Plancherel formula for ${{\,\mathrm{\mathrm {GL}}\,}}_n(F){\backslash } {{\,\mathrm{\mathrm {GL}}\,}}_n(E)$ and applications to the Ichino–Ikeda and formal degree conjectures for unitary groups

Abstract

We establish an explicit Plancherel decomposition for ${{\,\mathrm{\mathrm {GL}}\,}}_n(F){\backslash } {{\,\mathrm{\mathrm {GL}}\,}}_n(E)$ where E/F is a quadratic extension of local fields of characteristic zero by making use of a local functional equation for Asai $\gamma$-factors. We also give two applications of this Plancherel formula: first to the global Ichino–Ikeda conjecture for unitary groups by completing a comparison between local relative characters that was left open by Zhang (J Am Math Soc 27:541–612, 2014) and secondly to the Hiraga–Ichino–Ikeda conjecture on formal degrees (J Am Math Soc 21(1):283–304, 2008) in the case of unitary groups.

A Proof of Proposition 2.131

First, we recall the following elementary lemma (see [22, Proposition]).

Lemma A.01

(Sobolev lemma for compact groups) Let H be a Hilbert space, K be a compact group and C(K, H) be the space of continuous functions from K to H. Consider C(K, H) as a representation of K through the right regular action and for $\rho \in \widehat{K}$ denote by $C(K,H)[\rho ]$ its $\rho$-isotypic component. Then, for every $\rho \in \widehat{K}$ and $\varphi \in C(K,H)[\rho ]$ we have

$$\begin{aligned} \displaystyle \sup _{k\in K}\Vert \varphi (k)\Vert \leqslant \dim (\rho )\left( \int _K \Vert \varphi (k)\Vert ^2 dk\right) ^{1/2}. \end{aligned}$$

Now we proceed to the proof of Proposition 2.131.

Let $f\in {{\,\mathrm{\mathcal {S}}\,}}(G(F))$. The function $\pi \in {{\,\mathrm{Temp}\,}}_{{{\,\mathrm{ind}\,}}}(G)\mapsto f_\pi \in {{\,\mathrm{\mathcal {C}}\,}}^w(G(F))$ is smooth by [12, Lemma 2.3.1(ii)] together with [3, §3] in the Archimedean case and [73, Proposition VII.1.3] in the p-adic case. Moreover, in the p-adic case the function $\pi \mapsto f_\pi$ is compactly supported by [73, Théorème VIII.1.2]. It remains to check that f satisfies condition (2.9.1) in the Archimedean case and condition (2.9.2) in the p-adic case. We will concentrate on the Archimedean case, the p-adic case being similar and actually easier to handle. By definition of the topology on ${{\,\mathrm{\mathcal {C}}\,}}^w(G(F))$, it suffices to establish the following: for every Levi subgroup $M\subset G$, $D\in {{\,\mathrm{Sym}\,}}^\bullet ({{\,\mathrm{\mathcal {A}}\,}}_{M,\mathbb {C}}^*)$, $u,v\in {{\,\mathrm{\mathcal {U}}\,}}({{\,\mathrm{\mathfrak {g}}\,}})$ and $k\geqslant 0$ we have

$$\begin{aligned} \displaystyle N(\pi )^k\left|D(\lambda \mapsto (R(u)L(v)f_{\pi _\lambda })(g))_{\lambda =0}\right|\ll \varXi ^G(g)\sigma (g)^{\deg (D)} \end{aligned}$$

(A.0.1)

for $\tau \in \varPi _2(M)$ and $g\in G(F)$ where $\deg (D)$ stands for the degree of D and we have set $\pi _\lambda =i_M^G(\tau _\lambda )$ and $\pi =\pi _0$ for $\lambda \in i{{\,\mathrm{\mathcal {A}}\,}}_M^*$.

As $R(u)L(v)f_\pi =(R(u)L(v)f)_\pi$, up to replacing f by R(u)L(v)f we only need to establish (A.0.1) when $u=v=1$. Assume proved the slightly weaker inequality (for any $D\in {{\,\mathrm{Sym}\,}}^\bullet ({{\,\mathrm{\mathcal {A}}\,}}_{M,\mathbb {C}}^*)$)

$$\begin{aligned} \displaystyle \left|D(\lambda \mapsto f_{\pi _\lambda }(g))_{\lambda =0}\right|\ll \varXi ^G(g)\sigma (g)^{\deg (D)},\;\;\; \tau \in \varPi _2(M), g\in G(F).\nonumber \\ \end{aligned}$$

(A.0.2)

Then, we will show that (A.0.1) holds for any $k\geqslant 0$ (and $u=v=1$). We do this by induction on $\deg (D)$. Let $z\in {{\,\mathrm{\mathcal {Z}}\,}}({{\,\mathrm{\mathfrak {g}}\,}})$ be such that (2.6.2) is satisfied. Then we may as well assume that $N(\pi )=\chi _{\pi }(z)$. Since $(z^kf)_\pi =\chi _\pi (z)^kf_\pi$ for every $\pi \in {{\,\mathrm{Temp}\,}}_{{{\,\mathrm{ind}\,}}}(G)$ the result in degree 0 just follows from replacing f by $z^k f$ in (A.0.2). In the general case the difference between

$$\begin{aligned} \displaystyle D(\lambda \mapsto (z^kf)_{\pi _\lambda }(g))_{\lambda =0}=D(\lambda \mapsto \chi _{\pi _\lambda }(z)^kf_{\pi _\lambda }(g))_{\lambda =0} \end{aligned}$$

(A.0.3)

and

$$\begin{aligned} \displaystyle \chi _\pi (z)^k D(\lambda \mapsto f_{\pi _\lambda }(g))_{\lambda =0} \end{aligned}$$

can be written as a finite sum

$$\begin{aligned} \displaystyle \sum _{i=1}^nD_i(\lambda \mapsto \chi _{\pi _\lambda }(z)^k)_{\lambda =0} D_i'(\lambda \mapsto f_{\pi _\lambda }(g))_{\lambda =0} \end{aligned}$$

where $D_i,D'_i\in {{\,\mathrm{Sym}\,}}^\bullet ({{\,\mathrm{\mathcal {A}}\,}}_{M,\mathbb {C}})$ for all $1\leqslant i\leqslant n$ are of degree strictly less than D. Since the terms $D_i(\lambda \mapsto \chi _{\pi _\lambda }(z)^k)_{\lambda =0}$ are all essentially bounded by a power of $N(\pi )$, the above sum can be controlled by the induction hypothesis whereas (A.0.3) is controlled by (A.0.2) applied to $z^k f$. This shows (A.0.1) assuming (A.0.2).

Fix $D\in {{\,\mathrm{Sym}\,}}^\bullet ({{\,\mathrm{\mathcal {A}}\,}}_{M,\mathbb {C}}^*)$, a parabolic subgroup P with Levi component M, a maximal compact subgroup K of G(F) in good position relative to M and set $K_M=K\cap M(F)$. Then, by restriction to K we can identify $\pi _\lambda =i_P^G(\tau _\lambda )$ with $\pi _K=i_{K_M}^K(\tau _{\mid K_M})$ as a K-representation for every $\tau \in \varPi _2(M)$ and $\lambda \in i{{\,\mathrm{\mathcal {A}}\,}}_M^*$. Choosing an invariant scalar product (., .) on $\tau$, we endow $\pi _K$ with the K-invariant scalar product

$$\begin{aligned} \displaystyle (e,e')=\int _K (e(k),e'(k))dk. \end{aligned}$$

Finally, choose for any $\rho \in \widehat{K}$ an orthonormal basis ${{\,\mathrm{\mathcal {B}}\,}}_{\tau }(\rho )$ of the $\rho$-isotypic component $\pi _K[\rho ]$ of $\pi _K$. Then, we have

$$\begin{aligned} \displaystyle f_{\pi _\lambda }(g)=\sum _{\rho \in \widehat{K}}\sum _{e\in {{\,\mathrm{\mathcal {B}}\,}}_{\tau }(\rho )} (\pi _\lambda (g)\pi _\lambda (f^\vee )e,e) \end{aligned}$$

(A.0.4)

for all $\tau \in \varPi _2(M)$, $\lambda \in i{{\,\mathrm{\mathcal {A}}\,}}_M^*$ and $g\in G(F)$. Assume now that we can show the existence of $r>0$ such that

$$\begin{aligned} \displaystyle \left|D\left( \lambda \mapsto (\pi _\lambda (g)\pi _\lambda (f^\vee )e,e)\right) _{\lambda =0}\right|\ll N(\rho )^r \varXi ^G(g)\sigma (g)^{\deg (D)} \end{aligned}$$

(A.0.5)

for all $\tau \in \varPi _2(M)$, $g\in G(F)$, $\rho \in \widehat{K}$ and $e\in {{\,\mathrm{\mathcal {B}}\,}}_{\tau }(\rho )$. Let $z_K\in {{\,\mathrm{\mathcal {Z}}\,}}(\mathfrak {k})$ be such that (2.6.2) is satisfied for $G(F)=K$. Then we may as well assume that $N(\rho )=\rho (z_K)$ for all $\rho \in \widehat{K}$ and hence up to replacing f by $L(z_K)^{k+r}f$ in (A.0.5) we obtain the same inequality with $N(\rho )^r$ replaced by $N(\rho )^{-k}$. Therefore, by (A.0.4), we would get for any $k>0$ an inequality

$$\begin{aligned}&\displaystyle \left|D(\lambda \mapsto f_{\pi _\lambda }(g))_{\lambda =0}\right|\ll \varXi ^G(g)\sigma (g)^{\deg (D)}\sum _{\rho \in \widehat{K}} \frac{\dim (\pi _K[\rho ])}{N(\rho )^k},\nonumber \\&\quad \;\;\; \tau \in \varPi _2(M),\; g\in G(F). \end{aligned}$$

As for k large enough the sum $\sum _{\rho \in \widehat{K}} \frac{\dim (\pi _K[\rho ])}{N(\rho )^k}$ converges and is bounded independently of $\pi$ (see e.g. [12, (2.2.2)]). For such a k the above inequality implies (A.0.2).

Thus, it only remains to establish (A.0.5). Actually, it suffices to show the existence of $r>0$ such that

$$\begin{aligned} \displaystyle \left|D\left( \lambda \mapsto (\pi _\lambda (g)e,e)\right) _{\lambda =0}\right|\ll N(\rho )^r \varXi ^G(g)\sigma (g)^{\deg (D)} \end{aligned}$$

(A.0.6)

for all $\tau \in \varPi _2(M)$, $g\in G(F)$, $\rho \in \widehat{K}$ and $e\in {{\,\mathrm{\mathcal {B}}\,}}_{\tau }(\rho )$. Indeed, if this is the case we would get

$$\begin{aligned} \displaystyle&\left|D\left( \lambda \mapsto (\pi _\lambda (g)\pi _\lambda (f^\vee )e,e)\right) _{\lambda =0}\right|\nonumber \\&\quad =\left|D\left( \lambda \mapsto \int _G f^\vee (\gamma ) (\pi _\lambda (g\gamma )e,e)d\gamma \right) _{\lambda =0}\right|\\&\quad =\left|\int _G f^\vee (\gamma ) D\left( \lambda \mapsto (\pi _\lambda (g\gamma )e,e)\right) _{\lambda =0}d\gamma \right|\nonumber \\&\quad \ll N(\rho )^r\int _G |f^\vee (\gamma )|\varXi ^G(g\gamma )\sigma (g\gamma )^{\deg (D)}d\gamma \\&\quad \ll N(\rho )^r \sigma (g)^{\deg (D)} \int _G \sup _{k\in K}|f^\vee (k\gamma )|\int _K\varXi ^G(gk\gamma )dk\sigma (\gamma )^{\deg (D)}d\gamma \\&\quad =N(\rho )^r \varXi ^G(g)\sigma (g)^{\deg (D)}\int _G \sup _{k\in K}|f^\vee (k\gamma )|\varXi ^G(\gamma )\sigma (\gamma )^{\deg (D)}d\gamma \end{aligned}$$

for all $\tau \in \varPi _2(M)$, $g\in G(F)$, $\rho \in \widehat{K}$ and $e\in {{\,\mathrm{\mathcal {B}}\,}}_{\tau }(\rho )$, where the differentiation under the integral sign is justified by the absolute convergence of the resulting expression and in the last line we have used the well-known ‘doubling formula’ $\int _K \varXi ^G(gk\gamma )dk=\varXi ^G(g)\varXi ^G(\gamma )$ (see [72, Proposition 16.(iii) p. 329]).

Fix a decomposition $\gamma =m_P(\gamma )u_P(\gamma )k_P(\gamma )$ for every $\gamma \in G(F)$ where $m_P(\gamma )\in M(F)$, $u_P(\gamma )\in U_P(F)$ and $k_P(\gamma )\in K$ and set $H_P(\gamma ):=H_M(m_P(\gamma ))$. Then to prove (A.0.6) we first note that

$$\begin{aligned} \displaystyle (\pi _\lambda (g)e,e)=\int _K \delta _P(m_P(kg))^{1/2} e^{\langle \lambda , H_P(kg)\rangle } (\tau (m_P(kg))e(k_P(kg)),e(k)) dk \end{aligned}$$

so that (once again differentiation under the integral is easily justified)

$$\begin{aligned}&\displaystyle D\left( \lambda \mapsto (\pi _\lambda (g)e,e)\right) _{\lambda =0}\nonumber \\&\quad =\int _K \delta _P(m_P(kg))^{1/2} D(H_P(kg)) (\tau (m_P(kg))e(k_P(kg)),e(k)) dk \end{aligned}$$

for all $\tau \in \varPi _2(M)$, $e\in \pi _K$ and $g\in G(F)$ where when we write $D(H_P(kg))$ we consider D as a polynomial function on ${{\,\mathrm{\mathcal {A}}\,}}_M$. Clearly $|D(H_P(kg))|\ll \sigma (g)^{\deg (D)}$ for all $g\in G(F)$ and $k\in K$ and therefore

$$\begin{aligned}&\displaystyle \left|D\left( \lambda \mapsto (\pi _\lambda (g)e,e)\right) _{\lambda =0}\right|\nonumber \\&\quad \ll \sigma (g)^{\deg (D)} \int _K \delta _P(m_P(kg))^{1/2} |(\tau (m_P(kg))e(k_P(kg)),e(k))|dk\nonumber \\ \end{aligned}$$

(A.0.7)

for all $\tau \in \varPi _2(M)$, $e\in \pi _K$ and $g\in G(F)$. By [22, Theorem 2], we have

$$\begin{aligned} \displaystyle |(\tau (m)v,v')|\ll \dim (\tau (K_M)v)^{1/2} \dim (\tau (K_M)v')^{1/2} \varXi ^M(m) \Vert v\Vert \Vert v' \Vert \end{aligned}$$

for all $\tau \in \varPi _2(M)$, all $v,v'\in \tau$ and all $m\in M(F)$. Note that $\dim (\tau (K_M)e(k))\leqslant \dim (\pi (K)e)$ for all $\tau \in \varPi _2(M)$ and $e\in \pi _K$ and that by a new application of [22, Theorem 2] $\dim (\pi (K)e)\leqslant \dim (\rho )^2$ if $e\in \pi _K[\rho ]$. Combining this with (A.0.7), we obtain

$$\begin{aligned}&\displaystyle \left|D\left( \lambda \mapsto (\pi _\lambda (g)e,e)\right) _{\lambda =0}\right|\nonumber \\&\quad \ll \sigma (g)^{\deg (D)} \dim (\rho )^2 \int _K \delta _P(m_P(kg))^{1/2} \varXi ^M(m_P(kg))\nonumber \\&\quad \Vert e(k_P(kg))\Vert \Vert e(k)\Vert dk \\&\quad 0 \leqslant \sigma (g)^{\deg (D)} \dim (\rho )^2 \sup _{k\in K} \Vert e(k)\Vert ^2 \int _K \delta _P(m_P(kg))^{1/2} \varXi ^M(m_P(kg)) dk \\&\leqslant \sigma (g)^{\deg (D)} \dim (\rho )^3 \varXi ^G(g)\Vert e\Vert \end{aligned}$$

for all $\tau \in \varPi _2(M)$, $\rho \in \widehat{K}$, $e\in \pi _K[\rho ]$ and $g\in G(F)$ where in the last inequality we have used [72, Proposition 16(iv) p. 329] and Lemma A.01. Since $\Vert e\Vert =1$ and there exists $n\geqslant 1$ such that $\dim (\rho )\leqslant N(\rho )^n$ ([75, p. 291]), this gives (A.0.6) and ends the proof of the Proposition 2.131. $\square$