explicit spectral decomposition of the \(L^2\)-inner product \((\varphi_1, \varphi_2)_{Y_n}\) for \(\varphi_1, \varphi_2 \in \mathcal{S}(Y_n)\). The answer is given as follows. Fix an additive character \(\psi\) of \(E\) with \(\psi|_F = 1\), giving rise to the standard Whittaker datum \((N_n(E), \psi_n)\) for \(\mathrm{GL}_n(E)\). For all \(\pi \in \mathrm{Temp}(\mathrm{GL}_n(E))\), denote its Whittaker model as \(\mathcal{W}(\pi, \psi_n)\) and define an Hermitian form \((\cdot, \cdot)_{Y_n, \pi} \geq 0\) on \(\mathcal{S}(\mathrm{GL}_n(E))\) by \[ (f_1, f_2)_{Y_n, \pi} = \sum_{W} \beta(\pi(f_1) W) \overline{\beta(\pi(f_2)W)} \] where \(W\) ranges over an orthonormal basis of \(\mathcal{W}(\pi, \psi_n)\), with the scalar product induced from the Kirillov model (i.e. from \(L^2(N_n(E) \backslash P_n(E))\)), and \(\beta(W) = \int_{N_n(F) \backslash P_n(F)} W\), which turns out to converge. One shows that \((\cdot, \cdot)_{Y_n, \mathrm{BC}_n(\sigma)}\) factors through \(\mathcal{S}(Y_n)\). The explicit Plancherel formula (Theorem 2) asserts that \[ (\varphi_1, \varphi_2)_{Y_n} = \int_{\mathrm{Temp}(U(n))/\mathrm{st} } (\varphi_1, \varphi_2)_{Y_n, \mathrm{BC}_n(\sigma)} \cdot \frac{|\gamma^*(0, \sigma \mathrm{Ad}, \psi')|}{|\mathcal{S}_\sigma|} d\sigma. \] Here \(\mathcal{S}_\sigma\) is the usual component group in local Langlands correspondence, and \(\gamma^*(0, \cdots)\) is the value at \(s=0\) of \(\zeta_F(s)^{n_\sigma} \gamma(s, \cdots)\) where \(n_\sigma\) is the order of vanishing of \(\gamma(s, \cdots)\).
The overall strategy goes as follows. We may suppose that \(f_i \mapsto \varphi_i\) where \(f_i \in \mathcal{S}(\mathrm{GL}_n(E))\). Then \((\varphi_1, \varphi_2)_{Y_n} = \int_{\mathrm{GL}_n(F)} \overline{f_2} \star f_1^\vee\). We are thus led to the spectral decomposition of the form \(\int_{\mathrm{GL}_n(F)}\) on \(\mathcal{S}(\mathrm{GL}_n(E))\). The subsequent manipulations are arguably similar to the global ones in the factorization of Flicker–Rallis periods: a local unfolding leads to an integral \[ \int_{N_n(F) \backslash P_n(F)} dp \int_{N_n(F) \backslash \mathrm{GL}_n(F)} dh W_f(p, h) \] where \(W_f\) is a Whittaker function associated with \(f\), defined precisely in Section 2.14. Due to convergence issues, one cannot simply use the available spectral decomposition for \(W_f\). The workaround is to express the inner integral as the residue at \(s=0\) of some local Rankin-Selberg zeta integral for a test function \(\phi \in \mathcal{S}(F^n)\), perform the spectral decomposition in \(\operatorname{Re}(s) > 0\), and then use a local functional equation in terms of Asai \(\gamma\)-factors to reduce to a residue computabke via the available explicit formulas for Plancherel measures for \(\mathrm{GL}_n\).
The author gives three main applications. The first (Theorem 3) describes the Planchrel density for unitary groups in terms of adjoint \(\gamma\)-factors. The case \(F=\mathbb{R}\) is known by Harish-Chandra, whilst the special case of discrete series reduces to the formal degree conjecture of Hiraga-Ichino-Ikeda.
The second application (Theorem 4) establishes the matching of relative characters in the local unitary Gan-Gross-Prasad conjecture (codimension-one case). Compared to the earlier works, a remarkable feature is that the arguments here work uniformly for all \(F\). This confirms a conjecture of Wei Zhang.
The final application (Theorem 5) is that the Ichino-Ikeda conjecture in the local unitary Gan-Gross-Prasad case (codimension-one) holds when the representation \(\pi\) of \(G(\mathbb{A})\) is everywhere tempered and \(\mathrm{BC}(\pi)\) is supercuspidal at some place. These proofs are intertwined in a comparison of local Jacquet-Rallis trace formulas and their limits as \(\gamma \to 1\), and this is where the explicit Plancherel formula is applied.
Reviewer: Wen-Wei Li (Beijing)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
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