1.1 The base field and algebraic groups

Let F be a totally real number field of degree $\mathsf {r}_F = [F: {\mathbb Q}].$ Let $\Sigma _F := \mathrm {Hom}(F, {\mathbb R}) = \mathrm {Hom}(F,{\mathbb C})$ be the set of all embeddings of F into ${\mathbb R}$ or ${\mathbb C},$ and denote by $S_{\infty }$ the set of archimedean places of F. Of course, $S_\infty $ is in bijection with $\Sigma _F.$ The completion of F at any place v is denoted $F_v.$ For $v \in S_{\infty }$ , we identify $F_{v}$ with ${\mathbb R}$ . Let $n = 2r \geq 2$ be an even integer. Let $G_{0} = \mathrm {O}(2n+2) / F$ be the split orthogonal group defined by the equation ${}^\intercal g~ J_{2n+2}~ g = J_{2n+2} $ , where ${}^\intercal g$ is the transpose of g, and for any positive integer m, the matrix $J_{m}$ is given by

(4) $$ \begin{align} J_{m} = \left( \begin{array}{ccc} & & 1 \\ & \unicode{x22F0} \\ 1 & & \end{array} \right)_{m \times m}. \end{align} $$

When the size m is clear from context, we will denote $J_m$ simply by $J.$ The subgroup $B_{0}$ consisting of all upper triangular matrices in $G_{0}$ defines a Borel subgroup of $G_0$ . Let $T_{0} \subset B_{0}$ be a split torus of $G_0$ consisting of the diagonal matrices. Let $N_0$ be the unipotent radical of $T_0$ in $B_0$ . So we have $B_0 = T_0 N_0$ . Let $G = R_{F / {\mathbb Q}} (G_{0} / F)$ be the group obtained by the Weil restriction of scalars to ${\mathbb Q}$ . Similarly, for $B_0$ , $T_0,$ and $N_0$ , we denote the ${\mathbb Q}$ -groups obtained from by restriction of scalars by B, $T,$ and N, respectively.

1.2 The groups at infinity

The group of real points of G is given by

$$ \begin{align*}G({\mathbb R}) = \prod \limits_{v \in S_{\infty}} G(F_v) = \prod \limits_{v \in S_{\infty}} \mathrm{O}(n+1,n+1)({\mathbb R}). \end{align*} $$

The map $g \mapsto {}^\intercal g^{-1}$ defines a Cartan involution on $G({\mathbb R})$ , and its fixed points define a maximal compact subgroup $K_{\infty }$ of $G({\mathbb R})$ :

$$ \begin{align*}K_{\infty} \ \simeq \ \prod \limits_{v \in S_{\infty}} \mathrm{O}(n+1)({\mathbb R}) \times \mathrm{O}(n+1)({\mathbb R}). \end{align*} $$

To describe this isomorphism explicitly, consider the map

$$ \begin{align*}h : \mathrm{O}(n+1)({\mathbb R}) \times \mathrm{O}(n+1)({\mathbb R}) \ \longrightarrow \ \mathrm{O}(n+1,n+1)({\mathbb R}), \end{align*} $$

which is defined by

(5) $$ \begin{align} (k, l) \mapsto h(k, l) = \dfrac{1}{2} \begin{pmatrix} k + J l J & k J - J l \\ J k - l J & J k J + l \end{pmatrix}, \end{align} $$

where $k,l \in \mathrm {O}(n+1)({\mathbb R}),$ that is, they satisfy ${}^\intercal k \cdot k = I_{n+1} = {}^\intercal l \cdot l,$ and J in the formula above is $J_{n+1}.$ We leave it to the reader to check that h is a homomorphism and that the image of h lies in $\mathrm {O}(n+1,n+1)({\mathbb R})$ ; that is, the map h gives an embedding of $\mathrm {O}(n+1)({\mathbb R}) \times \mathrm {O}(n+1)({\mathbb R}) \hookrightarrow \mathrm {O}(n+1,n+1)({\mathbb R})$ . For $v \in S_{\infty }$ , let $K_{v}$ be the image of $\mathrm {O}(n+1)({\mathbb R}) \times \mathrm {O}(n+1)({\mathbb R})$ inside $G_{v}$ , and put $K_{\infty } = \prod \limits _{v \in S_{\infty }}K_{v}$ ; its connected component of the identity is $K^{\circ }_{\infty }$ , which is isomorphic to $\prod \limits _{v \in S_{\infty }} \mathrm {SO}(n+1)({\mathbb R}) \times \mathrm {SO}(n+1)({\mathbb R})$ and the component groups $ \pi _{0}(G({\mathbb R})) = \pi _{0}(K_{\infty }) \ \simeq \ ({\mathbb Z}/ 2{\mathbb Z} \times {\mathbb Z}/ 2{\mathbb Z})^{\mathsf {r_F}}.$ For any $m \geq 1$ , let’s write $\delta _m = \mathrm {diag}(-1,1,\dots ,1),$ and put $\mathrm {SO}(m)({\mathbb R})^\dagger := \mathrm {SO}(m)({\mathbb R})\delta _m.$ Then $\mathrm {SO}(m)({\mathbb R})$ and $\mathrm {SO}(m)({\mathbb R})^\dagger $ are the connected components of $\mathrm {O}(m)({\mathbb R}).$ For $v \in S_\infty $ , define a subgroup $\overline {K}_{v}$ of $K_v$ by

(6) $$ \begin{align} \overline {K}_{v} \ := \ h(\mathrm{SO}(n+1)({\mathbb R}) \times \mathrm{SO}(n+1)({\mathbb R})) \cup h(\mathrm{SO}(n+1)({\mathbb R})^\dagger \times \mathrm{SO}(n+1)({\mathbb R})^\dagger) \end{align} $$

and let $\overline {K}_{\infty } : = \prod \limits _{v \in S_{\infty }} \overline {K}_{v} $ . The component group of $\overline {K}_{\infty }$ will be described later and will be used in the context of boundary cohomology.

1.3 Adèlic locally symmetric spaces

Let ${\mathbb A}$ be the ring of adèles of ${\mathbb Q}$ , and ${\mathbb A}_f$ denote the finite adèles. Let $K_{f} = \prod \limits _{v \notin S_{\infty }} K_v \subset G({\mathbb A}_{f})$ be an open compact subgroup. We will work with the adèlic symmetric space for G defined by

$$ \begin{align*}{\mathcal S}^{G}: = G({\mathbb A}) / \overline{K}_{\infty} K_{f} = G({\mathbb R})/\overline{K}_{\infty} \times G({\mathbb A}_f)/K_{f}.\end{align*} $$

The discrete subgroup $G({\mathbb Q})$ of $G({\mathbb A})$ acts properly discontinuously on $G({\mathbb A}) / \overline {K}_{\infty } K_{f} $ . The adèlic locally symmetric space ${\mathcal S}^{G}_{K_{f}}$ , with level structure $K_{f} $ is the quotient space given by

$$ \begin{align*}G({\mathbb A}) / \overline{K}_{\infty} K_{f} \twoheadrightarrow G({\mathbb Q}) \backslash G({\mathbb A}) /\overline{K}_{\infty} K_{f}. \end{align*} $$

Note the technical artifice of dividing by $\overline {K}_{\infty }$ ; this will be useful later (see Section 6.6.6). In principle, we can divide by any subgroup in between $K_\infty ^\circ $ and $K_\infty .$ When we associate similar locally symmetric spaces for Levi subgroups of G, and compute the action of the group of components on the cohomology of a discrete series representation, in preparation to describe parts of the cohomology of the Borel–Serre boundary, then the virtue of dividing by $\overline {K}_\infty $ will become clear. The level structure $K_f$ will be assumed to be neat, which ensures that for any $\underline g_f \in G({\mathbb A}_f)$ , the discrete subgroup $G({\mathbb Q}) \cap \underline g_f K_f \underline g_f^{-1}$ of $G({\mathbb R})$ is torsion free. Given any open compact subgroup $K_f$ , we may pass to a subgroup of finite index which is neat. Later, we will also assume that $K_f$ is a principal congruence subgroup (see 7.2.2).

1.4 The character module and dominant integral weights

Let E be a ‘large enough’ finite Galois extension of ${\mathbb Q}$ such that $\mathrm {Hom}(F, {\mathbb R}) = \mathrm {Hom}(F, E)$ . The meaning of ‘large enough’ is usually clear from the context; we will want some Hecke summands in inner cohomology to split over E. Let $X^{\ast }(T \times E) = \mathrm {Hom}_{\mathrm {alg}} (T \times E, \mathbb G_m)$ be the group of characters of $T \times E.$ We have

$$ \begin{align*}X^*(T \times E) = \bigoplus_{\tau : F \to E} X^{\ast}(T_{0} \times_{F, \tau} E). \end{align*} $$

A character $\lambda \in X^*(T \times E),$ also called an integral weight for T, will be written in the above decomposition as $\lambda = (\lambda ^\tau )_{\tau : F \to E},$ with $\lambda ^\tau \in X^*(T_0 \times _{F,\tau } E) = \mathrm {Hom}_{\mathrm {alg}} (T_0 \times _{F,\tau } E, \mathbb G_m);$ furthermore, we write $\lambda ^{\tau } = (\lambda _0^{\tau }, \lambda _1^{\tau }, \ldots , \lambda _n^{\tau }),$ with integrality of $\lambda $ meaning that $\lambda ^\tau _j \in {\mathbb Z}$ , and describe its values on elements of the diagonal torus by

$$ \begin{align*}\lambda^{\tau}(\text{diag} [ t_0, \ldots, t_n, t_n^{-1}, \ldots, t_{0}^{-1} ]) \ = \ \prod \limits_{i=0}^{n} t_i^{\lambda_i^{\tau}}. \end{align*} $$

The Galois group $\mathrm {Gal} (E / {\mathbb Q})$ acts on $X^{\ast }(T \times E)$ by $ ({}^{\eta } \lambda )^{\tau } : = \lambda ^{\eta ^{-1} \circ \tau },$ for all $\lambda \in X^{\ast }(T \times E), \tau : F \to E$ , and $\eta \in \mathrm {Gal} (E / {\mathbb Q}).$ An integral weight $\lambda = (\lambda ^{\tau })_{\tau : F \rightarrow E} \in X^{\ast }(T \times E)$ is said to be a dominant with respect to Borel subgroup B if for all $\tau : F \to E$ , we have

$$ \begin{align*}\lambda_0^{\tau} \geq \lambda_1^{\tau} \geq \ldots \geq \lambda_{n-1}^{\tau} \geq |\lambda_{n}^{\tau}|. \end{align*} $$

We denote the set of dominant integral weights for T by $X^{\ast }_{+}(T \times E)$ .

1.5 The sheaf $\widetilde {\mathcal M}_{\lambda , E}$ and its cohomology

1.5.1

Let $G^{\circ } = \mathrm {Res}_{F/{\mathbb Q}}(\mathrm {SO}(2n+2)/F)$ denote the connected component of the identity of $G.$ Given $\lambda = (\lambda ^{\tau })_{\tau : F \rightarrow E} \in X^{\ast }_+(T \times E)$ as above, let $(\rho _{\lambda }, \mathcal N_{\lambda , E})$ be the absolutely irreducible finite-dimensional representation of the group $G^{\circ } \times E$ with highest weight $\lambda $ . Consider the induced representation $\mathrm {Ind}^{\, G \times E }_{G^{\circ } \times E} (\rho _{\lambda });$ we denote its representation space as $\mathcal M_{\lambda , E}$ . We may parse ${\mathcal M}_{\lambda , E}$ over the embeddings $\tau : F \to E$ as follows: let $\mathcal N_{\lambda ^{\tau }, E}$ be the absolutely irreducible finite-dimensional representation of $\mathrm {SO}(2n+2) \!\times _{F, \tau }\!E$ with highest weight $\lambda ^{\tau }$ ; induce $\mathcal N_{\lambda ^{\tau }, E}$ to a representation ${\mathcal M}_{\lambda ^{\tau }, E}$ of $\mathrm {O}(2n+2) \!\times _{F, \tau }\!E$ . Then

$$ \begin{align*} {\mathcal M}_{\lambda, E} = \bigotimes_{\tau: F \rightarrow E} \mathcal M_{\lambda^{\tau}, E}. \end{align*} $$

The tensor product is of E-vector spaces.

1.5.2

Consider the quotient map $ \pi : G(\mathbb A) / \overline {K}_{\infty } K_{f} \rightarrow G({\mathbb Q}) \backslash G(\mathbb A) / \overline {K}_{\infty } K_{f} $ . The representation ${\mathcal M}_{\lambda ,E}$ gives a sheaf $\widetilde {\mathcal M}_{\lambda ,E}$ of E-vector spaces on the space ${\mathcal {S}}^G_{K_{f}}$ defined as follows: the sections over any open subset V of ${\mathcal {S}}^G_{K_{f}}$ are given by

$$ \begin{align*} \widetilde{\mathcal M}_{\lambda,E}(V) \ := \ \{s: \pi^{-1}(V) \rightarrow \mathcal M_{\lambda,E} \ | \ s\text{ is smooth, and} \\ s(\gamma \underline x) = \rho_{\lambda}(\gamma) s(\underline x), \ \forall~\gamma \in G({\mathbb Q}), \ \underline x \in \pi^{-1}(V) \}, \end{align*} $$

where, smooth means locally constant at non-archimedean places, and infinitely differentiable at the archimedean places.

1.6 The cohomology of $\widetilde {\mathcal M}_{\lambda , E}$

1.6.2 A long exact sequence.

Let $ \bar {\mathcal {S}}^G_{K_f}$ be the Borel–Serre compactification of $\mathcal {S}^G_{K_f}$ ; that is, $ \bar {\mathcal {S}}^G_{K_f} = \mathcal {S}^G_{K_f} \cup \partial \mathcal {S}^G_{K_f}$ , where the boundary is stratified as $\partial \mathcal {S}^G_{K_f} = \cup _P \partial _P\mathcal {S}^G_{K_f},$ with P running through the $G({\mathbb Q})$ -conjugacy classes of proper parabolic subgroups defined over ${\mathbb Q}$ . (See Borel–Serre [Reference Borel and J-P7].) The sheaf $\widetilde {\mathcal {M}}_{\lambda , E}$ on $\mathcal {S}^G_{K_f}$ naturally extends to a sheaf on $ \bar {\mathcal {S}}^G_{K_f}$ that we also denote by $\widetilde {\mathcal {M}}_{\lambda , E}$ . The inclusion $\mathcal {S}^G_{K_f} \hookrightarrow \bar {\mathcal {S}}^G_{K_f}$ is a homotopy equivalence, and hence, the restriction from $ \bar {\mathcal {S}}^G_{K_f}$ to $\mathcal {S}^G_{K_f}$ induces an isomorphism in cohomology The cohomology of the boundary $H^\bullet (\partial \mathcal {S}^G_{K_f}, \widetilde {\mathcal {M}}_{\lambda , E})$ and the cohomology with compact supports $H^\bullet _c(\mathcal {S}^G_{K_f}, \widetilde {\mathcal {M}}_{\lambda , E})$ are naturally modules for $ \pi _0(G({\mathbb R})) \times \mathcal {H}^G_{K_f}.$ We have the following long exact sequence of $\pi _0(G({\mathbb R})) \times \mathcal {H}^G_{K_f}$ -modules:

$$ \begin{align*}\cdots \longrightarrow H^\bullet_c(\mathcal{S}^G_{K_f}, \widetilde{\mathcal{M}}_{\lambda, E})\stackrel{\mathfrak{i}^\bullet}{\longrightarrow} H^\bullet( \bar{\mathcal{S}}^G_{K_f}, \widetilde{\mathcal{M}}_{\lambda,E})\stackrel{\mathfrak{r}^\bullet}{\longrightarrow } &H^\bullet(\partial \mathcal{S}^G_{K_f}, \widetilde{\mathcal{M}}_{\lambda,E}) \\&\quad\ \ \ \xrightarrow{\partial^{\bullet}}H^{\bullet}_c(\mathcal{S}^G_{K_f}, \widetilde{\mathcal{M}}_{\lambda,E}) \longrightarrow \cdots . \end{align*} $$

1.6.4 Cohomology of a boundary stratum $\partial _P\mathcal {S}^G_{K_f}$ .

There is a spectral sequence whose $E_1^{pq}$ -term is built from the cohomology of various boundary strata $\partial _P\mathcal {S}^G_{K_f}$ that converges to the cohomology of $\partial {\mathcal {S}^G_{K_f}}$ (see [Reference Harder and Raghuram16, Sect. 4.1]). For the moment, P will denote any (representative of a $G({\mathbb Q})$ -conjugacy class of a) proper maximal parabolic subgroup of G, but from Section 1.7, it will denote a particular parabolic subgroup. The cohomology of any individual boundary strata $\partial _P\mathcal {S}^G_{K_f}$ is described in terms of representations induced from the cohomology of the Levi quotient of that boundary strata; we will briefly recall the essentials here while referring the reader to [Reference Harder and Raghuram16, Sect. 4.2] for more details. It is convenient to pass to the limit over all open-compact subgroups $K_f$ and define

$$ \begin{align*}H^\bullet(\partial_P\mathcal{S}^G, \widetilde{\mathcal{M}}_{\lambda,E}) \ := \ \varinjlim_{K_f} \, H^\bullet(\partial_P\mathcal{S}^G_{K_f}, \widetilde{\mathcal{M}}_{\lambda,E}),\\[-20pt] \end{align*} $$

on which there is an action of $G({\mathbb A}_f)$ ; for a given $K_f$ , we may retrieve $H^\bullet (\partial _P\mathcal {S}^G_{K_f}, \widetilde {\mathcal {M}}_{\lambda ,E})$ by taking the $K_f$ invariants: $H^\bullet (\partial _P\mathcal {S}^G, \widetilde {\mathcal {M}}_{\lambda ,E})^{K_f}.$ Let $\kappa _P : P \to P/U_P = M_P$ be the canonical map from P onto its Levi quotient $M_P$ by going modulo its unipotent radical $U_P.$ Define $K^{M_P}_\infty = \kappa _P(P({\mathbb R}) \cap \overline {K}_\infty );$ later, we will carefully analyse the inner structure of $K^{M_P}_\infty ,$ for a particular maximal parabolic subgroup. For any level structure $C_f \subset M_P({\mathbb A}_f)$ , define a locally symmetric space

(7) $$ \begin{align} \underline{\mathcal{S}}^{M_P}_{C_f} \ = \ M_P({\mathbb Q}) \backslash M_P({\mathbb A}) / K^{M_P}_\infty C_f. \end{align} $$

Again, it is convenient to pass to the limit over all open-compact subgroups $C_f;$ define

$$ \begin{align*}H^\bullet(\underline{\mathcal{S}}^{M_P}, \widetilde{\mathcal{M}}) \ := \ \varinjlim_{C_f} \, H^\bullet(\underline{\mathcal{S}}^{M_P}_{C_f}, \widetilde{\mathcal{M}}). \end{align*} $$

We recall Proposition 4.2 of loc.cit.:

Proposition 8. The cohomology of the boundary stratum for P is given by

$$ \begin{align*}H^\bullet(\partial_P\mathcal{S}^G, \widetilde{\mathcal{M}}_{\lambda,E}) \ = \ {}^{\mathrm{a}}\mathrm{Ind}_{\pi_0(P({\mathbb R})) \times P({\mathbb A}_f)}^{\pi_0(G({\mathbb R})) \times G({\mathbb A}_f)} \left( H^\bullet (\underline{\mathcal{S}}^{M_P}, \widetilde{H^\bullet(\mathfrak{u}_P, {\mathcal M}_{\lambda,E})} ) \right), \end{align*} $$

where the notations are as follows: The Lie algebra of $U_P$ is $\mathfrak {u}_P$ ; $H^\bullet (\mathfrak {u}_P, {\mathcal M}_{\lambda ,E})$ is the Lie algebra cohomology of ${\mathcal M}_{\lambda ,E}$ that is naturally an $M_P$ -module giving us a sheaf on $\underline {\mathcal {S}}^{M_P}_{C_f};$ the cohomology group on the right-hand side is a module for $\pi _0(M_P({\mathbb R})) \times M_P({\mathbb A}_f);$ note that $\pi _0(M_P({\mathbb R})) = \pi _0(P({\mathbb R}))$ ; ${}^{\mathrm {a}}\mathrm {Ind}$ stands for algebraic, or un-normalized, induction which is the usual notion for $P({\mathbb A}_f)$ to $G({\mathbb A}_f)$ , whereas induction from $\pi _0(P({\mathbb R}))$ to $\pi _0(G({\mathbb R}))$ means that we take invariants under the kernel of the canonical map $\pi _0(P({\mathbb R})) \to \pi _0(G({\mathbb R})).$

A celebrated theorem of Kostant [Reference Kostant23] asserts that as an $M_P \times E$ -module, one has a multiplicity-free decomposition

(9) $$ \begin{align} H^q(\mathfrak{u}_P, {\mathcal M}_{\lambda, E}) \ \simeq \ \bigoplus_{\substack{w \in W^P \\ l(w) = q}} {\mathcal M}_{w \cdot \lambda, E}, \end{align} $$

where $W^P$ is a special subset of the Weyl group of G consisting of all Kostant representatives (see Section 5.2 below), and for any element w in the Weyl group, $l(w)$ will denote its length defined in terms of simple reflections; the twisted action of w on a weight $\lambda $ is denoted $w \cdot \lambda ;$ see Section 5.1. When P is of the form $\mathrm {Res}_{F/{\mathbb Q}}(P_0)$ for a parabolic subgroup $P_0$ of $G_0$ , the elements of $W^P$ and the above decomposition of unipotent cohomology can be parsed over $\tau : F \to E$ as described in [Reference Harder and Raghuram16, Sect. 4.2.2]. Now, Proposition 8 together with (9) gives the following.

Proposition 10. The cohomology of $\partial _P\mathcal {S}^G$ is given by

$$ \begin{align*}H^q(\partial_P\mathcal{S}^G, \widetilde{\mathcal{M}}_{\lambda, E}) \ = \ \bigoplus_{w \in W^P} {}^{\mathrm{a}}\mathrm{Ind}_{\pi_0(P({\mathbb R})) \times P({\mathbb A}_f)}^{\pi_0(G({\mathbb R})) \times G({\mathbb A}_f)} \left(H^{q - l(w)}(\underline{\mathcal{S}}^{M_P}, \widetilde{\mathcal{M}}_{w \cdot \lambda, E}) \right). \end{align*} $$

We will be especially interested in certain summands in the cohomology in some special degrees of the boundary stratum $\partial _P\mathcal {S}^G$ for one particular maximal parabolic subgroup P of G that we describe in the next subsection.

1.7 The root system for $G_0$ and the maximal parabolic subgroup $P_0$

The root system for $G_0$ with respect to the split torus $T_0$ is given by $\Delta _{G_0}: = \left\{ \pm e_i \pm e_j: 0 \leq i < j \leq n \right\},$ where $e_i$ is defined by $e_i (\text {diag} [ t_0, \ldots , t_n, t_n^{-1}, \ldots , t_{0}^{-1} ])= t_i$ for each $0 \leq i \leq n$ . The positive roots with respect to the Borel subgroup $B_0$ are $\Delta ^{+}_{G_0} = \left\{ e_i \pm e_j: 0 \leq i < j \leq n \right\}$ . Half the sum of positive roots for $G_0$ is

$$ \begin{align*}{{\boldsymbol{\rho}}}_{G_{0}} = \sum \limits_{j=0}^{n} (n-j) e_j = ne_0 + (n-1) e_1 \ldots + 1e_{n-1} + 0e_n. \end{align*} $$

The simple roots are

$$\begin{align*}\Pi_{G_0} \ := \ \{ \alpha_0 = e_0 - e_1, \ \ \alpha_1 = e_1 - e_2, \ \ \ldots \ \, \ \alpha_{n-1} = e_{n-1} - e_n, \ \ \alpha_n = e_{n-1} + e_n\}. \end{align*}$$

Henceforth, we let $P_0$ be the maximal parabolic subgroup of $G_0$ corresponding to deleting the first simple root $\alpha _0 = e_0 - e_1$ . The unipotent radical of $P_0$ is given by

$$ \begin{align*}U_{P_0}= \left\{ u_{y_1, y_2, \ldots, y_{2n}} = \left( \begin{array}{ccccccc} 1 & y_1 & y_2 & \ldots & y_{2n} & 0 \\ & 1 & & & &-y_{2n} \\ & & 1 & & & -y_{2n-1} \\ & & & \ddots & & \vdots \ \\ & & & & \ddots & -y_{1} \\ & & & & & 1 \end{array}\right) \ \ | \ \ y_1,\dots,y_{2n} \in \mathbb G_{a} \right\}. \end{align*} $$

The Levi decomposition is $P_0 = M_{P_0} U_{P_0},$ where

$$ \begin{align*}M_{P_0} = P_0 / U_{P_0} \cong A_0 \times {}^{\circ}M_{P_0}, \quad A_0 \cong \mathrm{GL}_1/F, \quad {}^{\circ}M_{P_0} \cong \mathrm{O}(2n)/ F. \end{align*} $$

In matrix form, $M_{P_0}$ is given by

$$ \begin{align*}M_{P_0} = \left\{ \left( \begin{array}{ccc} x&& \\ &g& \\&&x^{-1} \end{array} \right) \ | \ x \in \mathrm{GL}(1), \ g \in \mathrm{O}(2n) \right\}. \end{align*} $$

We put $P = \mathrm {Res}_{F/{\mathbb Q}}(P_0).$ Then $M_P = \mathrm {Res}_{F/{\mathbb Q}}(M_{P_0})$ is its Levi quotient. The parabolic subgroup P of G being fixed, we often drop P from the notation. The diagonal torus $T_0$ of $G_0$ is contained in $M_0 := M_{P_0}$ , and we can write it as $T_0 = A_0 \times {}^\circ T_0$ , where ${}^\circ T_0$ is the diagonal torus in ${}^{\circ }M_0 := {}^\circ M_{P_0}.$ After restricting scalars from F to ${\mathbb Q}$ , we have $T = A \times {}^\circ T.$ The discussions about locally symmetric spaces, finite-dimensional algebraic representations, the corresponding sheaves and their cohomology for G are all applicable mutatis mutandis to ${}^\circ \! M := {}^\circ \! M_P = \mathrm {Res}_{F/{\mathbb Q}}({}^\circ \! M_{P_0}).$

1.8 Dimensions of certain adèlic locally symmetric spaces

Some piquant numerology involving various dimensions plays a role in our cohomology computations; for later use, we record some relevant dimensions. For notational simplicity, we will henceforth denote the groups $M_{P_0}/F$ and $^\circ M_{P_0}/F$ by $M_0$ and $^\circ M_{0}$ , respectively. The dimension of the locally symmetric space $\mathcal {S}^G_{K_f}$ for $G = \mathrm {Res}_{F/{\mathbb Q}}(\mathrm {O}(2n+2)/F)$ with level structure $K_f$ is

$$ \begin{align*}\mathrm{dim}(\mathcal S^G_{K_f}) \ = \ \mathsf{r_F} \cdot (n+1)^2. \end{align*} $$

The dimension of its Borel–Serre boundary $\partial \mathcal S^G_{K_f}$ is

$$ \begin{align*}\mathrm{dim}(\partial \mathcal S^G_{K_f}) \ = \ \mathsf{r_F} \cdot (n+1)^2 - 1. \end{align*} $$

Now consider $^{\circ }\!M = \mathrm {Res}_{F/{\mathbb Q}}(\mathrm {O}(2n)/F)$ ; the connected component of the identity of the maximal compact subgroup $K_{{}^\circ \!M({\mathbb R})}$ of ${}^\circ \!M({\mathbb R})$ is

$$ \begin{align*}K^{\circ}_{{}^\circ\!M({\mathbb R})} \ = \ \prod \limits_{\tau \in \Sigma_F} \mathrm{SO}(n) \times \mathrm{SO}(n), \end{align*} $$

and for any open-compact subgroup $K_{{}^{\circ }\!M_f}$ of ${}^{\circ }\!M({\mathbb A}_f)$ , we define

(11) $$ \begin{align} {\mathcal S}^{{}^\circ\!M}_{K_{{}^{\circ}\!M_f}} \ = \ {}^{\circ}\!M({\mathbb Q}) \backslash {}^{\circ}\!M({\mathbb A}) / K^{\circ}_{{}^\circ\!M({\mathbb R})} K_{{}^{\circ}\!M_f}. \end{align} $$

Whereas for $\mathcal {S}^G_{K_f}$ we divided by $\overline {K}_\infty $ which is in between the full maximal compact subgroup at infinity and its connected component, for $\mathrm {dim}(S^{{}^{\circ }\!M}_{K_{{}^{\circ }\!M_f}})$ , we divide by the connected component of the maximal compact subgroup at infinity. We have

$$ \begin{align*}\mathrm{dim}({\mathcal S}^{{}^{\circ}\!M}_{K_{{}^{\circ}\!M_f}}) \ = \ \mathsf{r_F} \cdot n^2, \end{align*} $$

and its middle dimension (since n is even) is

$$ \begin{align*}q_m \ := \ \dfrac{1}{2} ~\mathrm{dim} ~\mathcal S^{^{\circ}M}_{K_{{}^{\circ}M,f}} \ = \ \dfrac{ \mathsf{r_F} \cdot n^2}{2}. \end{align*} $$

This particular middle-dimension is interesting since it is only in this middle-degree that a discrete series representation of ${}^{\circ }M({\mathbb R})$ has nontrivial cohomology.

Write $M = A \times {}^{\circ }\!M$ , where $A = \mathrm {Res}_{F/{\mathbb Q}}(\mathrm {GL}_1)$ . Note that A contains a copy of $\mathbb G_m/{\mathbb Q}$ , which we denote by S. We have the group of real points:

$$ \begin{align*}A({\mathbb R}) = \prod \limits_{\tau \in \Sigma_F} {\mathbb R}^{\times}, \quad {}^{\circ}\!M({\mathbb R}) = \prod \limits_{\tau \in \Sigma_F} \mathrm{O}(n,n)({\mathbb R}), \quad M({\mathbb R}) = A({\mathbb R}) \times {}^{\circ}\!M({\mathbb R}). \end{align*} $$

The group $S({\mathbb R}) = {\mathbb R}^\times $ sits diagonally in $A({\mathbb R})$ . Let $\kappa _P : P \to P/U_P = M$ be the canonical surjection. Let $C_f$ be the open compact subgroup of $M({\mathbb A}_f)$ defined by

$$ \begin{align*}C_f = \kappa_P (K_f \cap P({\mathbb A}_f)). \end{align*} $$

We suppose that $K_f$ is such that we can write $C_f = B_f \times {}^{\circ }C_f,$ for an open compact subgroup $B_f$ of $A({\mathbb A}_f)$ and an open compact subgroup ${}^{\circ }C_f$ of ${}^{\circ }M({\mathbb A}_f)$ . Define the locally symmetric space for A as

$$ \begin{align*}{\mathcal S}^{A}_{B_f} \ := \ A({\mathbb Q}) \backslash A({\mathbb A}) / S({\mathbb R})^{\circ}B_f \ = \ F^{\times} \backslash {\mathbb A}_{F}^{\times} / S({\mathbb R})^{\circ}B_f; \quad \dim({\mathcal S}^{A}_{B_f}) = \mathsf{r}_F-1, \end{align*} $$

and the locally symmetric space for M as

(12) $$ \begin{align} {\mathcal S}^{M}_{C_f} \ := \ {\mathcal S}^{A}_{B_f} \times {\mathcal S}^{{}^{\circ}\!M}_{{}^{\circ}C_f}. \end{align} $$

Hence, we see

$$ \begin{align*}\dim({\mathcal S}^{M}_{C_f}) \ = \ \mathsf{r}_F -1 + \mathsf{r}_F \cdot n^2 \ = \ 2q_m + \mathsf{r}_F -1. \end{align*} $$

For later use, we denote

(13) $$ \begin{align} \begin{aligned} q_b \ & := \ q_m \ = \ \mathsf{r}_F \cdot n^2/2, \\ q_t \ & := \ q_m + \mathsf{r}_F -1 \ = \ \mathsf{r}_F \cdot n^2/2 + \mathsf{r}_F -1. \end{aligned} \end{align} $$

Note: $q_b + q_t = \dim ({\mathcal S}^{M}_{C_f}).$ Furthermore, let’s denote

(14) $$ \begin{align} \begin{aligned} \mathfrak{q}_b \ & := \ q_b + \tfrac{1}{2}\dim(U_P) \ = \ \mathsf{r}_F(n^2/2 + n), \\ \mathfrak{q}_t \ & := \ q_t + \tfrac{1}{2}\dim(U_P) \ = \ \mathsf{r}_F(n^2/2 + n +1)-1. \end{aligned} \end{align} $$

Note: $\mathfrak {q}_b + \mathfrak {q}_t = \mathrm {dim}(\partial \mathcal S^G_{K_f}).$ The subscripts $b, m$ and t are meant to suggest ‘bottom’, ‘middle’ and ‘top’.

1.9 Cohomology of spaces associated to the Levi subgroup $M_P$

There is a delicate interplay between the various ‘locally symmetric spaces’ attached to the Levi quotient $M_P$ of our parabolic subgroup $P.$ As before, we abbreviate $M_P$ as $M.$ Here, we clarify the mutual relationship between these spaces and their cohomology. Fix a level structure $C_f \subset M({\mathbb A}_f)$ which we assume to be of the form $B_f \times {}^\circ \!C_f$ as in the previous subsection. Define $\tilde {{\mathcal S}}^{A}_{B_f} := A({\mathbb Q}) \backslash A({\mathbb A}) /B_f,$ and let $\theta _{A} : \tilde {{\mathcal S}}^{A}_{B_f} \to {\mathcal S}^{A}_{B_f}$ denote the canonical map which is a $S({\mathbb R})^\circ $ -fibration. Now define

(15) $$ \begin{align} \tilde{{\mathcal S}}^{M}_{C_f} \ := \ \tilde{{\mathcal S}}^{A}_{B_f} \times {\mathcal S}^{{}^{\circ}\!M}_{{}^{\circ}C_f}, \end{align} $$

which comes with the fibration $\theta _{M} : \tilde {{\mathcal S}}^{M}_{C_f} \to {\mathcal S}^{M}_{C_f},$ defined as $\theta _M = \theta _A \times 1_{{\mathcal S}^{{}^{\circ }\!M}_{{}^{\circ }C_f}}.$ We have the following spaces:

• • ${\mathcal S}^{M}_{C_f}$ ; defined in (12); it is the space that we can best refer to as the locally symmetric space attached to M.

• • $\tilde {{\mathcal S}}^{M}_{C_f}$ , defined in (15); it comes with the fibration $\theta _M : \tilde {{\mathcal S}}^{M}_{C_f} \to {\mathcal S}^{M}_{C_f}.$

• • $\underline {\mathcal {S}}^{M}_{C_f}$ , defined in (7); note that we have a finite-cover $\tilde {{\mathcal S}}^{M}_{C_f} \to \underline {\mathcal {S}}^{M_P}_{C_f}$ whose fibre is the kernel of the map $\pi _0(M({\mathbb R})) \to \pi _0(G({\mathbb R})).$

Let us clarify the relation between the cohomology groups attached to these spaces and a sheaf $\widetilde {\mathcal {M}}$ (which for us will be of the form $\widetilde {\mathcal {M}}_{\mu ,E}$ for a dominant integral weight $\mu $ for M) that is defined compatibly on all these three spaces. Denote

(16) $$ \begin{align} \overline{\pi_0}(M({\mathbb R})) \ := \ \mathrm{Kernel}\left(\pi_0(M({\mathbb R})) \to \pi_0(G({\mathbb R})) \right). \end{align} $$

In the description of the cohomology of a boundary stratum (see Proposition 10), one is naturally led to the cohomology of $\underline {\mathcal {S}}^{M}_{C_f}$ , which is related to the cohomology of $\tilde {{\mathcal S}}^{M}_{C_f}$ via

(17) $$ \begin{align} H^\bullet(\underline{\mathcal{S}}^{M}_{C_f}, \widetilde{\mathcal{M}}) \ = \ H^\bullet(\tilde{{\mathcal S}}^{M}_{C_f}, \widetilde{\mathcal{M}})^{\overline{\pi_0}(M({\mathbb R}))}. \end{align} $$

Next, the cohomology of the locally symmetric space ${\mathcal S}^{M}_{C_f}$ can be pulled back, via the map induced by $\theta _{M}$ , to the cohomology of $\tilde {{\mathcal S}}^{M}_{C_f}$ ; that is, we have a map

(18) $$ \begin{align} \theta_M^* : H^\bullet({\mathcal S}^{M}_{C_f}, \widetilde{\mathcal{M}}) \ \to \ H^\bullet(\tilde{{\mathcal S}}^{M}_{C_f}, \widetilde{\mathcal{M}}). \end{align} $$

2.1 Discrete series representations for a connected semisimple Lie group

We briefly review the general theory which is due to Harish-Chandra, and we refer the reader to Borel–Wallach [Reference Borel and Wallach8, II.5] for more details and references. Just for this subsection, we let G stand for a connected semisimple Lie group, and K a maximal compact subgroup of $G.$ Let $\mathfrak g$ and $\mathfrak k$ be the Lie algebras of G and K, respectively. Denote their complexfications as $\mathfrak {g}_{\mathbb C}$ and $\mathfrak {k}_{\mathbb C},$ respectively. Suppose that $\mathrm {rank}(G) = \mathrm {rank}(K)$ . Let $\mathfrak h$ be a Cartan subalgebra of both $\mathfrak g$ and $\mathfrak k$ , and $\mathfrak h_{{\mathbb C}}$ its complexification. Let $\Phi $ be the root system of $\mathfrak g_{{\mathbb C}}$ with respect to $\mathfrak h_{{\mathbb C}}$ and let $P(\Phi )$ be the weight lattice for $\Phi $ . A weight $\Lambda \in P(\Phi )$ is called regular if $\langle \Lambda , \alpha \rangle \neq 0 ~\text {for any} ~\alpha \in \Phi .$ A regular weight $\Lambda $ gives a positive system of roots and their half-sum:

$$ \begin{align*}\Phi^+: = \{ \alpha \in \Phi: \langle \Lambda, \alpha \rangle> 0 \}, \quad {\boldsymbol{\rho}}: = \dfrac{1}{2} \sum \limits_{\alpha \in \Phi^+ } \alpha. \end{align*} $$

Denote the set of compact roots by $\Phi _K$ ; a compact root is a root whose rootspace lies in $\mathfrak k_{{\mathbb C}}$ . The positive system $\Phi ^+$ gives a positive system for compact roots and their half-sum:

$$ \begin{align*}\Phi^+_K: = \Phi_K \cap \Phi^+; \quad {\boldsymbol{\rho}}_K: = \frac{1}{2} \sum \limits_{\alpha \in \Phi^+_K } \alpha. \end{align*} $$

Let be the Harish-Chandra isomorphism for the center $Z(\mathfrak U(\mathfrak g_{{\mathbb C}}))$ of the enveloping algebra $\mathfrak U(\mathfrak g_{{\mathbb C}})$ of $\mathfrak g_{{\mathbb C}};$ the Weyl group of G is denoted $W_G.$ For each regular weight $\Lambda \in P(\Phi )$ , we can associate a discrete series representation $\pi _{\Lambda }$ of G such that

1. (1) The infinitesimal character of $\pi _{\Lambda }$ is $\chi _{\Lambda }$ defined by $\chi _{\Lambda }(z) = \Lambda (\gamma (z)), \forall z \in Z(\mathfrak U(\mathfrak g_{{\mathbb C}})).$ Thus, $\pi _{\Lambda }$ and $\pi _{\Lambda '}$ have same infinitesimal character if and only if there is $w \in W_G$ such that $w\lambda = \lambda '$ .

2. (2) The lowest K-type of ${\pi _{\Lambda }}|_{K}$ has multiplicity one and its highest weight (called the Blattner parameter) is given by $ \Lambda + {\boldsymbol {\rho }} - 2{\boldsymbol {\rho }}_K$ .

3. (3) The representations $\pi _{\Lambda }$ and $\pi _{\Lambda '}$ are equivalent if there is $w \in W_K,$ the Weyl group of K, such that $w\Lambda = \Lambda '$ .

The following proposition describes the cohomology of a discrete series representation; see [Reference Borel and Wallach8, Prop. II.5.3].

2.2 Discrete series representations of orthogonal groups

Fix an archimedean place v of $F.$ Just for this Section 2.2, for brevity, we introduce these notations:

${}^\circ \!M := {}^{\circ }\!M_{P_0 v} = {}^{\circ }\!M_{P_0}(F_v) = \mathrm {O}(n,n)({\mathbb R}),$ the real split even orthogonal group;

${}^\circ \! M^\circ := {}^{\circ }\!M_{P_0 v}^\circ = \mathrm {SO}(n,n)({\mathbb R})^\circ ,$ the connected component of identity in ${}^\circ \!M$ ;

$K := K_{{}^{\circ }\!M_{P_0 v}} = \{g \in M \ | \ {}^{\intercal }g\,g = I_{2n} \}$ , the maximal compact subgroup of ${}^\circ \!M$ ;

$K^\circ := K_{{}^{\circ }\!M_{P_0 v}}^\circ ,$ the connected component of identity in $K,$

${}^{\circ }\mathfrak m$ the Lie algebra of ${}^\circ \!M$ or of ${}^\circ \!M^\circ $ , and $\mathfrak k$ the Lie algebra of K or of $K^\circ $ .

2.2.1 Representatives for the group of connected components.

The compact group K is isomorphic to $\mathrm {O}(n)({\mathbb R}) \times \mathrm {O}(n)({\mathbb R})$ , the isomorphism being the map h defined in (5), but after replacing $n+1$ by n. Inclusion induces an equality $\pi _0(K) = \pi _0({}^{\circ }\!M)$ of component groups; we give explicit representatives for elements of these component groups. Recall that $\delta _n = \mathrm {diag}(-1,1,\dots ,1)$ . Then $\pi _0(\mathrm {O}(n)({\mathbb R})) = \{I_n, \delta _n\}.$ Further, if $g \in \mathrm {O}(n)({\mathbb R})$ , then clearly, $J_n\,g\,J_n \in \mathrm {O}(n)({\mathbb R}).$ Hence, $J_n \, \delta _n \, J_n$ also represents the nontrivial element of $\pi _0(\mathrm {O}(n)({\mathbb R})).$ We take as representatives for the connected components of $\mathrm {O}(n)({\mathbb R}) \times \mathrm {O}(n)({\mathbb R})$ the elements

$$ \begin{align*}\{(I_n,I_n), \ (I_n, \delta_n), \ (J_n\delta_n J_n, I_n), \ (J_n\delta_n J_n, \delta_n)\}. \end{align*} $$

For any integer $m \geq 1$ , let

(20) $$ \begin{align} s_{2m}: = \begin{pmatrix} I_{m-1} & & \\ & -I_2 & \\ & & I_{m-1} \end{pmatrix}, \quad \text{and} \quad \kappa_{2m} : = \begin{pmatrix} I_{m-1} & & \\ & J_2 & \\ & & I_{m-1} \end{pmatrix}. \end{align} $$

One can check that $h(I_n,\delta _n) = \kappa _{2n}$ , and $h(J_n\delta _n J_n, \delta _n) = s_{2n}.$ Furthermore, one sees

$$ \begin{align*}\kappa_{2n} \in \mathrm{O}(n,n)({\mathbb R}) \setminus \mathrm{SO}(n,n)({\mathbb R}), \quad s_{2n} \in \mathrm{SO}(n,n)({\mathbb R}) \setminus \mathrm{SO}(n,n)({\mathbb R})^{\circ}. \end{align*} $$

The component group $\pi _0(K)$ of K (as also $\pi _0({}^{\circ }M)$ of ${}^{\circ }M$ ) is isomorphic to ${\mathbb Z}/2{\mathbb Z} \times {\mathbb Z}/2{\mathbb Z},$ and as representatives, we take

$$ \begin{align*}\left\{I_{2n}, \ s_{2n}, \ \kappa_{2n}, \ s_{2n} \kappa_{2n} \right\}. \end{align*} $$

Note that $s_{2n} \kappa _{2n} = \kappa _{2n}s_{2n}.$

2.2.2 Compact Cartan subgroups and subalgebras.

At the level of Lie algebras, the same map h gives us $\mathfrak t_K$ , a compact Cartan subalgebra that is shared by both ${}^\circ \mathfrak {m}$ and $\mathfrak k_{{}^\circ \!M}$ , the Lie algebras of ${}^{\circ }\!M$ and $K,$ respectively. Let $a_{\theta }: = \left(\begin {smallmatrix} 0 & -\theta \\ \theta & 0 \end {smallmatrix}\right)$ for $\theta \in {\mathbb R}$ . Given $\theta _1, \ldots , \theta _n \in {\mathbb R}$ , let k and l be elements of ${\mathfrak s \mathfrak o}(n)$ , the Lie algebra of the compact $\mathrm {SO}(n)({\mathbb R})$ , defined by

$$ \begin{align*}k = \begin{pmatrix} a_{\theta_1} && \\ &\ddots& \\ && a_{\theta_{r}} \end{pmatrix}, \quad l = \begin{pmatrix} a_{\theta_{r +1}} && \\ &\ddots& \\ && a_{\theta_{n}} \end{pmatrix}; \quad \text{recall that }n = 2r. \end{align*} $$

Then, we check that

$$ \begin{align*}h(k,l) = \begin{pmatrix} a_{(\theta_1-\theta_n)/2} &&&&& a_{(\theta_1+\theta_n)/2}J_{2} \\ & \ddots && &\unicode{x22F0} & \\ && a_{(\theta_r-\theta_{r+1})/2} & \quad a_{(\theta_r+\theta_{r+1})/2}J_2 && \\ \\ && a_{-(\theta_r+\theta_{r+1})/2} J_2 & \quad a_{(\theta_{r+1}-\theta_{r})/2} && \\ & \unicode{x22F0} && & \ddots & \\ a_{-(\theta_1+\theta_n)/2}J_2 &&&&& a_{(\theta_n-\theta_1)/2} \end{pmatrix}. \end{align*} $$

Reparameterizing our variables,

$$ \begin{align*}t_1 = \frac{\theta_1-\theta_n}{2}, \ \ \ t_2 = \frac{\theta_1+\theta_n}{2}, \ \ \dots, \ \ \ t_{n-1} = \frac{\theta_r-\theta_{r+1}}{2}, \ \ \ t_n = \frac{\theta_r+\theta_{r+1}}{2}, \end{align*} $$

we can describe the elements of the compact Cartan subalgebra as

$$ \begin{align*}\mathfrak{t}_K \ = \ \left\{X_{t_1,\dots,t_n} := \begin{pmatrix} a_{t_1} &&&&& a_{t_2}J_{2} \\ & \ddots && &\unicode{x22F0} & \\ && a_{t_{n-1}} & \quad a_{t_n}J_2 && \\ && a_{-t_n} J_2 & \quad a_{-t_{n-1}} && \\ & \unicode{x22F0} && & \ddots & \\ a_{-t_2}J_2 &&&&& a_{-t_1} \end{pmatrix} \ | \ t_1, \dots, t_n \in {\mathbb R} \right\}. \end{align*} $$

Let $T_K$ be the analytic subgroup associated to $\mathfrak t_K$ inside $K^{\circ }$ . Consider the complexification $\mathfrak t_{K,{\mathbb C}}$ of $\mathfrak t_K$ ; its elements are of the form $ X_{t_1, \ldots , t_n},$ where $t_j \in {\mathbb C}$ . It can be observed that all $ X_{t_1,\dots ,t_n} \in \mathfrak t_{K,{\mathbb C}}$ are simultaneously conjugate to elements of the complexified diagonal algebra $\mathfrak t_{{\mathbb C}}$ . The conjugating matrix can be taken in $\mathrm {O}(n,n)({\mathbb C})$ . Under this correspondence, the matrix $ X_{t_1,\dots ,t_n} \in \mathfrak t_{K,{\mathbb C}}$ is conjugate to the diagonal matrix $\mathrm {diag}[{\mathbf i}\theta _1, \ldots , {\mathbf i}\theta _n, -{\mathbf i}\theta _n, \ldots , -{\mathbf i}\theta _1] \in \mathfrak t_{{\mathbb C}}.$

2.2.3 Roots and compact roots for $\mathrm {O}(n,n)({\mathbb R})^\circ $ .

Recalling that ${\mathbf i}{\mathbb R}$ is the Lie algebra of the circle group $S^1$ , we define the linear functionals $e_i^K : \mathfrak t_K \to {\mathbf i} {\mathbb R},$ for $1 \leq i \leq n,$ as

(21) $$ \begin{align} \begin{aligned} e_1^K (X_{t_1,\dots,t_n}) & := \ {\mathbf i}(t_1 + t_2) = {\mathbf i} \theta_1, \quad e_2 ^K(X_{t_1,\dots,t_n}) : = \ {\mathbf i}(t_1 - t_2) = -{\mathbf i} \theta_n, \\ & \vdots \\ e_{n-1}^K (X_{t_1,\dots,t_n}) &: = \ {\mathbf i}(t_{n-1}+t_n) = {\mathbf i} \theta_r, \quad e_n^K (X_{t_1,\dots,t_n}) := {\mathbf i}(t_{n-1}-t_n) = -{\mathbf i} \theta_{r+1}. \end{aligned} \end{align} $$

It is clear then that $\{e_1^K,e_3^K,\dots ,e_{n-1}^K\}$ (resp., $\{e_2^K,e_4^K,\dots ,e_n^K\}$ ) are the linear functionals corresponding to the first (resp., second) summand of $\mathfrak {so}(n) \oplus \mathfrak {so}(n).$ The root system $\Phi $ of ${}^{\circ }\mathfrak {m}_{\mathbb C}$ with respect to $\mathfrak t_{K,{\mathbb C}}$ is given by

$$ \begin{align*}\Phi = \{ \pm e_i^K \pm e_j^K \ | \ 1 \leq i < j \leq n \}. \end{align*} $$

The set of all compact roots $\Phi _K$ is

$$ \begin{align*}\Phi_K = \{ \pm e_i^K \pm e_j^K \ | \ 1 \leq i < j \leq n, \ \ i \equiv j \ (\text{mod} \ 2)\}. \end{align*} $$

The weight lattice is given by

$$ \begin{align*}P_{\Phi}: = \{ \sum \limits_{1 \leq j \leq n} a_j e_j^K \ | \ a_j \in {\mathbb Z} \}. \end{align*} $$

An element $\Lambda = \sum \limits _{1 \leq j \leq n} a_j e_j^K \in P_{\Phi }$ is regular if and only if $\langle \Lambda , \alpha \rangle \neq 0$ for all $\alpha \in \Phi $ , which is equivalent to $ a_i \neq \pm a_j$ for all $i \neq j.$

2.2.6 Dominant weights, discrete series and cohomology.

Let $\mu _1, \ldots , \mu _n$ be integers such that $\mu _1 \geq \mu _2 \geq \ldots \geq \mu _{n-1} \geq | \mu _{n} |$ . Furthermore, we will assume that $|\mu _n| \geq 1.$ Define $\Lambda $ by

$$ \begin{align*}\Lambda = \sum \limits_{1 \leq j \leq n} (\mu_j + n - j) \, e_j^K. \end{align*} $$

Thus, $ \Lambda $ is a regular element of $P_{\Phi }$ and the set of roots $\alpha $ for which $\langle \Lambda , \alpha \rangle> 0$ is precisely the set $\{ e_i ^K\pm e_j^K\ | \ 1 \leq i < j \leq n \}$ . So ${\boldsymbol {\rho }}^c = \sum \limits _{1 \leq j \leq n} ( n - j) \, e_j^K$ and

$$ \begin{align*}\Lambda - {\boldsymbol{\rho}}^c = \sum \limits_{1 \leq j \leq n} \mu_j \, e_j^K \ =: \ \mu. \end{align*} $$

Let $D_{\mu }$ be a discrete series representation of ${}^\circ \! M^{\circ } = \mathrm {SO}(n,n)({\mathbb R})^\circ $ equivalent to $\pi _{\Lambda }$ . By an abuse of notation, we also denote the $({}^{\circ }\mathfrak {m}, K^\circ )$ -module of $K^\circ $ -finite vectors by $D_{\mu }$ . Let $\mathcal N_{\mu }$ be the finite-dimensional irreducible representation of ${}^\circ \!M = \mathrm {SO}(n,n)({\mathbb R})$ with highest weight $\mu $ with respect to the maximal compact torus $T_K$ . Let us bear in mind that the representation ${\mathcal N}_\mu $ admits a ${\mathbb Q}$ -structure since it comes from a purely algebraic theory of highest weight modules for the real points of the split algebraically connected algebraic group $\mathrm {SO}(n,n)$ . For any even positive integer $2m$ , since the dimension of $\mathrm {SO}(2m)$ is $m(2m-1),$ we see that the dimension of the symmetric space $\mathrm {O}(n,n)({\mathbb R})/(\mathrm {O}(n) \times \mathrm {O}(n))$ is $n^2.$ From Proposition 19, we conclude that

$$ \begin{align*}H^{q}({}^{\circ}{\mathfrak m}, {\mathfrak k};\, D_{\mu} \otimes \mathcal N_{\mu}^{\mathsf{v}} ) = \begin{cases} 0 & \text{if }q \neq q_0 := \dfrac{n^2}{2}\text{, and} \\ {\mathbb C} & \text{if }q = q_0. \end{cases} \end{align*} $$

Note that $\mathcal {N}_\mu $ is a self-dual representation (i.e., $\mathcal N_{\mu }^{\mathsf {v}} = \mathcal N_{\mu });$ see, for example, [Reference Balasubramanyam and Raghuram3, Lem. 5.0.2].

2.2.7 The action of the group of connected components on cohomology.

As noted in Section 2.2.1, $s_{2n} \in \mathrm {SO}(n,n)({\mathbb R}) \backslash \mathrm {SO}(n,n)({\mathbb R})^{\circ }$ and $\kappa _{2n} \in \mathrm {O}(n,n)({\mathbb R}) \backslash \mathrm {SO}(n,n)({\mathbb R})$ generate the group of connected components of $K.$ The actions of $s_{2n}$ and $\kappa _{2n}$ on $\mathfrak t_K$ and on the weights $e_j^K$ are given by

• • $\mathrm {Ad}(\kappa _{2n})$ maps $X_{t_1, \ldots , t_n}$ to $X_{t_1, \ldots , t_{n-2}, t _{n}, t_{n-1}},$ and thus,

  $$ \begin{align*}\mathrm{Ad}(\kappa_{2n}) \cdot e_j^K = e_j^K, \ 1 \leq j \leq n-1, \quad \mathrm{Ad}(\kappa_{2n}) \cdot e_n^K = - e_n^K. \end{align*} $$

• • $\mathrm {Ad}(s_{2n})$ maps $X_{t_1, \ldots , t_n}$ to $X_{t_1, \ldots , t_{n-2}, -t _{n-1}, -t_{n}},$ and thus,

  $$ \begin{align*}\mathrm{Ad}(s_{2n}) \cdot e_j ^K= e_j^K, \ 1 \leq j \leq n-2, \quad \mathrm{Ad}(s_{2n}) \cdot e_{n-1}^K = - e_{n-1}^K, \quad \mathrm{Ad}(s_{2n}) \cdot e_n^K = - e_n^K. \end{align*} $$

In particular, $\mathrm {Ad}(\kappa _{2n})$ is not the effect of an element of $W_{{}^\circ \!M}.$ Also, $\mathrm {Ad}(s_{2n})$ is obtained by the action of an element of $W_{{}^{\circ }\!M}$ ; however, from Section 2.2.3, it follows that it is not realizable via $W_{K}.$ Since $|\mu _n| \geq 1$ , we conclude that the discrete series representations

$$ \begin{align*}\{D_{\mu}, \quad {}^{s_{2n}}\!D_{\mu}, \quad {}^{\kappa_{2n}}\!D_{\mu}, \quad {}^{ s_{2n} \, \kappa_{2n}}\!D_{\mu}\} \end{align*} $$

obtained by conjugating by representatives of the group of connected components of ${}^\circ \!M$ are pairwise not equivalent to each other. It follows that

$$ \begin{align*}\mathbb D_{\mu} \ := \ \mathrm{Ind}^{\mathrm{O}(n,n)({\mathbb R})}_{\mathrm{SO}(n,n)({\mathbb R})^{\circ}}(D_{\mu}) \end{align*} $$

is an irreducible representation of $\mathrm {O}(n,n)({\mathbb R})$ . Furthermore, its restriction to $\mathrm {SO}(n,n)({\mathbb R})^{\circ }$ is given by

$$ \begin{align*}{\mathbb D_{\mu}}|_{\mathrm{SO}(n,n)({\mathbb R})^{\circ}} \ = \ D_{\mu} \ \oplus \ {}^{s_{2n}}\!D_{\mu} \ \oplus \ D_{^{\kappa_{2n}} \mu} \ \oplus \ {}^{s_{2n}}\!D_{^{\kappa_{2n}} \mu}, \end{align*} $$

after noting that ${}^{\kappa _{2n}}\!D_{\mu } = D_{^{\kappa _{2n}} \mu }$ by using the action of the torus on a highest weight vector, where if $\mu = \mu _1 \, e_1^K + \cdots + \mu _n \, e_n^K ,$ then ${}^{\kappa _{2n}}\mu = \mu _1 \, e_1^K + \cdots + \mu _{n-1} \, e_{n-1}^K - \mu _n \, e_n^K.$ If we restrict ${\mathbb D}_\mu $ to the intermediate subgroup $\mathrm {SO}(n,n)({\mathbb R})$ , the four representations fuse two at a time to give a multiplicity free direct sum of two irreducible representations of $\mathrm {SO}(n,n)({\mathbb R}):$

$$ \begin{align*}{\mathbb D_{\mu}}|_{\mathrm{SO}(n,n)({\mathbb R})} \ = \ \mathcal D_{\mu} \oplus \mathcal D_{^{\kappa_{2n}}\mu}, \end{align*} $$

where,

$$ \begin{align*}\mathcal D_{\mu}|_{\mathrm{SO}(n,n)({\mathbb R})^{\circ}} \ := \ D_{\mu} \oplus {}^{s_{2n}}\!D_{\mu}, \ \ \text{and} \ \ \mathcal D_{{}^{\kappa_{2n}}\mu}|_{\mathrm{SO}(n,n)({\mathbb R})^{\circ}} \ := \ D_{{}^{\kappa_{2n}}\mu} \oplus {}^{s_{2n}}\!D_{{}^{\kappa_{2n}}\mu}. \end{align*} $$

For the finite-dimensional coefficient systems, let’s define

$$ \begin{align*}\mathcal M_{\mu} \ := \ \mathrm{Ind}^{\mathrm{O}(n,n)({\mathbb R})}_{\mathrm{SO}(n,n)({\mathbb R})^{\circ}} (\mathcal N_{\mu}). \end{align*} $$

Restricting back to $\mathrm {SO}(n,n)({\mathbb R})^{\circ }$ , we get

$$ \begin{align*}{\mathcal M}_{\mu}|_{\mathrm{SO}(n,n)({\mathbb R})^{\circ}} \ = \ {\mathcal N}_\mu \ \oplus \ {}^{\kappa_{2n}}{\mathcal N}_\mu \end{align*} $$

since ${\mathcal N}_\mu $ is an irreducible representation of $\mathrm {SO}(n,n)({\mathbb R}).$ Now we observe that the cohomology $H^{q_0}({}^{\circ }{\mathfrak m}, {\mathfrak k};\, \mathbb D_{\mu } \otimes \mathcal M_{\mu })$ is a direct sum of four one-dimensional cohomology spaces:

(22) $$ \begin{align} H^{q_0}({}^{\circ}{\mathfrak m}, {\mathfrak k};\, D_{\mu} \otimes \mathcal N_{\mu} ) &\ \oplus \ H^{q_0}({}^{\circ}{\mathfrak m}, {\mathfrak k};\, {}^{s_{2n}}D_{\mu} \otimes \mathcal N_{\mu} ) \ \oplus \nonumber\\ &H^{q_0}({}^{\circ}{\mathfrak m}, {\mathfrak k};\,D_{{}^{\kappa_{2n}} \mu} \otimes \mathcal N_{^{\kappa_{2n}} \mu} ) \ \oplus \ H^{q_0}({}^{\circ}{\mathfrak m}, {\mathfrak k};\, {}^{s_{2n}} D_{^{\kappa_{2n}} \mu} \otimes \mathcal N_{^{\kappa_{2n}} \mu} ). \end{align} $$

2.2.8 The parameter of the discrete series representation ${\mathbb D_{\mu }}$ .

Let us recall some preliminaries on the Weil group of ${\mathbb R}$ : $W_{\mathbb R} = {\mathbb C}^\times \cup j\cdot {\mathbb C}^\times ,$ with $jzj^{-1} = \bar {z}$ for $z \in {\mathbb C}^\times .$ For an integer $\ell $ , define $\chi _\ell : {\mathbb C}^\times \to {\mathbb C}^\times $ as $\chi _\ell (z) = (z/\bar {z})^{\ell /2}$ or $\chi _\ell (re^{i\theta }) = e^{i\ell \theta }$ for $z = re^{i\theta } \in {\mathbb C}^\times .$ Define $I(\ell )$ to be the $2$ -dimensional induced representation $\mathrm {Ind}_{{\mathbb C}^\times }^{W_{\mathbb R}}(\chi _\ell ).$ Then $I(\ell ) \simeq I(-\ell )$ ; $I(\ell )$ is irreducible if $\ell \neq 0$ ; $I(0) = 1\!\!1 \oplus \mathrm {sgn}.$

Now, given $\mu $ as above with $\mu _1 \geq \cdots \geq \mu _{n-1} \geq |\mu _n|$ , put

(23) $$ \begin{align} \ell_1 = 2(\mu_1+n-1), \ \ \ell_2 = 2(\mu_2+n-2), \ \ \dots \ \ , \ \ell_{n-1} = 2(\mu_{n-1}+1), \ \ \ell_n = 2|\mu_n|. \end{align} $$

The Langlands parameter of ${\mathbb D_{\mu }}$ is the homomorphism $\varphi ({\mathbb D}_\mu ) : W_{\mathbb R} \to \mathrm {O}(2n)({\mathbb C})$ defined by

$$ \begin{align*}{\mathbb C}^\times \ni z \mapsto (a_1,\dots,a_n,a_n^{-1},\dots,a_1^{-1}), \quad a_i = (z/\bar{z})^{\ell_i/2}; \quad j \mapsto J_{2n}. \end{align*} $$

This follows from from the very general [Reference Borel6, Ex. 10.5]; for the particular example at hand, see also [Reference Raghuram and Sarnobat30, Sect. 5.4]. The parameter is a direct sum of n induced representations:

(24) $$ \begin{align} \varphi({\mathbb D}_\mu) \ = \ I(\ell_1) \oplus \cdots \oplus I(\ell_{n-1}) \oplus I(\ell_n). \end{align} $$

Since $|\mu _n| \geq 1,$ each $\ell _i> 0,$ ensuring the irreducibility of the n-summands.

Let us add a comment that will be relevant in the next section. The parameter $\varphi ({\mathbb D}_\mu )$ thought of as a parameter of a representation of $\mathrm {GL}_{2n}({\mathbb R})$ , does not parametrise a cohomological representation. However, after any half-integral Tate twist – namely, for any $k \in {\mathbb Z}$ – the parameter $\varphi ({\mathbb D}_\mu ) \otimes |\ |^{(2k+1)/2}$ is indeed a parameter of a cohomological representation of $\mathrm {GL}_{2n}({\mathbb R}).$ See, for example, [Reference Raghuram27, 2.4.1.2].

3.1 Strongly inner cohomology

Let $\mu \in X^*_+({}^\circ T \times E)$ be a dominant integral weight for ${}^\circ \!M.$ Recall that $\mu = (\mu ^\tau )_{\tau \in \mathrm {Hom}(F,E)},$ where $\mu ^\tau $ is an n-tuple of integers ordered as $\mu ^\tau _1 \geq \mu ^\tau _2 \geq \cdots \geq \mu ^\tau _{n-1} \geq |\mu ^\tau _n|.$ Define

$$ \begin{align*}\mu_{\mathrm{min}} \ := \ \mathrm{min} \{ \, |\mu_n^{\tau} | \ : \ \tau \in \mathrm{Hom}(F,E) \}. \end{align*} $$

We henceforth assume that $\mu _{\mathrm {min}} \geq 1.$ For an open-compact subgroup ${}^\circ C_f$ of ${}^\circ \!M({\mathbb A}_f)$ , consider the sheaf $ \widetilde {\mathcal M}_{\mu ,E}$ on ${\mathcal S}^{{}^{\circ }\!M}_{{}^\circ C_f}.$

The inner cohomology $H^{\bullet }_{!} (\mathcal S^{{}^\circ \! M}_{{}^{\circ }C_{f}}, \widetilde {\mathcal M}_{\mu ,E})$ is a module for the Hecke algebra ${\mathcal H}^{^{\circ }\!M}_{^{\circ }C_{f}} = \otimes _{v \notin S_\infty } {\mathcal H}^{^{\circ }\!M_{v}}_{^{\circ }C_{v}}$ over ${\mathbb Q}$ . Let S be a finite set of places containing $S_\infty $ , the ramified places of $F,$ and any place where ${}^{\circ }C_v$ is a proper subgroup of the maximal compact subgroup. Then, ${\mathcal H}^{{}^{\circ }\!M, S} = \mathbin {\mathop {\otimes }\limits _{v \notin S}} {\mathcal H}^{^{\circ }\!M_{v}}_{^{\circ }C_{v}}$ is a central subalgebra of ${\mathcal H}^{^{\circ }\!M}_{^{\circ }C_{f}}.$ We consider inner cohomology as a ${\mathcal H}^{^{\circ }\!M}_{^{\circ }C_{f}}$ -module. It is a general fact that inner cohomology is semi-simple as a Hecke module. Define $\mathrm {Coh}_{!}(^{\circ }\!M, \mu /E, {}^{\circ }C_{f})$ as the isomorphism classes of absolutely simple modules $\sigma _f$ over E (these are necessarily finite-dimensional E-vector spaces) which appear in $H^{\bullet }_{!} (\mathcal S^{^{\circ }\!M}_{^{\circ }C_{f}}, \widetilde {\mathcal M}_{\mu ,E}),$ and inner cohomology is decomposed as the sum of such isotypic components

$$ \begin{align*}H^{\bullet}_{!} (\mathcal S^{{}^{\circ}\!M}_{{}^{\circ}C_{f}}, \widetilde{\mathcal M}_{\mu,E}) \ = \ \bigoplus_{\sigma_f \in \mathrm{Coh}_{!}(M, \mu/E, {}^{\circ}C_{f})} H^{\bullet}_{!} (\mathcal S^{{}^{\circ}\!M}_{{}^{\circ}C_{f}}, \widetilde{\mathcal M}_{\mu,E}) (\sigma_f); \end{align*} $$

by taking E large enough we can guarantee that the full isotypic component of a particular $\sigma _f$ appears as a summand. Each $\sigma _f$ is isotypic as a ${\mathcal H}^{{}^{\circ }\!M, S}$ -module.

Towards defining the strongly inner spectrum $\mathrm {Coh}_{!!}(^{\circ }\!M, \mu /E, {}^{\circ }C_{f})$ , we fix once and for all a Whittaker datum $\psi $ for ${}^{\circ }\!M$ which is a nontrivial additive character on the unipotent radical of the standard Borel subgroup which is nontrivial on all simple root spaces. We also recall Arthur’s classification, as finessed by Atobe–Gan [Reference Atobe and Gan2], of the discrete (and hence cuspidal) spectrum of $^{\circ }\!M({\mathbb A}) = \mathrm {O}(2n)({\mathbb A}_F)$ in terms of automorphic representations of $\mathrm {GL}_{2n}({\mathbb A}_F)$ ; the Arthur parameter of an irreducible automorphic representation $\varsigma $ appearing in the discrete spectrum of $^{\circ }\!M({\mathbb A})$ will be denoted $\Psi (\varsigma ).$ Let us also recall some notation from Section 1.6.1: an embedding $\iota : E \to {\mathbb C}$ gives an identification $X^*({}^\circ T \times E) \to X^*({}^\circ T \times {\mathbb C})$ that we denote $\mu \mapsto {}^\iota \mu .$ For $\mu = (\mu ^\tau )_{\tau : F \to E},$ identifying $\mathrm {Hom}(F,{\mathbb C})$ with $S_\infty ,$ and since $\iota $ induces a bijection $\mathrm {Hom}(F,E) \to \mathrm {Hom}(F,{\mathbb C}) = S_\infty $ via composition, for $v \in S_\infty $ , the v-th component ${}^\iota \mu _v$ is $\mu ^{\iota ^{-1}v}.$

Since the Arthur parameter $\Psi (\varsigma )$ is cuspidal, we know from [Reference Atobe and Gan2, Prop. 7.2] that $\varsigma $ appears in the cuspidal spectrum of ${}^\circ \!M$ with multiplicity one, and we let $V_\varsigma $ denote the representation space of $\varsigma $ inside of the space of cusp forms of ${}^\circ \!M$ . In the above definition, condition (i) pins down every local component $\varsigma _v$ in its local A-packet; however, there can be other cuspidal automorphic representations in the same global A-packet as $\varsigma $ and satisfying (iv). This is further elaborated upon in Section 3.3 below.

Define strongly inner cohomology as the space spanned by the isotypic components of strongly-inner Hecke-modules:

$$ \begin{align*}H^{\bullet}_{!!}(\mathcal S^{^{\circ}\!M}_{^\circ C_{f}}, \widetilde {\mathcal M}_{\mu,E}) \ := \ \bigoplus_{\sigma_f \in \mathrm{Coh}_{!!}(^{\circ} \!M, \mu, {}^\circ C_{f})} H^{\bullet}_{!}(\mathcal S^{^{\circ}\!M}_{^\circ C_{f}}, \widetilde {\mathcal M}_{\mu,E}) (\sigma_f). \end{align*} $$

For brevity, denote $\mathrm {Coh}_{*}(^{\circ }\!M, \mu /E, {}^{\circ }C_{f})$ by $\mathrm {Coh}_{*}(^{\circ }\!M, \mu ),$ for $* \in \{!, !!\}$ , with the level structure ${}^{\circ }C_{f}$ and the field of coefficients E being understood.

The definition of strongly inner spectrum (and so also strongly inner cohomology) seemingly needs an embedding $\iota : E \to {\mathbb C}$ to invoke concepts of automorphic representation theory; however, we show independence of $\iota $ in the next subsection.

3.2 Galois equivariance

Let $\mu \in X^*({}^\circ T \times E)$ with $\mu _{\mathrm {min}} \geq 1,$ and $\sigma _f \in \mathrm {Coh}_{!!}(^{\circ }\!M, \mu ).$ For $\iota : E \to {\mathbb C}$ , let $\varsigma $ be the cuspidal automorphic representation of $^\circ \!M({\mathbb A}) = \mathrm {O}(2n)({\mathbb A}_F)$ as in Definition 25 above. The image of an $\iota : E \to {\mathbb C}$ lands in $\bar {{\mathbb Q}}$ , and we may further compose this with $\eta \in \mathrm {Gal}(\bar {{\mathbb Q}}/ {\mathbb Q}).$ We wish to understand what happens when we replace $\iota $ by $\eta \circ \iota $ ; for this we need to be able to consider the Galois conjugate of a cuspidal representation of $\mathrm {O}(2n)({\mathbb A}_F)$ satisfying $(i)-(iii)$ of Definition 25. We show in this subsection that there is a cuspidal automorphic representation ${}^\eta \varsigma $ of $^\circ \!M({\mathbb A}) = \mathrm {O}(2n)({\mathbb A}_F)$ which is globally generic with respect to the same Whittaker datum $\psi ,$ and $({}^\eta \varsigma )_v = \varsigma _{\eta ^{-1}\circ v}$ for $v \in S_{\infty }$ , and whose Arthur parameter is $\Psi ({}^\eta \varsigma ) = {}^\eta (\Psi (\varsigma ) \otimes |\ |^{-1/2}) \otimes |\ |^{1/2}.$

Since $\varsigma $ is cuspidal and for $v \in S_\infty $ the local representation $\varsigma _v$ being a discrete series representation of $\mathrm {O}(n,n)({\mathbb R})$ , it follows that $\varsigma _f$ contributes to cuspidal cohomology (i.e., $\varsigma _f$ appears as a Hecke-summand of $H^\bullet _{\mathrm {cusp}}(\mathcal S^{^{\circ }\!M}_{{}^\circ C_{f}}, \widetilde {\mathcal M}_{{}^\iota \mu ,{\mathbb C}})).$ (The reader is referred to [Reference Harder and Raghuram16, Sect. 3.2] for basics on cuspidal and square-integrable cohomology; although the reference is for general linear groups, the discussion in there applies mutatis mutandis to orthogonal groups.) But, cuspidal cohomology is contained inside inner cohomology which admits a natural rational structure since it is sheaf-theoretically defined, and so we may consider the effect of a Galois element on the latter. Furthermore, inner cohomology is contained in square-integrable cohomology which is captured by the discrete spectrum of $\mathrm {O}(2n)/F$ . This is summarised in the diagram:

that is, the $\eta $ -conjugate of $\varsigma _f,$ denoted ${}^\eta \varsigma _f,$ appears in inner-cohomology for the conjugated sheaf, and so also in square-integrable cohomology. Hence, there exists an automorphic representation $\delta $ that contributes to the global discrete spectrum of $\mathrm {O}(2n)/F$ such that

$$ \begin{align*}\delta_f \ = \ {}^\eta\varsigma_f, \end{align*} $$

and furthermore, $\delta _\infty $ has nonvanishing relative Lie algebra cohomology with respect to the coefficient system ${\mathcal M}_{{}^{\eta \circ \iota }\mu ,{\mathbb C}},$ giving us the following equality of infinitesimal characters:

$$ \begin{align*}\chi_{\delta_v} \ = \ \chi_{{\mathcal M}_{({}^{\eta\circ\iota}\mu)_v,{\mathbb C}}} \ = \ \chi_{{\mathcal M}_{({}^{\iota}\mu)_{\eta^{-1}v,{\mathbb C}}}} = \chi_{\varsigma_{\eta^{-1}v}}. \end{align*} $$

Since $\delta $ is in the discrete spectrum, we have its Arthur parameter $\Psi (\delta ).$ We claim that $\Psi (\delta )$ is a cuspidal representation of $\mathrm {GL}_{2n}({\mathbb A}_F).$ Now, $\Psi (\varsigma )$ is not of cohomological type on $\mathrm {GL}_{2n}({\mathbb A}_F)$ ; however, as remarked in Section 2.2.8, after a half-integral Tate-twist, it becomes cohomological, and we know from Clozel [Reference Clozel11] that there is a Galois action on cohomological cuspidal representations of $\mathrm {GL}_{2n}({\mathbb A}_F)$ (i.e., we have the cuspidal representation ${}^\eta (\Psi (\varsigma ) \otimes |\ |^{-1/2})).$ For almost all v (where the local representations are unramified), we have

$$ \begin{align*} \Psi( \delta)_v \ &= \ \Psi(\delta_v) \ = \ \Psi({}^\eta\varsigma_v) \\ &= \ \left({}^\eta(\Psi(\varsigma)_v \otimes |\ |^{-1/2})_v \otimes |\ |_v^{1/2}\right) \ = \ \left({}^\eta(\Psi(\varsigma) \otimes |\ |^{-1/2}) \otimes |\ |^{1/2}\right)_v, \end{align*} $$

where the third equality is an easy calculation with unramified principal series representations of $\mathrm {GL}_{2n}(F_v)$ . Hence, we have two Arthur parameters $\Psi _1 := \Psi ( \delta )$ and $\Psi _2 := {}^\eta (\Psi (\varsigma ) \otimes |\ |^{-1/2}) \otimes |\ |^{1/2}$ such that $\Psi _2$ cuspidal and $\Psi _{1,v} = \Psi _{2,v}$ for almost all v. Hence $\Psi _1 = \Psi _2,$ and in particular, $\Psi _1$ is also cuspidal. Let us record this as

(26) $$ \begin{align} \Psi( \delta) \ = \ {}^\eta(\Psi(\varsigma) \otimes |\ |^{-1/2}) \otimes |\ |^{1/2}. \end{align} $$

The Arthur parameter $\Psi ( \delta )$ for $\mathrm {O}(2n)$ is cuspidal, and so a partial symmetric square L-function, $L^S(s, \mathrm {Sym}^2, \Psi ( \delta )),$ has a pole at $s=1$ , and by the backward lifting (see Soudry [Reference Soudry36] and the references therein), there exists a unique globally $\psi $ -generic cuspidal representation $ \delta '$ of $\mathrm {O}(2n)$ whose local components transfer almost everywhere to the local components of $\Psi ( \delta ).$ Let $\Psi ( \delta ')$ be the Arthur parameter of $ \delta '$ . Once again, we have two Arthur parameters, $\Psi ( \delta )$ and $\Psi ( \delta '),$ that agree almost everywhere, and one of them is cuspidal, and hence so is the other, and they are equal; that is, we have

(27) $$ \begin{align} \Psi( \delta) \ = \ \Psi( \delta'). \end{align} $$

Fix $v \in S_\infty $ . Consider the local component $\delta ^{\prime }_v$ . We have

$$ \begin{align*}\Psi(\delta^{\prime}_v) = \Psi(\delta')_v =\left(^{\eta}\!(\Psi(\varsigma) \otimes |\ |^{-1/2}) \otimes |\ |^{1/2}\right)_{v} = \Psi(\varsigma)_{\eta^{-1}v} = \Psi(\varsigma_{\eta^{-1}v}). \end{align*} $$

Hence, $\delta ^{\prime }_v$ and $\varsigma _{\eta ^{-1}v} = \mathbb {D}^{\dagger }_{{}^\iota \!\mu _{\eta ^{-1}v}}$ are in the same A-packet and indeed in the same L-packet; but $\delta ^{\prime }_v$ is also locally generic with respect to $\psi _v$ ; hence, $\delta ^{\prime }_v = \mathbb {D}^{\dagger }_{{}^\iota \!\mu _{\eta ^{-1}v}} = \mathbb {D}^{\dagger }_{{}^{\eta \circ \iota }\!\mu _{v}}.$

Definition 28. Let $\varsigma $ be a cuspidal automorphic representation of $\mathrm {O}(2n)({\mathbb A}_F)$ that satisfies $(i)-(iii)$ of Definition 25. For $\eta \in \mathrm {Gal}(\bar {{\mathbb Q}}/ {\mathbb Q}),$ define the $\eta $ conjugate of $\varsigma $ , denoted ${}^\eta \varsigma $ , as the cuspidal representation $\delta '$ as above. Then ${}^\eta \varsigma $ satisfies $(i)-(iii)$ of Definition 25, where the highest weight for ${}^\eta \varsigma $ in $(iii)$ is the $\eta $ -conjugate of the highest weight for $\varsigma .$ The Arthur parameters are related by

$$ \begin{align*}\Psi({}^\eta\varsigma) \ = \ {}^\eta(\Psi(\varsigma) \otimes |\ |^{-1/2}) \otimes |\ |^{1/2}. \end{align*} $$

Now we can see the independence of $\iota $ in Definition 25, in that if the conditions $(i)-(iv)$ hold for one embedding $\iota : E \to {\mathbb C}$ , then they hold for any embedding $\iota : E \to {\mathbb C}.$ Indeed, given some other $\iota ' : E \to {\mathbb C}$ , there is an $\eta \in \mathrm {Gal}(\bar {{\mathbb Q}}/ {\mathbb Q}),$ such that $\iota ' = \eta \circ \iota .$ Given a cuspidal representation $\varsigma $ attached to the data $\{\sigma _f, \iota \}$ , then the cuspidal representation ${}^\eta \varsigma $ is attached to the data $\{\sigma _f, \iota ' = \eta \circ \iota \}.$

3.3 The inner structure of an isotypic component of strongly inner cohomology

For $\sigma _f \in \mathrm {Coh}_{!!}(^\circ M, \mu )$ and $\iota : E \rightarrow {\mathbb C}$ , we wish to describe

$$ \begin{align*}H^{\bullet}_{!!}(\mathcal S^{^{\circ}\!M}_{^\circ C_{f}}, \widetilde {\mathcal M}_{\mu,E}) (\sigma_f)\otimes_{E, \iota} {\mathbb C}. \end{align*} $$

Let ${\mathcal A}_2({}^\circ M) = {\mathcal A}_2\left( {}^{\circ } M({\mathbb Q}) \backslash {}^{\circ } M(\mathbb A) \right)$ denote the automorphic discrete spectrum of ${}^\circ M;$ see [Reference Atobe and Gan2, Sec. 6.7] for the definition of ${\mathcal A}_2({}^\circ M)$ . Inside, ${\mathcal A}_2({}^\circ M)$ is the cuspidal spectrum which we denote ${\mathcal A}_{\mathrm {cusp}}({}^\circ M) = {\mathcal A}_{\mathrm {cusp}} \left( {}^{\circ } M({\mathbb Q}) \backslash {}^{\circ } M(\mathbb A) \right).$ Consider the representation ${}^{\iota } \sigma : = \varsigma $ in Definition 25. There exist finitely many inequivalent cuspidal automorphic representations ${}^\iota \sigma = {}^\iota \sigma _{1},\ldots , {}^\iota \sigma _{l}$ of ${}^\circ \!M({\mathbb A})$ all having the same Arthur parameter as ${}^\iota \sigma ,$ and so appearing with multiplicity one in ${\mathcal A}_2({}^\circ M)$ ([Reference Atobe and Gan2, Prop. 7.2]), indeed, appearing with multiplicity one in ${\mathcal A}_{\mathrm {cusp}}$ , and whose finite-parts are uniquely determined since ${}^\iota \sigma _{j,f}^{{}^\circ C_{f}} \cong \sigma _f \otimes _{E,\iota } {\mathbb C};$ however, for the archimedean components (i.e., for $v \in S_\infty $ ), ${}^\iota \sigma _{j, v}$ can vary in the A-packet, or in this case, L-packet of $\varsigma _v$ which consists of discrete series representations determined by ${}^\iota \mu _v.$ For each $1 \leq j \leq l$ , we fix

(29) $$ \begin{align} \Phi_j \in \mathrm{Hom}_{(^\circ \mathfrak m({\mathbb R}), K^\circ_{{}^\circ\!M({\mathbb R})}) \times ^\circ M(\mathbb A_f)} \left( ^\iota \sigma_j, \ {\mathcal A}^2_{\mathrm{cusp}} \left( {}^{\circ} M({\mathbb Q}) \backslash ^{\circ} M(\mathbb A) \right) \right), \end{align} $$

which is unique up to scalars by the multiplicity-one property; denote $\Phi _j({}^\iota \sigma _j) = V_{{}^\iota \sigma _j}.$ The base-change to ${\mathbb C}$ via $\iota $ of the strongly-inner isotypic component of $\sigma _f$ takes the shape

(30) $$ \begin{align} H^{\bullet}_{!!}(\mathcal S^{^{\circ}\!M}_{^\circ C_{f}}, \widetilde {\mathcal M}_{\mu,E}) (\sigma_f)\otimes_{E, \iota} {\mathbb C} \ \ \cong \ \ \bigoplus_{j=1}^l H^\bullet({}^\circ \mathfrak m({\mathbb R}), K^\circ_{{}^\circ\!M({\mathbb R})}; \, V_{{}^\iota \sigma_j}^{{}^\circ C_{f}} \otimes {\mathcal M}_{{}^\iota\mu, {\mathbb C}}). \end{align} $$

Using the inverse of $\Phi _j$ in the j-th summand, we also have

(31) $$ \begin{align} H^{\bullet}_{!!}(\mathcal S^{^{\circ}\!M}_{^\circ C_{f}}, \widetilde {\mathcal M}_{\mu,E}) (\sigma_f)\otimes_{E, \iota} {\mathbb C} \ \cong \ \bigoplus_{j=1}^l H^\bullet({}^\circ \mathfrak m({\mathbb R}), K^\circ_{{}^\circ\!M({\mathbb R})}; \, {}^\iota \sigma_{j, \infty} \otimes {\mathcal M}_{{}^\iota\mu, {\mathbb C}}) \otimes {}^\iota \sigma_j^{{}^\circ C_{f}}. \end{align} $$

Note that for all $v \in S_{\infty }$ and $1 \leq j \leq l$ , we have $\Psi (^\iota \sigma _{j,v}) = \Psi ({\mathbb D}^{\dagger }_{\mu ^{v}})$ . From (31), it is clear that an absolutely simple Hecke-module represented by $\sigma _f$ , after base-change to ${\mathbb C}$ via $\iota $ , appears with finite-multiplicity in cuspidal cohomology. Later, in the context of the standard intertwining operator, we will simultaneously consider the cuspidal representations $^{\iota } \sigma _j$ and $^{\kappa _{2n}} { ^{\iota } \sigma _j}$ . Since $\kappa _{2n} \in {}^\circ M,$ we have $^{\iota } \sigma _j \cong {}^{\kappa _{2n}} { ^{\iota } \sigma _j}$ , and their representation spaces in the space of cusp forms are the same: $\Phi _j({}^{\iota } \sigma _j) = \Phi _j(^{\kappa _{2n}} { ^{\iota } \sigma _j}),$ or $V_{{}^{\iota } \sigma _j} = V_{{}^{\kappa _{2n}\iota } \sigma _j}.$ Also, for $1 \leq j,j' \leq l$ , if $j \neq j'$ , then the representations ${}^\iota \sigma _{j}$ and ${}^\iota \sigma _{j'}$ being inequivalent, implies that there is some $v \in S_\infty $ such that the local representations ${}^\iota \sigma _{j, v}$ and ${}^\iota \sigma _{j', v}$ are inequivalent. In particular, ${}^{\kappa _{2n} \iota } \sigma _j$ is not equivalent to ${}^{\iota } \sigma _{j'}.$

4.1 L-functions for orthogonal groups

Let $\mu \in X^*({}^\circ T \times E)$ be a dominant integral weight for ${}^\circ \!M$ with $\mu _{\mathrm {min}} \geq 1,$ and $\sigma _f \in \mathrm {Coh}_{!!}(^{\circ }M, \mu ).$ For $\iota : E \to {\mathbb C}$ , the cuspidal representation $\varsigma $ of Definition 25 will be henceforth denoted ${}^\iota \sigma .$ Let $\chi ^{\circ }$ be a finite order Hecke character of F taking values in $E,$ and d be an integer. Let $\chi $ stand for the algebraic Hecke character of F with values in E which gives a continuous homomorphism ${}^\iota \chi : F^\times \backslash {\mathbb A}_F^\times \to {\mathbb C}^\times $ of the form ${}^\iota \chi = {}^\iota \chi ^\circ \otimes |\ |^{-d}.$ The purpose of this section is to understand the critical set for the degree- $2n$ Langlands L-function $L(s, {}^\iota \chi \times {}^\iota \sigma ).$

For a place v of F, the local representation ${}^\iota \sigma _v$ is a representation of $\mathrm {O}(2n)(F_v)$ , and its restriction to $\mathrm {SO}(2n)(F_v)$ can be irreducible or a sum of two irreducible constituents that are mutually $\kappa _{2n}$ -conjugates since $\kappa _{2n} \in \mathrm {O}(2n)(F_v) \setminus \mathrm {SO}(2n)(F_v)$ . From Theorems 3.5, 3.6, 3.9 and 3.10 of Atobe–Gan [Reference Atobe and Gan2], we get an L-parameter for $\mathrm {O}(2n)(F_v)$ . Then we consider the degree- $2n$ local Langlands–Shahidi L-function $L_v(s, {}^\iota \chi _v \times {}^\iota \sigma _v)$ attached to the data ${}^\iota \chi _v$ and this L-parameter (corresponding to case ( $D_{n,i}$ ) of [Reference Shahidi33, App. A]). The global L-function is defined as a Euler product: $L(s, {}^\iota \chi \times {}^\iota \sigma ) = \prod _v L_v(s, {}^\iota \chi _v \times {}^\iota \sigma _v).$ The analytic properties of this L-function are well-understood by the Langlands–Shahidi machinery for which we refer the reader to Shahidi’s book [Reference Shahidi33].

4.2 Critical points for Rankin–Selberg L-functions for orthogonal groups

The infinite part of $L(s, {}^\iota \chi \times {}^\iota \sigma )$ is a product over archimedean places:

$$\begin{align*}L_\infty(s, {}^\iota\chi \times {}^\iota\sigma) \ := \ \prod \limits_{v \in S_\infty} L(s, {}^\iota\chi_{v} \times {}^\iota\sigma_v). \end{align*}$$

An integer m is said to be a critical for $L(s, {}^\iota \chi \times {}^\iota \sigma )$ if both

$$ \begin{align*}L_\infty(s, {}^\iota\chi \times {}^\iota\sigma) \ \ \mathrm{and}\ \ L_\infty(1-s, {}^\iota\chi^{-1} \times {}^\iota\sigma^{\mathsf{v}}) \end{align*} $$

are holomorphic at $s=m$ , where ${}^\iota \sigma ^{\mathsf {v}}$ is the contragredient of ${}^\iota \sigma .$ The purpose of this subsection is to determine all such critical integers.

Proposition 32. Let the notations be as in the first paragraph of Section 4.1. The critical set for the degree- $2n L$ -function $L(s, {}^\iota \chi \times {}^\iota \sigma )$ is the contiguous set of $2\mu _{\min }$ integers:

$$ \begin{align*}\left\{ 1+d-\mu_{\mathrm{min}}, ~ 2+d-\mu_{\mathrm{min}}, \ \dots \ , \ d+\mu_{\mathrm{min}}\right\}. \end{align*} $$

Since $\mu _{\mathrm {min}} \geq 1$ , there are at least two successive critical integers. The critical set is independent of $\iota $ .

Proof. For $v \in S_\infty $ , the local representation ${}^\iota \sigma _v$ is the discrete series representation ${\mathbb D}_{{}^\iota \!\mu ^{v}}$ of $\mathrm {O}(n,n)({\mathbb R}).$ As before, identifying $S_\infty $ with $\mathrm {Hom}(F,{\mathbb C})$ , we have ${}^\iota \!\mu ^{v} = \mu ^{\iota ^{-1}v}.$ For given v and $\iota ,$ let’s denote $\iota ^{-1}v$ by $\tau .$ Recall that $\mu ^\tau _1 \geq \cdots \geq \mu ^\tau _{n-1} \geq |\mu ^\tau _n| \geq 1.$ We know the Langlands parameter of ${}^\iota \sigma _v$ from (24). Also, the local component at v of the character $\chi $ may be expressed as $\chi _v = |\ |^{-d} \mathrm {sgn}^{\epsilon _v}$ with $\epsilon _v \in \{0,1\}.$ Recall that the $2$ -dimensional irreducible representation $I(\ell )$ of $W_{\mathbb R}$ is invariant under twisting by the sign character (i.e., $I(\ell ) \otimes \mathrm {sgn} = I(\ell )).$ Recall also ([Reference Knapp22]) that the local L-factor attached to $I(\ell )$ is $2(2{\boldsymbol {\pi }})^{-(s+|\ell |/2)}\Gamma (s+|\ell |/2).$ Putting these together, and recalling $\ell ^\tau _j$ in terms of $\mu ^\tau _j$ from (23), we get the local L-factor at v as

(33) $$ \begin{align} L(s, {}^\iota\chi_{v} \times {}^\iota\sigma_v) &= \ L(s-d, {}^\iota\chi^\circ_{v} \times {}^\iota\sigma_v) \nonumber\\ & = 2^{n} \prod \limits_{j=1}^{n} (2 {\boldsymbol{\pi}})^{-(s-d+ |\mu_j^\tau| + n - j)} ~\Gamma(s-d+ |\mu_j^\tau| + n - j). \end{align} $$

To compute the critical integers, powers of $2$ and exponential terms – since they are holomorphic and nonvanishing everywhere – are not relevant. Ignoring such terms, we write $L(s, {}^\iota \chi _{v} \times {}^\iota \sigma _v) \approx \prod _{j=1}^{n} \Gamma (s-d+ |\mu _j^\tau | + n - j).$ Taking the product over $v \in S_\infty $ , which on the right-hand side will be taking the product over all $\tau : F \to E$ , we get

$$ \begin{align*}L_\infty(s, {}^\iota\chi \times {}^\iota\sigma) \ \approx \ \prod_{\tau : F \to E} \, \prod_{j=1}^{n} \, \Gamma(s-d+ |\mu_j^\tau| + n - j). \end{align*} $$

Similarly, on the dual side we get

$$ \begin{align*}L_\infty(1-s, {}^\iota\chi^{-1} \times {}^\iota\sigma^{\mathsf{v}}) \ \approx \ \prod_{\tau : F \to E} \, \prod_{j=1}^{n} \, \Gamma(1-s+d+ |\mu_j^\tau| + n - j). \end{align*} $$

We see that m is critical if and only if for all $\tau : F \to E$ and all $1 \leq j \leq n$ , both the conditions are satisfied:

• • $m-d+ | \mu _j^\tau |+ n - j \geq 1$ ;

• • $ 1 - m + d + |\mu _j^\tau | + n - j \geq 1$ .

Since $ \mu _1^\tau +n-1 \> \ \mu _2^\tau +n- 2 \ > \ \cdots \ > \ |\mu _n^\tau | \ \geq \ \mu _{\mathrm {min}},$ we see that the conditions are equivalent to $1 - \mu _{\mathrm {min}} \ \leq \ m-d \ \leq \ \mu _{\mathrm {min}}.$

We draw the following two easy corollaries of Proposition 32:

Corollary 34. The integers $-n$ and $1-n$ are critical for $L(s, {}^\iota \chi \times {}^\iota \sigma )$ if and only if

$$ \begin{align*} 1 - \mu_{\mathrm{min}} \ \leq \ -(d+n) \ \leq \ \mu_{\mathrm{min}} - 1. \end{align*} $$

Corollary 35. If the inequalities in Corollary 34 hold, then

$$ \begin{align*}\frac{L_\infty(-n, {}^\iota\chi \times {}^\iota\sigma)}{L_\infty(1-n, {}^\iota\chi \times {}^\iota\sigma)} \ \approx_{{\mathbb Q}^{\times}} \ {\boldsymbol{\pi}}^{n \textsf{r}_{\textsf{F}}}, \end{align*} $$

where $\approx _{{\mathbb Q}^{\times }}$ means equality up to a nonzero rational number.

5.1 Weyl group

Recall our notation that $n = 2r$ , and $G_0 = \mathrm {O}(2n+2)/ F$ . The Weyl group $W_{G_0}$ is isomorphic to the subgroup of the group $\mathfrak {S}_{n+1} \rtimes ({\mathbb Z}/2{\mathbb Z})^{n+1}$ of signed permutations consisting of all signed permutations of $n+1$ symbols with an even number of sign changes. An element $w \in W_{G_0}$ permutes the $e_j$ ’s according to the corresponding signed permutation.

Recall the maximal parabolic subgroup $P_0$ of $G_0$ from Section 1.7 corresponding to deleting the simple root $\alpha _0.$ Let $w_{P_{0}} $ be the unique element of $W_{G_0}$ such that $w_{{P_{0}}}( \Pi _{G_{0}} \setminus \{ \alpha _0\}) \subset \Pi _{G_{0}}$ and $w_{P_0}( \alpha _{0}) < 0$ . Observe that ${P_{0}}$ is self associate, and we have

$$ \begin{align*}w_{{P_{0}}}(\Pi_{G_{0}} \setminus \{ \alpha_0 \}) = \Pi_{G_{0}} \setminus \{ \alpha_0\} ~ \text{and}~w_{P_{0}}(\Delta_{U{_{0}}}) = - \Delta_{U_{0}}, \end{align*} $$

where $\Delta _{U_{0}}$ is the set of positive roots with root spaces in $\mathfrak {u}_{P_0} = \mathrm {Lie}(U_{P_0}).$ The element $w_{P_0}$ is the signed permutation:

$$ \begin{align*}e_0 \mapsto -e_0, \quad e_i \mapsto e_i \ \ \mathrm{for} \ \ 1 \leq i \leq n-1, \quad e_n \mapsto -e_n. \end{align*} $$

The element $w_{P_0}$ can be factorised as a product in terms of the simple reflections $\mathsf {s}_{\alpha _{j}}$ as

(36) $$ \begin{align} w_{P _{0}} \ = \ \mathsf{s}_{\alpha_{0}} ~\mathsf{s}_{\alpha_{1}} ~\cdots ~ \mathsf{s}_{\alpha_{n-3}} ~ \mathsf{s}_{\alpha_{n-2}} ~\mathsf{s}_{\alpha_{n}} ~\mathsf{s}_{\alpha_{n-1}} \mathsf{s}_{\alpha_{n-2}} ~\cdots~ \mathsf{s}_{\alpha_{1}} ~\mathsf{s}_{\alpha_{0}}. \end{align} $$

The length of $w_{P _{0}}$ is $l(w_{P _{0}}) = 2n = \dim (U_{P_0}).$

The Weyl group of $G \times E$ factors as a product

$$ \begin{align*}W_{G \times E} \ = \ \prod \limits_{\tau: F \rightarrow E} \, W_{G_0 \times_{F, \tau} E}, \end{align*} $$

where each $W_{G_0 \times _{F, \tau } E}$ is isomorphic to $W_{G_{0}}$ . Sometimes, we abbreviate $W_{G_0 \times _{F, \tau } E}$ as $W^\tau .$ The element $w_P \in W_{G \times E}$ is $(w_{P_0}^\tau )_{\tau : F \to E}$ , where each $w_{P_0}^\tau $ is the copy of $w_{P_0}$ in $W_{G_0 \times _{F, \tau } E}.$ The length of $w_{P}$ is $l(w_{P}) = 2n \mathsf {r}_F = \dim (U_P).$

The twisted action of $w \in W_{G \times E}$ on $\lambda \in X^{\ast }(T \times E)$ is given by $w \cdot \lambda : = (w^{\tau } \cdot \lambda ^\tau )_{\tau : F \rightarrow E},$ where

$$ \begin{align*}w^{\tau} \cdot \lambda^{\tau} \ := \ w^{\tau} (\lambda^{\tau} + {\boldsymbol{\rho}}_{G^{\tau}_{0}}) - {\boldsymbol{\rho}}_{G^{\tau}_0}. \end{align*} $$

5.2 Kostant representatives

Let $\Pi _{M_{P_{0}}}$ denote the set of simple roots of the Levi quotient $M_{P_0}$ of $P_0.$ The set of Kostant representatives in $W_{G_0}$ corresponding to the parabolic subgroup $P_0$ is defined as

$$ \begin{align*}W^{P_{0}} \ := \ \{ w \in W_{G_0} \ : \ w^{-1} \alpha> 0 \ \ \forall \alpha \in \Pi_{M_{P_{0}}} \}. \end{align*} $$

Now define the set of Kostant representatives in $W_{G \times E}$ corresponding to the parabolic subgroup P as

$$ \begin{align*}W^{P} \ = \ \left\{ w = (w^{\tau}) \ : \ w^{\tau} \in {W^{\tau}}^{P_{0}^{\tau}} \right\}. \end{align*} $$

If $w_0 \in W^{P_{0}}$ , then for its length, we have $0 \leq \ell (w_0) \leq \dim (U_{P_0})$ , and hence, for $w \in W^P$ , we have $0 \leq \ell (w) \leq \dim (U_{P}).$ Recall that $\dim (U_{P_0}) = 2n$ is even.

In the self-associate case of our parabolic subgroup P, a couple of self-bijections of $W^P$ with complementary lengths will be important. We record these in the following two propositions. (See Lemma 5.6 and Lemma 5.7 of [Reference Harder and Raghuram16], the proofs of which are totally general.)

Let $w_{G_0}$ be the element of $W_{G_0}$ with longest length. As a signed permutation, it is

$$ \begin{align*}e_i \mapsto -e_i \ \ \mathrm{for} \ \ 0 \leq i \leq n-1, \ \ \mathrm{and} \quad e_n \mapsto e_n. \end{align*} $$

(Since n is even, we have an even number of sign changes.) The element $w_G \in W_{G \times E}$ is $(w_{G_0}^\tau )_{\tau : F \to E}$ , where each $w_{G_0}^\tau $ is the copy of $w_{G_0}$ in $W_{G_0 \times _{F, \tau } E}.$ Similarly, let $w_{M_0}$ be the element of $W_{M_{P_0}}$ of longest length which is given by

$$ \begin{align*}e_i \mapsto -e_i \quad \mathrm{for} \ 1 \leq i \leq n. \end{align*} $$

We have $w_{M} = (w_{M_0}^\tau )_{\tau : F \to E},$ where $w_{M_0}^\tau $ is the copy of $w_{M_0}$ in $W_{G_0 \times _{F, \tau } E}.$

In the set of Kostant representatives $W^{P_0}$ , one can check that there are only two balanced elements $w^+$ and $w^{-}$ which are described below:

$$ \begin{align*} \begin{aligned} &w^+: \quad e_i \mapsto e_{i+1} \quad \mathrm{for} \ 0 \leq i \leq n-1, \quad e_n \mapsto e_0; \\ &w^-: \quad e_i \mapsto e_{i+1} \quad \mathrm{for} \ 0 \leq i \leq n-2, \quad e_{n-1} \mapsto -e_n, \quad e_n \mapsto -e_0. \end{aligned} \end{align*} $$

It follows using the definitions of the self-bijections in the propositions above that

$$ \begin{align*} \begin{aligned} & w^+{}' = w^-, \quad w^-{}' = w^+; \\ & w^+{}^{\mathsf{v}} = w^+, \quad w^-{}^{\mathsf{v}} = w^-. \end{aligned} \end{align*} $$

5.3 A Combinatorial lemma

Let the notations be as in the first paragraph of Section 4.1. The weight $\mu = (\mu ^\tau )_{\tau : F \to E} \in X^*_+({}^\circ \!T \times E)$ and the integer d give a weight $de_0 + \mu \in X^*(T\times E)$ whose $\tau $ -component is $de_0 + \mu ^\tau .$ Note that $de_0 + \mu $ need not be dominant as a weight for $G.$

Proof. The equivalence $(i) \iff (ii)$ is exactly Corollary 34. It suffices to prove $(ii) \iff (iii)$ .

Suppose $(ii)$ holds. A calculation shows that

$$ \begin{align*} \begin{aligned} (w^{+})^{-1} \cdot (de_0 + \mu^{\tau}) = (\mu_1^{\tau} -1)~ e_0 + \ldots + (\mu_n ^{\tau}- 1)~ e_{n-1} + (d+n)~ e_n, \\ (w^{-})^{-1} \cdot (de_0 + \mu^{\tau}) = (\mu^{\tau}_1 -1)~ e_0 + \ldots + (-\mu^{\tau}_n - 1)~ e_{n-1} - (d+n)~ e_n. \end{aligned} \end{align*} $$

For $\tau : F \to E$ , if $\mu ^{\tau }_n \geq 1$ , then take $w^{\tau } = w^+$ , and if $\mu ^\tau _n \leq -1$ , then take $w^{\tau } = w^-$ . Now put $w = (w^\tau )_{\tau : F \to E}$ , which is a balanced element of $W^P.$ For each $\tau $ , observe that $(w^\tau )^{-1} \cdot (de_0 + \mu ^\tau )$ is dominant; indeed, if $\mu ^{\tau }_n \geq 1$ , then $(ii)$ implies that $\mu _n ^{\tau }- 1 \geq |d+n|$ , rendering $(w^{+})^{-1} \cdot (de_0 + \mu ^{\tau })$ dominant, and if $\mu ^{\tau }_n \leq -1$ , then $(ii)$ implies that $-\mu _n ^{\tau }- 1 \geq |d+n|$ , rendering $(w^{-})^{-1} \cdot (de_0 + \mu ^{\tau })$ dominant. Hence $(iii)$ .

Conversely, if $(iii)$ holds for some balanced element w of $W^P,$ then for a given $\tau ,$ the above formulae tell us that the sign of $\mu _n^\tau $ determines if $w^\tau $ is $w^+$ or $w^-,$ and furthermore, that $1- |\mu ^{\tau }_n| \leq -(n+d) \leq |\mu ^{\tau }_n|-1.$ But this is true for all $\tau .$ Hence, $(ii)$ .

5.4 Consequences of the combinatorial lemma

Under the notations and conditions imposed by the Combinatorial Lemma (Lemma 40), let’s record some consequences on the effect of Kostant representatives related to $w,$ such as $w'$ , $w^{\mathsf {v}}$ and $w^{\mathsf {v}}{}'$ (see Proposition 38 and Proposition 39), on some weights related to $\lambda : = w^{-1} \cdot (de_0 + \mu ).$ For $\mu \in X^*({}^\circ \!T \times E)$ , define $\mu ^{\mathsf {v}} := -w_{M_{P}} \mu .$ Observe that $\mu ^{\mathsf {v}} = \mu ,$ reflecting the fact that the representation ${\mathcal M}_{\mu ,E}$ is self-dual. Similarly, for $\lambda \in X^*(T \times E)$ , define $\lambda ^{\mathsf {v}} : = -w_{G} \lambda .$ When $\lambda $ is dominant-integral, then $\lambda ^{\mathsf {v}}$ is the the highest weight of the dual of ${\mathcal M}_{\lambda ,E}$ . We omit the proof of the following proposition which is a straightforward calculation. The information in this proposition will be relevant when we deal with Poincaré duality for the cohomology of the Borel–Serre boundary.

6.1 Induced representations

We begin by recalling the induced representations as set up in the Langlands–Shahidi machinery [Reference Shahidi33]. Recall that the Levi-quotient $M_0 = M_{P_0}$ of the maximal parabolic subgroup $P_0$ of $G_0$ corresponding to deleting the simple root $\alpha _0 = e_0 - e_1$ is explicitly given by

$$ \begin{align*}M_0 = \left\{ m_{x,g} = \left(\begin{matrix} x & & \\ & g & \\ & & x^{-1}\end{matrix}\right) \ : \ x \in A_0 = \mathrm{GL}_1/F, \ \ g \in {}^{\circ}\!M_0 = \mathrm{O}(2n)/F \right\} = A_0 \times {}^\circ\!M_0. \end{align*} $$

We have the dual of the real Lie algebra ${\mathfrak a}^* = X^*(M_0) \otimes {\mathbb R} = X^*(A_0) \otimes {\mathbb R} = {\mathbb R} \, e_0.$ The set of roots whose root spaces appear in $U_{P_0}$ is $\{e_0 \pm e_i : 1 \leq i \leq n\}$ , and half their sum is ${\boldsymbol {\rho }}_{P_0} = n\, e_0.$ We have $\langle {\boldsymbol {\rho }}_{P_0}, \alpha _0 \rangle = 2 ({\boldsymbol {\rho }}_{P_0}, \alpha _0)/(\alpha _0, \alpha _0) = n.$ The fundamental weight $\tilde {\alpha }$ corresponding to $\alpha _0$ is

$$ \begin{align*}\tilde{\alpha} \ = \ \langle {\boldsymbol{\rho}}_{P_0}, \alpha_0 \rangle^{-1} {\boldsymbol{\rho}}_{P_0} \ = \ e_0. \end{align*} $$

For any place v of F, the modular character $\delta _{P_0}$ at v given by the adjoint action of $M_{P_0}(F_v)$ on $U_{P_0}(F_v)$ is $\delta _{P_0, v}(m_{x,g}) = |x|_v^{2n}.$ At an adèlic level, it has the same form: $\delta _{P_0}(m_{x,g}) = |x|^{2n}.$ Hence, $\delta _{P_0}^{1/2} = |{\boldsymbol {\rho }}_{P_0}|.$ For $s \in {\mathbb C}$ , in the Langlands–Shahidi method, we start with the induced representation $I(s\tilde {\alpha }, \chi \otimes \sigma ),$ which we will denote simply as $I(s, \chi \times \sigma ),$ consisting of all smooth functions $f: G_0({\mathbb A}_F) \longrightarrow V_{\sigma }$ such that

$$ \begin{align*}f(m_{x,g}uh ) \ = \ |x|^{n+s} \, \chi(x) \, \sigma(g) \, f(h), \end{align*} $$

for all $m_{x,g} \in M_{P_0}({\mathbb A}_F), \ u \in U_{P_0}({\mathbb A}_F), \ h \in G_0({\mathbb A}_F).$ In terms of normalised parabolically induced representations, we have

$$ \begin{align*}I(s, \chi \times \sigma) \ = \ \mathrm{Ind}_{P({\mathbb A})}^{G({\mathbb A})} \left(\chi[s] \times \sigma \right), \end{align*} $$

where $\chi [s] = \chi \otimes |\ |^s.$ In terms of algebraically (un-normalised) parabolically induced representations, we have

$$ \begin{align*}I(s, \chi \times \sigma) \ = \ {}^{\mathrm{a}}\mathrm{Ind}_{P({\mathbb A})}^{G({\mathbb A})} \left(\chi[s+n] \times \sigma \right). \end{align*} $$

In particular, observe that

$$ \begin{align*}I(s, \chi \times \sigma)|_{s = -n} \ = \ {}^{\mathrm{a}}\mathrm{Ind}_{P({\mathbb A})}^{G({\mathbb A})} \left(\chi \times \sigma \right). \end{align*} $$

We designate $s = s_0 = -n$ as our point of evaluation. The algebraic induction works well for cohomological purposes, whereas the analytic theory of Langlands–Shahidi L-functions works well with normalised induction. In the arithmetic theory of L-functions, it is a necessary evil to go back and forth between these two forms of induction.

6.2 The standard intertwining operator

The weyl group element $w_{P_{0}}$ defined in Section 5.1 is realised by conjugation by the matrix in $\mathrm {O}(2n+2)$ :

$$ \begin{align*}\tilde{w}_{P_0} \ = \ \begin{pmatrix} & & & & 1 \\ & I_{n-1} & & & \\ & & J_2 & & \\ & & & I_{n-1} & \\ 1 & & & & \end{pmatrix}. \end{align*} $$

We will denote $\tilde {w}_{P_0} $ simply as $w_{P_0}$ itself. The conjugation action of $w_{P_0}$ on the inducing representation is given by $w_{P_0}(\chi [s] \times \sigma ) = \chi ^{-1}[-s] \times {}^{\kappa _{2n}} \sigma ,$ where the element $\kappa _{2n}$ was defined in (20). The standard global intertwining operator

$$ \begin{align*}T_{\mathrm{st}}(s, w_{P_0}, \chi \times \sigma) \ : \ I(s, \chi \times \sigma) \ \longrightarrow \ I(-s, \chi^{-1} \times {}^{\kappa_{2n}} \sigma) \end{align*} $$

is defined by the integral

$$ \begin{align*}T_{\mathrm{st}}(s, w_{P_0}, \chi \times \sigma)(f)(g) \ = \ \int \limits_{U_P({\mathbb A})} f(w_{P_0}^{-1} \, u\, g) \, du. \end{align*} $$

We will denote $T_{\mathrm {st}}(s, w_{P_0}, \chi \times \sigma )$ simply by $T_{\mathrm {st}}(s, w_{P_0})$ or even just $T_{\mathrm {st}}(s).$ The intertwining operator $T_{\mathrm {st}}(s)$ is a product of local intertwining operators

$$ \begin{align*}T_{\mathrm{st}}(s) \ = \ \otimes^{\prime}_v \, T_{\mathrm{st}, v}(s), \end{align*} $$

where the local operators are defined by analogous local integrals for each place $v $ of F. The global measure $du$ on $U_P({\mathbb A}) = U_{P_0}({\mathbb A}_F)$ is a product over the coordinates of $U_{P_0}$ of the additive measure $dx$ on ${\mathbb A}_F$ , which in turn is a product $\prod _v dx_v$ of local additive measures $dx_v$ on $F_v^+$ ; for a finite place v, we normalise $dx_v$ by $\mathrm {vol}({\mathcal O}_v) = 1$ , where ${\mathcal O}_v$ is the ring of integers of $F_v$ , and for an archimedean v, we take $dx_v$ as the Lebesgue measure on ${\mathbb R}.$

6.3 Local intertwining operators at a non-archimedean place: unramified case

Since we are dealing with an algebraically disconnected group $\mathrm {O}(2n)$ and not the connected algebraic group $\mathrm {SO}(2n)$ , some brief comments are in order on local unramified representations. The reader is referred to [Reference Atobe and Gan2, Sect. 2.3] for further details. For a finite place v of F, let ${\mathcal O}_v$ be the ring of integers of $F_v$ , $\varpi _v$ a uniformizer, and $q_v$ the cardinality of the residue field ${\mathcal O}_v/\varpi _v{\mathcal O}_v$ . Assume that v is such that $\chi _v$ and $\sigma _v$ are unramified. The orthogonal space we consider is $F^{2n}$ with a basis $\{w_1,\dots ,w_n, w_n^*,\dots ,w_1^*\}$ , and the nondegenerate symmetric bilinear form on $F^{2n}$ is represented by the matrix $J_{2n}$ with respect to this basis. Consider the ${\mathcal O}_v$ -lattice $L_v$ in $F^{2n} \otimes F_v = F_v^{2n}$ spanned by this basis: