1.1.1 The base field

Let F stand for a totally imaginary finite extension of $\mathbb {Q}$ of degree $d_F = [F:\mathbb {Q}].$ Let $\Sigma _{F} = \mathrm {{Hom}}(F,\mathbb {C})$ be the set of all complex embeddings, and $\mathrm {S}_\infty $ denote the set of archimedean places of F; denote the cardinality of $\mathrm {S}_\infty $ by ${\mathrm r}$ ; hence, $d_F = 2{\mathrm r}.$ There is a canonical surjection $\Sigma _{F} \to \mathrm {S}_\infty ;$ the fibre over $v \in \mathrm {S}_\infty $ is a pair $\{\eta _v, \bar {\eta }_v\}$ of conjugate embeddings; via such a non-canonical choice of $\eta _v$ , fix the identification $F_v \simeq \mathbb {C}.$ Let $\mathbb {A} = \mathbb {A}_{\mathbb {Q}}$ be the adèle ring of $\mathbb {Q}$ , and $\mathbb {A}_f = \mathbb {A}^\infty $ the ring of finite adèles. Then $\mathbb {A}_F = \mathbb {A} \otimes _{\mathbb {Q}} F$ , and $\mathbb {A}_{F,f} = \mathbb {A}_f \otimes _{\mathbb {Q}} F$ . When F is a CM field (i.e., a totally imaginary quadratic extension of a totally real extension $F^+$ (say) of $\mathbb {Q},$ then $\Sigma _{F^+} = \mathrm {{Hom}}(F^+,\mathbb {C}) = \mathrm {{Hom}}(F^+,\mathbb {R}),$ and the restriction from F to $F^+$ gives a canonical surjection $\Sigma _F \to \Sigma _{F^+}$ ; the fiber over $\eta \in \Sigma _{F^+}$ is a pair of conjugate embeddings that will be denoted as $\{\eta , \bar {\eta }\}$ , with the understanding that the choice of $\eta $ in $\{\eta , \bar {\eta }\}$ though not canonical is nevertheless fixed once and for all. If $\Sigma _F = \{\nu _1, \dots , \nu _{d_F}\}$ , $\{\omega _1,\dots ,\omega _{d_F}\}$ is a $\mathbb {Q}$ -basis of F, and $\theta _F = \det [\sigma _i(\omega _j)]$ , then $\theta _F^2$ is the absolute discriminant $\delta _{F/\mathbb {Q}}$ of F. The square root of the absolute value of the discriminant, $|\delta _{F/\mathbb {Q}}|^{1/2},$ as an element of $\mathbb {R}^{\times }/\mathbb {Q}^{\times },$ is independent of the enumeration and the choice of basis. Let ${\mathfrak {i}}$ denote a fixed choice of $\sqrt {-1}.$ Since F is totally imaginary, ${\mathfrak {i}}^{d_F/2} \cdot \theta _F$ is a real number whose absolute value is $|\delta _{F/\mathbb {Q}}|^{1/2}.$

1.1.2 The groups

For an integer $N \geq 2$ , let $G_{0} = \mathrm {{GL}}_N/F$ , and put $G = \mathrm {Res}_{F/\mathbb {Q}}(G_{0})$ as the $\mathbb {Q}$ -group obtained by the Weil restriction of scalars. To emphasize the dependence on N, $G_0$ will also be denoted $G_{N,0}$ and similar notation will be adopted for other groups to follow. Let $B_0$ be the subgroup of $G_0$ of upper-triangular matrices, $T_0$ the diagonal torus in $B_0$ , and $Z_0$ the center of $G_0$ ; the corresponding $\mathbb {Q}$ -groups via $\mathrm {Res}_{F/\mathbb {Q}}$ will be denoted $B, T$ and Z, respectively. Let S stand for the maximal $\mathbb {Q}$ -split torus of Z; note that $S \simeq \mathbb {G}_m.$ Let n and $n'$ be positive integers such that $n + n' = N,$ and let $P_0$ be the maximal parabolic subgroup of $G_0$ containing $B_0$ of type $(n,n').$ The unipotent radical of $P_0$ is denoted $U_{P_0}$ and Levi quotient of $P_0$ is $M_{P_0} = \mathrm {{GL}}_n \times \mathrm {{GL}}_{n'}.$ Put $P = \mathrm {Res}_{F/\mathbb {Q}}(P_0)$ , and similarly, $U_P$ and $M_P.$ The dimension of $U_P$ is $nn'd_F = 2nn'{\mathrm r}.$

1.2.1 Locally symmetric spaces

Note that $ G(\mathbb {R}) \ = \ G_0(F \otimes _{\mathbb {Q}} \mathbb {R}) \ = \ \prod _{v \in \mathrm {S}_\infty } \mathrm {{GL}}_N(F_v) \ \simeq \ \prod _{v \in \mathrm {S}_\infty } \mathrm {{GL}}_N(\mathbb {C}).$ Similarly, $Z(\mathbb {R}) = Z_0(F \otimes _{\mathbb {Q}} \mathbb {R}) \simeq \prod _{v \in \mathrm {S}_\infty } \mathbb {C}^{\times } 1_N,$ where $1_N$ is the identity $N \times N$ -matrix; $S(\mathbb {R}) = \mathbb {R}^{\times }$ sits diagonally in $Z(\mathbb {R})$ . The maximal compact subgroup of $G(\mathbb {R})$ will be denoted $C_\infty $ ; we have $C_\infty \ = \ \prod _{v \in \mathrm {S}_\infty } \mathrm {{U}}(N),$ where $\mathrm {{U}}(N),$ the usual compact unitary group in N-variables, is a maximal compact group of $\mathrm {{GL}}_N(\mathbb {C})$ . Put $K_\infty = C_\infty S(\mathbb {R})$ and note that $K_\infty = C_\infty S(\mathbb {R})^\circ $ is a connected group, since $-1 \in S(\mathbb {R})$ gets absorbed into $C_\infty .$ Define the symmetric space of G as $\mathcal {S}^G \ := \ G(\mathbb {R})/K_\infty .$ For any open compact subgroup $K_f \subset G(\mathbb {A}_f)$ , define the adèlic symmetric space: $G(\mathbb {A})/K_\infty K_f \ = \ \mathcal {S}^G \times (G(\mathbb {A}_f)/K_f).$ On this space, $G(\mathbb {Q})$ acts properly discontinuously and we get a quotient

(1.1) $$ \begin{align} \pi \, : \, G(\mathbb{R})/K_\infty \times G(\mathbb{A}_f)/K_f \ \longrightarrow \ G(\mathbb{Q}) \backslash \left( G(\mathbb{R})/K_\infty \times G(\mathbb{A}_f)/K_f \right). \end{align} $$

The target space, called the adèlic locally symmetric space of G with level structure $K_f$ , is denoted $\mathcal {S}^G_{K_f} = G(\mathbb {Q}) \backslash G(\mathbb {A}) / K_\infty K_f.$ A typical element in the adèlic group $G(\mathbb {A}) = G(\mathbb {R}) \times G(\mathbb {A}_f)$ will be denoted $\underline g = g_\infty \times { \underline g}_f.$ As in [Reference Harder and Raghuram27, Sect. 2.1.4], one has $\mathcal {S}^G_{K_f} \ \cong \ \coprod _{i=1}^m \ \Gamma _i \backslash G(\mathbb {R}) / K_\infty $ ; if necessary, replacing $K_f$ by a subgroup of finite-index, assume that each $\Gamma _i$ is torsion-free. It is easy to see that $\mathrm {dim}(\mathcal {S}^G_{K_f}) \ = \ \mathrm {dim}(G(\mathbb {R})/K_\infty ) \ = \ \mathrm { dim}(G(\mathbb {R})/C_\infty ) - 1 \ = \ {\mathrm r}N^2 - 1.$

1.2.2 The field of coefficients E

Throughout this paper, let $E/\mathbb {Q}$ be a ‘large enough’ finite Galois extension that takes a copy of $F.$ (The meaning of E being large enough will depend on the context: for example, large enough so that some Hecke summand in inner-cohomology would split over E. To relate cohomology groups with automorphic forms, one could drop finiteness and take $E = \mathbb {C}$ , or anticipating p-adic interpolation of the L-values considered here, E could be a large enough p-adic field.) An embedding $\iota : E \to \mathbb {C}$ gives a bijection $\iota _* : \mathrm {{Hom}}(F,E) \to \mathrm {{Hom}}(F,\mathbb {C})$ given by composition: $\iota _*\tau = \iota \circ \tau .$ If $E = \mathbb {C}$ , then there is a natural notion of complex-conjugation on $\mathrm {{Hom}}(F,\mathbb {C})$ defined by $\bar {\eta }(x) = \overline {\eta (x)}.$ But, on $\mathrm {{Hom}}(F,E)$ , there is no natural notion of complex-conjugation; however, using $\iota : E \to \mathbb {C}$ , we can consider the conjugate $\overline {\tau }^\iota $ of $\tau $ defined as $\iota _*(\overline {\tau }^\iota ) = \overline {\iota _*\tau }.$ If F is a CM field, then let $\{1, c\}$ denote the Galois group of $F/F^+;$ restriction $\tau \mapsto \tau |_{F^+}$ gives a surjective map $\mathrm {{Hom}}(F, E) \twoheadrightarrow \mathrm {{Hom}}(F^+,E);$ for $\tau \in \mathrm {{Hom}}(F, E)$ , define $\tau ^c$ by $\tau ^c(x) = \tau (c(x))$ for all $x \in F$ , then $\{\tau , \tau ^c\}$ is the fiber above $\tau |_{F^+}.$ If $E = \mathbb {C}$ , then $\tau ^c = \bar {\tau }.$

1.2.3 Characters of the torus T

For E as above, let $X^{*}(T \times E) := \mathrm {{Hom}}_{E-\mathrm {alg}}(T \times E, \mathbb {G}_m),$ where $ \mathrm {{Hom}}_{E-\mathrm {alg}}$ is to mean homomorphisms of E-algebraic groups. There is a natural action of $\mathrm {Gal}(E/\mathbb {Q})$ on $X^{*}(T \times E)$ . Since $T = \mathrm {Res}_{F/\mathbb {Q}}(T_0)$ , one has

$$ \begin{align*}X^{*}(T \times E) \ = \ \bigoplus_{\tau : F \to E} X^{*}(T_0 \times_{F,\tau} E) \ = \ \bigoplus_{\tau : F \to E} X^{*}(T_0), \end{align*} $$

where the last equality is because $T_0$ is split over F. Let $X^{*}_{\mathbb {Q}}(T \times E) = X^{*}(T \times E) \otimes \mathbb {Q}.$ The weights are parametrized as in [Reference Harder and Raghuram27]: $\lambda \in X^{*}_{\mathbb {Q}}(T \times E)$ will be written as $\lambda = (\lambda ^\tau )_{\tau : F \to E}$ with

where is the i-th fundamental weight for $\mathrm {{SL}}_N$ extended to $\mathrm {{GL}}_N$ by making it trivial on the center, and is the determinant character of $\mathrm {{GL}}_N.$ If $r_{\lambda } := (Nd - \sum _{i=1}^{N-1} i (a_i-1))/N,$ then $b_1 = a_1 + a_2 + \dots + a_{N-1} - (N-1) + r_{\lambda }, \ b_2 = a_2 + \dots + a_{N-1} - (N-2) + r_{\lambda }, \dots , b_{N-1} = a_{N-1} - 1 + r_{\lambda }, \ b_N = r_{\lambda },$ and conversely, $a_i - 1 = b_i - b_{i+1}, \ d = (b_1+\dots +b_N)/N.$ A weight is an integral weight if and only if

$$ \begin{align*}\lambda \in X^{*}(T_0) \ \Longleftrightarrow \ b_i \in \mathbb{Z}, \ \forall i \ \Longleftrightarrow \ \left\{\begin{array}{l} a_i \in \mathbb{Z}, \quad 1 \leq i \leq N-1, \\ Nd \in \mathbb{Z}, \\ Nd \equiv \sum_{i=1}^{N-1} i (a_i-1) \quad\pmod{N}. \end{array}\right. \end{align*} $$

A weight $\lambda = (\lambda ^\tau )_{\tau : F \to E} \in X^{*}_{\mathbb {Q}}(T \times E)$ is integral if and only if each $\lambda ^\tau $ is integral. Next, an integral weight $\lambda \in X^{*}(T_0)$ is dominant, for the choice of the Borel subgroup being $B_0$ , if and only if

$$ \begin{align*}b_1 \geq b_2 \geq \dots \geq b_N \ \Longleftrightarrow \ a_i \geq 1 \ \mbox{for } 1 \leq i \leq N-1. \ (\text{There is no condition on } d.) \end{align*} $$

A weight $\lambda = (\lambda ^\tau )_{\tau : F \to E} \in X^{*}_{\mathbb {Q}}(T \times E)$ is dominant-integral if and only if each $\lambda ^\tau $ is dominant-integral. Let $X^+(T \times E)$ stand for the set of all dominant-integral weights.

1.2.4 The sheaf $\widetilde {\mathcal {M}}_{\lambda , E}$

For $\lambda \in X^+(T \times E)$ , put $\mathcal {M}_{\lambda , E} \ = \ \bigotimes _{\tau : F \to E} \mathcal {M}_{\lambda ^\tau },$ where $\mathcal {M}_{\lambda ^\tau }/E$ is the absolutely-irreducible finite-dimensional representation of $G_0 \times _\tau E = \mathrm {{GL}}_n/F \times _\tau E$ with highest weight $\lambda ^\tau .$ Denote this representation as $(\rho _{\lambda ^\tau }, \mathcal {M}_{\lambda ^\tau })$ . The group $G(\mathbb {Q}) = \mathrm {{GL}}_n(F)$ acts on $\mathcal {M}_{\lambda , E}$ diagonally; that is, $a \in G(\mathbb {Q})$ acts on a pure tensor $\otimes _\tau m_\tau $ via $a \cdot (\otimes _\tau m_\tau ) \ = \ \otimes _\tau \rho _{\lambda ^\tau }(\tau (a))(m_\tau ).$ This representation gives a sheaf $\widetilde {\mathcal {M}}_{\lambda , E}$ of E-vector spaces on $\mathcal {S}^G_{K_f}$ : the sections over an open subset $V\subset \mathcal {S}^G_{K_f}$ are the locally constant functions $s: \pi ^{-1}(V) \to \mathcal {M}_{\lambda , E}$ such that $s(a v) = \rho (a) s(v)$ for all $a \in G(\mathbb {Q}),$ where $\pi $ is as in (1.1).

Let us digress for a moment to clarify a certain point that seemingly causes some confusion. In the definition of $\mathcal {S}^G_{K_f},$ one could have divided by $Z(\mathbb {R}) C(\mathbb {R})$ instead of $K_\infty $ ; that is, one can consider $G(\mathbb {Q}) \backslash G(\mathbb {A}) / Z(\mathbb {R}) C(\mathbb {R}) K_f.$ Over this space, the same construction of the sheaf $\widetilde {\mathcal {M}}_{\lambda , E}$ carries through; however, for it to be nonzero, the central character of $\rho _\lambda $ has to have the type of an algebraic Hecke character of F (see [Reference Harder22, 1.1.3]). Let $\lambda = (\lambda ^\tau )_{\tau : F \to E}\in X^+(T \times E)$ , and suppose the condition on the central character means $d^{\iota \circ \tau } + d^{\overline {\iota \circ \tau }}$ is a constant independent every embedding $\iota : E \to \mathbb {C}$ , and every $\tau \in \mathrm {{Hom}}(F,E).$ Define $X^+_{\mathrm {alg}}(T \times E)$ to be the subset of $X^+(T \times E)$ consisting of all dominant-integral weights which satisfy the algebraicity condition that ‘ $d^{\iota \circ \tau } + d^{\overline {\iota \circ \tau }} = \mathrm {constant}$ ’ for all $\tau \in \mathrm {{Hom}}(F,E)$ and for all $\iota : E \hookrightarrow \mathbb {C}$ . To end the digression, for the sheaf $\widetilde {\mathcal {M}}_{\lambda ,E}$ on $\mathcal {S}^G_{K_f}$ , at this moment we do not need to impose this algebraicity condition; however, later on for the sheaf to support interesting cohomology, such as cuspidal cohomology, we will be needing the condition of strong-purity that will imply algebraicity.

If $\lambda \in X^+_{\mathrm {alg}}(T \times E)$ and $K_f$ small enough as in 1.2.1, then every stalk of $\widetilde {\mathcal {M}}_{\lambda , E}$ is isomorphic to the E-vector space $\mathcal {M}_{\lambda ,E},$ in which case the sheaf $\widetilde {\mathcal {M}}_{\lambda ,E}$ is a local system.

2.3.1 Strongly-pure weights over $\mathbb {C}$

If a weight $\lambda = (\lambda ^\eta )_{\eta :F \to \mathbb {C}} \in X^+_{\mathrm {alg}}(T \times \mathbb {C})$ supports cuspidal cohomology (i.e., if $H_{\mathrm {cusp}}^{\bullet }(\mathcal {S}^G_{K_f}, \widetilde {\mathcal {M}}_{\lambda , \mathbb {C}}) \neq 0$ ), then $\lambda $ satisfies the purity condition

(2.3) $$ \begin{align} a_i^\eta = a_{N-i}^{\bar{\eta}} \ \mbox{for all } \eta : F \to \mathbb{C}\ \iff \ \exists \,\mathbf{{w}} \mbox{ such that } b^\eta_i + b^{\bar{\eta}}_{N-i+1} = \mathbf{{w}} \text{ for all } \eta \text{ and } i, \end{align} $$

which follows from the purity lemma [Reference Clozel7, Lem. 4.9]. The integer $\mathbf {{w}}$ is called the purity weight of $\lambda .$ The weight $\lambda $ is said to be pure if it satisfies (2.3), and denote by $X^+_{0}(T \times \mathbb {C})$ the set of all such pure weights. Next, recall a theorem of Clozel that says that cuspidal cohomology for $\mathrm {{GL}}_N/F$ admits a rational structure [Reference Clozel7, Thm. 3.19], from which it follows that any $\varsigma \in \mathrm {{Aut}}(\mathbb {C})$ stabilizes cuspidal cohomology, that is, ${}^\varsigma \lambda $ also satisfies the above purity condition, where if $\lambda = (\lambda ^\eta )_{\eta : F \to \mathbb {C}}$ and $\varsigma \in \mathrm {{Aut}}(\mathbb {C})$ , then ${}^\varsigma \lambda $ is the weight $({}^\sigma \lambda ^\eta )_{\eta : F \to \mathbb {C}}$ where ${}^\varsigma \lambda ^\eta = \lambda ^{\varsigma ^{-1}\circ \eta }$ . A pure weight $\lambda $ will be called strongly-pure if ${}^\varsigma \lambda $ is pure with purity-weight $\mathbf {{w}}$ for every $\varsigma \in \mathrm {{Aut}}(\mathbb {C});$ denote by $X^+_{00}(T \times \mathbb {C})$ the set of all such strongly-pure weights. For $\lambda \in X^+_{00}(T \times \mathbb {C}),$ note that

$$ \begin{align*}b^{\varsigma^{-1} \circ \eta}_j + b^{\varsigma^{-1} \circ \overline{\eta}}_{N-j+1} = \mathbf{{w}}, \ \mbox{for all} \ 1 \leq j \leq N, \ \eta : F \to \mathbb{C}, \ \varsigma \in \mathrm{{Aut}}(\mathbb{C}). \end{align*} $$

We have the following inclusions inside the character group of $T \times \mathbb {C},$ which are all, in general, strict inclusions:

$$ \begin{align*}X^+_{00}(T \times \mathbb{C}) \ \subset \ X^+_{0}(T \times \mathbb{C}) \ \subset \ X^+_{\mathrm{alg}}(T \times \mathbb{C}) \ \subset \ X^+(T \times \mathbb{C}) \ \subset \ X^{*}(T \times \mathbb{C}). \end{align*} $$

2.3.2 Strongly-pure weights over E

The set of strongly-pure weights may be defined at an arithmetic level. Recall the standing assumption on E that is a finite Galois extension of $\mathbb {Q}$ that takes a copy of F; in particular, any embedding $\iota : E \to \mathbb {C}$ factors as $\iota : E \to \bar {\mathbb {Q}} \subset \mathbb {C}.$ Furthermore, $\iota : E \to \mathbb {C}$ gives a bijection $\iota _* : \mathrm {{Hom}}(F,E) \to \mathrm {{Hom}}(F,\mathbb {C})$ as $\iota _*(\tau ) = \iota \circ \tau $ , which in turn gives a bijection $X^{*}(T \times E) \to X^{*}(T \times \mathbb {C})$ that maps $\lambda = (\lambda ^\tau )_{\tau : F \to E}$ to ${}^\iota \lambda = ({}^\iota \lambda ^\eta )_{\eta : F \to \mathbb {C}} = (\lambda ^{\iota ^{-1} \circ \eta })_{\eta : F \to \mathbb {C}}.$

Proof. Fix $\iota _0 : E \to \mathbb {C}$ . Since $E/\mathbb {Q}$ is a finite Galois extension, the inclusions

$$ \begin{align*}\{\gamma \circ \iota_0 \ | \ \gamma \in \mathrm{{Gal}}(\bar{\mathbb{Q}}/\mathbb{Q})\} \ \subset \ \{\gamma \circ \iota \ | \ \gamma \in \mathrm{{Gal}}(\bar{\mathbb{Q}}/\mathbb{Q}), \iota : E \to \mathbb{C}\} \ \subset \ \mathrm{{Hom}}(E,\mathbb{C}) \end{align*} $$

are all equalities.

The set of strongly-pure weights over E, denoted $X^+_{00}(T \times E)$ , consists of the algebraic dominant integral weights $\lambda \in X^{*}(T \times E)$ that satisfy any one, and hence all, of the conditions in the above proposition. It is most convenient to work with the characterization in $(iii)$ . There are the following inclusions within the character group of $T \times E$ , which are all, in general, strict inclusions:

$$ \begin{align*}X^+_{00}(T \times E) \ \subset \ X^+_{\mathrm{alg}}(T \times E) \ \subset \ X^+(T \times E) \ \subset \ X^{*}(T \times E). \end{align*} $$

The existence of a strongly-pure weight over a totally imaginary base field F depends on the internal structure of F; this is explained over the course of the next four paragraphs.

2.3.3 Interlude on (strongly-)pure weights for a CM field

When the base field F is a CM field, then a pure weight is also strongly-pure. Given any $\varsigma \in \mathrm {{Aut}}(\mathbb {C})$ , one can check that $\varsigma _*(X^+_{0}(T \times \mathbb {C})) \ = \ X^+_{0}(T \times \mathbb {C}).$

Lemma 2.5. Let $\eta : F \to \mathbb {C}$ and $\varsigma : \mathbb {C} \to \mathbb {C}$ be field homomorphisms, and let ${\mathfrak {c}} : \mathbb {C} \to \mathbb {C}$ stand for complex conjugation. Then

$$ \begin{align*}\varsigma \circ {\mathfrak{c}} \circ \eta \ = \ {\mathfrak{c}} \circ \varsigma \circ \eta, \end{align*} $$

that is, complex conjugation and any automorphism of $\mathbb {C}$ commute on the image of a CM field.

Let $\lambda = (\lambda ^\eta )_{\eta : F \to \mathbb {C}} \in X^+_{0}(T \times \mathbb {C})$ ; hence, $d^\eta + d^{\bar {\eta }} = \mathbf {{w}}$ for all $\eta : F \to \mathbb {C}.$ Take any $\varsigma \in \mathrm {{Aut}}(\mathbb {C})$ and consider ${}^\varsigma \lambda $ ; to see its purity, note that

$$ \begin{align*}b_{j}^{\varsigma^{-1} \circ \eta} + b_{N-j+1}^{\varsigma^{-1} \circ \bar{\eta}} \ = \ b_{j}^{\varsigma^{-1} \circ \eta} + b_{N-j+1}^{\varsigma^{-1} \circ {\mathfrak{c}} \circ \eta} \ = \ b_{j}^{\varsigma^{-1} \circ \eta} + b_{N-j+1}^{{\mathfrak{c}} \circ \varsigma^{-1} \circ \eta} \ = \ b_{j}^{\varsigma^{-1} \circ \eta} + b_{N-j+1}^{\overline{\varsigma^{-1} \circ \eta}} = \mathbf{{w}}, \end{align*} $$

where the second equality is from Lemma 2.5 above. Hence, $\lambda $ is strongly-pure.

2.3.4 Interlude on strongly-pure weights for a general totally imaginary field

When F is totally imaginary but not CM, there may exist weights that are pure but not strongly-pure. The following example is instructive: take $F = \mathbb {Q}(2^{1/3},\omega ),$ where $2^{1/3}$ is the real cube root of $2$ and $\omega = e^{2\pi i/3}.$ Then $\Sigma _F = \mathrm {{Gal}}(F/\mathbb {Q}) \simeq S_3$ the permutation group in 3 letters taken to be $\{2^{1/3}, \, 2^{1/3}\omega , \, 2^{1/3}\omega ^2\}.$ Let $s \in S_3$ correspond to $\eta _s : F \to \mathbb {C}.$ Consider the weights $\lambda = (\lambda ^{\eta _s})_{s \in S_3}$ and $\mu = (\mu ^{\eta _s})_{s \in S_3}$ for $\mathrm {Res}_{F/\mathbb {Q}}(\mathrm {{GL}}(1)/F)$ described in the table:

$$ \begin{align*} \begin{array}{|c||c|c|c|c|c|c||} \hline s & \ e \ & (12) & (23) & (13) & (123) & (132) \\ \hline \hline \lambda^{\eta_s} & a & b & \mathbf{{w}}-a & c & \mathbf{{w}}-c & \mathbf{{w}}-b \\ \hline \mu^{\eta_s} & a & \mathbf{{w}}-a & \mathbf{{w}}-a & \mathbf{{w}}-a & a & a \\ \hline \end{array} \end{align*} $$

where $a,b,c,\mathbf {{w}} \in \mathbb {Z}$ . For the tautological embedding $F \subset \mathbb {C},$ the set $\Sigma _F$ is paired into complex conjugates as $\{(\eta _{e}, \eta _{(23)}), (\eta _{(12)}, \eta _{(132)}), (\eta _{(13)}, \eta _{(123)})\},$ from which it follows that $\lambda $ is a pure weight. All other possible pairings of $\Sigma _F$ into conjugates via automorphisms of $\mathbb {C}$ are given by: $\{(\eta _{e}, \eta _{(12)}), (\eta _{(23)}, \eta _{(123)}), (\eta _{(13)}, \eta _{(132)})\},$ and $\{(\eta _{e}, \eta _{(13)}), (\eta _{(23)}, \eta _{(132)}), (\eta _{(12)}, \eta _{(123)})\};$ (F being Galois this simply boils down to composing these embeddings $\eta _s$ by a fixed one $\eta _{s_0},$ and using $\eta _{s_0} \circ \eta _{s} = \eta _{s_0s}$ ). It follows that $\lambda $ is not strongly-pure if $\mathbf {{w}}-a, b$ and c are not all equal, but $\mu $ is strongly-pure and has purity weight $\mathbf {{w}}.$

2.3.5 On the internal structure of a general totally imaginary field

Let F be a totally imaginary field as before. Let $F_0$ be the largest totally real subfield in F. Then there is at most one totally imaginary quadratic extension $F_1$ of $F_0$ inside F. (See, for example, Weil [Reference Weil51].) If $\alpha $ and $\beta $ are two totally negative elements of $F_0$ giving two possible such extensions $F_0(\sqrt {\alpha })$ and $F_0(\sqrt {\beta })$ , then by maximality of $F_0$ , one has $\sqrt {\alpha \beta } \in F_0$ , that is, $\alpha = t^2 \beta $ for $t \in F_0$ , whence $F_0(\sqrt {\alpha }) = F_0(\sqrt {\beta }).$ There are two distinct cases to consider:

1. (i) CM: when there is indeed such an imaginary quadratic extension $F_1$ of $F_0$ , then $F_1$ is the maximal CM subfield of F; of course, $[F_1:F_0] = 2$ . For example, if $F = \mathbb {Q}(2^{1/3},\omega )$ as in 2.3.4, then $F_0 = \mathbb {Q}$ and $F_1 = \mathbb {Q}(\omega )$ .

2. (ii) TR: when there is no imaginary quadratic extension of $F_0$ inside F, then put $F_1 = F_0$ for the maximal totally real subfield of F. For example, take $F_0$ to be a cubic totally real field (e.g., $F_0 = \mathbb {Q}(\zeta _7+\zeta _7^{-1})$ , $\zeta _7 = e^{2\pi i/7}$ ), and take non-square elements $a,b \in F_0$ whose conjugates $a,a',a"$ and $b,b',b"$ are such that $a>0, \ a'<0, a"<0$ and $b<0, \ b'<0,\ b">0$ ; such a and b exist by weak-approximation; take $F = F_0(\sqrt {a}, \sqrt {b})$ . Then there is no intermediate CM-subfield between $F_0$ and F; hence, $F_1 = F_0.$

As will be explained later on, that in case TR, asking for a critical point for a Rankin–Selberg L-function for $\mathrm {{GL}}(n) \times \mathrm {{GL}}(n')/F$ will impose the restriction $nn'$ is even. This should not be surprising because, as is well-known, for an algebraic Hecke character $\chi $ over F, if the L-function $L(s,\chi )$ has critical points, then that forces us to be in case CM (see [Reference Raghuram42]).

2.3.6 Strongly-pure weights over F are base-change from $F_1$

For brevity, the conclusion will be denoted as $\lambda = \mathrm {BC}_{F/F_1}(\kappa ).$

Proof. It suffices to prove the proposition over $\mathbb {C}$ , that is, if $'\lambda \in X^+_{00}(\mathrm {Res}_{F/\mathbb {Q}}(T_{N,0}) \times \mathbb {C})$ , then it suffices to show the existence $'\kappa \in X^+_{00}(\mathrm {Res}_{F_1/\mathbb {Q}}(T_{N,0}) \times \mathbb {C})$ such that $'\lambda = \mathrm { BC}_{F/F_1}('\kappa );$ because, then given the $\lambda $ in the proposition, take an embedding $\iota : E \to \mathbb {C}$ , and let $'\lambda = {}^\iota \lambda $ , to which using the statement over $\mathbb {C}$ one gets $'\kappa $ , which defines a unique $\kappa $ via ${}^\iota \kappa = {}'\kappa .$ It is clear that $\lambda = \mathrm {BC}_{F/F_1}(\kappa )$ because this is so after applying $\iota $ .

To prove the statement over $\mathbb {C}$ , take $\lambda \in X^+_{00}(\mathrm {Res}_{F/\mathbb {Q}}(T_{N,0}) \times \mathbb {C})$ , and suppose $\lambda = (\lambda ^\eta )_{\eta : F \to \mathbb {C}}$ with $\lambda ^\eta = (b^\eta _1 \geq b^\eta _2 \geq \cdots \geq b^\eta _N).$ Strong-purity gives

$$ \begin{align*}b^{\gamma \circ \eta}_{N-j+1} + b^{\gamma \circ \bar{\eta}}_j = \mathbf{{w}}, \quad \forall \gamma \in \mathrm{{Gal}}(\bar{\mathbb{Q}}/\mathbb{Q}), \ \forall \eta \in \Sigma_F, \ 1 \leq j \leq N. \end{align*} $$

Also, one has

$$ \begin{align*}b^{\gamma \circ \eta}_{N-j+1} + b^{\overline{\gamma \circ \eta}}_j = \mathbf{{w}}, \quad \forall \gamma \in \mathrm{{Gal}}(\bar{\mathbb{Q}}/\mathbb{Q}), \ \forall \eta \in \Sigma_F, \ 1 \leq j \leq N. \end{align*} $$

Hence, we get $b^{\gamma \circ \bar {\eta }}_j \ = \ b^{\overline {\gamma \circ \eta }}_j.$ Exactly as explicated in the proof of Proposition 26 in [Reference Raghuram42], one gets $b^{\gamma \circ \eta }_j = b^{\eta }_j$ for all $\gamma $ in the normal subgroup of $\mathrm {{Gal}}(\bar {\mathbb {Q}}/\mathbb {Q})$ generated by the commutators $\{g {\mathfrak {c}} g^{-1} {\mathfrak {c}} : g \in \mathrm {{Gal}}(\bar {\mathbb {Q}}/\mathbb {Q})\}$ , and all $\eta : F \to \mathbb {C}.$ This means that $b^{\eta }_j$ depends only on $\eta |_{F_1}.$

2.4.1 Tate twists

Let $m \in \mathbb {Z}.$ For $\lambda \in X^+(T_N \times E)$ , define by the rule that if $\lambda ^\tau = (b_1^\tau ,\dots , b_N^\tau )$ , then for $\tau : F \to E.$ It is clear that preserves each of the properties: dominant, integral, algebraic and (strongly-)pure. As in [Reference Harder and Raghuram27, Sect. 5.2.4], cupping with the m-th power of the fundamental class gives us an isomorphism that maps (strongly-)inner cohomology to (strongly-)inner cohomology, and suppose $\pi _f \in \mathrm {{Coh}}_{!!}(G,\lambda )$ . Then $T^{\bullet }_{\mathrm {Tate}}(m)(\pi _f) \ = \ \pi _f(-m),$ where, $\pi _f(-m)$ is defined by $\pi _f(-m)(g_f) = \pi _f(g_f) \otimes |\!|g_f|\!|^{-m}.$

2.5.1 Cuspidal parameters and cohomological representations of $\mathrm {{GL}}_N(\mathbb {C})$

Given a weight $\lambda = (\lambda ^\eta )_{\eta :F \to \mathbb {C}} \in X^+_{00}(T \times \mathbb {C}),$ for each $v \in \mathrm {S}_\infty $ (recall that v corresponds to a pair of complex embeddings $\{\eta _v, \bar {\eta }_v\}$ of F into $\mathbb {C}$ , with $\eta _v$ used to identify $F_v$ with $\mathbb {C}$ ), define the cuspidal parameters of $\lambda $ at v by

$$ \begin{align*} \alpha^v \ := \ -w_0\lambda^{\eta_v} + \rho \ \ \mathrm{and} \ \ \beta^v \ := \ - \lambda^{\bar{\eta}_v} - \rho. \end{align*} $$

If $\lambda ^\eta = (b^\eta _1,\dots ,b^\eta _N)$ , then

(2.8) $$ \begin{align} \alpha^v \ = \ (\alpha^v_1, \dots, \alpha^v_N) \ = \ \left(-b^{\eta_v}_N+ \tfrac{(N-1)}{2}, \, -b^{\eta_v}_{N-1}+ \tfrac{(N-3)}{2}, \, \dots, \, -b^{\eta_v}_1 - \tfrac{(N-1)}{2}\right), \end{align} $$

and similarly,

(2.9) $$ \begin{align} \beta^v \ := \ (\beta^v_1, \dots, \beta^v_N) \ = \ \left(-b^{\bar{\eta}_v}_1 - \tfrac{(N-1)}{2}, \, -b^{\bar{\eta}_v}_2 - \tfrac{(N-3)}{2}, \, \dots, \, -b^{\bar{\eta}_v}_N + \tfrac{(N-1)}{2}\right). \end{align} $$

Purity implies that $\alpha ^v_j + \beta ^v_j = -\mathbf {{w}}.$ Define a representation of $\mathrm {{GL}}_N(F_v) \simeq \mathrm {{GL}}_N(\mathbb {C})$ as

(2.10) $$ \begin{align} \mathbb{J}_{\lambda_v} \ := \ \mathbb{J}(\lambda^{\eta_v}, \lambda^{\bar{\eta}_v}) \ := \ \mathrm{{Ind}}_{B_N(\mathbb{C})}^{\mathrm{{GL}}_N(\mathbb{C})} \left(z^{\alpha^v_1} \bar{z}^{\beta^v_1} \otimes \cdots \otimes z^{\alpha^v_N} \bar{z}^{\beta^v_N} \right), \end{align} $$

where $B_N$ is the subgroup of all upper-triangular matrices in $\mathrm {{GL}}_N,$ and by $\mathrm {{Ind}}$ we mean normalized (i.e., unitary) parabolic induction. Now define a representation of $G(\mathbb {R}) = \prod _v \mathrm {{GL}}_N(F_v)$ :

(2.11) $$ \begin{align} \mathbb{J}_{\lambda} \ := \ \bigotimes_{v \in \mathrm{S}_\infty} \mathbb{J}_{\lambda_v}. \end{align} $$

Some basic properties of $\mathbb {J}_\lambda $ are described in the following two propositions.

These are well-known results for $\mathrm {{GL}}_N(\mathbb {C})$ and are all easily seen from this elementary observation: for $z \in \mathbb {C}$ , let $|z|_{\mathbb {C}} = z \bar {z}$ ; then the representation

$$ \begin{align*}\mathbb{J}_{\lambda_v} \otimes |\ |_{\mathbb{C}}^{\mathbf{{w}}/2} \ = \ \mathrm{{Ind}}_{B_N(\mathbb{C})}^{\mathrm{{GL}}_N(\mathbb{C})} \left(z^{\alpha^v_1+ \mathbf{{w}}/2} \bar{z}^{\beta^v_1+\mathbf{{w}}/2} \, \otimes \, \cdots \, \otimes \, z^{\alpha^v_N+\mathbf{{w}}/2} \bar{z}^{\beta^v_N+\mathbf{{w}}/2} \right) \end{align*} $$

is unitarily induced from unitary characters (because of purity) and hence irreducible. A representation irreducibly induced from essentially discrete series representation is essentially tempered. Admitting a Whittaker model is a hereditary property. Nonvanishing of cohomology follows from Delorme’s Lemma (Borel–Wallach [Reference Borel and Wallach5, Thm. III.3.3]), and that relative Lie algebra cohomology satisfies a Künneth theorem. Finally, among all representations with given infinitesimal character, there is at most one that is essentially tempered.

Define the following numbers:

(2.14) $$ \begin{align} \begin{aligned} b_N^{\mathbb{C}} & \ = \ N(N-1)/2, \\ t_N^{\mathbb{C}} & \ = \ (N^2-1) - b_N^{\mathbb{C}}, \\ b_N^F & \ = \ {\mathrm r} \cdot b_N^{\mathbb{C}}, \\ t_N^F & \ = \ \dim(\mathcal{S}^G) - b_N^F. \end{aligned} \end{align} $$

The relation between $b_N^F$ and $t_N^F$ is mitigated by an appropriate version of Poincaré duality, which is the reason why the ‘top-degree’ is defined in terms of the ‘bottom-degree’ and the dimension of the intervening symmetric space.

Proposition 2.15. Let $\lambda \in X^+_{00}(T \times \mathbb {C})$ and $\mathbb {J}_\lambda $ as above. Then

$$ \begin{align*}H^q(\mathfrak{g}, K_\infty; \, \mathbb{J}_{\lambda} \otimes \mathcal{M}_{\lambda, \mathbb{C}}) \neq 0 \ \ \iff \ \ b_N^F \ \leq \ q\ \leq \ t_N^F. \end{align*} $$

Furthermore, for extremal degrees $q \in \{b_N^F, \, t_N^F\},$ we have $\dim (H^q(\mathfrak {g}, K_\infty; \, \mathbb {J}_{\lambda } \otimes \mathcal {M}_{\lambda , \mathbb {C}})) = 1.$

There is a piquant numerological relation between the bottom or top degee for the cuspidal cohomology of Levi subgroup $\mathrm {{GL}}_n \times \mathrm {{GL}}_{n'}$ of a maximal parabolic subgroup P of an ambient $\mathrm {{GL}}_N$ , the corresponding bottom or top degree for $\mathrm {{GL}}_N$ , and the dimension of the unipotent radical of P given in the following proposition that has a crucial bearing on certain degree-considerations for Eisenstein cohomology. For any positive integer r, define $b_r^{\mathbb {C}}, \, t_r^{\mathbb {C}}, \, b_r^F$ and $t_r^F$ as in (2.14) replacing N by r.

Proof. Keeping in mind that $N = n+ n'$ , (1) follows from the identity

$$ \begin{align*}{\mathrm r} \frac{n(n-1)}{2} + {\mathrm r} \frac{n'(n'-1)}{2} + {\mathrm r} nn' \ = \ {\mathrm r} \frac{(n+n')(n+n'-1)}{2}. \end{align*} $$

For (2), observe that $t_n^F \ = \ (n^2{\mathrm r} - 1) - {\mathrm r}n(n-1)/2 = {\mathrm r}n(n+1)/2 - 1.$ Now (2) follows from

$$ \begin{align*}\left({\mathrm r} \frac{n(n+1)}{2} -1\right) + \left({\mathrm r} \frac{n'(n'+1)}{2} - 1\right)+ {\mathrm r} nn' \ = \ \left({\mathrm r} \frac{(n+n')(n+n'+1)}{2} - 1\right) - 1.\\[-47pt] \end{align*} $$

2.5.2 Archimedean constituents: CM-case

If $\pi _\infty = \otimes _{v \in \mathrm {S}_\infty } \pi _v$ is an irreducible representation of $G(\mathbb {R}) = \prod _{v \in \mathrm {S}_\infty } \mathrm {{GL}}_N(\mathbb {C})$ ,then the set $\{\pi _v : v \in \mathrm {S}_\infty \}$ of the irreducible factors, up to equivalence, will be called as the set of constituents of $\pi _\infty $ . Let $\lambda \in X^+_{00}(\mathrm {Res}_{F/\mathbb {Q}}(T_{N,0}) \times E)$ , $\pi _f \in \mathrm {{Coh}}_{!!}(G,\lambda ),$ and $\iota : E \to \mathbb {C}$ . The set of constituents of ${}^\iota \pi _\infty $ may be explicitly described.

CM-case

Recall from Proposition 2.6 that $\lambda = \mathrm {BC}_{F/F_1}(\kappa );$ that is, $\lambda ^\tau = \kappa ^{\tau |_{F_1}};$ after applying $\iota $ , one has ${}^\iota \lambda ^\eta = {}^\iota \kappa ^{\eta |_{F_1}},$ which is the same as $\lambda ^{\iota ^{-1}\circ \eta } = \kappa ^{\iota ^{-1}\circ \eta |_{F_1}}.$ Using the notations fixed in 2.3.5, for any place $v_{jl} \in \mathrm {S}_\infty (F)$ above $w_j \in \mathrm {S}_\infty (F_1),$ the ordered pair $(\eta _{v_{jl}}, \bar {\eta }_{v_{jl}})$ of conjugate embeddings of F restricts to the ordered pair $(\nu _{w_j}, \bar \nu _{w_j})$ of conjugate embeddings of $F_1$ ; hence, the ordered pair of weights $({}^\iota \lambda ^{\eta _{v_{jl}}}, {}^\iota \lambda ^{\bar {\eta }_{v_{jl}}})$ is equal to the ordered pair $({}^\iota \kappa ^{\nu _{w_j}}, {}^\iota \kappa ^{\bar \nu _{w_j}}),$ whence the archimedean component ${}^\iota \pi _{v_{jl}}$ is equivalent to $\mathbb {J}({}^\iota \kappa ^{\nu _{w_j}}, {}^\iota \kappa ^{\bar \nu _{w_j}})$ . Just for the moment, for brevity, denoting $\mathbb {J}({}^\iota \kappa ^{\nu _{w_j}}, {}^\iota \kappa ^{\bar \nu _{w_j}})$ by $\mathbb {J}_{w_j}$ , one concludes that the constituents of ${}^\iota \pi _\infty $ is the multi-set $\{\mathbb {J}_{w_1},\dots , \mathbb {J}_{w_1}, \ \dots \ ,\mathbb {J}_{w_{{\mathrm r}_1}}, \dots , \mathbb {J}_{w_{{\mathrm r}_1}} \},$ with each $\mathbb {J}_{w_j}$ appearing $k = [F:F_1]$ -many times; this multi-set may also be variously written as $\{[F:F_1] \cdot \mathbb {J}_{w} \ | \ w \in \mathrm {S}_\infty (F_1) \} \ = \ \{[F:F_1] \cdot \mathbb {J}({}^\iota \kappa ^{\nu _{w}}, {}^\iota \kappa ^{\bar \nu _{w}}) \ | \ w \in \mathrm {S}_\infty (F_1) \}.$ Putting these together one has

(2.17) $$ \begin{align} {}^\iota\pi_\infty \ = \ \bigotimes_{v \in \mathrm{S}_\infty(F)} {}^\iota\pi_v \ &= \ \bigotimes_{v \in \mathrm{S}_\infty(F)} \mathbb{J}({}^\iota\lambda^{\eta_{v}}, {}^\iota\lambda^{\bar{\eta}_{v}}) \notag\\ & \ = \ \bigotimes_{w \in \mathrm{S}_\infty(F_1)} \bigotimes_{v|w} \mathbb{J}({}^\iota\kappa^{\nu_{w}}, {}^\iota\kappa^{\bar\nu_{w}}) \ = \ \bigotimes_{w \in \mathrm{S}_\infty(F_1)} \bigotimes_{v|w} \mathbb{J}(\kappa^{\iota^{-1} \circ \nu_{w}}, \kappa^{\iota^{-1} \circ \bar\nu_{w}}). \end{align} $$

TR-case

We still have from Proposition 2.6 that $\lambda = \mathrm {BC}_{F/F_1}(\kappa );$ that is, $\lambda ^\tau = \kappa ^{\tau |_{F_1}};$ after applying $\iota $ , one has ${}^\iota \lambda ^\eta = {}^\iota \kappa ^{\eta |_{F_1}},$ which is the same as $\lambda ^{\iota ^{-1}\circ \eta } = \kappa ^{\iota ^{-1}\circ \eta |_{F_1}}.$ Using the notations fixed in 2.3.5, for any place $v_{jl} \in \mathrm {S}_\infty (F)$ above $w_j \in \mathrm {S}_\infty (F_1),$ both the embeddings in the ordered pair $(\eta _{v_{jl}}, \bar {\eta }_{v_{jl}})$ restrict to $\nu _{w_j}$ . Hence, the ordered pair of weights $({}^\iota \lambda ^{\eta _{v_{jl}}}, {}^\iota \lambda ^{\bar {\eta }_{v_{jl}}})$ is equal to the ordered pair $({}^\iota \kappa ^{\nu _{w_j}}, {}^\iota \kappa ^{\nu _{w_j}})$ – note that both weights in the ordered pair are the same, whence the archimedean component ${}^\iota \pi _{v_{jl}}$ is equivalent to $\mathbb {J}({}^\iota \kappa ^{\nu _{w_j}}, {}^\iota \kappa ^{\nu _{w_j}})$ . Once again, for brevity, denoting $\mathbb {J}({}^\iota \kappa ^{\nu _{w_j}}, {}^\iota \kappa ^{\nu _{w_j}})$ by $\mathbb {J}_{w_j}$ , one concludes that the constituents of ${}^\iota \pi _\infty $ are elements of the multi-set $\{\mathbb {J}_{w_1},\dots , \mathbb {J}_{w_1}, \ \dots \ ,\mathbb {J}_{w_{{\mathrm r}_1}}, \dots , \mathbb {J}_{w_{{\mathrm r}_1}} \},$ with each $\mathbb {J}_{w_j}$ appearing $k_1 = [F:F_1]/2$ -many times; putting these together, one has

(2.18) $$ \begin{align} {}^\iota\pi_\infty &= \bigotimes_{v \in \mathrm{S}_\infty(F)} {}^\iota\pi_v \ = \ \bigotimes_{v \in \mathrm{S}_\infty(F)} \mathbb{J}({}^\iota\lambda^{\eta_{v}}, {}^\iota\lambda^{\eta_{v}})\notag\\& = \bigotimes_{w \in \mathrm{S}_\infty(F_1)} \bigotimes_{v|w} \mathbb{J}({}^\iota\kappa^{\nu_{w}}, {}^\iota\kappa^{\nu_{w}}) \ = \ \bigotimes_{w \in \mathrm{S}_\infty(F_1)} \bigotimes_{v|w} \mathbb{J}(\kappa^{\iota^{-1} \circ \nu_{w}}, \kappa^{\iota^{-1} \circ \nu_{w}}) \ = \ \bigotimes_{w \in \mathrm{S}_\infty(F_1)} \bigotimes_{v|w} \mathbb{J}_w, \end{align} $$

and each of these $\mathbb {J}_w$ is self-conjugate from Remark 2.12.

2.5.3 Galois action on archimedean constituents

Let $\gamma \in \mathrm {{Gal}}(\bar {\mathbb {Q}}/\mathbb {Q}).$ The archimedean constituents of ${}^{\gamma \circ \iota }\pi $ is a permutation of the archimedean constituents of ${}^\iota \pi $ , possibly up to replacing a local component by its conjugate which will only be relevant when F is in the CM-case. This is made more precise in the following paragraphs.

The case when F is itself a CM field

If $F = F_1$ is a CM field, and $F_0$ its maximal totally real quadratic subfield, for $\gamma \in \mathrm {{Gal}}(\bar {\mathbb {Q}}/\mathbb {Q})$ and $\nu \in \Sigma _{F_1}$ from Lemma 2.5, one has $\gamma \circ \bar {\nu } = \overline { \gamma \circ \nu };$ this means that $\gamma $ permutes the set of pairs of conjugate embeddings $\left \{\{\nu _w, \bar \nu _w\} \ | \ w \in \mathrm {S}_\infty (F_1)\right \}$ , giving an action of $\gamma $ on $\mathrm {S}_\infty (F_1)$ . If we identify $\mathrm {S}_\infty (F_1) = \mathrm {S}_\infty (F_0) = \Sigma _{F_0}$ , then the action of $\gamma $ on $\mathrm {S}_\infty (F_1)$ is the same as action of $\gamma $ on $\Sigma _{F_0}$ via composition. It is important to note that $\gamma $ need not map the distinguished embedding corresponding to w to the distinguished embedding corresponding to $\gamma \cdot w$ ; all one can say is that $\gamma \circ \nu _w \in \{\nu _{\gamma \cdot w}, \bar {\nu }_{\gamma \cdot w}\}.$ Suppose $\kappa \in X^+_{00}(\mathrm {Res}_{F_1/\mathbb {Q}}(T_{N,0}) \times E)$ , $\pi _{1,f} \in \mathrm {{Coh}}_{!!}(\mathrm {Res}_{F_1/\mathbb {Q}}(\mathrm {{GL}}_N/F_1,\kappa ))$ , $\iota : E \to \mathbb {C}$ , and ${}^\iota \pi _1$ the corresponding cuspidal automorphic representation of $\mathrm {{GL}}_N(\mathbb {A}_{F_1})$ . Then ${}^\iota \pi _{1,\infty } = \otimes _{w \in \mathrm {S}_\infty (F_1)} {}^\iota \pi _{1,w}$ , where

$$ \begin{align*}{}^\iota\pi_{1,w} \ = \ \mathbb{J}({}^\iota\kappa^{\nu_w}, {}^\iota\kappa^{\bar\nu_w}) \ = \ \mathbb{J}(\kappa^{\iota^{-1} \circ \nu_w}, \kappa^{\iota^{-1} \circ \bar\nu_w}). \end{align*} $$

By the same token, replacing $\iota $ by $\gamma \circ \iota $ , one has

$$ \begin{align*}{}^{\gamma \circ \iota} \pi_{1,w} \ = \ \mathbb{J}(\kappa^{\iota^{-1} \circ \gamma^{-1} \circ \nu_w}, \kappa^{\iota^{-1} \circ \gamma^{-1} \circ \bar\nu_w}) \ = \ \mathbb{J}(\kappa^{\iota^{-1} \circ \gamma^{-1} \circ \nu_w}, \kappa^{\iota^{-1} \circ \overline{\gamma^{-1} \circ \nu_w}}) \end{align*} $$

Depending on whether $\gamma ^{-1} \circ \nu _w = \nu _{\gamma ^{-1} \cdot w}$ or $\overline {\nu _{\gamma ^{-1} \cdot w}}$ , from Remark 2.12, it follows that

(2.19) $$ \begin{align} {}^{\gamma \circ \iota} \pi_{1,w} \ = \ \begin{cases} {}^\iota\pi_{1, \gamma^{-1} \cdot w} & \mbox{if } \gamma^{-1} \circ \nu_w = \nu_{\gamma^{-1} \cdot w}, \\ \\ \overline{{}^\iota\pi_{1, \gamma^{-1} \cdot w}} & \mbox{if } \gamma^{-1} \circ \nu_w = \overline{\nu_{\gamma^{-1} \cdot w}}. \end{cases} \end{align} $$

Hence, the archimedean components of ${}^{\gamma \circ \iota } \pi _1$ is a permutation of the archimedean components of ${}^{\iota } \pi _1$ up to taking conjugates; this paragraph fixes a mistake in [Reference Gan and Raghuram12, Prop. 3.2, (i)].

When F is totally imaginary in the CM-case.

Let $\lambda \in X^+_{00}(\mathrm {Res}_{F/\mathbb {Q}}(T_{N,0}) \times E)$ , $\lambda = \mathrm { BC}_{F/F_1}(\kappa )$ , $\pi _f \in \mathrm {{Coh}}_{!!}(G,\lambda )$ , $\iota : E \to \mathbb {C}$ , and $\gamma \in \mathrm {{Gal}}(\bar {\mathbb {Q}}/\mathbb {Q})$ . The Galois action on $\Sigma _F$ and $\Sigma _{F_1}$ preserves the fibers of the restriction map $\Sigma _F \to \Sigma _{F_1}$ . Suppose $w_1, w_j \in \mathrm {S}_\infty (F_1)$ and $\nu _1, \nu _j \in \Sigma _{F_1}$ are the corresponding distinguished elements, and suppose $\gamma \circ \{\nu _1, \bar \nu _1\} = \{\nu _j, \bar \nu _j\}$ . Suppose the fiber over $\nu _1$ is $\{\eta _{11}, \eta _{12}, \dots , \eta _{1 k}\}$ (recall $k = [F:F_1]$ ). Then the fiber over $\bar \nu _1$ is $\{\bar {\eta }_{11}, \bar {\eta }_{12}, \dots , \bar {\eta }_{1 k}\}$ ; and similarly, if the fiber over $\nu _j$ is $\{\eta _{j1}, \eta _{j2}, \dots , \eta _{j k}\}$ and then the fiber over $\bar \nu _j$ is $\{\bar {\eta }_{j1}, \bar {\eta }_{j2}, \dots , \bar {\eta }_{j k}\}.$ There are two cases:

1. 1. $\gamma \circ \nu _1 = \nu _j$ . Then necessarily, $\gamma \circ \bar \nu _1 = \bar \nu _j$ , $\gamma \circ \{\eta _{11}, \dots , \eta _{1 k}\} = \{\eta _{j1}, \dots , \eta _{j k}\}$ and $\gamma \circ \{\bar {\eta }_{11}, \dots , \bar {\eta }_{1 k}\} = \{\bar {\eta }_{j1}, \dots , \bar {\eta }_{j k}\}.$

2. 2. $\gamma \circ \nu _1 = \bar \nu _j$ . Then necessarily, $\gamma \circ \bar \nu _1 = \nu _j$ , $\gamma \circ \{\eta _{11}, \dots , \eta _{1 k}\} = \{\bar {\eta }_{j1}, \dots , \bar {\eta }_{j k}\} $ and $\gamma \circ \{\bar {\eta }_{11}, \dots , \bar {\eta }_{1 k}\} = \{\eta _{j1}, \dots , \eta _{j k}\}.$

Since $F = F_1$ is already discussed in 2.5.3 above, suppose that $k> 1$ . Suppose $\gamma \circ \eta _{11} = \eta _{j1}$ . Then it is possible that $\gamma \circ \{\eta _{11}, \bar {\eta }_{11}\} \neq \{\eta _{j1}, \bar {\eta }_{j1}\}.$ In particular, the Galois action on $\Sigma _F$ does not descend to give a Galois action of $\gamma $ on $S_\infty (F)$ . Similarly, also in case (2). Nevertheless, using (2.17), it follows that

(2.20) $$ \begin{align} {}^{\gamma\circ\iota}\pi_\infty \ = \ \bigotimes_{w \in \mathrm{S}_\infty(F_1)} \bigotimes_{v|w} \mathbb{J}(\kappa^{\iota^{-1} \circ \gamma^{-1} \circ \nu_{w}}, \kappa^{\iota^{-1} \circ \gamma^{-1} \circ \bar\nu_{w}}), \end{align} $$

and as in (2.19), the inner constituent is given by

(2.21) $$ \begin{align} \mathbb{J}(\kappa^{\iota^{-1} \circ \gamma^{-1} \circ \nu_{w}}, \kappa^{\iota^{-1} \circ \gamma^{-1} \circ \bar\nu_{w}}) \ = \ \begin{cases} \mathbb{J}(\kappa^{\iota^{-1} \circ \nu_{\gamma^{-1} \cdot w}}, \kappa^{\iota^{-1} \circ \bar\nu_{\gamma^{-1} \cdot w}}) & \mbox{if } \gamma^{-1} \circ \nu_w = \nu_{\gamma^{-1} \cdot w}, \\ \\ \overline{ \mathbb{J}(\kappa^{\iota^{-1} \circ \nu_{\gamma^{-1} \cdot w}}, \kappa^{\iota^{-1} \circ \bar\nu_{\gamma^{-1} \cdot w}})} & \mbox{if } \gamma^{-1} \circ \nu_w = \overline{\nu_{\gamma^{-1} \cdot w}}. \end{cases} \end{align} $$

Hence, the archimedean components of ${}^{\gamma \circ \iota } \pi $ is a permutation of the archimedean components of ${}^{\iota } \pi $ up to taking conjugates.

When F is totally imaginary in the TR-case.

The Galois action on $\Sigma _F$ and $\Sigma _{F_1}$ preserves the fibers of the restriction map $\Sigma _F \to \Sigma _{F_1}$ , and since $F_1$ is totally real, identify the Galois-sets $\Sigma _{F_1} = \mathrm {S}_\infty (F_1).$ Using the notations of (2.18), if

$$ \begin{align*} {}^\iota\pi_\infty \ = \ \bigotimes_{w \in \mathrm{S}_\infty(F_1)} \bigotimes_{v|w} \ \mathbb{J}(\kappa^{\iota^{-1} \circ \nu_{w}}, \kappa^{\iota^{-1} \circ \nu_{w}}), \end{align*} $$

then for $\gamma \in \mathrm {{Gal}}(\bar {\mathbb {Q}}/\mathbb {Q})$ , one has

$$ \begin{align*} ({}^{\gamma \circ \iota} \pi)_\infty \ = \ \bigotimes_{w \in \mathrm{S}_\infty(F_1)} \bigotimes_{v|w} \ \mathbb{J}(\kappa^{\iota^{-1} \circ \nu_{\gamma^{-1}\circ w}}, \kappa^{\iota^{-1} \circ \nu_{\gamma^{-1}\circ w}}). \end{align*} $$

Hence, the archimedean components of ${}^{\gamma \circ \iota } \pi $ is a permutation of the archimedean components of ${}^{\iota } \pi $ .

3.1.1 Definition of the critical set

For any two half-integers $\alpha $ and $\beta ,$ the local L-factor (see [Reference Knapp35]) of the character $z \mapsto z^\alpha \bar {z}^\beta $ of $\mathbb {C}^{\times }$ is given by

(3.3) $$ \begin{align} L(s, z^\alpha \bar{z}^\beta) = 2(2\pi)^{-\left(s + \frac{\alpha+\beta}{2} + \frac{|\alpha - \beta|}{2}\right)} \Gamma\left(s + \frac{\alpha+\beta}{2} + \frac{|\alpha - \beta|}{2}\right) \sim \Gamma\left(s + \frac{\alpha+\beta}{2} + \frac{|\alpha - \beta|}{2}\right), \end{align} $$

where, by $\sim $ , we mean up to nonzero constants and exponential functions, which are entire and nonvanishing everywhere and hence are irrelevant to the computation of critical points; see Definition 3.4 below. For any $v \in \mathrm {S}_\infty $ , let $\{\eta _v, \bar {\eta }_v\}$ be the pair of conjugate embeddings of F to $\mathbb {C}$ as before. Let

be the cuspidal parameters of $\mu $ at v; see (2.8) and (2.9). Similarly, let

be the cuspidal parameters of $\mu '$ at $v.$ Note that

$$ \begin{align*}\alpha := \alpha_i^v + \alpha^{\prime v}_j \ \in \ \frac{n-1}{2} + \mathbb{Z} + \frac{n'-1}{2} + \mathbb{Z} \ = \ \frac{N}{2} + \mathbb{Z}, \quad \mathrm{and} \quad \beta := \beta_i^v + \beta^{\prime v}_j \ \in \ \frac{N}{2} + \mathbb{Z}. \end{align*} $$

Then, it is clear that the quantity $\tfrac {\alpha +\beta }{2} + \tfrac {|\alpha - \beta |}{2}$ inside the argument of the $\Gamma $ -function above is in $\tfrac {N}{2} + \mathbb {Z}.$ This tells us that the critical set for $L(s, \sigma \times \sigma ')$ will be a subset of $\tfrac {N}{2} + \mathbb {Z}.$

Let $\sigma $ and $\sigma '$ be cuspidal automorphic representations of $G_n(\mathbb {A})$ and $G_{n'}(\mathbb {A})$ , respectively. The set of critical points for $L(s, \sigma \times \sigma ^{\prime \mathrm {v}})$ is defined to be

(3.4) $$ \begin{align} &\mathrm{{Crit}}(L(s, \sigma \times \sigma^{\prime \mathrm{v}})) \ := \ \notag\\ &\qquad\qquad\quad\left\{m \in \tfrac{N}{2} + \mathbb{Z} \ : \mbox{both } L_\infty(s, \sigma \times \sigma^{\prime \mathrm{v}}) \text{ and } L_\infty(1-s, \sigma^{\mathrm{v}} \times \sigma') \text{ are finite at } s = m \right\}. \end{align} $$

If $\sigma $ and $\sigma '$ are cohomological with respect to $\mu $ and $\mu '$ , then we denote

(3.5) $$ \begin{align} \mathrm{{Crit}}(\mu, \mu') \ := \ \mathrm{{Crit}}(L(s, \sigma \times \sigma^{\prime \mathrm{v}})). \end{align} $$

3.1.2 Computing the critical set

Recall the purity conditions

$$ \begin{align*}\alpha^v_i + \beta^v_i = - \mathbf{{w}}, \ \ \mathrm{and} \ \ \alpha^{\prime v}_i + \beta^{\prime v}_i = - \mathbf{{w}}'. \end{align*} $$

We define a quantity $a(\mu ,\mu ')$ , and call it the abelian width between $\mu $ and $\mu '$ , as

(3.6) $$ \begin{align} a(\mu,\mu') \ := \ \frac{\mathbf{{w}} - \mathbf{{w}}'}{2} \ = \ \frac{(d^\eta + d^{\bar{\eta}}) - (d^{\prime \eta} + d^{\prime \bar{\eta}})}{2}. \end{align} $$

From the local Langlands correspondence and (3.3) on abelian local L-factors, we get

(3.7) $$ \begin{align} L_\infty(s, \sigma \times \sigma^{\prime \mathrm{v}}) \ \sim \ \prod_{v \in \mathrm{S}_\infty} \, \prod_{i=1}^n \, \prod_{j=1}^{n'} \ \Gamma\left(s - a(\mu,\mu') + \frac{|\alpha^v_i - \alpha^{\prime v}_j - \beta^v_i + \beta^{\prime v}_j|}{2} \right). \end{align} $$

And similarly,

(3.8) $$ \begin{align} L_\infty(1-s, \sigma^{\mathrm{v}} \times \sigma') \ \sim \ \prod_{v \in \mathrm{S}_\infty} \, \prod_{i=1}^n \, \prod_{j=1}^{n'} \ \Gamma\left(1-s + a(\mu,\mu') + \frac{|\alpha^v_i - \alpha^{\prime v}_j - \beta^v_i + \beta^{\prime v}_j|}{2} \right). \end{align} $$

Let $m \in \tfrac {N}{2} + \mathbb {Z}.$ Then $m \in \mathrm {{Crit}}(\mu ,\mu ')$ if and only if

(3.9) $$ \begin{align} m - a(\mu,\mu') + \frac{|\alpha^v_i - \alpha^{\prime v}_j - \beta^v_i + \beta^{\prime v}_j|}{2} \ \geq \ 1, \quad \forall v \in \mathrm{S}_\infty, \forall i, \forall j, \end{align} $$

which is the condition that $L_\infty (m, \sigma \times \sigma ^{\prime \mathrm {v}})$ is finite from (3.7), and

(3.10) $$ \begin{align} 1-m + a(\mu,\mu') + \frac{|\alpha^v_i - \alpha^{\prime v}_j - \beta^v_i + \beta^{\prime v}_j|}{2} \ \geq \ 1, \quad \forall v \in \mathrm{S}_\infty, \forall i, \forall j, \end{align} $$

which is the condition that $L_\infty (1-m, \sigma ^{\mathrm {v}} \times \sigma ')$ is finite from (3.8). Define the cuspidal width $\ell (\mu ,\mu ')$ between $\mu $ and $\mu '$ as

(3.11) $$ \begin{align} \ell(\mu,\mu') \ := \ \mathrm{min} \left\{|\alpha^v_i - \alpha^{\prime v}_j - \beta^v_i + \beta^{\prime v}_j| \ : \ v \in \mathrm{S}_\infty, 1 \leq i \leq n, \ 1\leq j \leq n' \right\}. \end{align} $$

Then (3.9) and (3.10) together gives us the following

Proposition 3.12. Let $\mu \in X^+_{00}(T_n \times \mathbb {C})$ and $\mu ' \in X^+_{00}(T_{n'} \times \mathbb {C}).$ For $\sigma _f \in \mathrm {{Coh}}_{!!}(G_n, \mu )$ and $\sigma ^{\prime }_f \in \mathrm {{Coh}}_{!!}(G_{n'}, \mu ')$ , the critical set for the Rankin–Selberg L-function $L(s, \sigma \times \sigma ^{\prime \mathrm {v}})$ is given by

$$ \begin{align*}\mathrm{{Crit}}(\mu, \mu') \ = \ \left\{m \in \tfrac{N}{2} + \mathbb{Z} \ : \ 1 - \frac{\ell(\mu,\mu')}{2} + a(\mu, \mu') \ \leq \ m \ \leq \ \frac{\ell(\mu,\mu')}{2} + a(\mu, \mu') \right\}. \end{align*} $$

This is contiguous string of integers or half-integers (depending on whether N is even or odd), centered around $\tfrac 12+ a(\mu , \mu ')$ , of length $\ell (\mu ,\mu ').$

Corollary 3.13. With notations as in Proposition 3.12, the points $s = -N/2$ and $s = 1-N/2$ are both critical for $L(s, \sigma \times \sigma ^{\prime \mathrm {v}})$ if and only if

$$ \begin{align*}- \frac{N}{2} + 1 - \frac{\ell(\mu,\mu')}{2} \ \leq \ a(\mu, \mu') \ \leq \ - \frac{N}{2} -1 + \frac{\ell(\mu,\mu')}{2}. \end{align*} $$

Of course, for this to be possible, one needs $\ell (\mu ,\mu ') \geq 2$ (i.e., that there at least two critical points). The corollary, which is one part of a combinatorial lemma below (Lemma 3.16), is to be viewed like this: the two successive L-values at $s = -N/2$ and $s = 1-N/2$ are critical if and only if the abelian width is bounded in absolute value in terms of the cuspidal width.

3.1.3 Critical set at an arithmetic level

Let $\mu \in X^+_{00}(T_n \times E)$ and $\mu ' \in X^+_{00}(T_{n'} \times E),$ and take $\sigma _f \in \mathrm {{Coh}}_{!!}(G_n, \mu )$ and $\sigma ^{\prime }_f \in \mathrm {{Coh}}_{!!}(G_{n'}, \mu ').$ For any $\iota : E \to \mathbb {C}$ , Proposition 3.12 gives the critical set $\mathrm {{Crit}}({}^\iota \mu , {}^\iota \mu ')$ for the Rankin–Selberg L-function $L(s, {}^\iota \sigma \times {}^\iota \sigma ^{\prime \mathrm {v}}).$

Corollary 3.15. The critical set $\mathrm {{Crit}}({}^\iota \mu , {}^\iota \mu ') = \mathrm {{Crit}}(L(s, {}^\iota \sigma \times {}^\iota \sigma ^{\prime \mathrm {v}}))$ is independent of $\iota :$

$$ \begin{align*}\mathrm{{Crit}}(L(s, {}^\iota\sigma \times {}^\iota\sigma^{\prime \mathrm{v}})) \ = \ \mathrm{{Crit}}(L(s, {}^{\gamma \circ\iota}\sigma \times {}^{\gamma \circ\iota}\sigma^{\prime \mathrm{v}})), \quad \forall \ \iota : E \to \mathbb{C}, \ \ \forall \gamma \in \mathrm{{Gal}}(\bar{\mathbb{Q}}/\mathbb{Q}). \end{align*} $$

3.2.1 Statement of the lemma

We have already proved $(1) \iff (2)$ . It remains to prove $(2) \iff (3).$ It is clear that

$$ \begin{align*}l(w^\eta) + l(w^{\bar{\eta}}) = \dim(U_{P_0}), \ \forall \eta : F \to \mathbb{C} \ \implies \ l(w) = \tfrac12 \dim(U_P). \end{align*} $$

However, if the degree of F is greater than $2$ (i.e., if ${\mathrm r}> 1$ ), then the converse is not true in general.

Definition 3.17. A Kostant representative $w \in W^P$ is said to be balanced if

$$ \begin{align*}l(w^\eta) + l(w^{\bar{\eta}}) = \dim(U_{P_0}), \ \forall \eta : F \to \mathbb{C}.\end{align*} $$

For the benefit of the reader, we will make two passes over the proof of (2) $\iff $ (3) in simpler situations, because the proof in the general case is intricate in details and somewhat tedious; it is the sort of proof that makes one believe the dictum ‘der Teufel steckt im Detail’.

3.2.2 Explicating (2) $\iff $ (3) in the simplest nontrivial example

Proof. Let us consider the case of $n = n' = 1$ and so $N = 2.$ Take F to be an imaginary quadratic field with $\mathrm {{Hom}}(F,\mathbb {C}) = \{\eta , \bar {\eta }\}.$ The weights $\mu $ and $\mu '$ are both a pair of integers indexed by $\mathrm {{Hom}}(F,\mathbb {C});$ we will write $ \mu \ = \ ((a), (a^{*})), \quad \mu ' \ = \ ((b), (b^{*})), $ with $a,a^{*}, b, b^{*} \in \mathbb {Z},$ with the convention that $\mu ^\eta = (a), \mu ^{\bar {\eta }} = (a^{*})$ and similarly for $\mu '.$ Note that purity of $\mu $ and $\mu '$ is automatic, and the purity weights are $\mathbf {{w}} = a+a^{*}, \ \mathbf {{w}}' = b+b^{*}.$ The abelian width is $a(\mu , \mu ') = \frac {a+a^{*} - b-b^{*}}{2}.$ The cuspidal parameters at the only complex place v of F are $\alpha ^v = (-a), \ \beta ^v = (-a^{*}), \ \alpha ^{\prime v} = (-b), \ \beta ^{\prime v} = (-b^{*}).$ The cuspidal width is $\ell (\mu , \mu ') = |-a+a^{*} +b-b^{*}|.$ The weight $\mu +\mu '$ which we would like make dominant using a balanced Kostant representative has the shape $\mu +\mu ' \ = \ ((a,b), \, (a^{*}, b^{*})).$ For simplicity, let us denote $p := a-b, \ p^{*} := a^{*} - b^{*}.$ Hence, $\mu +\mu '$ is dominant if and only if $p \geq 0$ and $p^{*} \geq 0.$ The inequalities in (2) now take the shape

(3.18) $$ \begin{align} - \frac{|p^{*}-p|}{2} \ \leq \ \frac{p+p^{*}}{2} \ \leq \ \frac{|p^{*}-p|}{2} - 2. \end{align} $$

Since, $P_0 = B_0$ is the Borel subgroup, the Levi subgroup $M_P$ is a torus; hence, $W_{M_P}$ is trivial and $W^P = W_G.$ If $W_{G_0}$ is written as $\{1,s\}$ with s the nontrivial element, then the elements of $W^P$ may be written as $ W_G = (\{1,s\}, \{1^{*}, s^{*}\}). $ The dimension of $U_P$ is $2$ ; hence, the balanced elements (of length $1$ ) of $W^P$ are $(1,s^{*})$ and $(s,1)$ . Now, consider three cases depending on the sign of $p - p^{*}$ :

• ○ $p=p^{*}.$ In this case, (3.18) reads $0 \leq p \leq -2,$ which is absurd; hence, (2) is violated. If $p \geq 0$ , then the only $w \in W^P$ such that $w^{-1}\cdot (\mu +\mu ')$ is dominant is $w = (1,1^{*})$ which has length $0$ ; hence, (3) is violated. Similarly, if $p < 0$ , then the only $w \in W^P$ such that $w^{-1}\cdot (\mu +\mu ')$ is dominant is $w = (s,s^{*}),$ which has length $2$ ; hence, (3) is violated again. So, both (2) and (3) are false.

• ○ $p>p^{*}.$ In this case, (3.18) simplifies to $p^{*} - p \leq p + p^{*} \leq p - p^{*} - 4,$ which implies that $p \geq 0> -2 \geq p^{*}.$ The only $w \in W^P$ such that $w^{-1}\cdot (\mu +\mu ')$ is dominant is $w = (1,s^{*})$ which has length $1$ ; hence, (3) is satisfied.

• ○ $p<p^{*}.$ In this case, $p^{*} \geq 0> -2 \geq p$ and the only $w \in W^P$ that works is $(s,1)$ which is of length $1$ .

In all cases, either both (2) and (3) are satisfied, or both are violated. Hence, $(2) \iff (3).$

In the second case ( $p>p^{*}$ ), one might ask what happens in the degenerate case of $p=0$ and $p^{*}=-1.$ (So we are violating (2) but keeping $p>p^{*}$ .) This means that $\mu +\mu '$ has the shape $((a,a), (b^{*}-1,b^{*}))$ . The $\eta $ component $(a,a)$ is dominant, but one has to make the $\bar {\eta }$ -component $(b^{*}-1,b^{*})$ dominant. This can only be done using $s^{*}$ ; however, the reader can easily check that $s^{*} \cdot (b^{*}-1,b^{*}) = (b^{*}-1,b^{*})$ . In other words, there is no element w such that $w^{-1}\cdot (\mu +\mu ')$ is dominant.

3.2.3 Proof of (2) $\iff $ (3) for $\mathrm {{GL}}_n \times \mathrm {{GL}}_1$

It is most convenient to first understand the case when F is an imaginary quadratic field. Then $\Sigma _F = \{\eta , \bar {\eta }\}$ (for a non-canonical choice of $\eta : F \to \mathbb {C}$ that is fixed once and for all). As above, we will follow a notational artifice that all quantities indexed by $\bar {\eta }$ will be designated with a $*$ . A weight $\mu \in X^+_{0}(T_n \times \mathbb {C})$ may be written as $\mu = \{ \mu ^\eta , \mu ^{\bar {\eta }} \}$ with $\mu ^\eta = (\mu _1 \geq \mu _2 \geq \cdots \geq \mu _n)$ and $\mu ^{\bar {\eta }} = (\mu _1^{*} \geq \mu _2^{*} \geq \cdots \geq \mu _n^{*}),$ with $\mu _i, \mu _j^{*} \in \mathbb {Z},$ and purity implies $\mathbf {{w}} \ = \ \mu _i + \mu _{n-i+1}^{*}.$ A weight $\mu ' \in X^+_{0}(T_1 \times \mathbb {C})$ is simply a pair of integers $\mu ' = \{b, b^{*}\}$ with purity weight $\mathbf {{w}}' = b+b^{*}$ . The weight $\mu +\mu '$ is given by

$$ \begin{align*}\mu + \mu' \ = \ \{(\mu_1, \mu_2, \dots, \mu_n, b), \quad (\mu_1^{*}, \mu_2^{*}, \dots, \mu_n^{*}, b^{*}) \}. \end{align*} $$

We are seeking to understand when we can find a Kostant representative $w \in W^P$ which is balanced ( $l(w^\eta ) + l(w^{\bar {\eta }})= \dim (U_{P_0}) = n$ ) and such that $w^{-1}\cdot (\mu + \mu ')$ is a dominant weight. For this, first identify the Kostant representatives for $P_0$ in $G_0$ ; the simple roots of $M_{P_0}$ are $\mathbf {\Pi }_{M_{P_0}} = \{e_1-e_2, e_2-e_3, \, \dots , e_{n-1} - e_n\}.$ The Weyl group of $G_0$ is $W_{G_0} = \mathfrak {S}_{n+1}$ the symmetric group on $n+1$ letters. We have

$$ \begin{align*} w \in W^{P_0} & \iff w^{-1}(e_1-e_2)> 0, \ w^{-1}(e_2-e_3) > 0, \ \dots, w^{-1}(e_{n-1} - e_n) > 0 \\ & \iff w^{-1}(1) < w^{-1}(2) < \cdots < w^{-1}(n). \end{align*} $$

The elements of $W^{P_0}$ and their lengths are listed below:

$$ \begin{align*}\begin{array}{|l||c||} \hline w^{-1} \ \ (w\in W^{P_0}) & l(w) \\ \hline s_0 := 1 & 0 \\ s_1 := (n, n+1) & 1 \\ s_2 := (n-1, n, n+1) & 2 \\ \quad \quad \vdots & \vdots \\ s_{n-1} := (2,3,\dots, n+1) & n-1 \\ s_n := (1,2,3,\dots, n+1) & n \\ \hline \hline \end{array} \end{align*} $$

Note that the $(n+1)$ -cycle $(1,2,\dots ,n+1) = (1,2)(2,3)\cdots (n,n+1)$ as a product of n simple transpositions giving its length which applies to the last row and a similar calculation gives all the other lengths. The Kostant representatives for P are $W^P \ = \ \{(w, w^{*}) \ : \ w , w^{*} \in W^{P_0}\},$ where $l(w,w^{*}) = l(w) + l(w^{*}).$ Hence, the inverses of the balanced Kostant representatives are

$$ \begin{align*}\{(s_0, s_n^{*}), \ (s_1, s_{n-1}^{*}), \dots, (s_n, s_0^{*}) \}. \end{align*} $$

The twisted action of the Kostant representatives on the weight are given in the table below:

(3.19) $$ \begin{align} \begin{array}{||c|l||} \hline w^{-1} \ \ (w\in W^{P_0}) & \quad w^{-1} \cdot (\mu_1,\mu_2,\dots, \mu_n,b) \\ \hline 1 & \quad (\mu_1,\mu_2,\dots, \mu_n, b) \\ (n, n+1) & \quad (\mu_1,\mu_2,\dots, \mu_{n-1}, b-1, \mu_n +1) \\ (n-1, n, n+1) & \quad (\mu_1,\mu_2,\dots, \mu_{n-2}, b-2, \mu_{n-1} +1, \mu_n +1) \\ \quad \quad \vdots & \quad\quad\quad \quad\quad\vdots \\ (2,3,\dots, n+1) & \quad (\mu_1, b-n+1, \mu_2 +1 ,\dots, \mu_{n-1} +1, \mu_n +1) \\ (1,2,3,\dots, n+1) & \quad (b-n, \mu_1+1, \dots, \mu_{n-1} +1, \mu_n +1)\\ \hline \hline \end{array} \end{align} $$

For the combinatorial lemma (Lemma 3.16), the abelian width is given by

$$ \begin{align*}a(\mu, \mu') \ = \ \frac{\mathbf{{w}} - \mathbf{{w}}'}{2} \ = \ \frac{\mu_i + \mu_{n-i+1}^{*} - b - b^{*}}{2} \ = \ \frac{(\mu_i - b) + (\mu_{n-i+1}^{*} - b^{*})}{2}, \end{align*} $$

and for the cuspidal width, the cuspidal parameters are given by

$$ \begin{align*} \alpha = \ (\alpha_1,\dots, \alpha_n) \ = & \ (-\mu_n+ \tfrac{(n-1)}{2}, \ -\mu_{n-1}+\tfrac{(n-3)}{2},\ \dots, \ -\mu_1-\tfrac{(n-1)}{2}), \\ \beta = \ (\alpha_1,\dots, \alpha_n) \ = & \ (-\mu_1^{*}-\tfrac{(n-1)}{2}, \ -\mu_2^{*}-\tfrac{(n-3)}{2},\ \dots, \ -\mu_n^{*}+\tfrac{(n-1)}{2}), \end{align*} $$

and similarly, $ \alpha ' = -b, \ \beta ' = -b^{*},$ from which the cuspidal width is

$$ \begin{align*}\ell(\mu,\mu') = \min\left\{ \begin{array}{l} |\mu_1^{*}-\mu_n+(n-1) +b -b^{*}|, \\ |\mu_2^{*}-\mu_{n-1}+(n-3) +b -b^{*}|, \\ \quad \quad \quad \vdots \\ |\mu_n^{*}-\mu_1-(n-1) +b -b^{*}| \end{array}\right\}. \end{align*} $$

From the shape of $a(\mu ,\mu ')$ and $\ell (\mu , \mu ')$ , it is convenient to introduce the quantities $c_i := \mu _i - b$ and $c_i^{*} := \mu _i^{*}-b^{*}.$ (These are the p and $p^{*}$ when $n=1$ .) Then we have $a(\mu , \mu ') = \frac {c_i + c_{n-i+1}^{*}}{2},$ and

$$ \begin{align*}\ell(\mu,\mu') \ = \ \min\{ \ |c_1^{*}-c_n + (n-1)|, \ |c_2^{*}-c_{n-1}+(n-3)|, \ |c_n^{*} - c_1 - (n-1)| \ \}. \end{align*} $$

From the dominance of the weights $\mu $ and $\mu '$ , we have the inequalities

$$ \begin{align*}c_1^{*}-c_n + (n-1) \> \ c_2^{*}-c_{n-1}+(n-3) \ > \ \cdots \ > c_n^{*} - c_1 - (n-1). \end{align*} $$

The proof conveniently breaks into $(n+1)$ disjoint cases depending on the relative position of $0$ in the above decreasing sequence.

Case 0: $0> c_1^{*}-c_n + (n-1) \ > \ c_2^{*}-c_{n-1}+(n-3) \ > \ \cdots \ > c_n^{*} - c_1 - (n-1),$

Case j ( $1\ \leq j \leq n-1$ ): $c_j^{*}-c_{n-j+1} + (n-2j+1) \> 0 \ > \ c_{j+1}^{*}-c_{n-j}+(n-2j-1),$

Case n: $c_1^{*}-c_n + (n-1) \> \ c_2^{*}-c_{n-1}+(n-3) \ > \ \cdots \ > c_n^{*} - c_1 - (n-1) > 0.$

In Case 0, we have $\ell (\mu ,\mu ') = -c_1^{*}+c_n - (n-1).$ Keeping in mind that $N = n+1$ , the inequalities in (2) of the lemma read

$$ \begin{align*}- \frac{(n+1)}{2} + 1 - \frac{(-c_1^{*}+c_n-(n-1))}{2} \ \leq \ \frac{c_1^{*}+c_n}{2} \ \leq \ - \frac{(n+1)}{2} -1 + \frac{(-c_1^{*}+c_n-(n-1))}{2} \end{align*} $$

This simplifies to

$$ \begin{align*}c_1^{*}-c_n \ \leq \ c_1^{*} + c_n \ \leq \ -c_1^{*} + c_n - 2n - 2. \end{align*} $$