2.1 The group $U(m,m)$

We fix notation for the unitary group $U(m,m)$ .

Let $A \rightarrow B$ be a finite locally free morphism of (commutative) rings, and suppose B is given an involution $b \mapsto \overline {b}$ (“conjugation”) over A. We are mostly interested in the case where $F/F^+$ is a CM extension of number fields (with $F^+$ the index $2$ totally real subfield) and $B/A = \mathcal {O}_F/\mathcal {O}_{F^+}$ for the corresponding rings of integers (also the local analogues) etc.

Fix an integer $m \geq 0$ . Write $1_m$ for the $m \times m$ identity matrix (sometimes we drop the subscript m), and let $H = U(m,m)$ be the unitary group

(2.1.1) $$ \begin{align} H = U(m,m) {:=}q \left \{ h \in \operatorname{\mathrm{Res}}_{B/A} \operatorname{\mathrm{GL}}_{2m} : {}^t \overline{h} \begin{pmatrix} 0 & 1_m \\ -1_m & 0 \end{pmatrix} h = \begin{pmatrix} 0 & 1_m \\ -1_m & 0 \end{pmatrix} \right \} \end{align} $$

where ${}^t \overline {h}$ denotes conjugate transpose (with H the trivial group if $m = 0$ , by convention). Equivalently, H consists of block matrices

(2.1.2) $$ \begin{align} \begin{pmatrix} a & b \\ c & d \end{pmatrix} \quad \text{satisfying} \quad {}^t \overline{a} c = {}^t \overline{c} a \quad \quad {}^t \overline{a} d - {}^t \overline{c} b = 1_m \quad \quad {}^t \overline{b} d = {}^t \overline{d} b \end{align} $$

with $a, b, c, d \in \mathrm {Res}_{B/A} M_{m \times m}$ . We refer to H as the group $U(m,m)$ (for signature reasons when $B/A$ is $\mathbb {C} / \mathbb {R}$ ).

Given an integer j with $0 \leq j \leq m$ , we consider the injection

(2.1.3) $$ \begin{align} \mu_j^m \colon U(j,j) \rightarrow U(m,m) \quad \quad \begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \begin{pmatrix} 1_{m-j} & 0 & 0 & 0 \\ 0 & a & 0 & b \\ 0 & 0 & 1_{m-j} & 0 \\ 0 & c & 0 & d \end{pmatrix}. \end{align} $$

Consider the subgroups

(2.1.4) $$ \begin{align} P & {:=}q \left \{ h = \begin{pmatrix} * & * \\ 0 & * \end{pmatrix} \in H \right \} \end{align} $$

(2.1.5) $$ \begin{align} M & {:=}q \left \{ m(a) = \begin{pmatrix} a & 0 \\ 0 & {}^t \overline{a}^{-1} \end{pmatrix} : a \in \operatorname{\mathrm{Res}}_{B/A} \operatorname{\mathrm{GL}}_m \right \} \end{align} $$

(2.1.6) $$ \begin{align} N & {:=}q \left \{ n(b) = \begin{pmatrix} 1_m & b \\ 0 & 1_m \end{pmatrix} : b \in \mathrm{Herm}_m \right \} \end{align} $$

of H. We have $P(R) = M(R) N(R)$ for all A-algebras R. We occasionally write $P_m, M_m, N_m$ to emphasize dependence on m.

Set

(2.1.7) $$ \begin{align} w_j = \begin{pmatrix} 1_{m - j} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1_j \\ 0 & 0 & 1_{m-j} & 0 \\ 0 & -1_j & 0 & 0 \end{pmatrix} \end{align} $$

for j with $0 \leq j \leq m$ . We also write $w = w_m$ when $j = m$ and m is understood.

Let $F_v$ be a finite étale algebra of degree $2$ over a local field $F^+_v$ . Consider $B / A = \mathcal {O}_{F_v} / \mathcal {O}_{F^+_v}$ for the respective rings of integers (with $\mathcal {O}_{F^+_v} {:=}q F^+_v$ and $\mathcal {O}_{F_v} {:=}q F_v$ if $F^+_v$ is Archimedean).

If $F_v/F^+_v = \mathbb {C}/\mathbb {R}$ , we consider the standard maximal compact subgroup

(2.1.8) $$ \begin{align} K_v^{\circ} {:=}q \left \{ \begin{pmatrix} a & b \\ - b & a \end{pmatrix} \in H(\mathbb{R}) : a {}^t \overline{a} + b {}^t \overline{b} = 1_m \text{ and } a {}^t \overline{b} = b {}^t \overline{a} \right \} \subseteq H(\mathbb{R}) \end{align} $$

We write $U(m) \subseteq \operatorname {\mathrm {GL}}_m(\mathbb {C})$ for (the real points of) the unitary group for the usual positive definite rank m complex Hermitian space (specified by the Gram matrix $1_m$ ). There is an isomorphism $K_{v}^{\circ } \rightarrow U(m) \times U(m)$ sending the displayed matrix to $(a + i b, a - i b) \in U(m) \times U(m)$ (see, e.g., [Reference Garcia and SankaranGS19, §2.5.1]).

If $F^+_v$ is non-Archimedean, we consider the standard open compact subgroup

(2.1.9) $$ \begin{align} K_v^{\circ} {:=}q H(\mathcal{O}_{F^+_v}) \subseteq H(F^+_v). \end{align} $$

If $F_v/F^+_v = \mathbb {C} / \mathbb {R}$ or if $F^+_v$ is non-Archimedean, we have $H(F^+_v) = P(F^+_v) K_v^{\circ }$ . If $F^+_v$ is non-Archimedean, and given any $n(b) \in N(F^+_v)$ with any decomposition

(2.1.10) $$ \begin{align} w^{-1} n(b) w = m(a) n(b') k \end{align} $$

with $m(a) \in M(F^+_v)$ and $n(b') \in N(F^+_v)$ and $k \in K_v^{\circ }$ , we may assume $\det a \in F^+_v$ (it a priori lies in $F_v$ ) after possibly changing k, as follows from [Reference ShimuraShi97, (13.4.1)] and the explanation at the end of [Reference ShimuraShi97, §3.6].

If $F/F^+$ is a CM extension of number fields and $B/A = \mathcal {O}_F/\mathcal {O}_{F^+}$ , we write

(2.1.11) $$ \begin{align} K^{\circ} = \prod_v K_v^{\circ} \quad \quad K_{\infty}^{\circ} = \prod_{v \mid \infty} K_v^{\circ} \quad \quad K_f^{\circ} = \prod_{v < \infty} K_v^{\circ} \end{align} $$

where the products run over places v of $F^+$ .

For places v of $F^+$ , we use the notation $F_v {:=}q \prod _{w \mid v} F_w$ where w runs over places of F, similarly $\mathcal {O}_{F_v} {:=}q \prod _{w \mid v} \mathcal {O}_{F_w}$ , as well as $F^+_{\infty } = \prod _{v \mid \infty } F^+_v$ and $F_{\infty } = \prod _{w \mid \infty } F_w$ , etc.

2.2 Adèlic and classical Eisenstein series

Characters are assumed continuous and unitary unless specified otherwise. Let $F_v$ be a degree $2$ étale algebra over a local field $F^+_v$ , and form the corresponding unitary group $H = U(m,m)$ as in Section 2.1. If $F^+_v$ is Archimedean, we assume in Section 2.2 that $F_v/F^+_v$ is $\mathbb {C}/\mathbb {R}$ .

Given a character $\chi _v \colon F_v^{\times } \rightarrow \mathbb {C}^{\times }$ and $s \in \mathbb {C}$ , we may form the local degenerate principal series

(2.2.1) $$ \begin{align} I(s, \chi_v) {:=}q \mathrm{Ind}_{P(F^+_v)}^{H(F^+_v)}(\chi_v | - |_{F_v}^{s + m/2}). \end{align} $$

This is an unnormalized induction, consisting of smooth and $K_v^{\circ }$ -finite functions $\Phi _v \colon H(F^+_v) \rightarrow \mathbb {C}$ satisfying

(2.2.2) $$ \begin{align} \Phi_v(m(a) n(b) h, s) = \chi_v(a) |\det a|_{F_v}^{s + m/2} \Phi_v(h, s) \end{align} $$

for all $m(a) \in M(F^+_v)$ and $n(b) \in N(F^+_v)$ and $h \in H(F^+_v)$ . Here we wrote $\chi _v(a) {:=}q \chi _v(\det a)$ for short. A section $\Phi _v(h,s)$ of $I(s, \chi _v)$ is standard if $\Phi (k,s)$ is independent of s for any fixed $k \in K_v^{\circ }$ . We say $\Phi _v$ is spherical if $\Phi _v(h k,s) = \Phi _v(h, s)$ for any $k \in K_v^{\circ }$ . If $F_v / F^+_v$ is non-Archimedean, then $I(s,\chi _v)$ contains nonzero spherical $\Phi _v$ if and only if $\chi _v$ is unramified. In this case, we write $\Phi _v^{\circ }$ for the unique spherical standard section satisfying $\Phi _v^{\circ }(1,s) = 1$ for all s, and call $\Phi ^{\circ }_v$ the normalized spherical section.

Next, suppose $F/F^+$ is a CM extension of number fields. We write $\mathbb {A}_F$ for the adèle ring of F and $\mathbb {A}$ for the adèle ring of $F^+$ . Given a character $\chi \colon F^\times \backslash \mathbb {A}_F^{\times } \rightarrow \mathbb {C}^{\times }$ and $s \in \mathbb {C}$ , we similarly form the global degenerate principal series

(2.2.3) $$ \begin{align} I(s, \chi) {:=}q \mathrm{Ind}_{P(\mathbb{A})}^{H(\mathbb{A})}(\chi | - |_F^{s + m/2}) \end{align} $$

which is an unnormalized induction, consisting of smooth and $K^{\circ }$ -finite functions $\Phi \colon H(\mathbb {A}) \rightarrow \mathbb {C}$ satisfying

(2.2.4) $$ \begin{align} \Phi(m(a) n(b) h, s) = \chi(a) |\det a|_F^{s + m/2} \Phi(h, s) \end{align} $$

for all $m(a) \in M(\mathbb {A})$ and $n(b) \in N(\mathbb {A})$ and $h \in H(\mathbb {A})$ . Given characters $\chi _f \colon \mathbb {A}_{F,f}^{\times } \rightarrow \mathbb {C}^{\times }$ and $\chi _{\infty } \colon \mathbb {A}_{F,\infty }^{\times } \rightarrow \mathbb {C}^{\times }$ , we similarly form $I(s, \chi _f)$ and $I(s, \chi _{\infty })$ . We also speak of spherical sections and the spherical standard section, as above. We sometimes write $I_m(s, \chi )$ etc. to indicate dependence on m.

Given a standard section $\Phi (h,s)$ of the global degenerate principal series $I(s, \chi )$ , we form the Siegel Eisenstein series

(2.2.5) $$ \begin{align} E(h, s, \Phi) = \sum_{\gamma \in P(F^+) \backslash H(F^+)} \Phi(\gamma h, s) \end{align} $$

which is absolutely convergent for $ (s)> m/2$ . We also form $E(h,\Phi ,s)$ when $\Phi $ is a finite meromorphic linear combination of standard sections by extending linearly.

Define another character $\check {\chi } \colon F^{\times } \backslash \mathbb {A}^{\times }_F \rightarrow \mathbb {C}^{\times }$ as $\check {\chi }(a) {:=}q \chi (\overline {a})^{-1}$ . There is a functional equation

(2.2.6) $$ \begin{align} E(h, -s, M(\chi, s) \Phi) = E(h, s, \Phi) \end{align} $$

where $M(\chi ,s) \colon I(s, \chi ) \rightarrow I(-s, \check {\chi })$ is the intertwining operator

(2.2.7) $$ \begin{align} (M(s, \chi) \Phi)(h) = \int_{N(\mathbb{A})} \Phi(w^{-1} n(b) h, s) ~ d n(b) \end{align} $$

for $ (s)> m/2$ (see, e.g., [Reference TanTan99]). Here the Haar measure on $N(\mathbb {A})$ should be the one which is self-dual with respect to any nontrivial additive character $\psi \colon F^+ \backslash \mathbb {A} \rightarrow \mathbb {C}^{\times }$ , and the pairing $b,b' \mapsto \psi (\operatorname {tr}(b b'))$ . We occasionally write $M_m(s, \chi )$ to emphasize the understood m (in $U(m,m)$ ).

Fix an identification of $F^+_v$ -algebras $F_v \cong \mathbb {C}$ for each Archimedean place v of $F^+$ . We consider classical Eisenstein series on the Hermitian upper half-space

(2.2.8) $$ \begin{align} \mathcal{H}_m & {:=}q \{ z \in M_{m, m}(F_{\infty}) : (2i)^{-1} (z - {}^t \overline{z})> 0 \}\end{align} $$

(2.2.9) $$ \begin{align} & = \{ z = x + i y : x,y \in \mathrm{Herm}_m(F^+_{\infty}) \text{ with } y> 0 \}, \end{align} $$

where the latter expression means that x and y are $m \times m$ Hermitian matrices with y positive definite (at every place $v \mid \infty $ of $F^+_v$ ). Given $z = x + i y \in \mathcal {H}_m$ , we write $h_z \in H(F^+_{\infty }) \subseteq H(\mathbb {A})$ for any element $h_z = n(x) m(a)$ where $a \in \mathrm {GL}_m(F_{\infty })$ satisfies $a {}^t \overline {a} = y$ . Note $h_z \cdot i 1_m = z$ .

We restrict to $\Phi = \Phi _{\infty } \otimes \Phi _f$ for standard sections $\Phi _{\infty } \in I(s, \chi _{\infty })$ and $\Phi _f \in I(s, \chi _f)$ . Fix an integer $n_v$ for each place $v \mid \infty $ of $F^+_v$ , and assume $\chi _v|_{F^{+ \times }_v} = \mathrm {sgn}(-)^{n_v}$ for every $v \mid \infty $ . We also let $k(\chi _v) \in \mathbb {Z}$ be the integer satisfying

(2.2.10) $$ \begin{align} \chi_{v}(z) = (z / |z|^{1/2}_{F_v} )^{k(\chi_v)} \quad \quad \text{where } z \in F_v, \end{align} $$

for each place $v \mid \infty $ of $F^+_v$ . For such v, we let $\Phi _v = \Phi _v^{(n_v)}$ be the unique standard section of $I(s, \chi _v)$ of scalar weight

(2.2.11) $$ \begin{align} (n_1, n_2) \quad \quad \text{where} \quad \quad n_1 = \frac{n_v + k(\chi)}{2} \quad \text{and} \quad n_2 = \frac{-n_v + k(\chi)}{2} \end{align} $$

such that $\Phi _{v}^{(n_v)}(1,s) = 1$ (as in [Reference Garcia and SankaranGS19, §3.2, §3.3]). The scalar weight condition means that $\Phi _{v}^{(n_v)}(h k,s) = \det (k_1)^{n_1} \det (k_2)^{n_2} \Phi _{v}^{(n_v)}(h,s)$ for all $h \in H(F^+_v)$ and $k \in K_{v}^{\circ }$ where

(2.2.12) $$ \begin{align} k = \begin{pmatrix} a & b \\ - b & a \end{pmatrix} \in K_{v}^{\circ} \quad \quad k_1 = a + i b \quad \quad k_2 = a - i b. \end{align} $$

Note that $\Phi _v^{(n_v)}$ does not depend on the choice of identification $F_v \cong \mathbb {C}$ .

If $y = a {}^t \overline {a}$ for some $a \in \operatorname {\mathrm {GL}}_m(F_v)$ , a computation (omitted) shows

(2.2.13) $$ \begin{align} \chi_v(a)^{-1} (\det y)^{- n_v/2} \Phi^{(n_v)}_v(w^{-1}n(b)m(a)) = (\det y)^{s - s_0} \det(-i y + b)^{-(s - s_0)} \det(iy + b)^{-(s - s_0) - n_v} \end{align} $$

for any $b \in \mathrm {Herm}_m(F^+_v)$ , where $s_0 = (n_v - m)/2$ (reduce to the case $a = 1_m$ and write $w^{-1} n(b) = n(-b(1_m + b^2)^{-1}) m(b + i 1_m)^{-1} k$ for $k \in K_v^{\circ }$ ). Equation (2.2.13) may be used to translate various statements from [Reference ShimuraShi82] to statements about Archimedean Whittaker functions, etc. (see [Reference ChenChe24b, Section 4.3] for more on this).

Remark 2.2.1. Given $g = x_g + i y_g \in M_{m,m}(\mathbb {C})$ with $x_g, y_g$ Hermitian and $x_g$ positive definite, we define $\log \det (g)$ by the “principal branch” (such that $g \mapsto \log \det g$ is holomorphic, and $\log \det g \in \mathbb {R}$ if $y_g = 0$ ) as in [Reference ShimuraShi82, (1.11)] and the surrounding discussion of loc. cit. If $y_g$ is positive definite and $x_g$ is only assumed Hermitian, we also take

(2.2.14) $$ \begin{align} \log \det g = \log \det (-i g) + m \log i \quad \quad \log \det \overline{g} = \log \det (i g) - m \log i \end{align} $$

where $\log i {:=}q \pi i/2$ (as in [Reference ShimuraShi82, (1.11)]). This convention is implicit in (2.2.13).

We take $\Phi _{\infty } = \otimes _{v \mid \infty } \Phi _v^{(n_v)}$ . We write $n = (n_v)_{v \mid \infty }$ for the collection of Archimedean weights (and will eventually focus on the case where all $n_v$ are equal to some fixed integer n). In the above situation, we write $E(h,s,\Phi )_n {:=}q E(h,s,\Phi )$ and consider an associated classical Eisenstein series

(2.2.15) $$ \begin{align} E(z, s, \Phi)_n {:=}q E(z, s, \Phi) {:=}q \chi_{\infty}(a)^{-1} \det(y)^{-n/2} E(h_z, s, \Phi)_n \end{align} $$

where $z = x + i y$ and $h_z = n(x) m(a)$ with $a {}^t \overline {a} = y$ as above, and where $\det (y)^{-n/2}$ stands for $\prod _{v \mid \infty } \det (y_v)^{-n_v/2}$ . This does not depend on the choice of $h_z$ , that is, $E(h_z k_{\infty }, s, \Phi )_n = E(h_z, s, \Phi )_n$ for any $k_{\infty } \in K_{\infty }^{\circ }$ .

When $F^+ = \mathbb {Q}$ and $s_0 {:=}q (n - m)/2$ (setting $n = n_{\infty }$ and $k(\chi ) = k(\chi _{\infty })$ ), a computation (omitted) gives the more classical form

(2.2.16) $$ \begin{align} E(z, s, \Phi)_n & = \sum_{\gamma \in P(\mathbb{Z}) \backslash U(m,m)(\mathbb{Z})} \frac{\det(y)^{s - s_0} \det(\gamma)^{(n + k(\chi))/2}}{\det (c z + d)^n |\det ( c z + d )|^{2(s - s_0)}} \Phi_f(\gamma, s) \end{align} $$

(2.2.17) $$ \begin{align} & = \sum_{\gamma \in P_1(\mathbb{Z}) \backslash SU(m,m)(\mathbb{Z})} \frac{\det(y)^{s - s_0}}{\det (c z + d)^n |\det ( c z + d )|^{2(s - s_0)}} \Phi_f(\gamma, s) \end{align} $$

where

(2.2.18) $$ \begin{align} \gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \end{align} $$

where $SU(m,m) \subseteq U(m,m)$ is the determinant $1$ subgroup, and $P_1 {:=}q SU(m,m) \cap P$ . We have $P(\mathbb {Q}) \backslash H(\mathbb {Q}) = P(\mathbb {Z}) \backslash H(\mathbb {Z}) = P_1(\mathbb {Q}) \backslash H_1(\mathbb {Q}) = P_1(\mathbb {Z}) \backslash H_1(\mathbb {Z})$ (e.g., [Reference IkedaIke08, Proposition 12.6]). When $m = 1$ , the exceptional isomorphism $\operatorname {\mathrm {SL}}_2 \rightarrow SU(1,1)$ (over $\operatorname {\mathrm {Spec}} \mathbb {Q}$ ) implies that the above expression is a classical Eisenstein series for $\operatorname {\mathrm {SL}}_2$ on the upper half-plane.

Our main theorem (Theorem 9.1.1) concerns Fourier coefficients of $E(z,s,\Phi )_n$ (normalized as in Section 6.1), but the variant

(2.2.19) $$ \begin{align} \tilde{E}(a, s, \Phi)_n {:=}q \chi(a)^{-1} |\det(a)|_F^{- n / 2} E(m(a), s, \Phi)_n \quad \text{for } a \in \operatorname{\mathrm{GL}}_m(\mathbb{A}_F) \end{align} $$

will be useful for studying Fourier coefficients of $E(z,s,\Phi )_n$ for singular T (see below). If $a \in \operatorname {\mathrm {GL}}_m(F_{\infty })$ is any element satisfying $a {}^t \overline {a} = y$ for $y \in \mathrm {Herm}_m(F^+_{\infty })_{>0}$ , we have

(2.2.20) $$ \begin{align} E(i y, s, \Phi)_n = \tilde{E}(a, s, \Phi)_n. \end{align} $$

2.3 Fourier expansion and local Whittaker functions

Take notation as in Section 2.2, for example, $F/F^+$ is a CM extension of number fields. Choose a nontrivial additive character $\psi \colon F^+ \backslash \mathbb {A} \rightarrow \mathbb {C}^\times $ . We have a Fourier expansion

(2.3.1) $$ \begin{align} E(h,s,\Phi) = \sum_{T \in \mathrm{Herm}_m(F^+)} E_T(h, s, \Phi) \end{align} $$

where

(2.3.2) $$ \begin{align} E_T(h, s, \Phi) = \int_{N(F^+) \backslash N(\mathbb{A})} E( n(b) h, s, \Phi) \psi(-\mathrm{tr}(Tb)) ~ d n(b) \end{align} $$

for $\mathrm {Re}(s)> m/2$ , and where $d n(b)$ is the Haar measure on $N(\mathbb {A})$ which is self-dual with respect to the pairing $(b, b') \mapsto \psi (\mathrm {tr}(b b'))$ . We refer to $E_T(h,s,\Phi )$ as the T-th Fourier term.

For any $a \in \operatorname {\mathrm {GL}}_m(F)$ , a change of variables gives

(2.3.3) $$ \begin{align} E_T(m(a) h, s, \Phi) = E_{{}^t \overline{a} T a}(h, s, \Phi). \end{align} $$

We also have

(2.3.4) $$ \begin{align} E_T(m(a) h, s, \Phi) = E_T(h, s, \Phi) \quad \text{for any} \quad \begin{pmatrix} 1_{m - m^{\flat}} & * \\ 0 & 1_{m^{\flat}} \end{pmatrix} \in \operatorname{\mathrm{GL}}_m(\mathbb{A}_F) \quad \text{if }\quad T = \begin{pmatrix} 0 & 0 \\ 0 & T^{\flat} \end{pmatrix} \end{align} $$

with the block matrix $T^{\flat } \in \mathrm {Herm}_{m^{\flat }}(F^+)$ having $\det T^{\flat } \neq 0$ (here $m^{\flat }$ is arbitrary) (follows from [Reference Garcia and SankaranGS19, Lemma 5.4, (5.56)]).

Allowing arbitrary T again, assume there is a factorization $\Phi = (\otimes _{v \mid \infty } \Phi _v) \otimes \Phi _f$ . For each $v \mid \infty $ , assume $\Phi _v = \Phi _v^{(n_v)}$ is the scalar weight standard section as in Section 2.2, for some $n_v \in \mathbb {Z}$ . Write $n = (n_v)_{v \mid \infty }$ for the resulting tuple of integers.

Consider $a = a_{\infty } a_f \in \operatorname {\mathrm {GL}}_m(\mathbb {A}_F)$ , with $a_{\infty } \in \operatorname {\mathrm {GL}}_m(F_{\infty })$ and $a_f \in \operatorname {\mathrm {GL}}_m(\mathbb {A}_{F,f})$ . Set $y = a_{\infty } {}^t \overline {a}_{\infty }$ (temporary). We then have T-th Fourier coefficients $E_T(y,s,\Phi )_n$ and $\tilde {E}_T(a, s, \Phi )_n$ characterized by the relations

(2.3.5) $$ \begin{align} E_T(y,s,\Phi)_n q^T & = \chi_{\infty}(a_{\infty})^{-1} \det(y)^{-n / 2} E_T(n(x) m(a_{\infty}), s, \Phi) \end{align} $$

(2.3.6) $$ \begin{align} \tilde{E}_T(a,s,\Phi)_n \psi_f(\mathrm{tr}(T b)) q^T & = \chi(a)^{-1} |\det a|_F^{-n / 2} E_T(n(x + b) m(a), s, \Phi) \end{align} $$

for any $x \in \mathrm {Herm}_m(F_{\infty })$ and $b \in \mathrm {Herm}_m(\mathbb {A}_f)$ , with $z {:=}q x + i y$ , and with $q^T {:=}q \psi _{\infty }(\mathrm {tr}(Tz))$ . These correspond to the classical Eisenstein series and its variants in (2.2.15) and (2.2.19).

When $\det T \neq 0$ and $\Phi = \otimes _v \Phi _v$ is factorizable over all places, we have a factorization

(2.3.7) $$ \begin{align} E_T(h, s, \Phi) = \prod_v W_{T,\psi_v}(h_v, s, \Phi_v) \end{align} $$

into local Whittaker functions defined below (2.3.8).

We switch to local notation: let $F_v$ be a degree $2$ étale algebra over a local field $F^+_v$ , with nontrivial involution $a \mapsto \overline {a}$ . We assume $F^+_v$ has characteristic $0$ (because Karel assumes this [Reference KarelKar79]). If $F^+_v$ is Archimedean, we also assume $F_v / F^+_v = \mathbb {C} / \mathbb {R}$ .

Let $\chi _v \colon F_v^{\times } \rightarrow \mathbb {C}^{\times }$ and $\psi _v \colon F^+_v \rightarrow \mathbb {C}^{\times }$ be characters with $\psi _v$ nontrivial, and suppose $\Phi _v \in I(s, \chi _v)$ is a standard section. Given $T \in \mathrm {Herm}_m(F^+_v)$ with $\det T \neq 0$ , there is a local Whittaker function defined by the absolutely convergent integral

(2.3.8) $$ \begin{align} W_{T,\psi_v}(h, s, \Phi_v) {:=}q \int_{N(F^+_v)} \Phi_v(w^{-1} n(b) h, s) \psi_v(- \mathrm{tr}(Tb)) ~dn(b) \end{align} $$

for $h \in H(F^+_v)$ and $s \in \mathbb {C}$ with $\mathrm {Re}(s)> m/2$ , where $d n(b)$ is the Haar measure which is self-dual with respect to the pairing $(b, b') \mapsto \psi _v(\mathrm {tr}(b b'))$ on $\mathrm {Herm}_m(F^+_v) \cong N(F^+_v)$ . For each fixed h, the function $W_{T,\psi _v}(h, s, \Phi _v)$ admits holomorphic continuation to $s \in \mathbb {C}$ [Reference KarelKar79, Corollary 3.6.1][Reference Kudla and SweetKS97][Reference IchinoIch04, §6]. Extending linearly defines $W_{T,\psi _v}(h, s, \Phi _v)$ whenever $\Phi _v$ is a finite meromorphic linear combination of standard sections. For any $a \in \operatorname {\mathrm {GL}}_m(F_v)$ , a change of variables shows

(2.3.9) $$ \begin{align} W_{T,\psi_v}(m(a) h, s, \Phi_v) = \check{\chi}_v(a) |\det a|_{F_v}^{-s + m / 2} W_{{}^t \overline{a} T a, \psi_v}(h, s, \Phi_v) \end{align} $$

for $\check {\chi }_v(a) {:=}q \chi _v(\overline {a})^{-1}$ as above. We use the shorthand $W_{T,\psi _v}(s, \Phi _v) {:=}q W_{T,\psi _v}(1, s, \Phi _v)$ .

If $F_v / F^+_v$ is $\mathbb {C} / \mathbb {R}$ , we write $W_{T,v}$ instead of $W_{T,\psi _v}$ when $\psi _v(x) = e^{2 \pi i x}$ is the standard additive character. If $F^+_v$ is non-Archimedean, we write $W_{T,v}$ instead of $W_{T,\psi _v}$ if $\psi _v$ is an understood unramified additive character (suppressed from notation).

Proof. Claim (1) follows from a general result of Karel [Reference KarelKar79, Corollary 3.6.1], which implies that $W_{T,\psi _v}(h, s, \Phi _v) \in \mathbb {C}[q_v^{-s}, q_v^s]$ , and that $W_{T,\psi _v}(h, s, \Phi _v)$ may be computed for all s as the integral over a sufficiently large open compact subgroup of $N(F^+_v)$ .

Next, given any $b \in N(F^+_v)$ , write $w^{-1} n(b) w = m(a) n(b') k$ for $m(a) \in M(F^+_v)$ and $n(b') \in N(F^+_v)$ and $k \in K^{\circ }_v$ , as in (2.1.10). As explained in loc. cit., we can moreover assume $\det a \in F^+_v$ . We then have $\Phi _v(w^{-1} n(b) h, s) = \Phi _v(m(a) n(b') k w^{-1} h, s) = \chi _v(\det a) |\det a|_{F_v}^{s + m/2} \Phi _v(k w^{-1} h, s)$ . Since $\det a \in F^+_v$ , we must have $|\det a|_{F_v}^s$ being a nonnegative power of $q_v^{-2s}$ . Substituting into (2.3.8) (and replacing the integral over $N(F^+_v)$ with an integral over a sufficiently large open compact subgroup) now proves claim (2). The claims about algebraicity and integrality in (2)(i) and (2)(ii) follow because $\psi _v$ and $\chi |_{F^{+ \times }_v}$ are both valued in roots of unity, under the given hypotheses. The claim in (2)(iii) follows because, in the above notation, we have $|\det a|_{F_v} = 1$ if and only if $b \in \mathrm {Herm}_m(\mathcal {O}_{F^+_v})$ by inspection, or see, e.g., the discussion at the end of [Reference ShimuraShi97, §3.6] or [Reference ShimuraShi97, §13.5]. The volume $\operatorname {vol}(\mathrm {Herm}_m(\mathcal {O}_{F^+_v}))$ should be taken with respect to the self-dual Haar measure associated to $\psi _v$ , as above.

For (3), a similar computation gives $\Psi _v(w^{-1} n(b) h, s) = \chi ^{\prime }_v(\det a) |\det a|^{s + m / 2}_{F_v} \Psi _v(k w^{-1} h, s)$ , for the same $a \in \operatorname {\mathrm {GL}}_m(F_v)$ as above. If $v_0$ is the valuation on $F^+_v$ normalized with $v_0(\varpi _0) = 1$ , we thus have

(2.3.10) $$ \begin{align} \Phi_v(w^{-1} n(b) h, s) & = \chi_v(\det a) (q_v^{-2s})^{v_0(\det a)} \cdot q_v^{-m \cdot v_0(\det a)} \Phi_v(k w^{-1} h, s) \end{align} $$

(2.3.11) $$ \begin{align} \Psi_v(w^{-1} n(b) h, s) & = \chi_v(\det a) (\xi_v(\varpi_0) q_v^{-2s})^{v_0(\det a)} \cdot q_v^{-m \cdot v_0(\det a)} \Psi_v(k w^{-1} h, s). \end{align} $$

Claim (3) of the lemma now follows upon substitution into (2.3.8) (and again replacing the integral over $N(F^+_v)$ with an integral over a sufficiently large open compact subgroup).

In the case where $F^+_v$ is non-Archimedean, consider the case where $\chi _v$ is unramified and $\chi _v|_{F^{+ \times }_v} = \eta _v^n$ for some integer n, where $\eta _v \colon F^{+ \times }_v \rightarrow \{ \pm 1\}$ is the quadratic character associated to $F_v / F^+_v$ . Consider the normalized spherical standard section $\Phi ^{\circ }_v \in I(s, \chi _v)$ . We temporarily write $W_{T,\psi _v}(h, s, \Phi _v^{\circ })_n$ for the associated local Whittaker function, emphasizing the possible dependence on n. By Lemma 2.3.1(3), the implicit $\chi _v$ -dependence of $W_{T,\psi _v}(h, s, \Phi _v^{\circ })_n$ is only on the restriction $\chi _v|_{F^{+ \times }_v}$ . If $F_v / F^+_v$ is not inert, then $W_{T,\psi _v}(h, s, \Phi _v^{\circ })_n$ does not depend on n (note n must be even if $F_v / F^+_v$ is ramified). If $F_v / F^+_v$ is inert, then $W_{T,\psi _v}(h, s, \Phi _v^{\circ })_n$ depends only on the parity of n. The $\mathbb {C}$ -algebra endomorphism of $\mathbb {C}[q_v^{-2s}]$ sending $q_v^{-2s} \mapsto -q_v^{-2s}$ swaps $W_{T,\psi _v}(h, s, \Phi _v^{\circ })_n$ and $W_{T,\psi _v}(h, s, \Phi _v^{\circ })_{n+1}$ , by Lemma 2.3.1(3).

Remark 2.3.2 (Change of additive character)

We sometimes fix convenient choices of additive character $\psi $ or $\psi _v$ . This recovers the general case, as follows. Given $\psi $ , we write $E_{T, \psi }$ as needed to emphasize dependence on the choice of additive character, and similarly for normalized variants like $\tilde {E}_{T,\psi }$ , etc.

Given $u \in (F^+)^{\times }$ or $u \in (F^+_v)^{\times }$ , consider $\psi '(x) {:=}q \psi (u \cdot x)$ and $\psi _{v}'(x) {:=}q \psi _v(u \cdot x)$ . We have

(2.3.12) $$ \begin{align} E_{u T,\psi}(h,s,\Phi) &= E_{T, \psi'}(h,s,\Phi) && E_{u T,\psi}(y,s,\Phi)_n = E_{T, \psi'}(y,s,\Phi)_n \end{align} $$

(2.3.13) $$ \begin{align}\tilde{E}_{u T,\psi}(a,s,\Phi)_n &= \tilde{E}_{T, \psi'}(a,s,\Phi)_n && |u|_{F^+_v}^{m^2/2} W_{u T,\psi_v}(h,s,\Phi_v) = W_{T, \psi_{v}'}(h,s,\Phi_v)\end{align} $$

with notation as above. The additional factor $|u|^{m^2/2}_{F^+_v}$ arises from the change of Haar measure on $N(F^+_v)$ , from the requirements of self-duality with respect to $\psi _v$ or $\psi ^{\prime }_v$ .

2.4 Singular Fourier coefficients

Retain notation from Section 2.3 (switching back to global notation). The Fourier terms $E_{T}(h,s,\Phi )$ for singular $T \in \mathrm {Herm}_m(F^+)$ are known to be closely related with Fourier terms of Eisenstein series on smaller groups (e.g., [Reference Garcia and SankaranGS19, §5.2]). We focus on the case where $\operatorname {\mathrm {rank}} T = m - 1$ (assume this throughout Section 2.4). On account of (2.3.3), it will be enough to describe the case where T is block diagonal of the form

(2.4.1) $$ \begin{align} T = \begin{pmatrix} 0 & 0 \\ 0 & T^{\flat} \end{pmatrix} \end{align} $$

with $\det T^{\flat } \neq 0$ .

Assume $m \geq 1$ , and fix an integer $n \in \mathbb {Z}$ . Let $\chi \colon F^\times \backslash \mathbb {A}_F^{\times } \rightarrow \mathbb {C}^{\times }$ be a character satisfying $\chi |_{\mathbb {A}^{\times }} = \eta ^n$ , where $\eta $ is the quadratic character associated with $F / F^+$ . Note $\check {\chi } = \chi $ in this case.

Take a factorizable standard section $\Phi = \otimes _v \Phi _v \in I(s, \chi )$ , and assume $\Phi _v = \Phi _v^{(n)}$ is the normalized scalar weight standard section (Section 2.2) for every Archimedean place v, with n the fixed integer from above (same for every $v \mid \infty $ ).

Take T as in (2.4.1). Given $a \in \operatorname {\mathrm {GL}}_m(\mathbb {A}_F)$ , we study the Fourier coefficient $\tilde {E}_T(a, s, \Phi )_n$ . By the Iwasawa decomposition, every $a \in \operatorname {\mathrm {GL}}_m(\mathbb {A}_F)$ admits a decomposition

(2.4.2) $$ \begin{align} a = \begin{pmatrix} 1_1 & * \\ 0 & 1_{m - 1} \end{pmatrix} \begin{pmatrix} a^{\#} & 0 \\ 0 & a^{\flat} \end{pmatrix} k \end{align} $$

with $a^{\#} \in \operatorname {\mathrm {GL}}_1(\mathbb {A}_F)$ , with $a^{\flat } \in \operatorname {\mathrm {GL}}_{m - 1}(\mathbb {A}_F)$ , and with $k \in \prod _{v \mid \infty } U(m) \times \prod _{v < \infty } \operatorname {\mathrm {GL}}_m(\mathcal {O}_{F_v})$ . We will be eventually interested in the case when $\Phi _f$ is spherical, which implies $\tilde {E}_T(a k, s, \Phi )_n = \tilde {E}_T(a, s, \Phi )$ for any $k \in \prod _{v \mid \infty } U(m) \times \prod _{v < \infty } \operatorname {\mathrm {GL}}_m(\mathcal {O}_{F_v})$ (also using the fact that $\Phi _v$ is a scalar weight standard section for each $v \mid \infty $ ). In light of the invariance property in (2.3.4), it is thus harmless to restrict to the case of block diagonal $a = \mathrm {diag}(a^{\#}, a^{\flat })$ . Assume this for the rest of Section 2.4 (but we do not assume $\Phi _f$ is spherical for now).

Set $m^{\flat } {:=}q m - 1$ and $s_0 {:=}q (n - m)/2$ . Consider the operators

(2.4.3)

(with $\mu ^m_{m^{\flat }} \colon U(m^{\flat }, m^{\flat }) \rightarrow U(m,m)$ as in Section 2.1), and where

(2.4.4)

for $\mathrm {Re}(s)> m/2$ (with meromorphic continuation to $s \in \mathbb {C}$ ). A calculation shows

(2.4.5) $$ \begin{align} M_{m^{\flat}}(s - 1/2, \chi) \circ U^m_{m^{\flat}}(s, \chi) = \mu_{m^{\flat}}^{m *}(-s, \chi) \circ M_m(s, \chi); \end{align} $$

compare [Reference Garcia and SankaranGS19, Lemma 5.5(iii)] (this will not be used until Section 6.2).

Lemma 2.4.1. We have

(2.4.6) $$ \begin{align} \tilde{E}_{T}(a, s, \Phi)_n & = |\det a^{\#}|_F^{s - s_0} \tilde{E}_{T^{\flat}}(a^{\flat}, s + 1/2, \mu_{m^{\flat}}^{m*}(s, \chi) \Phi)_n \\ & \hphantom{=} + |\det a^{\#}|_F^{-s - s_0} \tilde{E}_{T^{\flat}}(a^{\flat}, s - 1/2, U_{m^{\flat}}^m(s, \chi) \Phi)_n \notag. \end{align} $$

Proof. We claim that arguing as in the proof of [Reference Kudla and RallisKR88, Lemma 2.4] (see also [Reference Garcia and SankaranGS19, Lemma 5.4] and [Reference He, Shi and YangHSY21, Theorem 2.2]) gives

(2.4.7) $$ \begin{align} E_{T}(a, s, \Phi) & = \chi(\det a^{\#}) |\det a^{\#}|_F^{s + m/2} E_{T^{\flat}}(a^{\flat}, s + 1/2, \mu_{m^{\flat}}^{m*}(s, \chi) \Phi) \\ \notag & \hphantom{=} + \chi(\det \overline{a}^{\#})^{-1} |\det a^{\#}|_F^{-s + m/2} E_{T^{\flat}}(a^{\flat}, s - 1/2, U_{m^{\flat}}^m(s, \chi) \Phi). \end{align} $$

The formula in the lemma statement follows from this, upon using our normalization convention in (2.2.19) and using the equality $\check {\chi } = \chi $ .

The special case of (2.4.7) with $a = 1$ and $F^+ = \mathbb {Q}$ is explained in [Reference He, Shi and YangHSY21, Proposition 2.4] (taking their a to equal $1$ , with their $w_n$ being our $w^{-1}$ , with their g being our $m(a)$ , and noting that their $T_a$ need only be nonsingular), and the case of general $F / F^+$ is similar. In the general case (for arbitrary block diagonal $a = \operatorname {\mathrm {diag}}(a^{\#}, a^{\flat })$ and arbitrary $F / F^+$ ), we claim that the first line (resp. second line) of our (2.4.7) corresponds to $W^{(n-1)}_{T^{\flat }}(m(a), s, \Phi )$ and $W^{(n)}_{T}(m(a), s, \Phi )$ , in the notation of [Reference He, Shi and YangHSY21, Theorem 2.2]. Reformulated in our notation, we have $W^{(r)}_T(m(a), s, \Phi ) = \int _{N_{m-1}(\mathbb {A})} \Phi (w^{-1}_{m-1} n(\operatorname {diag}(0,b)) m(a), s) \psi (-\operatorname {tr}(T^{\flat } b)) ~dn(b)$ . We have $w^{-1}_{m-1} n(\operatorname {diag}(0,b)) m(a) = m(\operatorname {\mathrm {diag}}(a^{\#}, 1)) w^{-1}_{m-1} n(\operatorname {diag}(0,b)) m(\operatorname {\mathrm {diag}}(1, a^{\flat }))$ , so the first line of (2.4.7) now follows upon applying (2.2.2), with $m(a)$ of loc. cit. replaced by $m(\operatorname {\mathrm {diag}}(a^{\#}, 1))$ . For the second line of (2.4.7), consider the third-to-last line of the last displayed equation in the proof of [Reference He, Shi and YangHSY21, Proposition 2.4]. Taking $a = 1$ in loc. cit., with their $w_n$ being our $w^{-1}$ , and taking their g to be our $m(a)$ , the second line of (2.4.7) now follows from the formula

(2.4.8) $$ \begin{align} & w^{-1} n(b) m(a) = m(\operatorname{\mathrm{diag}}({}^t\overline{a}^{\# -1},1)) w^{-1} n(b') m(\operatorname{\mathrm{diag}}(1, a^{\flat})) \end{align} $$

(2.4.9) $$ \begin{align} & \text{where } n(b') {:=}q m(\operatorname{\mathrm{diag}}(a^{\#}, 1)) \cdot n(b) \cdot m(\operatorname{\mathrm{diag}}(a^{\#}, 1))^{-1} \end{align} $$

for any $b \in N(\mathbb {A})$ , noting that b and $b'$ have the same lower-right $(m-1) \times (m-1)$ block, and again applying (2.2.2), with $m(a)$ of loc. cit. replaced by $m(\operatorname {\mathrm {diag}}(\overline {a}^{\# -1}, 1))$ . To obtain the second line of (2.4.7), one should additionally note that the change of variables $b' \mapsto b$ introduces a factor of $|\det a^{\#}|_F^{m}$ , so that we obtain $|\det a^{\#}|_F^{-s + m/2}$ rather than $|\det a^{\#}|_F^{-s - m/2}$ .

In Corollary 6.2.2, we rewrite (2.4.6) more explicitly when $\Phi _v$ is the normalized spherical standard section for every non-Archimedean v.

3.1 Weil index

We recall Weil indices, which are certain constants appearing in the Weil representation and other calculations below. We compute the instances which we need.

Suppose $F^+_v$ is a local field (arbitrary characteristic) with nontrivial additive character $\psi _v \colon F^+_v \rightarrow \mathbb {C}^\times $ , and suppose $V_v$ is a (finite dimensional) $F^+_v$ vector space equipped with a non-degenerate quadratic form $Q(-)$ . The map $V_v \rightarrow \mathbb {C}^{\times }$ given by $x \mapsto \psi _v(Q(x))$ is a “non-degenerate character of the second degree” in the sense of [Reference WeilWei64] [Reference Ranga RaoRao93, Appendix], so there is an associated Weil index $\gamma _{\psi _v}(V_v) \in \mathbb {C}^\times $ (which is an eighth root of unity). The quantity $\gamma _{\psi _v}(V_v)$ depends only on $\psi _v$ and the isomorphism class of $V_v$ , and we have

(3.1.1) $$ \begin{align} \gamma_{\overline{\psi}_v}(V_v) = \overline{\gamma_{\psi_v}(V_v)} \quad \quad \gamma_{\psi_v}(V_v \oplus V^{\prime}_v) = \gamma_{\psi_v}(V_v) \gamma_{\psi_v}(V^{\prime}_v) \end{align} $$

for orthogonal direct sums $V_v \oplus V^{\prime }_v$ (follows from the definition, see [Reference Ranga RaoRao93, Theorem A.2]). The Weil index also satisfies a global product formula [Reference WeilWei64, Proposition 5].

When $F^+_v$ has characteristic $\neq 2$ and $V_v$ has a bilinear pairing $(-,-)$ , our convention is that $x \mapsto (x,x)$ is the associated quadratic form (and vice-versa).

Next, let $F_v$ be an étale algebra of degree $2$ over a local field $F^+_v$ of characteristic $\neq 2$ (residue characteristic $2$ allowed). Write $\eta _v \colon F^{+ \times }_v \rightarrow \{ \pm 1 \}$ for the quadratic character associated to $F_v / F^+_v$ (trivial if $F_v/F^+_v$ is split), and write $a \mapsto \overline {a}$ for the nontrivial involution of $F_v$ over $F^+_v$ . If $F^+_v$ is non-Archimedean, we write $\mathfrak {d}$ (resp. $\Delta $ ) for the different (resp. discriminant) ideal for the extension $F_v/F^+_v$ (where $\mathfrak {d} = \mathcal {O}_{F_v}$ and $\Delta = \mathcal {O}_{F^+_v}$ in the split case). We sometimes abuse notation and write $\mathfrak {d}$ and $\Delta $ for understood/chosen generators of these ideals. We write $q_v$ for the residue cardinality of $F^+_v$ if $F^+_v$ is non-Archimedean.

Any non-degenerate $F_v/F^+_v$ Hermitian space $V_v$ with pairing $(-,-)$ has an associated $F^+_v$ -bilinear pairing $\frac {1}{2} \mathrm {tr}_{F_v/F^+_v}(-,-)$ and quadratic form $x \mapsto \frac {1}{2} \mathrm {tr}_{F_v/F^+_v}(x,x)$ . (Elsewhere, we typically normalize the trace bilinear pairing without the factor of $\frac {1}{2}$ .) We write $\gamma _{\psi _v}(V_v)$ for the Weil index of this quadratic space with respect to a nontrivial additive character $\psi _v \colon F^+_v \rightarrow \mathbb {C}^{\times }$ . We know $\gamma _{\psi _v}(V_v)^4 = 1$ (see, e.g., [Reference Ranga RaoRao93, Corollary A.5(4)] and [Reference KudlaKud94, Theorem 3.1]).

We write $\gamma _{\psi _v}(F_v)$ for the Weil index associated to the one-dimensional Hermitian space $F_v$ with pairing $(x,y) = \overline {x} y$ . We write $\epsilon _v(s, \xi _v, \psi _v)$ for the local epsilon factor associated to a quasi-character $\xi _v \colon F^{+ \times }_v \rightarrow \mathbb {C}^{\times }$ (as in [Reference TateTat79, §3][Reference TateTat67], for the quasi-character $\xi _v |-|^s$ and the self-dual Haar measure for $\psi _v$ ).

If $F^+_v$ is non-Archimedean with uniformizer $\varpi _0$ , we have

(3.1.3) $$ \begin{align} \epsilon_v(s, \eta_v, \psi_v) = |\varpi_0^{c(\psi_v)} \Delta|_{F^+_v}^{s - 1/2} \gamma_{\psi_v}(F_v) \end{align} $$

where

(3.1.4) $$ \begin{align} c(\psi_v) = \max \{j \in \mathbb{Z} : \psi_v|_{\varpi_0^{-j} \mathcal{O}_{F^+_v}} \text{ is trivial} \}. \end{align} $$

If $F^+_v$ is Archimedean, we have

(3.1.5) $$ \begin{align} \epsilon_v(s, \eta_v, \psi_v) = |u|_{F^+_v}^{s - 1/2} \gamma_{\psi_v}(F_v) \end{align} $$

where $u \in F^{+ \times }_v$ is such that

(3.1.6) $$ \begin{align} \psi_v(x) = e^{2 \pi i u x} \quad \text{if } F^+_v = \mathbb{R} \quad \text{and} \quad \psi_v(z) = e^{2 \pi i \mathrm{tr}_{\mathbb{C}/\mathbb{R}} (u z)} \quad \text{if } F^+_v = \mathbb{C}. \end{align} $$

In both non-Archimedean and Archimedean cases, if $\psi ^{\prime }_v(x) = \psi _v(ux)$ for some $u \in F^{+ \times }_v$ , we have

(3.1.7) $$ \begin{align} \epsilon_v(s, \eta_v, \psi^{\prime}_v) = \eta_v(u) |u|^{s - 1/2} \epsilon(s, \eta_v, \psi_v) \quad\quad \gamma_{\psi^{\prime}_v}(F_v) = \eta_v(u) \gamma_{\psi_v}(F_v). \end{align} $$

These identities follow from [Reference Jacquet and LanglandsJL70, Lemma 1.2(iii), (iv)] and properties of epsilon factors as in [Reference TateTat79, §3.2]. For the reader’s convenience, we recall $\gamma _{\psi _v}(\mathbb {C}) = i$ if $F^+_v = \mathbb {R}$ and $\psi _v(x) = e^{2 \pi i x}$ .

In all cases, we have

(3.1.8) $$ \begin{align} \gamma_{\psi_v}(F_v)^2 = \epsilon_v(1/2, \eta_v, \psi_v)^2 = \eta_v(-1). \end{align} $$

If $F^+_v$ is non-Archimedean, recall that $\epsilon _v(s,\xi _v, \psi _v) = 1$ if $\xi _v$ and $\psi _v$ are unramified. If $F^+_v = \mathbb {R}$ and $\psi _v(x) = e^{2 \pi i x}$ , recall $\epsilon _v(s, \mathrm {sgn}^j, \overline {\psi }_v) = 1$ (resp. $= -i$ ) if j is even (resp. odd) where $\mathrm {sgn} \colon \mathbb {R}^{\times } \rightarrow \{ \pm 1\}$ is the sign character (these formulas will be used implicitly in Section 5.2). Recall our convention that self-duality for Hermitian lattices is understood with respect to the trace pairing (unless otherwise specified), see [Reference ChenChe24a, Section 2.2]. If $F_v/F^+_v$ is ramified and L is a self-dual Hermitian lattice, then L must have even rank (explained in loc. cit.).

By the standard additive character $\psi _{\mathrm {tr},v} \colon F^+_v \rightarrow \mathbb {C}^{\times }$ for Archimedean $F^+_v$ , we mean the characters of (3.1.6) with $u = 1$ . By the standard additive character for non-Archimedean $F^+_v$ , we mean $\psi _{\mathrm {tr}}(x) {:=}q \psi _p(\operatorname {tr}_{F^+_v / \mathbb {Q}_p}(x))$ , where the residue characteristic is p, and where $\psi _p \colon \mathbb {Q}_p^{\times } \rightarrow \mathbb {C}^{\times }$ is the unramified character satisfying $\psi _p(x) = e^{-2 \pi i x}$ for any $x \in \mathbb {Z}[1/p]$ . In the non-Archimedean case, we write $\mathfrak {d}_{F^+_v / \mathbb {Q}_p}$ for the different ideal of $F^+_v / \mathbb {Q}_p$ . An element $u \in F^+_v$ is a generator for $\mathfrak {d}_{F^+_v / \mathbb {Q}_p}$ if and only if the additive character $x \mapsto \psi _{\mathrm {tr},v}(u^{-1} x)$ is unramified.

3.2 Weil representation

Let $F_v/F^+_v$ and accompanying notation be as in Section 3.1. Assume $F_v/F^+_v = \mathbb {C} / \mathbb {R}$ if $F^+_v$ is Archimedean. We also assume $F^+_v$ has characteristic $0$ (because [Reference KudlaKud94] assumes this).

Let $V_v$ be a non-degenerate $F_v/F^+_v$ Hermitian space of dimension $n \geq 0$ . Choose a nontrivial additive character $\psi _v \colon F^+_v \rightarrow \mathbb {C}^\times $ , and let $\chi _v \colon F_v^{\times } \rightarrow \mathbb {C}^{\times }$ be a character such that $\chi _v|_{F^{+ \times }_v} = \eta _v^{n}$ . There is a local Weil representation $\omega _v = \omega _{\chi _v,\psi _v}$ of $U(m,m)(F^+_v) \times U(V_v)(F^+_v)$ on the space of Schwartz functions $\mathcal {S}(V_v^m)$ (the Schrödinger model [Reference KudlaKud94, §5]),Footnote ⁷ which we normalize as

$$ \begin{align*} (\omega_v(m(a)) \varphi_v)(\underline{x}) & = \chi_v(\det a) |\det a|^{n/2}_{F_v} \varphi_v(\underline{x} \cdot a) && m(a) \in M(F^+_v) \\ (\omega_v(n(b)) \varphi_v)(\underline{x}) & = \psi_v(\operatorname{tr} (b \cdot (\underline{x}, \underline{x}))) \varphi_v(\underline{x}) && n(b) \in N(F^+_v) \\ (\omega_v(w) \varphi_v)(\underline{x}) & = \gamma_{\psi_v}(V_v)^m \widehat{\varphi}_v(\underline{x}) && \\ (\omega_v(h) \varphi_v)(\underline{x}) & = \varphi_v(h^{-1} \cdot \underline{x}) && h \in U(V_v)(F^+_v) \end{align*} $$

for $\varphi _v \in \mathcal {S}(V_v^m)$ and $\underline {x} \in V_v^m$ (viewed as $n \times m$ matrices), where

(3.2.1) $$ \begin{align} \widehat{\varphi}_v(\underline{x}) = \int_{V_v^m} \varphi_v(\underline{\smash{y}}) \psi_v(\operatorname{tr}_{F_v/F^+_v} \operatorname{tr}(\underline{x}, \underline{\smash{y}})) ~d\underline{\smash{y}} \end{align} $$

is the Fourier transform for the corresponding self-dual Haar measure on $V_v^m$ , and where we should view $\underline {x}$ as a length m row vector with elements in $V_v$ in the above formulas. The constant $\gamma _{\psi _v}(V_v)$ is the Weil index from Section 3.1. The symbol $(\underline {x}, \underline {\smash {y}})$ denotes the matrix of inner products with $i,j$ -th entry $(x_i, y_j)$ .

With $s_0 {:=}q (n - m)/2$ , there is a map $\mathcal {S}(V_v^m) \rightarrow I(\chi _v, s_0)$ sending $\varphi _v \in \mathcal {S}(V_v^m)$ to the function $h \mapsto (\omega _v(h) \varphi _v)(0)$ . The associated standard section $\Phi _{\varphi _v} \in I(\chi _v,s)$ is the Siegel–Weil section for $\varphi _v$ [Reference Garcia and SankaranGS19, §5.1].

If $F^+_v$ is non-Archimedean, choose a generator $\mathfrak {d}$ of the different ideal of $F_v / F^+_v$ , and let $M^{\circ }_2$ be the rank $2$ Hermitian $\mathcal {O}_{F_v}$ -lattice admitting a basis with Gram matrix

(3.2.2) $$ \begin{align} \begin{pmatrix} 0 & \mathfrak{d}^{-1} \\ \overline{\mathfrak{d}}^{-1} & 0 \end{pmatrix}. \end{align} $$

Note that $M_2^{\circ } = M_2^{\circ *}$ is self-dual (with respect to the $F^+_v$ -bilinear pairing $\operatorname {tr}_{F_v/F^+_v}(-,-)$ ).

In the statement of Lemma 3.2.2, condition (1) implies condition (2) if M has even rank (the ramified case is [Reference Li and LiuLL22, Lemma 2.12]).

Next, consider the case where $F_v/F^+_v = \mathbb {C}/\mathbb {R}$ . Suppose the n-dimensional Hermitian space $V_v$ is positive definite, with Hermitian pairing $(-,-)$ . If $\psi _v(x) = e^{2 \pi i u x}$ for some $u> 0$ , the Gaussian

(3.2.3) $$ \begin{align} \varphi_v(\underline{x}) = e^{-2 \pi u \cdot \mathrm{tr}(\underline{x}, \underline{x})} \in \mathcal{S}(V_v^m) \end{align} $$

for $\underline {x} = (x_1, \ldots , x_m) \in V_v^m$ has associated Siegel–Weil section

(3.2.4) $$ \begin{align} \Phi_{\varphi_v} = \Phi_v^{(n)} \end{align} $$

where $\Phi _v^{(n)}$ is the scalar weight standard section described surrounding (2.2.11), see [Reference Garcia and SankaranGS19, (2.68)] (there in the case $u = 1$ , and the general case follows from Remark 3.2.1).

4.1 Local L-factors

We use the following (standard) local factors as in [Reference TateTat79, §3].

If $F^+_v$ is a local field (allowing arbitrary characteristic in Section 4.1) and $\xi _v \colon F^{+ \times }_v \rightarrow \mathbb {C}^{\times }$ is a quasi-character, we write $L_v(s, \xi _v)$ for the corresponding local L-factor (for the quasi-character $\xi _v |-|_{F^+_v}^s$ ). Given any nontrivial additive character $\psi _v \colon F^+_v \rightarrow \mathbb {C}^{\times }$ , we write $\epsilon _v(s, \xi _v, \psi _v)$ for the corresponding local epsilon factor (as appeared in Section 3.1) and $\rho _v(s, \xi _v, \psi _v)$ for the local factor from Tate’s thesis [Reference TateTat67, Theorem 2.4.1], which satisfies

(4.1.1) $$ \begin{align} \rho_{v}(s, \xi_v, \psi_v) = \epsilon_v(s, \xi_v, \psi_v)^{-1} L_v(1-s, \xi_v^{-1})^{-1} L_v(s, \xi_v). \end{align} $$

If $F^+$ is a global field with a quasi-character $\xi \colon F^{+ \times } \backslash \mathbb {A}_{F^+} \rightarrow \mathbb {C}^{\times }$ and nontrivial additive character $\psi \colon F^+ \backslash \mathbb {A}_{F^+} \rightarrow \mathbb {C}^{\times }$ , we write

(4.1.2) $$ \begin{align} \Lambda(s, \xi) = \prod_v L_v(s, \xi_v) \quad \quad L(s, \xi) = \prod_{v < \infty} L_v(s, \xi) \quad \quad \epsilon(s, \xi) = \prod_v \epsilon_v(s, \xi_v, \psi_v) \end{align} $$

and have $\Lambda (s, \xi ) = \epsilon (s, \xi ) \Lambda (1-s, \xi ^{-1})$ . For the reader’s convenience, we recall the formulas

$$ \begin{align*} L_v(s, \xi_v) & = \begin{cases} (1 - \xi_v(\varpi_0) |\varpi_0|^s_{F^+_v})^{-1} & \text{if } \xi_v \text{ is unramified} \\ 1 & \text{if } \xi_v \text{ is ramified} \end{cases} \quad \quad \text{ if } F^+_v \text{ is non-Archimedean}\\ &\hspace{74.3pt}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{with uniformizer } \varpi_0 \in F^+_v \\ L_v(s, \mathrm{sgn}^j) & = \begin{cases} \pi^{-s / 2} \Gamma(s/2) & \text{if } j \text{ is even} \\ \pi^{-(s+1)/2} \Gamma((s+1)/2) & \text{if } j \text{ is odd} \end{cases} \quad \quad \text{ if } F^+_v = \mathbb{R} \text{ and } \mathrm{sgn} \text{ denotes}\\[-4pt] &\hspace{40pt}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{ the sign character.} \end{align*} $$

4.2 Normalized Archimedean Whittaker functions

With notation as above, assume $F_v/F^+_v$ is $\mathbb {C} / \mathbb {R}$ . The symbol h will denote an element of $U(m,m)(F^+_v)$ . Fix integers $n,m$ with $m \geq 0$ .

Let $\psi _v \colon F^+_v \rightarrow \mathbb {C}^{\times }$ be an arbitrary nontrivial additive character, which is necessarily of the form $\psi _v(x) = e^{2\pi i u x}$ for some $u \in \mathbb {R}^{\times }$ . Consider $T \in \mathrm {Herm}_m(F^+_v)$ with $\det T \neq 0$ . With $s_0 {:=}q (n-m)/2$ as above, we define the normalizing factor

(4.2.1) $$ \begin{align} \Lambda_{T,\psi_v}(s)^{\circ}_n {:=}q \frac{(2 \pi)^{m(m-1)/2}}{(-2 \pi i)^{nm}} \pi^{m(- s + s_0)} \left ( \prod_{j = 0}^{m - 1} \Gamma(s - s_0 + n - j) \right ) |\det T|^{- s - s_0}_{F^+_v} |u|_{F^+_v}^{-m (s + s_0) - m^2/2} \end{align} $$

(compare [Reference Garcia and SankaranGS19, (3.3.14)], also Shimura [Reference ShimuraShi82]) where $\Gamma $ is the usual gamma function. If $\psi _v(x) = e^{2 \pi i x}$ is the standard additive character, we write $\Lambda _{T,v}(s)^{\circ }_n {:=}q \Lambda _{T,\psi _v}(s)^{\circ }_n$ . For general $\psi _v(x) = e^{2\pi i u x}$ , we have $\Lambda _{T,\psi _v}(s)^{\circ }_n = |u|_{F^+_v}^{-m^2/2} \Lambda _{u T,v}(s)^{\circ }_n$ .

For arbitrary $\psi _v$ , we define a normalized Archimedean Whittaker function

(4.2.2) $$ \begin{align} W^{\ast}_{T,\psi_v}(h, s)^{\circ}_n {:=}q \Lambda_{T,\psi_v}(s)^{\circ}_n W_{T,\psi_v}(h, s, \Phi^{(n)}_v). \end{align} $$

For $a \in \operatorname {\mathrm {GL}}_m(F_v)$ , we also consider the variant

(4.2.3) $$ \begin{align} \tilde{W}^{\ast}_{T,\psi_v}(a, s)^{\circ}_n {:=}q \chi_v(a)^{-1} |\det a|_{F_v}^{-n / 2} W^{\ast}_{T,\psi_v}(m(a), s)^{\circ}_n \cdot q^{-T} \quad \quad q^{-T} {:=}q \psi_v(-\operatorname{tr}(i T y)) \end{align} $$

with $y {:=}q a {}^t \overline {a}$ (temporary notation). This is a (normalized) local analogue of (2.3.6). When $\psi _v(x) = e^{2 \pi i x}$ is the standard additive character, we write $W^{\ast }_{T,v}$ and $\tilde {W}^{\ast }_{T,v}$ instead of $W^{\ast }_{T,\psi _v}$ and $\tilde {W}^{\ast }_{T,\psi _v}$ . For general $\psi _v(x) = e^{2\pi i u x}$ , we have $W^{\ast }_{T,\psi _v} = W^{\ast }_{u T, v}$ and $\tilde {W}^{\ast }_{T, \psi _v} = \tilde {W}^{\ast }_{u T, v}$ (consider Remark 2.3.2).

For any $a \in \operatorname {\mathrm {GL}}_m(F_v)$ and $k \in U(m)$ , we have the “linear invariance” properties

(4.2.4) $$ \begin{align} \tilde{W}^{\ast}_{T,\psi_v}(a, s)^{\circ}_n = \tilde{W}_{{}^t \overline{a} T a, \psi_v}^{\ast}(1, s)^{\circ}_n \quad \quad \tilde{W}^{\ast}_{T,\psi_v}(1,s)^{\circ}_n = \tilde{W}^{\ast}_{T,\psi_v}(k,s)^{\circ}_n. \end{align} $$

The first expression follows from (2.3.9), and the second expression follows from the scalar weight property of $\Phi ^{(n)}_v$ . Given $y \in \mathrm {Herm}_m(\mathbb {R})_{>0}$ , we also set $W^{\ast }_{T,\psi _v}(y,s)^{\circ }_n {:=}q \tilde {W}^{\ast }_{T,\psi _v}(a, s)^{\circ }_n$ for any $a \in \operatorname {\mathrm {GL}}_m(F_v)$ satisfying $a {}^t \overline {a} = y$ (does not depend on the choice of a).Footnote ⁹ We use the shorthand $W_{T,\psi _v}^{\ast }(s)^{\circ }_n {:=}q \tilde {W}^{\ast }_{T,\psi _v}(1,s)^{\circ }_n$ . We continue to use our usual convention of using the subscript “v” instead of “ $\psi _v$ ” if $\psi _v$ is the standard character.

To simplify exposition, assume $\psi _v(x) = e^{2 \pi i u x}$ with $u> 0$ for the moment. For all $n \in \mathbb {Z}$ , we have the functional equation

(4.2.5) $$ \begin{align} W_{T,\psi_v}^{\ast}(h, s)^{\circ}_n = \eta_v(\det T)^{n - m - 1} W_{T,\psi_v}^{\ast}(h,-s)^{\circ}_n. \end{align} $$

The case when T is positive definite follows from [Reference ShimuraShi82, Theorem 3.1] (via (2.2.13), see also [Reference Garcia and SankaranGS19, (3.54)]). The case of general T (still with $\det T \neq 0$ ) should follow from [Reference ShimuraShi82, Theorem 4.2, (4.34.K)], though we will give an alternative proof (Lemma 5.2.1). Here $\eta _v$ is the sign character $\operatorname {sgn}(-)$ .

Write $(r_1,r_2)$ for the signature of T (temporary notation). If either $n \geq r_1$ or $r_2 = 0$ , then the function $W_{T,\psi _v}^{\ast }(h,s)^{\circ }_n$ is holomorphic for all $s \in \mathbb {C}$ , for fixed h (follows from [Reference ShimuraShi82, Theorem 4.2, (4.34.K)]). For any $a \in \operatorname {\mathrm {GL}}_m(F_v)$ , we also have

(4.2.6) $$ \begin{align} \tilde{W}^{\ast}_{T,\psi_v}(a, s_0)^{\circ}_n = \begin{cases} 1 & \text{if } T \text{ is positive definite} \\ 0 & \text{if } m \leq n \text{ and } T \text{ is not positive definite.} \end{cases} \end{align} $$

For the case when T is positive definite, see [Reference ShimuraShi97, (3.15)] (also the proof of [Reference Garcia and SankaranGS19, Proposition 3.2]). The case of T not positive definite with $m \leq n$ follows from [Reference ShimuraShi82, Theorem 4.2, (4.34.K)] (see also [Reference Garcia and SankaranGS19, Proposition 3.3(i)]).

For the general case of $\psi _v(x) = e^{2 \pi i u x}$ with possibly $u < 0$ , the formula in (4.2.5) remains valid if “ $\eta _v(\det T)$ ” is replaced with “ $\eta _v(\det uT)$ ”, and (4.2.6) remains valid if both instances of “T” on the right are replaced by “ $uT$ ”.

4.3 Normalized non-Archimedean Whittaker functions

With n, $\chi _v$ , $\psi _v$ , $\eta _v$ , etc. as at the beginning of Section 4, assume $F^+_v$ is non-Archimedean. For the moment, we only assume $F^+_v$ has characteristic $\neq 2$ , and allow both $\chi _v$ and $\psi _v$ to be possibly ramified. We can also allow $F_v/F^+_v$ to be ramified with $F^+_v$ of residue characteristic $2$ in Section 4.3. The symbol h will denote an element of $U(m,m)(F^+_v)$ .

Let $\varpi _0$ be any uniformizer of $F^+_v$ , and let $q_v$ be the residue cardinality of $F^+_v$ . Consider $T \in \mathrm {Herm}_m(F^+_v)$ with $\det T \neq 0$ . We define the local normalizing factor

(4.3.1) $$ \begin{align} \Lambda_{T,\psi_v}(s)^{\circ}_n {:=}q |\Delta|_{F^+_v}^{- m(m-1)/4} \left ( \prod_{j = 0}^{m - 1} L_v(2s + m - j, \eta_v^{j + n}) \right ) |(\det T)\Delta^{\lfloor m/2 \rfloor}|_{F^+_v}^{-s - s_0} (q_v^{c(\psi_v)})^{m(s + s_0) + m^2/2} \end{align} $$

where $c(\psi _v)$ is the conductor of $\psi _v$ , as in (3.1.4). The local L-factors appearing in $\Lambda _{T,\psi _v}(s)^{\circ }_n$ should be compared with, for example, [Reference Harris, Kudla and SweetHKS96, §6]. If $\psi _v$ is any unramified nontrivial additive character (i.e., $c(\psi _v) = 0$ ), we write $\Lambda _{T,v}(s)^{\circ }_n {:=}q \Lambda _{T,\psi _v}(s)^{\circ }_n$ . For general $\psi ^{\prime }_v(x) = \psi _v(u x)$ for some unramified $\psi _v$ and $u \in F^{+ \times }_v$ , we have $\Lambda _{T,\psi ^{\prime }_v}(s)^{\circ }_n = |u|_{F^+_v}^{-m^2 / 2}\Lambda _{u T,v}(s)^{\circ }_n$ .

Suppose $n \geq 0$ , and suppose $V_v$ is an n-dimensional non-degenerate $F_v / F^+_v$ Hermitian space. Consider a full-rank lattice $L_v \subseteq V_v$ , and take the Schwartz function $\varphi _v = {\pmb {1}}_{L_v}^m \in \mathcal {S}(V_v^m)$ . Form the associated Siegel–Weil standard section $\Phi _{\varphi _v} \in I(\chi _v, s)$ , which could depend on $\psi _v$ . Let S be the Gram matrix of any basis for $L_v$ .