2.1 Induced representations and “the point of evaluation”

Let $\delta _P$ be the modulus character of P; it is trivial on $N_P$ and on $M_P$ it is given by:

$$ \begin{align*}\delta_P(m) \ = \ |\det(\mathrm{Ad}_{N_P}(m))|, \quad m \in M_P, \end{align*} $$

where $\mathrm {Ad}_{N_P} : M_P \to {\textrm {GL}}({\textrm {Lie}}(N_P))$ is the adjoint representation of $M_P$ on the Lie algebra of $N_P$ , and $|\ |$ is the normalized absolute value on F. Let $Z(M_P)$ be the center of $M_P$ , $A_P$ the maximal split torus in $Z(M_P)$ . Let $X^*(A_P) = \mathrm {Hom}(A_P, F^*)$ and $X^*(M_P) = \mathrm {Hom}(M_P, F^*)$ denote the group of rational characters of $A_P$ and $M_P$ . Restriction from $M_P$ to $A_P$ gives an inclusion $X^*(M_P) \hookrightarrow X^*(A_P)$ , which induces an isomorphism $X^*(M_P) \otimes _{\mathbb {Z}} \mathbb {Q} \cong X^*(A_P) \otimes _{\mathbb {Z}} \mathbb {Q}.$ The modulus character $\delta _P$ is naturally an element of $\mathfrak {a}_P^* := X^*(A_P) \otimes _{\mathbb {Z}} \mathbb {R}.$ Fix a Weyl group invariant inner product (,) on $X^*(A_P) \otimes _{\mathbb {Z}} \mathbb {R}.$

Let $\mathbf {\Delta }_G$ be the set of all roots which are naturally in $X^*(T)$ ; for the choice of the Borel subgroup B, let $\mathbf {\Delta }_G^+$ be the set of positive roots, and $\mathbf {\Pi }_G$ the set of simple roots. Let $\rho _P$ be half the sum of all positive roots whose root spaces are in ${\textrm {Lie}}(N_P)$ ; via the restriction from T to $A_P$ we have $\rho _P \in X^*(A_P) \otimes _{\mathbb {Z}} \mathbb {Q}.$ We have the equality:

$$ \begin{align*}|2 \rho_P(m)| = \delta_P(m), \quad \forall m \in M. \end{align*} $$

Let $\mathfrak {a}_P = \mathrm {Hom}(X^*(A_P), \mathbb {R}) = \mathrm {Hom}(X^*(M_P), \mathbb {R})$ denote the real Lie algebra of $A_P,$ and $H_P : M_P \to \mathfrak {a}_P$ denote the Harish-Chandra homomorphism defined by:

$$ \begin{align*}\exp\langle \chi, H_P(m)\rangle \ := \ |\chi(m)|, \quad \forall \chi \in X^*(A_P), \ \forall m \in M. \end{align*} $$

In particular, taking $\chi = \rho _P$ , we get:

$$ \begin{align*}\exp\langle \rho_P, H_P(m) \rangle \ = \ \delta_P(m)^{1/2}, \quad \forall m \in M. \end{align*} $$

Let $(\pi , V_\pi )$ be an irreducible admissible representation of $M_P;$ the representation space $V_\pi $ is a vector space over $\mathbb {C}$ . For $\nu \in \mathfrak {a}_P^* \otimes _{\mathbb {R}} \mathbb {C}$ , define the induced representation

$$ \begin{align*}I(\nu,\pi) \ := \ \mathrm{Ind}_P^G(\pi \otimes \exp(\langle \nu, H_P(\ )\rangle) \otimes 1_U), \end{align*} $$

where $\mathrm {Ind}$ means normalized parabolic induction. The representation space $V(\nu , \pi )$ is the vector space of all smooth (i.e., locally constant) functions $f : G \to V_\pi $ such that

$$ \begin{align*}f(mng) \ = \ \pi(m) \exp(\langle \nu + \rho_P, H_P(m)\rangle ) f(g), \quad \forall g \in G, \ m\in M, \ n\in N. \end{align*} $$

Recall that P is a maximal parabolic subgroup, defined by a simple root $\alpha _P$ which is the unique simple root whose root space is in ${\textrm {Lie}}(N_P).$ Set $\langle \rho _P, \alpha _P \rangle = 2\frac {(\rho _P, \alpha _P)}{(\alpha _P, \alpha _P)}$ and put

$$ \begin{align*}\gamma_P \ := \ \tilde \alpha_P \ := \ \frac{1}{\langle \rho_P, \alpha_P \rangle} \, \rho_P. \end{align*} $$

In the Langlands–Shahidi machinery the notation $\tilde \alpha $ is commonly used; elsewhere in the arithmetic theory of automorphic forms the notation $\gamma _P$ is commonly used; it is the fundamental weight corresponding to the simple root $\alpha _P.$ For $s \in \mathbb {C}$ , define $\nu _s$ as:

$$ \begin{align*}\nu_s \ := \ s \tilde\alpha_P \ = \ \frac{s}{\langle \rho_P, \alpha_P \rangle} \, \rho_P. \end{align*} $$

Let $I(s, \pi ) := I(s\tilde \alpha _P, \pi )$ , whose representation space $V(s, \pi ) := V(s\tilde \alpha _P, \pi )$ consists of all locally constant functions $f : G \to V_\pi $ such that

$$ \begin{align*}f(mng) \ = \ \pi(m) \, \delta_P(m)^{\frac12 + \frac{s}{2 \langle \rho_P, \alpha_P \rangle}} \, f(g), \quad \forall g \in G, \ m\in M, \ n\in N. \end{align*} $$

Definition 2.1.1 (Point of evaluation)

Define the point of evaluation k as

$$ \begin{align*}k \ := \ - \langle \rho_P, \alpha_P \rangle, \end{align*} $$

which depends only on P and G, and has the property that $I(k, \pi ) = {}^{\mathrm {a}}\mathrm {Ind}_P^G(\pi )$ which is the algebraic (i.e., un-normalized) parabolic induction from P to G of the representation $\pi .$

The point of evaluation k is half-integral, i.e., $k \in \mathbb {Z}$ or $k \in \tfrac 12 + \mathbb {Z}$ , or more succinctly $2k \in \mathbb {Z}$ , since $\langle \beta , \alpha _P \rangle = 2(\beta , \alpha _P)/(\alpha _P, \alpha _P) \in \mathbb {Z}$ for any root $\beta $ . In general, k can be integral or a genuine half-integer; for example, if $G = {\textrm {GL}}(N)$ with $N \geq 2$ , and P is any maximal parabolic subgroup, then $k = -N/2$ ; see Section 5.1.

2.2 The standard intertwining operator: definition and analytic properties

For the maximal parabolic subgroup $P = P_{\Theta }$ , where $\Theta = \mathbf {\Pi } \setminus \{\alpha _P\}$ , recall that $w_0$ is the unique element in the Weyl group such that $w_0(\Theta ) \subset \mathbf {\Pi }$ and $w_0(\alpha _P) < 0;$ let $Q = P_{w_0(\Theta )}$ be the maximal parabolic subgroup associate to P. Then $M_Q = w_0 M_P w_0^{-1},$ and let ${}^{w_0}\pi $ be the representation of $M_Q$ given by conjugation.

Let $f \in I_P^G(s, \pi )$ and $g \in G$ . Suppose there exists a vector v in the inducing representation of $I_Q^G(-s, {}^{w_0}\pi )$ , such that for all $\check {v}$ in the contragredient of this inducing representation the integral $\int _{N_Q} \langle f(w_0^{-1} n g), \check {v} \rangle dn$ converges absolutely to $\langle v, \check {v} \rangle $ then define $\int _{N_Q} \langle f(w_0^{-1} n g) \, dn = v.$ If this is verified for all $f \in I_P^G(s, \pi )$ and all $g \in G$ then define an intertwining operator for G-modules

$$ \begin{align*}T_{\mathrm{st}}(s) : \ I_P^G(s, \pi) \ \longrightarrow \ I_Q^G(-s, {}^{w_0}\pi) \end{align*} $$

by the integral

(2.2.1) $$ \begin{align} T_{\mathrm{st}}(s)(f)(g) \ = \ \int_{N_Q} f(w_0^{-1} n g) \, dn; \quad f \in V(s, \pi), \ g \in G. \end{align} $$

Assume, here and henceforth, that the measures in such intertwining integrals are chosen to be $\mathbb {Q}$ -valued. The operator $T_{\mathrm {st}}(s)$ is denoted as $A(s, \pi , w_0)$ in Shahidi [Reference Shahidi23, Section 4.1] (see also Kim [Reference Kim, Cogdell, Kim and Ram Murty11, Section 4.3]). That $T_{\mathrm {st}}(s)(f) \in I_Q^G(-s, {}^{w_0}\pi )$ is verified in loc.cit. The following convergence statement is a special case of [Reference Shahidi23, Proposition 4.1.2]:

If $T_{\mathrm {st}}(s)(f)(g)$ converges absolutely for all $g \in G$ and all $f \in V(s, \pi ),$ then we will simply say that $T_{\mathrm {st}}(s)$ converges absolutely. One can be more specific about the domain of convergence in the tempered case.

The above convergence statements are contained in Harish-Chandra’s work on harmonic analysis on p-adic reductive groups; the reader is referred to [Reference Shahidi19, Section 2.2] and the references therein; see also [Reference Kim, Cogdell, Kim and Ram Murty11, Proposition 12.3].

Without worrying about convergence, let us see the shape of the standard intertwining operator at the point of evaluation $s = k.$ The domain of $T_{\mathrm {st}}(s)|_{s = k},$ as noted above, is $I(k, \pi ) = {}^{\mathrm {a}}\mathrm {Ind}_P^G(\pi )$ . The codomain is $I_Q^G(-s, {}^{w_0}\pi ) := I(-s\tilde \alpha _Q, {}^{w_0}\pi )$ , whose representation space consists of all locally constant functions $f' : G \to V_{{}^{w_0}\pi } = V_\pi $ such that

$$ \begin{align*}f'(m'n'g') \ = \ {}^{w_0}\pi(m') \, |\delta_Q(m')|^{\frac12 - \frac{s}{2 \langle \rho_Q, \alpha_Q \rangle}} \, f'(g'), \quad \forall g' \in G, \ m'\in M_Q, \ n'\in N_Q, \end{align*} $$

where ${}^{w_0}\pi (m') = \pi (w_0^{-1}m' w_0).$ Put $s = k = - \langle \rho _P, \alpha _P \rangle $ ; since $\langle \rho _Q, \alpha _Q \rangle = \langle \rho _P, \alpha _P \rangle $ we get:

$$ \begin{align*}f'(m'u'g') \ = \ {}^{w_0}\pi(m') \, \delta_Q(m') \, f'(g'). \end{align*} $$

Hence, at the point of evaluation, in terms of un-normalized induction we get:

(2.2.4) $$ \begin{align} T_{\mathrm{st}}(s)|_{s = k} : \ {}^{\mathrm{a}}\mathrm{Ind}_P^G(\pi) \ \longrightarrow \ {}^{\mathrm{a}}\mathrm{Ind}_Q^G({}^{w_0}\pi \otimes \delta_Q). \end{align} $$

2.3 Local factors and the local Langlands correspondence

A defining aspect of the Langlands program is Langlands’s computation [Reference Langlands14, Section 5] of the constant term of an Eisenstein series, which at a local unramified place boils down to computing the standard intertwining operator on “the” spherical vector which is a scalar multiple of the spherical vector on the other side, and this scalar multiple is an expression denoted $M(s)$ in loc.cit. Langlands says that Tits pointed out to him how to express $M(s)$ in a more convenient form. This is now an important ingredient in the Langlands–Shahidi machinery; see [Reference Shahidi20, Section 2].

Let ${}^L\!G^\circ $ be the complex reductive group which is the connected component of the Langlands dual ${}^L\!G$ of G; see [Reference Borel, Borel and Casselman4, I.2]; and let ${}^L\!P$ be the parabolic subgroup of ${}^L\!G$ corresponding to P, and ${}^L\!N$ its unipotent radical. The Levi quotient ${}^L\!M^\circ $ of ${}^L\!P^\circ $ acts on the Lie algebra ${}^L\mathfrak {n}$ of ${}^L\!N^\circ $ by the adjoint action. There is a positive integer m such that the set $\{\langle \tilde \alpha _P, \beta \rangle \}$ – as $\beta $ varies over positive roots such that the root space ${}^L\mathfrak {g}(\beta ^\vee )$ of the dual root $\beta ^\vee $ is in ${}^L\mathfrak {n}$ – is $\{1,\dots ,m\}.$ For each $1 \leq j \leq m$ put

(2.3.1) $$ \begin{align} V_j \ = \ \text{span of } {}^L\mathfrak{g}(\beta^\vee) \text{ for } \beta \text{ such that } \langle \tilde\alpha_P, \beta \rangle = j. \end{align} $$

Then the action of ${}^L\!M^\circ $ on ${}^L\mathfrak {n}$ stabilizes each $V_j$ and furthermore acts irreducibly on $V_j$ . Denote $r_j$ the action of ${}^L\!M^\circ $ on $V_j$ , and ${}^L\mathfrak {n} = \oplus _{j=1}^m r_j$ is a multiplicity free decomposition as an ${}^L\!M^\circ $ -representation. Let $\tilde r_j$ denote the contragredient of $r_j.$

Given a smooth irreducible admissible representation $\pi $ that is generic, i.e., has a Whittaker model, and for $1 \leq j \leq m$ , the local aspects of the Langlands–Shahidi machinery attaches a local L-factor $L(s, \pi , \tilde r_j)$ (see [Reference Shahidi21]) which is the inverse of a polynomial in $q^{-s}$ of degree at most $d_j := \dim (V_j);$ when $\pi $ is unramified this degree is $d_j$ .

Let $W_F$ be the Weil group of F, and $W_F' = W_F \times {\textrm {SL}}_2(\mathbb {C})$ the Weil–Deligne group. The local Langlands correspondence for G says that to $\pi $ corresponds its Langlands parameter which is an admissible homomorphism $\phi _\pi : W_F' \to {}^L\!M_P$ ; see Borel [Reference Borel, Borel and Casselman4, Section 8] for the requirements on the parameter $\phi _\pi .$ Composing with $\tilde r_j$ gives $\tilde r_j \circ \phi _\pi : W_F' \to {}^L {\textrm {GL}}_{d_j},$ an admissible homomorphism which parametrizes, via the local Langlands correspondence for ${\textrm {GL}}_{d_j}(F)$ , a smooth irreducible admissible representation of ${\textrm {GL}}_{d_j}(F)$ that we denote $\tilde r_j(\pi ).$ As in Shahidi [Reference Shahidi24, p. 3] we will impose the working hypothesis:

(2.3.2) $$ \begin{align} L(s, \pi, \tilde r_j) \ = \ L(s, \tilde r_j(\pi)), \end{align} $$

that is known in a number of instances; see the references in loc.cit.

2.4 The notion of being on the right of the unitary axis

Recall that $\pi $ is a smooth irreducible admissible representation of $M_P$ , which is to be a local component of a globally generic cuspidal automorphic representation (needed by the context in which we can evoke the Langlands–Shahidi machinery), and keeping the generalized Ramanujan conjecture in the back of our minds (see, for example, [Reference Shahidi22]), we will impose the condition that $\pi $ is essentially tempered, i.e., tempered mod the center.

Let $M_P^1 = \bigcap _{\chi \in X^*(M_P)} \mathrm {Ker}(|\chi |_F),$ the subgroup of $M_P$ generated by all compact subgroups. For the split center of $M_P$ , say $A_P \cong F^* \times \cdots \times F^*$ , and $\eta _i : A_P \to F^*$ is the projection to the ith copy. Define $A_P^1 \subset A_P$ similar to $M_P^1$ ; we have $A_P^1 = A_P \cap M_P^1.$ Let $X(M_P) = \mathrm {Hom}(M_P/M_P^1, \mathbb {C}^*)$ ; similarly, $X(A_P)$ . Restricting from $M_P$ to $A_P$ gives an isomorphism $X(M_P) \cong X(A_P).$ Given $\underline z = (z_1,\dots , z_l) \in \mathbb {C}^l$ , we get an unramified character $A_P \to \mathbb {C}^*$ given by $|\eta _1|^{z_1} \otimes \cdots \otimes |\eta _l|^{z_l}$ , and via $X(M_P) \cong X(A_P)$ an unramified character of $M_P$ which we will denote as $\underline \eta ^{\underline z}$ . Given $\pi $ as above, tempered modulo the center means that there exists an l-tuple of exponents $\underline e = (e_1, \dots , e_l) \in \mathbb {R}^l$ and a smooth irreducible unitary tempered representation $\pi ^t$ such that $\pi \ \cong \ \pi ^t \otimes \underline \eta ^{\underline e}.$ (Keeping global applications in mind, we will impose later a hypothesis that the exponents $e_i$ are [half-]integral.) The representation $\tilde r_j(\pi )$ of ${\textrm {GL}}_{d_j}(F)$ , obtained by functoriality, is also tempered modulo its center. A few words of explanation might be helpful. There is an exponent $\tilde f_j = f(\tilde r_j, e_1, \dots , e_l) \in \mathbb {R}$ , that depends on the representation $r_j$ and the exponents $e_1,\dots ,e_l$ , such that

$$ \begin{align*}\tilde r_j(\pi) \ \cong \ \tilde r_j(\pi)^t \otimes |\det |^{\tilde f_j} \end{align*} $$

with $\tilde r_j(\pi )^t = \tilde r_j(\pi ^t)$ being a unitary tempered irreducible representation of ${\textrm {GL}}_{d_j}(F)$ . Local functoriality preserves temperedness, by the desiderata in [Reference Borel, Borel and Casselman4, 10.4, (4)], and the one has to keep track of the central characters for which consider the diagram:

(2.4.1)

Since the representation $\tilde r_j$ is irreducible, the center ${}^L\!A_P^\circ \cong (\mathbb {C}^*)^l$ of ${}^L\!M_P^\circ $ acts via scalars, explaining the bottom horizontal arrow. The unramified character $\underline \eta ^{\underline e}$ of $A_P$ corresponds to its Satake parameter $\vartheta _{{\underline \eta }^{\underline e}}$ in ${}^L\!A_P^\circ $ ; we get $\tilde r_j(\vartheta _{{\underline \eta }^{\underline e}}) \in \mathbb {C}^*$ , which corresponds to an unramified character $|\ |^{\tilde f_j}$ of $F^*$ , or the character $|\det |^{\tilde f_j}$ of ${\textrm {GL}}_{d_j}(F)$ , for some exponent $\tilde f_j$ which, a priori, lives in $\mathbb {C}$ , but since $e_j \in \mathbb {R}$ and $\tilde r_j$ is an algebraic representation, it is clear that $\tilde f_j \in \mathbb {R}.$

Definition 2.4.3 Let $\pi $ be a smooth irreducible admissible generic representation of $M_P$ that is tempered modulo the center with exponents $e_1,\dots , e_l \in \mathbb {R}$ . We say $\pi $ is on the right of the unitary axis with respect to the ambient group G, if

$$ \begin{align*}- \langle \rho_P, \alpha_P \rangle + \tilde f_1> 0. \end{align*} $$

By Lemma 2.4.2 it follows that for each $1 \leq j \leq m$ we have: $- j \langle \rho _P, \alpha _P \rangle + \tilde f_j> 0.$

Corollary 2.4.4 Let $\pi $ be a smooth irreducible admissible generic representation of $M_P$ that is tempered modulo the center with exponents $e_1,\dots , e_l \in \mathbb {R}$ , and which is on the right of the unitary axis with respect to the ambient group G, then the local L-values

$$ \begin{align*}L(jk, \pi, \tilde r_j) \ \ \mathrm{and} \ \ L(jk+1, \pi, \tilde r_j) \end{align*} $$

are finite for each $1 \leq j \leq m$ , where, recall that $k = - \langle \rho _P, \alpha _P \rangle $ is the point of evaluation.

The condition of being on the right of the unitary axis is tailor-made to appeal to Shahidi’s tempered L-functions conjecture that is now a theorem after the work of many authors (see [Reference Shahidi23, p. 147]) culminating in [Reference Heiermann and Opdam9].

3.1 Digression on the adjectives: rationality, algebraicity, and arithmeticity

First of all, even among experts, there seems to be no universal agreement on the precise meaning of these adjectives. In this article, all three words are used, and it might help the reader to clarify their meanings. The word rationality has two meanings and the context usually makes it clear. First of all, a result of the form $``(\mathrm {L-value})/(\mathrm {periods}) \in \mathbb {Q}$ ” is often called a rationality result for L-values. Then there is a common abuse of terminology and a result of the form $``(\mathrm {L-value})/(\mathrm {periods}) \in \overline {\mathbb {Q}}$ ” is also called a rationality result. We will refer to the latter as an algebraicity result for L-values. A second usage of rationality, as in the context of Waldspurger’s result, comes from algebraic geometry and means that some function or operator at hand is a rational function on an algebraic variety. To explain our usage of the word arithmetic, suppose we have an L-value at hand, which is the value at $s = s_0$ of the L-function $L(s, \pi )$ attached to some object $\pi $ defined over $\mathbb {C}$ , for example, $\pi $ can be the finite part of a cuspidal automorphic representation. We may set up our context for the object $\pi $ to have an arithmetic origin, i.e., there is an object $\sigma $ defined over some coefficient field E, such that for some embedding of fields $\iota : E \to \mathbb {C}$ , the base-change ${}^\iota \sigma $ of $\sigma $ via $\iota $ is the object $\pi .$ In such a context, a result of the form $``L(s_0, {}^\iota \sigma )/(\mathrm {periods}) \in \iota (E)"$ is given the appellation of an arithmetic result for L-values. With this explanation of the words, the purpose of this section is to show that Waldspurger’s rationality result for intertwining operator has an arithmetic origin. We will use the word arithmeticity for the behaviour of an arithmetic result upon changing $\iota $ , or equivalently, by replacing $\iota $ by $\tau \circ \iota $ for any $\tau \in \mathrm {Gal}(\bar {\mathbb {Q}}/{\mathbb {Q}})$ ; this is compatible with the usage of arithmeticity as in [Reference Gan, Raghuram, Prasad, Rajan, Sankarnarayanan and Sengupta7].

3.2 A rationality result of Waldspurger

In this subsection, we will adumbrate the presentation in [Reference Waldspurger25, IV.1]. Recall the notations $M_P^1$ , $A_P^1$ , $X(M_P),$ and $X(A_P)$ from Section 2.4. When P is fixed we drop the subscript P from $M_P$ , $A_P$ , etc.

Note that $X(M)$ has the structure of an algebraic variety over $\mathbb {C}$ ; denote by $\mathcal {B}$ the $\mathbb {C}$ -algebra of polynomial functions on $X(M)$ . Let $(\pi , V)$ be a smooth admissible representation of M on a $\mathbb {C}$ -vector space V, and let $\mathcal O_{\mathbb {C}} = \{\pi \otimes \chi : \chi \in X(M)\}.$ A function $f : \mathcal O_{\mathbb {C}} \to \mathbb {C}$ is a polynomial if there exists $b \in \mathcal {B}$ such that $f(\pi \otimes \chi ) = b(\chi )$ . For an open set $\mathcal U \subset \mathcal O_{\mathbb {C}}$ , a function $f:\mathcal U \to \mathbb {C}$ is a rational function if there exists $b_1, b_2 \in \mathcal {B}$ such that $b_1(\chi ) f(\pi \otimes \chi ) = b_2(\chi )$ for all $\chi \in X(M)$ with $\pi \otimes \chi \in \mathcal U$ and $b_1(\chi ) \neq 0.$

Let $I_P^G(\pi \otimes \chi )$ be the normalized parabolically induced representation. Restriction from G to its maximal compact subgroup K sets up an isomorphism $I_P^G(\pi \otimes \chi ) \cong I_{K\cap P}^K(\pi ).$ Let $P'$ be a maximal parabolic subgroup of G that has the same Levi subgroup $M = M_{P'} = M_P.$ For each $\pi \otimes \chi \in \mathcal O_{\mathbb {C}}$ suppose we are given a G-equivariant operator $A(\pi \otimes \chi ) : I_P^G(\pi \otimes \chi ) \to I_{P'}^G(\pi \otimes \chi )$ that depends only on the equivalence class of $\pi \otimes \chi .$ We say that the operator $A(\pi \otimes \chi )$ is a polynomial if for all $f \in I_{K\cap P}^K(\pi )$ there exist finitely many $f_1,\dots ,f_r \in I_{K\cap P'}^K(\pi )$ and $b_1,\dots , b_r \in \mathcal {B}$ such that $A(\pi \otimes \chi )(f) \ = \ \sum _{i=1}^r b_i(\chi ) f_i$ for all $\chi \in X(M).$ Furthermore, we say $A(\pi \otimes \chi )$ is rational if there exists $b \in \mathcal {B}$ , such that for all $f \in I_{K\cap P}^K(\pi )$ there exist finitely many $f_1,\dots ,f_r \in I_{K\cap P'}^K(\pi )$ and $b_1,\dots , b_r \in \mathcal {B}$ such that

(3.2.1) $$ \begin{align} b(\chi) A(\pi \otimes\chi)(f) \ = \ \sum_{i=1}^r b_i(\chi) f_i, \quad \text{for all } \chi \in X(M) \text{ with } b(\chi) \neq 0. \end{align} $$

Rationality of the intertwining operators may be formulated in another way that is used in the proof of [Reference Waldspurger25, Theorem IV.1.1], and which will allow us to descend the statement and proof to an arithmetic level to give us Theorem 3.3.7. For $m \in M$ , let $b_m \in B$ be defined as $b_m(\chi ) = \chi (m).$ Define $V_{\mathcal {B}} = V \otimes _{\mathbb {C}} \mathcal {B}$ on which M acts as: $\pi _{\mathcal {B}}(m)(v \otimes b) = \pi (m)v \otimes b_mb.$ For $\chi \in X(M)$ , let $\mathcal {B}_\chi $ be the maximal ideal $\{b \in \mathcal {B} : b(\chi ) = 0\}.$ Then the action of M on $V_{\mathcal {B}} \otimes \mathcal {B}/\mathcal {B}_\chi $ is the representation $\pi \otimes \chi $ . Similarly, $I_P^G(V_{\mathcal {B}}) = I_P^G(V) \otimes _{\mathbb {C}} \mathcal {B}$ . Let $\mathrm {sp}_\chi : \pi _{\mathcal {B}} \to \pi _{\mathcal {B}} \otimes \mathcal {B}/\mathcal {B}_\chi = \pi \otimes \chi $ denote the specialization map; same notation also for $\mathrm {sp}_\chi : I_P^G(\pi _{\mathcal {B}}) \to I_P^G(\pi \otimes \chi )$ . The collection $\{A(\pi \otimes \chi )\}_{\pi \otimes \chi \in \mathcal O_{\mathbb {C}}}$ of operators is polynomial if and only if there exists a G-equivariant homomorphism of $\mathcal {B}$ -modules $A_{\mathcal {B}} : I_P^G(\pi _{\mathcal {B}}) \to I_{P'}^G(\pi _{\mathcal {B}})$ such that the following diagram commutes

(3.2.2)

for all $\chi $ , i.e., $\mathrm {sp}_\chi \circ A_{\mathcal {B}} = A(\pi \otimes \chi ) \circ \mathrm {sp}_{\chi }.$ Similarly, the collection $\{A(\pi \otimes \chi )\}_{\pi \otimes \chi \in \mathcal O_{\mathbb {C}}}$ of operators is rational if and only if there exists $A_{\mathcal {B}}$ as above and an element $b \in \mathcal {B}$ such that

(3.2.3) $$ \begin{align} A(\pi \otimes \chi) \circ \mathrm{sp}_{\chi} \circ (1 \otimes b) \ = \ \mathrm{sp}_\chi \circ A_{\mathcal{B}}. \end{align} $$

Suppose $\tilde f \in I_P^G(\pi _{\mathcal {B}})$ and $A_{\mathcal {B}}(\tilde f) = \sum _i \tilde f_i \otimes b_i$ , and if $\mathrm {sp}_\chi $ maps $\tilde f$ to f, and similarly, $\tilde f_i$ to $f_i$ , then (3.2.3) becomes $b(\chi ) A(\pi \otimes \chi )(f) = \sum _i b_i(\chi ) f_i$ as in (3.2.1). We may and shall talk about the collection $\{A(\pi \otimes \chi )\}_{\chi \in \mathcal U}$ of operators being rational on an open subset $\mathcal U \subset X(M).$

Let $N'$ be the unipotent radical of $P'$ . Let $f \in I_P^G(V)$ and $g \in G$ . Suppose, there exists $v \in V$ such that for all $\check {v} \in \check {V}$ —the representation space of the contragredient $\check {\pi }$ of $\pi $ —the integral $\int _{N' \cap N \backslash N'} \langle f(n'g), \check {v} \rangle dn'$ converges absolutely to $\langle v, \check {v} \rangle $ then define $\int _{N' \cap N \backslash N'} f(n'g) \, dn' = v.$ If this is verified for all $f \in I_P^G(\pi \otimes \chi )$ and all $g \in G$ then define an intertwining operator for G-modules $J(\pi \otimes \chi ) : I_P^G(\pi \otimes \chi ) \to I_{P'}^G(\pi \otimes \chi )$ as:

(3.2.4) $$ \begin{align} J(\pi \otimes \chi)(f)(g) \ = \ \int_{N' \cap N \backslash N'} f(n'g) \, dn'. \end{align} $$

Note the similarities and differences between the integrals in (2.2.1) and (3.2.4).

Let $\Sigma (A_P)$ denote the set of roots of $A_P$ in ${\textrm {Lie}}(G)$ ; identify $\Sigma (A_P)$ with a subset of $\mathfrak {a}_M^*$ . Denote by $\Sigma (P)$ the subset of $\Sigma (A_P)$ of those roots whose root spaces appear in ${\textrm {Lie}}(P).$ For $P'$ with the same Levi as P, let $\bar {P'}$ denote its opposing parabolic subgroup. (This $\bar {P}'$ is the Q from before.) The following theorem is contained in [Reference Waldspurger25, Theorem IV.1.1].

It is the rationality assertion in the above theorem that we are particularly interested in, since convergence in our context is already guaranteed by Proposition 2.2.2. We summarize the key steps of its proof, and refer the reader to [Reference Waldspurger25, IV.1] for all the details and also for some of the notations used below even if not defined here because it would take us too far to systematically define them.

1. (1) (Reduction step.) We need a G-equivariant homomorphism of $\mathcal {B}$ -modules

   $$ \begin{align*}J_{\mathcal{B}} : I_P^G(\pi_{\mathcal{B}})\to I_{P'}^G(\pi_{\mathcal{B}}) \end{align*} $$
   as in (3.2.2), that satisfies the requirement of (3.2.3). Frobenius reciprocity for Jacquet modules and parabolic induction gives:
   $$ \begin{align*}\mathrm{Hom}_{G, \mathcal{B}}(I_P^G(\pi_{\mathcal{B}}), \, I_{P'}^G(\pi_{\mathcal{B}})) = \mathrm{Hom}_{M, \mathcal{B}}(I_P^G(\pi_{\mathcal{B}})_{P'}, \, \pi_{\mathcal{B}}), \end{align*} $$
   where, $I_P^G(\pi _{\mathcal {B}})_{P'}$ is the Jacquet module of $I_P^G(\pi _{\mathcal {B}})$ with respect to $P'$ on which the action of M is the canonical action twisted by $\delta _{P'}^{-1/2}$ to account for normalized parabolic induction. It suffices then to construct
   $$ \begin{align*}j_{\mathcal{B}} \in \mathrm{Hom}_{M, \mathcal{B}}(I_P^G(\pi_{\mathcal{B}})_{P'}, \, \pi_{\mathcal{B}}) \end{align*} $$
   such that the associated map $J_{\mathcal {B}}$ via Frobenius reciprocity satisfies (3.2.3).

2. (2) (Exponents in the Jacquet module of an induced representation.) By the well-known results of Bernstein and Zelevinskii [Reference Bernstein and Zelevinsky2, 2.12] (see also [Reference Waldspurger25, I.3]), the Jacquet module $I_P^G(\pi _{\mathcal {B}})_{P'}$ is filtered by $(M,\mathcal {B})$ -submodules $\{\mathcal {F}_{w,P'}\}_{w \in {}^{P'}W^P},$ indexed by a certain totally ordered set ${}^{P'}W^P$ of representatives in the Weyl group, such that for the successive quotients we have an isomorphism

   $$ \begin{align*}q_w : \mathcal{F}_{w,P'}/\mathcal{F}_{w^+,P'} \to I_{M \cap w\cdot P}^M(w \cdot V_{\mathcal{B}, M \cap w^{-1}\cdot P'}), \end{align*} $$
   the right-hand side being a parabolically induced module of M. Consider these successive quotients for the action of the split center $A_P$ of M; and let $\mathcal {E}xp_w$ be the set of exponents, which are characters $A_P \to \mathcal {B}^\times $ , that appear in the (co-)domain of $q_w.$ We may suppose that $1 \in {}^{P'}W^P;$ the image of $q_1$ is $V_{\mathcal {B}}$ . For any $w \in {}^{P'}W^P$ , if $w \neq 1$ then $\mathcal {E}xp_w \cap \mathcal {E}xp_1 = \emptyset ;$ see [Reference Waldspurger25, p. 280].

3. (3) (Killing all subquotients except one.) Using the theory of resultants, Waldspurger constructs $R \in \mathcal {B}[A_P]$ and $b \in \mathcal {B}$ such that R maps the Jacquet module $I_P^G(\pi _{\mathcal {B}})_{P'}$ into $\mathcal {F}_{1,P'}$ and on each of the generalized eigenspace for $\mu \in \mathcal {E}xp_1$ appearing in $\mathcal {F}_{1,P'} / \mathcal {F}_{1^+,P'} \otimes \mathrm {Frac}(\mathcal {B})$ it acts as homothety by the element $b.$ The required element $j_{\mathcal {B}}$ as in (1) is the composition of R followed by $\mathcal {F}_{1,P'} \to \mathcal {F}_{1,P'} / \mathcal {F}_{1^+,P'} \stackrel {q_1}{\longrightarrow } V_{\mathcal {B}}.$

3.3 An arithmetic variant of Theorem 3.2.5

Let E be a “large enough” finite Galois extension of $\mathbb {Q}.$ The meaning of large enough will be explained in context. Let $X_E(M) = \mathrm {Hom}(M/M^1, E^*)$ ; similarly, $X_E(A)$ . Restriction from M to A gives an isomorphism $X_E(M) \cong X_E(A).$ If $A = F^* \times \cdots \times F^*$ , l-copies, then $A/A^1 = \varpi _F^{\mathbb {Z}} \times \cdots \times \varpi _F^{\mathbb {Z}},$ with $\varpi _F^{\mathbb {Z}}$ being the multiplicative infinite cyclic group generated by the uniformizer $\varpi _F.$ Also, $X_E(A) = E^* \times \cdots \times E^*$ , where $\underline t = (t_1, \dots , t_l) \in E^* \times \cdots \times E^*$ corresponds to the character $\chi _{\underline t}$ that maps $\underline a = (a_1,\dots ,a_l) \in A$ to $\prod t_i^{\mathrm {ord}_F(a_i)}.$ An embedding of fields $\iota : E \to \mathbb {C}$ , gives a map $\iota _* : X_E(M) \to X_{\mathbb {C}}(M)$ where $\iota _*\chi = \iota \circ \chi .$ The following diagram might help the reader:

For $\underline s =(s_1,\dots ,s_l) \in \mathbb {C}/(\mathbb {Z} \cdot \tfrac {2\pi i}{\log (q)}) \times \cdots \times \mathbb {C}/(\mathbb {Z} \cdot \tfrac {2\pi i}{\log (q)})$ put $\underline w := q^{\underline s}$ , i.e., $\underline w = (w_1,\dots ,w_l) = (q^{s_1},\dots , q^{s_l}) \in \mathbb {C}^* \times \cdots \times \mathbb {C}^*$ that corresponds to the character $\chi _{\underline w}$ of A given by $\underline a \mapsto \prod _i w_i^{\mathrm {ord}_F(a_i)}.$ Note that $X_E(M)$ has the structure of an algebraic variety over E; denote by $\mathcal {B}_E(M)$ the E-algebra of polynomial functions on $X_E(M)$ ; then $\mathcal {B}_E(M) = E[t_1,t_1^{-1},\dots ,t_l,t_l^{-1}].$ Similarly, $\mathcal {B}_{\mathbb {C}}(M) = \mathbb {C}[w_1,w_1^{-1},\dots , w_l,w_l^{-1}] = \mathbb {C}[q^{s_1}, q^{-s_1},\dots , q^{s_l}, q^{-s_l}].$ Base-change via the embedding $\iota $ gives: $\mathcal {B}_E(M) \otimes _{E,\iota } \mathbb {C} = \mathcal {B}_{\mathbb {C}}(M).$ To homogenize with the notations of Waldspueger [Reference Waldspurger25] as used in Section 3.2, abbreviate $X_{\mathbb {C}}(M)$ and $\mathcal {B}_{\mathbb {C}}(M)$ as $X(M)$ and $\mathcal {B}$ , respectively.

Hypotheses we impose on a representation in the main result have an arithmetic origin

Let $(\sigma , V_{\sigma , E})$ be a smooth absolutely irreducible admissible representation of M over an E-vector space $V_{\sigma , E}.$ For an embedding of fields $\iota : E \to \mathbb {C}$ , we have the irreducible admissible representation ${}^\iota \sigma $ of M on the $\mathbb {C}$ -vector space $V_{{}^\iota \sigma } := V_{\sigma , E} \otimes _{E, \iota } \mathbb {C}$ . We may apply the considerations of Section 2 to $({}^\iota \sigma , V_{{}^\iota \sigma }).$ We explicate below all the hypotheses we impose on the representation ${}^\iota \sigma $ in the main result Theorem 3.3.7; these hypotheses are motivated by our global applications, and are expected to have an arithmetic origin.

The global context of a cohomological cuspidal automorphic representation suggests, via purity considerations, the following hypothesis on $\sigma $ . Recall that for exponents $\underline e = (e_1,\dots ,e_l) \in \mathbb {R}^l$ , by $\underline \eta ^{\underline e} \in X(A) = X(M)$ defined as: $\underline \eta ^{\underline e}(\underline a) = \prod _i |a_i|^{e_i}$ for $\underline a = (a_1,\dots ,a_l) \in A.$

Hypothesis 3.3.1 (Arithmeticity for half-integral unitarity)

Let $(\sigma , V_{\sigma , E})$ be a smooth absolutely irreducible admissible representation of a reductive p-adic group M over an E-vector space $V_{\sigma , E}.$ If for one embedding $\iota : E \to \mathbb {C}$ , there exists an l-tuple of integers $\underline {\textsf {w}} = ({\textsf {w}}_1,\dots , {\textsf {w}}_l)$ such that the representation ${}^\iota \sigma \otimes \underline \eta ^{\underline {\textsf {w}}/2}$ is unitary then for every embedding $\iota : E \to \mathbb {C}$ , the representation ${}^\iota \sigma \otimes \underline \eta ^{\underline {\textsf {w}}/2}$ is unitary.

It makes sense to call a $\sigma $ satisfying the above hypothesis as half-integrally unitary.

Hypothesis 3.3.2 (Arithmeticity for essential-temperedness)

Let $(\sigma , V_{\sigma , E})$ be a smooth, absolutely irreducible, admissible, half-integrally unitary representation of a reductive p-adic group M over an E-vector space $V_{\sigma , E}.$ If for one embedding $\iota : E \to \mathbb {C}$ the representation ${}^\iota \sigma $ is essentially tempered, then for every embedding $\iota : E \to \mathbb {C}$ the representation ${}^\iota \sigma $ is essentially tempered.

Proof of Hypothesis 3.3.2 for ${\textrm {GL}}_n(F)$

For ${\textrm {GL}}_n(F)$ this follows from the considerations in Clozel [Reference Clozel5] while using Jacquet’s classification of tempered representations [Reference Jacquet, Carmona and Verne10]; such a proof is well-known to experts and so we will just sketch the details. The reader is also referred to [Reference Prasad and Raghuram16, Section 9.2] for a summary of the classification of tempered representations that we will use below. For a representation $\pi $ of ${\textrm {GL}}_n(F)$ and $t \in \mathbb {R}$ , $\pi (t)$ denotes $\pi \otimes |\ |^t$ .

1. (1) Any tempered representation $\pi $ of $G = {\textrm {GL}}_n(F)$ is fully induced from discrete series representations; it is of the form:

   $$ \begin{align*}\pi = \mathrm{Ind}_{P_{n_1,\dots,n_r}(F)}^{G}(\pi_1\otimes \cdots \otimes \pi_r), \end{align*} $$
   where $\pi _i$ is a discrete series representation of ${\textrm {GL}}_{n_i}(F)$ ; $\sum _i n_i = n$ ; $P_{n_1,\dots ,n_r}(F)$ is the parabolic subgroup of G with Levi subgroup ${\textrm {GL}}_{n_1}(F) \times \cdots \times {\textrm {GL}}_{n_r}(F)$ .

2. (2) A discrete series representation $\pi _i$ of ${\textrm {GL}}_{n_i}(F)$ is of the form

   $$ \begin{align*}\pi_i \ = \ Q(\Delta(\sigma_i, b_i)), \end{align*} $$
   where $n_i = a_ib_i$ , with $a_i, b_i \in \mathbb {Z}_{\geq 1}$ ; $\sigma _i$ is a supercuspidal representation of ${\textrm {GL}}_{a_i}(F)$ such that $\sigma _i(\frac {b_i-1}{2})$ is unitary, and $Q(\Delta (\sigma _i, b_i))$ is the unique irreducible quotient of a parabolically induced representation:
   $$ \begin{align*}\mathrm{Ind}_{P_{b_i,\dots,b_i}(F)}^{{\textrm{GL}}_{n_i}(F)}(\sigma_i \otimes \sigma_i(1) \otimes \cdots \otimes \sigma_i(b_i-1)) \ \twoheadrightarrow \ Q(\Delta(\sigma_i, b_i)). \end{align*} $$

In both the steps, the parabolic induction used is normalized induction which is not, in general, Galois equivariant. As in [Reference Clozel5], we may force Galois equivariance by considering a half-integral Tate twisted version of induction. Using the notations of (1), but letting for the moment $\pi _i$ be any irreducible admissible representation of ${\textrm {GL}}_{n_i}(F)$ , define:

$$ \begin{align*} &{}^{T}\mathrm{Ind}_{P_{n_1,\dots,n_r}(F)}^{G}(\pi_1\otimes \cdots \otimes \pi_r)\\ &\quad:= \mathrm{Ind}_{P_{n_1,\dots,n_r}(F)}^{G}\left(\pi_1(\frac{1-n_1}{2})\otimes \cdots \otimes \pi_r(\frac{1-n_r}{2})\right)\left(\frac{n-1}{2}\right). \end{align*} $$

Suppose $\tau \in \mathrm {Aut}(\mathbb {C})$ ; then one may verify that

$$ \begin{align*}{}^\tau({}^{T}\mathrm{Ind}_{P_{n_1,\dots,n_r}(F)}^{G}(\pi_1\otimes \cdots \otimes \pi_r)) \ = \ {}^{T}\mathrm{Ind}_{P_{n_1,\dots,n_r}(F)}^{G}({}^\tau\pi_1\otimes \cdots \otimes {}^\tau\pi_r). \end{align*} $$

Define a quadratic character of $F^*$ as $\varepsilon _\tau := (\tau \circ |\ |^{1/2})/|\ |^{1/2};$ it is trivial if and only if $\tau $ fixes $q^{1/2},$ where q is the cardinality of the residue field of F. Then:

$$ \begin{align*}{}^\tau\mathrm{Ind}_{P_{n_1,\dots,n_r}(F)}^{G}(\pi_1\otimes \cdots \otimes \pi_r) \ = \ \mathrm{Ind}_{P_{n_1,\dots,n_r}(F)}^{G}((\pi_1\otimes \varepsilon_\tau^{n-n_1}) \otimes \cdots \otimes (\pi_r \otimes \varepsilon_\tau^{n-n_r})). \end{align*} $$

Similarly, using Lemma 3.2.1 and the few lines following that lemma in [Reference Clozel5], we have:

$$ \begin{align*}{}^\tau Q(\Delta(\sigma_i, b_i)) \ = \ Q(\Delta({}^\tau\sigma_i \otimes \varepsilon_\tau^{a_i(b_i-1)}, b_i)). \end{align*} $$

Of course each ${}^\tau \sigma _i$ is supercuspidal and so also is any of its quadratic twists; furthermore, $({}^\tau \sigma _i \otimes \varepsilon _\tau ^{a_i(b_i-1)})(\frac {b_i-1}{2})$ is unitary. Hence, the $\tau $ -conjugate of the tempered representation $\pi $ is tempered.

Using the notations in the hypothesis, take $\pi = {}^\iota \sigma $ , and if $\iota ' : E \to \mathbb {C}$ is any other embedding then take $\tau \in \mathrm {Aut}(\mathbb {C})$ such that $\iota ' = \tau \circ \iota $ . By assumption $\pi = \pi ^t \otimes |\ |^{{\textsf {w}}/2}$ for a unitary tempered representation and an integral exponent ${\textsf {w}}$ . Then

(3.3.3) $$ \begin{align} {}^\tau\pi = {}^\tau\pi^t \otimes (\tau\circ |\, {|}^{{\textsf{w}}/2}) = ({}^\tau\pi^t \otimes \varepsilon_\tau^{\textsf{w}}) \otimes |\ {|}^{{\textsf{w}}/2}. \end{align} $$

By the above argument ${}^\tau \pi ^t$ is tempered, and hence so also is ${}^\tau \pi ^t \otimes \varepsilon _\tau ^{\textsf {w}}$ .▪

Remarks on the proof of Hypothesis 3.3.2 for classical groups

For classical groups, using a similar argument as in the case of ${\textrm {GL}}_n(F)$ , and also the result for ${\textrm {GL}}_n(F)$ , a proof follows from Mœglin and Tadic’s classification [Reference Mœglin and Tadić15] for discrete series and tempered representations. The proof is tedious. We will sketch the argument for even-orthogonal groups.

Consider $G = \mathrm {O}_{2n}(F) = \{g \in {\textrm {GL}}_{2n}(F): {}^tg \cdot J \cdot g = J\}$ the split even orthogonal group of rank n, where $J_{i,j} = \delta (i, 2n-j+1).$ Suppose $n = a_1+\cdots + a_q + n_0$ , with $a_1,\dots , a_q \geq 1$ and $n_0 \geq 0,$ and $P_{(a_1,\dots ,a_q; n_0)}$ is the parabolic subgroup of $\mathrm {O}_{2n}(F)$ with Levi subgroup

$$ \begin{align*}M_{(a_1,\dots,a_q; n_0)} = {\textrm{GL}}_{a_1}(F) \times \dots \times {\textrm{GL}}_{a_q}(F) \times \mathrm{O}_{2n_0}(F). \end{align*} $$

Let $\pi _0$ be a discrete series representation of $\mathrm {O}_{2n_0}(F)$ and $\theta _j$ an essentially discrete series representation of ${\textrm {GL}}_{a_j}(F)$ . For brevity, let

$$ \begin{align*}\theta_1 \times \cdots \times \theta_q \rtimes \pi_0 \ := \mathrm{Ind}_{P_{(a_1,\dots,a_q; n_0)}}^G(\theta_1 \otimes \cdots \otimes \theta_q \otimes \pi_0). \end{align*} $$

This induced representation is a multiplicity-free direct sum of tempered representations (see Mœglin–Tadic [Reference Mœglin and Tadić15, Theorem 13.1] and Atobe–Gan [Reference Atobe and Gan1, Desideratum 3.9, (6)]). Suppose $\pi $ is one such tempered representation: $\pi \hookrightarrow \theta _1 \times \cdots \times \theta _q \rtimes \pi _0$ . Suppose $m_{h_1,\dots , h_q,x} \in M$ with $h_j \in {\textrm {GL}}_{a_j}(F)$ and $x \in \mathrm {O}_{2n_0}(F)$ , then the absolute-value of the determinant of the adjoint action of $m_{h_1,\dots , h_q,x}$ on the Lie algebra of the unipotent radical of P is given by:

$$ \begin{align*} \delta_P(m_{h_1,\dots, h_q,x}) = \left(|\det(h_1)|^{2n- 2a_1}|\det(h_2)|^{2n- 2(a_1+a_2)}\cdots |\det(h_q)|^{2n_0}\right) \cdot \\ \cdot \left( |\det(h_1)|^{a_1-1} |\det(h_2)|^{a_2-1} \cdots |\det(h_q)|^{a_q-1} \right). \end{align*} $$

Using this, for $\tau \in \mathrm {Aut}(\mathbb {C}),$ one may verify that:

(3.3.4) $$ \begin{align} {}^\tau\pi \hookrightarrow {}^\tau(\theta_1 \times \cdots \times \theta_q \rtimes \pi_0) = ({}^\tau\theta_1 \otimes \varepsilon_\tau^{a_1-1}) \times \cdots \times ({}^\tau\theta_q \otimes \varepsilon_\tau^{a_q-1}) \rtimes {}^\tau\pi_0. \end{align} $$

By appealing to the above proof for ${\textrm {GL}}_{n_0}(F)$ , we know that each ${}^\tau \theta _j$ , and so also its quadratic twist ${}^\tau \theta _1 \otimes \varepsilon _\tau ^{a_1-1}$ , is an essentially discrete series representation. Hence, proof of arithmeticity for tempered representation of $\mathrm {O}_{2n}(F)$ boils down to proving arithmeticity for discrete series representation of $\mathrm {O}_{2n_0}(F)$ .

For the discrete series representation $\pi _0$ of $\mathrm {O}_{2n_0}(F)$ , there exist a and $n_1$ such that $n_0 = a + n_1$ , and there exist an essentially discrete series representation $\theta $ of ${\textrm {GL}}_a(F)$ and a discrete series representation $\pi _1$ of the smaller even-orthogonal group $\mathrm {O}_{2n_1}(F)$ such that $\pi _0$ is one of two possible subrepresentations of $\theta \rtimes \pi _1$ ; and both these sub-representations are in the discrete series; this being the crux of [Reference Mœglin and Tadić15]. Then, as above ${}^\tau \pi \hookrightarrow {}^\tau \theta \otimes \varepsilon _\tau ^{a-1} \otimes {}^\tau \pi _1$ . An induction argument (see [Reference Mœglin and Tadić15, p. 721]) concludes the proof as the reduction to a smaller even-orthogonal group ends with the case of $\pi _1$ being a supercuspidal representation (in loc.cit. called the weak cuspidal support of $\pi _0$ ), and clearly conjugation by $\tau $ preserves supercuspidality as it leaves the support of a matrix coefficient unchanged.

It is an interesting problem to prove this for a general p-adic group. Assuming that Hypothesis 3.3.2 is true, we can then formulate another hypothesis:

Hypothesis 3.3.5 (Arithmeticity for being on the right of the unitary axis)

Let $(\sigma , V_{\sigma , E})$ be a smooth, absolutely irreducible, admissible, half-integrally unitary, essentially tempered representation of a reductive p-adic group M over an E-vector space $V_{\sigma , E}.$ If for one embedding $\iota : E \to \mathbb {C}$ the representation ${}^\iota \sigma $ is to the right of the unitary axis, then for every embedding $\iota : E \to \mathbb {C}$ the representation ${}^\iota \sigma $ is to the right of the unitary axis.

For ${\textrm {GL}}_n(F)$ this follows from (3.3.3) since the exponent for $\pi $ and ${}^\tau \pi $ are equal. Similarly, the above hypothesis will follow from Hypothesis 3.3.2 that the half-integral exponents $\underline {\textsf {w}}/2$ for ${}^\iota \sigma $ are independent of $\iota $ ; in particular the exponent $f_1$ of $\tilde r_1({}^\iota \sigma )$ would be independent of $\iota $ .

Lemma 3.3.6 (Arithmeticity for genericity)

Let $(\sigma , V_{\sigma , E})$ be a smooth absolutely irreducible admissible representation of a reductive quasi-split p-adic group M over an E-vector space $V_{\sigma , E}.$ If for one embedding $\iota : E \to \mathbb {C}$ the representation ${}^\iota \sigma $ is generic, then for every embedding $\iota : E \to \mathbb {C}$ the representation ${}^\iota \sigma $ is generic.

Proof Suppose $\ell : {}^\iota \sigma \to \mathbb {C}$ is a Whittaker functional with respect to a character $\psi : U \to \mathbb {C}^*$ (that is nontrivial on all the root spaces corresponding to simple roots). Given another embedding $\iota ' : E \to \mathbb {C}$ , there exists $\tau \in \mathrm {Aut}(\mathbb {C})$ such that $\iota ' = \tau \circ \iota .$ Then $\tau \circ \ell $ is a Whittaker functional for ${}^{\iota '}\sigma $ with respect to the character $\tau \circ \psi $ of U.▪

After the above hypotheses and lemma, it makes sense to say that a smooth absolutely-irreducible admissible representation $(\sigma , V_{\sigma , E})$ of a reductive p-adic group M is half-integrally unitary, essentially tempered, to the right of the unitary axis, or generic, if for some, and hence any, embedding $\iota : E \to \mathbb {C}$ the representation ${}^\iota \sigma $ is half-integrally unitary, essentially tempered, to the right of the unitary axis, or generic, respectively. The first main theorem of this article is the following result.

An arithmetic variant of Theorem 3.2.5

Theorem 3.3.7 Let $P = MN$ be a maximal parabolic subgroup of a connected reductive p-adic group $G.$ Let $(\sigma , V_{\sigma , E})$ be a smooth absolutely-irreducible admissible representation of M over an E-vector space $V_{\sigma , E}.$ Assume that E is large enough to contain the values of the exponents of A that appear in the Jacquet module of $\;{}^{\mathrm {a}}\mathrm {Ind}_P^G(\sigma )$ with respect to the associate parabolic subgroup Q. Assuming Hypotheses 3.3.1, 3.3.2, and 3.3.5, we suppose that $\sigma $ is half-integrally unitary, essentially tempered, to the right of the unitary axis, and generic. Suppose that P satisfies the integrality condition: $\rho _P|_{A_P} \in X^*(A_P)$ , then so does Q and the modular character $\delta _Q$ takes values in $\mathbb {Q}^*.$ There exists an E-linear G-equivariant map

$$ \begin{align*}T_{st, E} : {}^{\mathrm{a}}\mathrm{Ind}_P^G(\sigma) \ \longrightarrow \ {}^{\mathrm{a}}\mathrm{Ind}_Q^G(\sigma \otimes \delta_Q) \end{align*} $$

such that for any embedding $\iota : E \to \mathbb {C}$ we have:

$$ \begin{align*}T_{\mathrm{st}, E} \otimes_{E,\iota} 1_{\mathbb{C}} \ = \ T_{\mathrm{st}, \iota}, \end{align*} $$

where $T_{\mathrm {st}, \iota } = T_{\mathrm {st}}(s, {}^\iota \sigma )|_{s = k} : {}^{\mathrm {a}}\mathrm {Ind}_P^G({}^\iota \sigma ) \to {}^{\mathrm {a}}\mathrm {Ind}_Q^G({}^\iota \sigma \otimes \delta _Q)$ is the standard intertwining operator at the point of evaluation.

Proof Fix an $\iota : E \to \mathbb {C}$ . For $\chi \in X(M)$ , we have the standard intertwining operator

$$ \begin{align*}T_{\mathrm{st}}({}^\iota\sigma, \chi) : I_P^G({}^\iota\sigma \otimes \chi) \to I_Q^G({}^{w_0}({}^\iota\sigma \otimes \chi)) \end{align*} $$

given by an integral where it converges. We will ultimately specialize to the point $\chi _k$ corresponding to the point of evaluation $k = -\langle \rho _P, \alpha _P\rangle $ ; note that $\chi _k = -\rho _P$ ; at this point our hypotheses guarantee convergence. Consider Theorem 3.2.5 with the small variation that we take the associate parabolic Q and not $P'$ which required $M_{P'} = M_P;$ for Q we have $M_Q$ is the $w_0$ -conjugate of $M_P$ . This causes no problem as long as we use the correct integral, i.e., we use (2.2.1) instead of (3.2.4). From Theorem 3.2.5 we get a $(G,\mathcal {B})$ -module map

$$ \begin{align*}T_{\mathcal{B}}: I_P^G({}^\iota\sigma \otimes_{\mathbb{C}} \mathcal{B}) \ \longrightarrow \ I_{Q}^G({}^{w_0\iota}\sigma \otimes_{\mathbb{C}} \mathcal{B}) \end{align*} $$

that satisfies (3.2.3) with a homothety element $b \in \mathcal {B}$ .

The main steps in the proof of Theorem 3.2.5 ((i) reduction via Frobenius reciprocity, (ii) Jacquet module calculation, and (iii) construction of an M-equivariant map using an element R in the group ring of A via the theory of resultants) are all purely algebraic in nature. The same proof, but now working with modules over E, gives us an E-linear map of $(G, \mathcal {B}_E)$ -modules:

$$ \begin{align*}T_{\mathcal{B}, E} : I_P^G(\sigma \otimes_E \mathcal{B}_E) \ \longrightarrow \ I_{Q}^G({}^{w_0}\sigma \otimes_E \mathcal{B}_E) \end{align*} $$

with a homothety $b_0 \in \mathcal {B}_E$ such that for any $\iota : E \to \mathbb {C}$ we have: $T_{\mathcal {B}, E} \otimes _{E,\iota } \mathbb {C} = T_{\mathcal {B}},$ and $b_0 \otimes _{E,\iota } 1 = b.$ Specialize at the point of evaluation in (3.2.3), i.e., take $\chi = \chi _k = -\rho _P \in X_E(A)$ ; hence, $b(\chi _k) = b_0(-\rho _P) \in E^*$ ; note that we have used $\rho _P|_{A_P}$ is an integral weight. We have:

$$ \begin{align*}\iota(b_0(-\rho_P)) T_{\mathrm{st}}({}^\iota\sigma, \chi_k) = \mathrm{sp}_{\chi_k} \circ (T_{\mathcal{B},E} \otimes_{E,\iota} 1_{\mathbb{C}}). \end{align*} $$

For $\chi _0 \in X_E(A),$ if $\mathrm {sp}_{\chi _0, E} : \mathcal {B}_E \to E$ denotes the specialization map at an arithmetic level, then clearly, $\mathrm {sp}_{\chi _0, E} \otimes _{E, \iota } 1_{\mathbb {C}} = \mathrm {sp}_{\chi _0}$ . Hence, $ T_{\mathrm {st}}({}^\iota \sigma , \chi _k) = (b_0(\rho _P) \mathrm {sp}_{\chi _k, E} \circ T_{\mathcal {B},E}) \otimes _{E,\iota } 1_{\mathbb {C}}. $ ▪

4.1 Criticality condition on the point of evaluation

In the context of Rankin–Selberg L-functions one takes $M_P = {\textrm {GL}}_n \times {\textrm {GL}}_{n'}$ as a Levi subgroup of an ambient $G = {\textrm {GL}}_N,$ where $N = n+ n'$ . The integrality condition on P forces $nn'$ to be even. For an inducing data $\pi \times \pi '$ of $M_P$ , the critical set for the L-function $L(s, \pi \times \pi ^{\prime \textsf {v}})$ consists of integers if $n \equiv n' \pmod {2}$ and consists of half-integers, i.e., elements of $\tfrac 12 + \mathbb {Z}$ if $n \not \equiv n' \pmod {2}$ ; see [Reference Harder and Raghuram8, Definition 7.3]. The purpose of this subsection is to formalize such parity constraints in the context of Langlands–Shahidi machinery.

Suppose $A_i = \eta _i(A) = F^*$ and A is the internal product $A_1 \times \dots \times A_l$ ; correspondingly, suppose $M = M_1\cdots M_l$ an almost direct product of reductive subgroups, with $A_i$ in the center of $M_i$ . Let $\rho _{M_i}$ be half the sum of positive roots for $M_i$ . If $\rho _{M_i}$ is integral, then put $\varepsilon _{M_i} = 0.$ If $\rho _{M_i}$ is not integral, then necessarily $2 \rho _{M_i}$ is integral, and put ${\varepsilon _{M_i} = 1}.$ Fix an unramified character $\chi _P^{\varepsilon _P/2} \in \mathrm {Hom}(M/M^1, \mathbb {C}^*) = \mathrm {Hom}(A/A^1, \mathbb {C}^*),$ defined by

$$ \begin{align*}\chi_P^{\varepsilon_P/2}(a_1,\dots,a_l) \ := \ |a_1|^{\varepsilon_{M_1}/2} \dots |a_l|^{\varepsilon_{M_l}/2}, \quad (a_1,\dots,a_l) \in A = F^* \times \cdots \times F^*. \end{align*} $$

Let $\vartheta _P \in {}^L\!A_P^\circ $ be the Satake parameter of $\chi _P^{\varepsilon _P/2}.$ Using (2.4.1), there exists $h_j \in \tfrac 12 \mathbb {Z}$ such that $\tilde r_j(\vartheta _P) = q^{-h_j}$ or that $\tilde r_j(\chi _P^{\varepsilon _P/2}) = |\ |^{h_j}.$ Let $\pi $ be an irreducible admissible half-integrally unitary, essentially tempered, generic representation of $M_P$ . Consider $\pi \otimes \chi _P^{\varepsilon _P/2}$ ; we have:

$$ \begin{align*}L(s, \pi, \tilde r_j) \ = \ L(s - h_j, \pi \otimes \chi_P^{\varepsilon_P/2}, \tilde r_j). \end{align*} $$

The idea is that given $\pi ,$ we algebrize it by considering the twist $\pi \otimes \chi _P^{\varepsilon _P/2}$ . For ${\textrm {GL}}_n$ this is equivalent to replacing $\pi $ by $\pi \otimes |\ |^{\varepsilon _n/2}$ , where $\varepsilon _n \in \{0,1\}$ and $\varepsilon _n \equiv n-1 \pmod {2}.$ Any point of evaluation of a global L-function attached to an algebraic data (think of a motivic L-function) should be an integer for the L-value to be critical in the sense of Deligne [Reference Deligne6]. This motivates the following definition which is independent of $\pi $ and depends only on $(G,P)$ .

Definition 4.1.1 Let G be a connected reductive p-adic group and P a maximal parabolic subgroup. We say that P is critical for G if the point of evaluation $k = - \langle \rho _P, \alpha _P \rangle $ satisfies the condition:

$$ \begin{align*}jk \in h_j + \mathbb{Z}, \quad \forall \ 1 \leq j \leq m. \end{align*} $$

4.2 Hypothesis on local critical L-values

We can now formulate the arithmeticity hypothesis for local critical L-values.

As already mentioned, this hypothesis can be verified in various concrete examples of interest. We briefly mention two examples below; these contexts are amplified in Section 5; the reader will readily appreciate that such examples may be generalized.

Example 4.2.2 (Local L-functions for ${\textrm {GL}}_n(F)$ )

If $\pi $ is an irreducible admissible representation of ${\textrm {GL}}_n(F)$ then it follows from Clozel [Reference Clozel5, Lemma. 4.6] that for any $k_0 \in \mathbb {Z}$ and any $\tau \in \mathrm {Aut}(\mathbb {C})$ one has:

$$ \begin{align*}\tau\left(L(k_0 + \tfrac{1-n}{2}, \pi)\right) \ = \ L(k_0 + \tfrac{1-n}{2}, {}^\tau\pi). \end{align*} $$

The reader can check that Hypothesis 4.2.1 follows from this Galois equivariance after appealing to the details in Section 5.1. Such a Galois equivariance can be reformulated as

$$ \begin{align*}\tau\left(L(k_0, \pi)\right) \ = \ L(k_0, {}^\tau\pi \otimes \varepsilon_\tau^{n-1}), \end{align*} $$

which is useful in other situations; see the next example below.

Example 4.2.3 (Local L-functions for $\mathrm {O}_{2n}(F)$ )

Suppose $\pi $ is an irreducible tempered representation of $\mathrm {O}_{2n}(F)$ as in the proof of Hypothesis 3.3.2 for orthogonal groups; in particular, $\pi \hookrightarrow \theta _1 \times \cdots \times \theta _q \rtimes \pi _0$ , with notations as therein. Then, the L-parameters are related as: $\phi _\pi \ = \ \phi _{\theta _1} + \cdots + \phi _{\theta _q} + \phi _{\pi _0} + \phi _{\theta _1}^{\textsf {v}} + \cdots + \phi _{\theta _q}^{\textsf {v}}$ (see [Reference Atobe and Gan1, Desideratum 3.9]). In particular, for L-functions, evaluating at $s = k \in \mathbb {Z}$ (see Section 5.2 for the fact that the point of evaluation is an integer), we get: $ L(k, \pi ) \ = \ L(k, \theta _1)\cdots L(k, \theta _q) \cdot L(k, \pi _0) \cdot L(k, \theta _1^{\textsf {v}})\cdots L(k, \theta _q^{\textsf {v}}). $ Apply $\tau \in \mathrm {Aut}(\mathbb {C})$ to both sides while using Example 4.2.2 to get $\tau (L(k, \pi ))$ is equal to

$$ \begin{align*} &L(k, {}^\tau\theta_1 \otimes \varepsilon_\tau^{a_1-1})\cdots L(k, {}^\tau\theta_q\otimes \varepsilon_\tau^{a_q-1}) \cdot \tau(L(k, \pi_0))\\ &\quad \cdot L(k, {}^\tau\theta_1^{\textsf{v}} \otimes \varepsilon_\tau^{a_1-1})\cdots L(k, {}^\tau\theta_q^{\textsf{v}}\otimes \varepsilon_\tau^{a_q-1}). \end{align*} $$

Now, use (3.3.4) for ${}^\tau \pi $ ; then take its L-function evaluated at $s = k$ to get:

$$ \begin{align*}\tau(L(k, \pi)) \ = \ L(k, {}^\tau\pi), \end{align*} $$

assuming by induction that $\tau (L(k, \pi _0)) \ = \ L(k, {}^\tau \pi _0)$ holds for a discrete series representation $\pi _0$ of $\mathrm {O}_{2n_0}(F)$ . The proof for a discrete series representation follows the same reduction strategy as in the proof of Hypothesis 3.3.2 for even orthogonal groups.

I expect that a proof of Hypothesis 4.2.1 in the general case should come from an arithmetic-analysis of Shahidi’s theory of local factors.

4.3 An arithmetic variant of Theorem 3.2.5 for normalized intertwining operator

We can now strengthen Theorem 3.3.7 for the normalized intertwining operator.

Theorem 4.3.1 Let the notations and hypotheses be as in Theorem 3.3.7. Assume furthermore that Hypothesis 4.2.1 holds. Let $T_{\mathrm {norm}} = T_{\mathrm {norm}}(s, {}^\iota \sigma )|_{s = k}$ be the normalized standard intertwining operator (see ( 1.0.1 )) at the point of evaluation $s = k$ . Then there exists an E-linear G-equivariant map

$$ \begin{align*}T_{\mathrm{norm}, E} : {}^{\mathrm{a}}\mathrm{Ind}_P^G(\sigma) \ \longrightarrow \ {}^{\mathrm{a}}\mathrm{Ind}_Q^G(\sigma \otimes \delta_Q) \end{align*} $$

such that for any embedding $\iota : E \to \mathbb {C}$ we have:

$$ \begin{align*}T_{\mathrm{norm}, E} \otimes_{E,\iota} 1_{\mathbb{C}} \ = \ T_{\mathrm{norm}}. \end{align*} $$

5.1 Rankin–Selberg L-functions

See case ( $A_{N-1}$ ) in [Reference Shahidi23, Appendix B].

1. (1) Ambient group: $G = {\textrm {GL}}(N)/F$ with $N \geq 2$ .

2. (2) Maximal parabolic subgroup: take $N = n+n'$ and let P be the maximal parabolic subgroup with Levi $M_P = {\textrm {GL}}(n) \times {\textrm {GL}}(n')$ ; the deleted simple root $\alpha _P = e_n-e_{n+1}.$

3. (3) The integrality condition $\rho _P \in X^*(A_P)$ holds if and only if $nn' \equiv 0 \! \pmod {2}.$ The set of roots with roots spaces appearing in the Lie algebra $\mathfrak {n}_P$ of the unipotent radical of P is $\{e_i - e_j : 1 \leq i \leq n, \ n+1 \leq j \leq n+n'\}.$ Hence $ \rho _P = \frac {n'}{2}(e_1+\dots + e_n) - \frac {n}{2}(e_{n+1}+\dots +e_{n+n'}).$ Whence, $\rho _P$ as a character of $A_P$ is given by: $\mathrm {diag}(t1_n, t'1_{n'}) \mapsto (t t')^{nn'/2},$ which is integral if and only if $nn'$ is even. It is curious that this very condition was imposed in [Reference Harder and Raghuram8] due to motivic considerations (the tensor product motive therein needed to be of even rank).

4. (4) At the level of dual groups, ${}^L\!M_P^\circ \cong {\textrm {GL}}_n(\mathbb {C}) \times {\textrm {GL}}_{n'}(\mathbb {C})$ acts irreducibly on the Lie algebra ${}^L\mathfrak {n}_P \cong M_{n \times n'}(\mathbb {C})$ of ${}^L\!N_P$ ; $m=1$ .

5. (5) Inducing data consists of $\pi $ and $\pi '$ which are essentially tempered irreducible generic representations of ${\textrm {GL}}_n(F)$ and ${\textrm {GL}}_{n'}(F)$ then $L(s, \pi \otimes \pi ', \tilde r_1) = L(s, \pi \times \pi ^{\textsf {v}})$ is the local Rankin–Selberg L-function attached to ${\textrm {GL}}_n(F) \times {\textrm {GL}}_{n'}(F).$

6. (6) The point of evaluation is $k = - \langle \rho _P, \alpha _P \rangle = -N/2$ .

7. (7) P is critical for G: $M = M_1M_2$ with $M_1 = {\textrm {GL}}_n(F)$ and $M_2 = {\textrm {GL}}_{n'}(F);$ $\varepsilon _{M_1} = \varepsilon _n$ and $\varepsilon _{M_2} = \varepsilon _{n'}$ (recall: $\varepsilon _n \in \{0,1\}$ by $\varepsilon _n \equiv n-1 \pmod {2}$ ); $h_1 = (\varepsilon _n - \varepsilon _{n'})/2$ ; and $k \in h_1 + \mathbb {Z}$ .

5.2 L-functions for orthogonal groups

See case ( $D_{n,i}$ ) in [Reference Shahidi23, Appendix A]; the corresponding global context is studied in [Reference Bhagwat and Raghuram3].

1. (1) Ambient group $G = \mathrm {O}(n+1, n+1) = \{g \in {\textrm {GL}}_{2n+2}(F) : {}^tg \cdot J_{2n+2} \cdot g = J_{2n+2}\}$ , where $J_{2n+2}(i,j) = \delta (i, 2n+3-j)$ ; this is the split even orthogonal group of rank $n+1$ ; the maximal torus consists of all diagonal matrices $\mathrm {diag}(t_0,t_1,\dots ,t_n, t_n^{-1},\dots , t_1^{-1}, t_0^{-1}).$

2. (2) Let P be the maximal parabolic subgroup described by the above Dynkin diagram; deleted simple root $\alpha _P = e_0-e_1;$ Levi subgroup is

   $$ \begin{align*}M_P = \left\{ m_{t,h} = \begin{pmatrix} t & & \\ & h & \\ & & t^{-1} \end{pmatrix} : t \in {\textrm{GL}}_1(F), \ h \in \mathrm{O}(n,n)\right\}; \end{align*} $$
   clearly, $A_P = \{m_{t,1} \in M_P : t \in {\textrm {GL}}_1\}$ ; the unipotent radical of P is:
   $$ \begin{align*}N_P = \left\{ u_{y_1, y_2, \ldots, y_{2n}} = \left( \begin{array}{ccccccc} 1 & y_1 & y_2 & \ldots & y_{2n} & 0 \\ & 1 & & & &-y_{2n} \\ & & 1 & & & -y_{2n-1} \\ & & & \ddots & & \vdots \ \\ & & & & \ddots & -y_{1} \\ & & & & & 1 \end{array}\right) \ \ | \ \ y_1,\dots,y_{2n} \in F \right\}. \end{align*} $$

3. (3) The integrality condition on $\rho _P$ holds for all $n.$ The set of roots with root spaces in the Lie algebra of $N_P$ is $ \{e_0-e_1, \, e_0-e_2,\dots , \, e_0-e_{2n}\}. $ Hence

   $$ \begin{align*}\rho_P = n e_0 - \tfrac12(e_1+e_3+\dots + e_{2n}); \end{align*} $$
   from the maximal torus one has $e_{n+1} = -e_n, e_{n+2} = -e_{n-1}, \dots , e_{2n} = -e_1$ from which it follows that $\rho _P = n e_0$ . Whence, $\rho _P|_{A_P}$ is the integral character $t = m_{t,1} \mapsto t^n.$

4. (4) At the level of dual groups, ${}^L\!M_P^\circ = \left \{ m_{t,h} : t \in \mathbb {C}^*, \ h \in \mathrm {O}(n,n)(\mathbb {C})\right \}$ acts irreducibly on the Lie algebra ${}^L\mathfrak {n}_P$ of ${}^L\!N_P^\circ $ ; $m=1$ and $r_1$ is the standard representation of $\mathrm {O}(n,n)(\mathbb {C})$ twisted by the $\mathbb {C}^*$ in the obvious way.

5. (5) Inducing data is of the form $\chi \otimes \pi $ for a character $\chi : F^* \to \mathbb {C}^*$ , and a tempered, irreducible, generic representation $\pi $ of $\mathrm {O}(n,n)(F).$ The local L-function $L(s, \chi \otimes \pi , \tilde r_1)$ is the local Rankin–Selberg L-function $L(s, \chi \otimes \tilde r_1(\pi ))$ for ${\textrm {GL}}_1 \times {\textrm {GL}}_{2n}.$

6. (6) The point of evaluation is $k = - \langle \rho _P, \alpha _P \rangle = -n.$

7. (7) P is critical for G, since $M = M_1M_2$ with $M_1 = {\textrm {GL}}_1(F)$ and $M_2 = \mathrm {O}(n,n)(F); \rho _{M_i}$ is integral; $h_1 = 0$ ; $k \in \mathbb {Z}$ .

5.3 Exterior square L-functions

See case ( $C_{n-1,ii}$ ) in [Reference Shahidi23, Appendix A].

1. (1) Ambient group $G = \mathrm {Sp}_{2n}(F) = \left \{ g \in {\textrm {GL}}_{2n}(F) : {}^tg \cdot \begin {pmatrix} & J_n \\ -J_n& \end {pmatrix} \cdot g = \begin {pmatrix} & J_n \\ -J_n& \end {pmatrix} \right \},$ where $J_n(i,j) = \delta (i, r-j+1)$ and ${}^tg$ is the transpose of g.

2. (2) Maximal parabolic subgroup as depicted by the above Dynkin diagram has Levi subgroup: $M_P = \left \{ \begin {pmatrix} h & \\ & {}^{(t)}h^{-1} \end {pmatrix} : h \in {\textrm {GL}}_n(F) \right \}$ where ${}^{(t)}h = J_n \cdot {}^t h \cdot J_n$ is the “other-transpose” of h defined by $({}^{(t)}h)_{i,j} = h_{n-j+1, n-i+1}.$ The deleted simple root $\alpha _P = 2 e_n.$

3. (3) The integrality condition $\rho _P \in X^*(A_P)$ holds if and only if $n \equiv 0,3 \pmod {4}.$ The Lie algebra of the unipotent radical of P is of the form

   $$ \begin{align*}\mathfrak{n}_P = \left\{ \begin{pmatrix} 0_n & X \\ 0_n & 0_n \end{pmatrix} : X \in M_n(F), \ {}^{(t)}X = X \right\}. \end{align*} $$
   The set of roots with root spaces appearing in $\mathfrak {n}_P$ is $ \{e_1-e_{n+1},\ \dots ,\ e_1-e_{2n}, \ e_2-e_{n+1},\ \dots ,\ e_2-e_{2n-1}, \ \dots , e_n-e_{n+1}\}. $ Keeping in mind that $e_j = -e_{2n-j+1}$ we get
   $$ \begin{align*}\rho_P = \frac{n+1}{2}(e_1+ \cdots + e_n). \end{align*} $$
   Whence, $\rho _P|_{A_P}$ is given by: $\mathrm {diag}(t1_n, t^{-1}1_n) \mapsto t^{n(n+1)/2},$ which is integral if and only if $n(n+1)/2$ is even, i.e., $n \equiv 0 \ \mathrm {or} \ 3 \! \pmod {4}.$

4. (4) Dual groups: ${}^L\!G^\circ = {\textrm {SO}}(2n+1,\mathbb {C}) = \left \{ g \in {\textrm {SL}}_{2n+1}(F) : {}^tg \cdot J_{2n+1} \cdot g = J_{2n+1} \right\},$

   $$ \begin{align*}{}^L\!M_P^\circ = \left\{ m_g = \begin{pmatrix} g & & \\ & 1 & \\ & & {}^{(t)}g^{-1} \end{pmatrix} : g \in {\textrm{GL}}_n(\mathbb{C}) \right\} \cong {\textrm{GL}}_n(\mathbb{C}); \end{align*} $$
   $$ \begin{align*}{}^L\mathfrak{n}_P \!=\! \left\{\!n_{y,X} = \begin{pmatrix} 0_n & y & X \\ 0_{1 \times n} & 0 & {-}^ty J_n \\ 0_n & 0_{n \times 1} & 0_n \end{pmatrix}{:}\; y \in M_{n \times 1}(\mathbb{C}),\; X \in M_{n \times n}(\mathbb{C}),\; {}^{(t)}X \!=\! -X \!\right\}. \end{align*} $$

   The adjoint action of ${}^L\!M_P^\circ $ on ${}^L\mathfrak {n}_P$ is the direct sum of two irreducible representations with representation spaces $V_1 = \{n_{y,0} \in {}^L\mathfrak {n}_P\}$ and $V_2 = \{n_{0,X} \in {}^L\mathfrak {n}_P\}$ of dimensions n and $n(n-1)/2$ , respectively; $r_1$ is the standard representation and $r_2$ is the exterior square representation; $m=2$ . The center ${}^L\!A_P^\circ $ of ${}^L\!M_P^\circ $ consists of elements $a_t = m_{t\cdot I_n}$ for $t \in \mathbb {C}^\times $ ; then $a_t$ acts on $V_1$ by the scalar t and on $V_2$ by the scalar $t^2$ .

5. (5) The inducing data is a half-integrally unitary, irreducible, essentially tempered, generic representation $\pi $ of ${\textrm {GL}}_n(F);$ for the L-functions we have:

   1. (a) $L(s, \pi , \tilde r_1) = L(s, \pi ),$ the standard L-function for ${\textrm {GL}}(n)$ and

   2. (b) $L(s, \pi , \tilde r_2) = L(s, \pi , \wedge ^2),$ the exterior square L-function for ${\textrm {GL}}(n).$

6. (6) The point of evaluation is $k = - \langle \rho _P, \alpha _P \rangle = - \frac {n+1}{2}.$

7. (7) P is critical for G. Since $\varepsilon _M = \varepsilon _n, h_1 = \varepsilon _n/2,$ and $h_2 = \varepsilon _n$ ; hence $jk \in h_j + \mathbb {Z}$ holds for $j = 1, 2.$

5.4 Explicit intertwining calculation for the case of ${\textrm {GL}}(2)$

Some essential features of main results are already visible for the example of ${\textrm {GL}}(2)$ from first principles; although the reader is warned of the well-known dictum that ${\textrm {GL}}(2)$ is misleadingly simple and it is difficult to carry out a straightforward generalization of such calculations.

Let $E/\mathbb {Q}$ be a finite extension, and for $i = 1,2$ , let $\chi _i : F^\times \to E^\times $ be a smooth character, and $\chi _i^\circ $ its restriction to $\mathcal O_F^\times .$ Let $\iota : E \to \mathbb {C}$ be an embedding of fields, and ${}^\iota \chi _i = \iota \circ \chi _i$ be the corresponding $\mathbb {C}$ -valued character of $F^\times .$ Let $G = {\textrm {GL}}_2(F)$ , $K = {\textrm {GL}}_2(\mathcal O_F),$ and for $m \geq 0$ let $K(m)$ be the principal congruence subgroup of K of level m; $K(0) = K.$ The standard intertwining operator $T_{\mathrm {st}}(s)$ at the point of evaluation $s = -1$ between the $K(m)$ -invariants of algebraically induced representations has the shape:

$$ \begin{align*}T_{\mathrm{st}}(s)|_{s = -1} : {}^{\mathrm{a}}\mathrm{Ind}_B^G({}^\iota\chi_1 \otimes {}^\iota\chi_2)^{K(m)} \longrightarrow {}^{\mathrm{a}}\mathrm{Ind}_B^G({}^\iota\chi_2(1) \otimes {}^\iota\chi_1(-1))^{K(m)}. \end{align*} $$

The standing assumptions that ${}^\iota \chi _i$ is half-integrally unitary, essentially tempered, and $\pi = {}^\iota \chi _1 \otimes {}^\iota \chi _2$ is on the right of the unitary axis with respect to G implies that $T := T_{\mathrm {st}}(s)|_{s = -1}$ is finite. A function in ${}^{\mathrm {a}}\mathrm {Ind}_B^G({}^\iota \chi _1 \otimes {}^\iota \chi _2)^{K(m)}$ is completely determined by its restriction to K. This gives us the following diagram:

Working with $K(m)$ -invariants is not strictly necessary; it has the virtue of making the spaces finite-dimensional and G-action is replaced by action of the Hecke-algebra $\mathcal C^\infty _c(G/\!/K(m)).$ Let $f^\circ \mapsto \tilde {f^\circ }$ denote the inverse of $f \mapsto f|_K.$ Let $f \in {}^{\mathrm {a}}\mathrm {Ind}_{B}^G({}^\iota \chi _1 \otimes {}^\iota \chi _2)^{K(m)}$ and for brevity let $f^\circ = f|_K.$ Since $T(f)$ is determined by its restriction to K, we have:

$$ \begin{align*}T^\circ(f^\circ)(k) = T(f)(k) = \int_F f\left( \begin{pmatrix} & -1\\ 1 & \end{pmatrix} \begin{pmatrix} 1 & x \\ & 1\end{pmatrix}k\right) \, dx, \quad k \in K. \end{align*} $$

Break up the integral over $x \in \mathcal {P}^{-m}$ and $x \notin \mathcal {P}^{-m}.$ Note that

$$ \begin{align*}\int_{\mathcal{P}^{-m}} f\left( \begin{pmatrix} & -1\\ 1 & \end{pmatrix} \begin{pmatrix} 1 & x \\ & 1\end{pmatrix}k\right) \, dx = \sum_{a \in \mathcal{P}^{-m}/\mathcal{P}^m} \int_{y \in \mathcal{P}^m} f\left( \begin{pmatrix} & -1\\ 1 & \end{pmatrix} \begin{pmatrix} 1 & a + y \\ & 1\end{pmatrix} k\right) \, dy. \end{align*} $$

We write

$$ \begin{align*}\begin{pmatrix} 1 & a + y \\ & 1\end{pmatrix} k = \begin{pmatrix} 1 & a \\ & 1\end{pmatrix} \begin{pmatrix} 1 & y \\ & 1\end{pmatrix} k = \begin{pmatrix} 1 & a \\ & 1\end{pmatrix} k \cdot k^{-1} \begin{pmatrix} 1 & y \\ & 1\end{pmatrix} k \end{align*} $$

and use that $K(m)$ is a normal subgroup of K and $\widetilde {f^\circ }$ is right $K(m)$ -invariant to get

(5.4.1) $$ \begin{align} \int_{\mathcal{P}^{-m}} f\left( \begin{pmatrix} & -1\\ 1 & \end{pmatrix} \begin{pmatrix} 1 & x \\ & 1\end{pmatrix}k\right) \, dx = \mathrm{vol}(\mathcal{P}^m) \sum_{a \in \mathcal{P}^{-m}/\mathcal{P}^m} f\left( \begin{pmatrix} & -1\\ 1 & \end{pmatrix} \begin{pmatrix} 1 & a \\ & 1\end{pmatrix} k\right), \end{align} $$

which is a finite-sum. For the integral over $x \notin \mathcal {P}^{-m}$ use:

$$ \begin{align*}\begin{pmatrix} & -1\\ 1 & \end{pmatrix} \begin{pmatrix} 1 & x \\ & 1\end{pmatrix} = \begin{pmatrix}x^{-1} & \\ & x\end{pmatrix} \begin{pmatrix}1 & -x\\ & 1\end{pmatrix} \begin{pmatrix}1 & \\ x^{-1} & 1\end{pmatrix}; \end{align*} $$

break up $\int _{x \notin \mathcal {P}^{-m}}$ as $\sum _{r = m}^{\infty } \int _{\varpi ^{-r}\mathcal O^\times }$ to get

$$ \begin{align*}&\int_{x \notin \mathcal{P}^{-m}} f\left( \begin{pmatrix} & -1\\ 1 & \end{pmatrix} \begin{pmatrix} 1 & x \\ & 1\end{pmatrix}k\right) \, dx \\ &\quad= \sum_{r = m}^\infty \int_{\varpi^{-r}\mathcal O^\times} f\left( \begin{pmatrix}x^{-1} & \\ & x\end{pmatrix} \begin{pmatrix}1 & -x\\ & 1\end{pmatrix} \begin{pmatrix}1 & \\ x^{-1} & 1\end{pmatrix}k\right) \, dx. \end{align*} $$

Since $x^{-1} \in \mathcal {P}^m$ , using the equivariance of f, the right hand side simplifies to:

$$ \begin{align*}\sum_{r = m}^\infty \int_{\varpi^{-r}\mathcal O^\times} {}^\iota \chi_1(x^{-1}){}^\iota\chi_2(x) f(k) \, dx. \end{align*} $$

Make the substitution $x = \varpi ^{-r}u$ with $u \in \mathcal O^\times $ ; then $dx = q^r du = q^r d^\times u$ , and one gets:

$$ \begin{align*}f(k) \sum_{r = m}^\infty {}^\iota \chi_1(\varpi^{r}){}^\iota\chi_2(\varpi^{-r}) q^r \int_{\mathcal O^\times} {}^\iota \chi_1(u^{-1}){}^\iota\chi_2(u)\, d^\times u. \end{align*} $$

The inner integral is nonzero if and only if ${}^\iota \chi _1(u) = {}^\iota \chi _2(u)$ for all $u \in \mathcal O^\times $ ; assuming this to be the case we get:

(5.4.2) $$ \begin{align} &\int_{x \notin \mathcal{P}^{-m}} f\left( \begin{pmatrix} & -1\\ 1 & \end{pmatrix} \begin{pmatrix} 1 & x \\ & 1\end{pmatrix}k\right) \, dx \nonumber\\ &\quad=\mathrm{vol}(\mathcal O^\times) \cdot {}^\iota \chi_1(\varpi^{m}){}^\iota\chi_2(\varpi^{-m}) q^m \cdot \left(1 - {}^\iota \chi_1(\varpi){}^\iota\chi_2(\varpi^{-1}) q\right)^{-1} \cdot f(k). \end{align} $$

For $G = {\textrm {GL}}(2)$ , the point of evaluation $k = -1$ , and $\left (1 - {}^\iota \chi _1(\varpi ){}^\iota \chi _2(\varpi ^{-1}) q\right )^{-1} $ is nothing but $L(s, {}^\iota \chi _1 \otimes {}^\iota \chi _2, \tilde {r}) = L(s, {}^\iota \chi _1 \otimes {}^\iota \chi _2^{-1})$ evaluated at this point of evaluation. Putting (5.4.1) and (5.4.2) together, one sees that $T(f)(k)$ is a finite-sum:

(5.4.3) $$ \begin{align} &\mathrm{vol}(\mathcal{P}^m) \sum_{a \in \mathcal{P}^{-m}/\mathcal{P}^m} f\left( \begin{pmatrix} & -1\\ 1 & \end{pmatrix} \begin{pmatrix} 1 & a \\ & 1\end{pmatrix} k\right) \nonumber\\ &\quad+\delta(\chi_1^\circ, \chi_2^\circ) \cdot \mathrm{vol}(\mathcal O^\times) \cdot {}^\iota \chi_1(\varpi^{m}){}^\iota\chi_2(\varpi^{-m}) q^m \cdot L(-1, {}^\iota \chi_1 \otimes {}^\iota\chi_2^{-1}) \cdot f(k). \end{align} $$

For brevity, let $\mathfrak {I} = {}^{\mathrm {a}}\mathrm {Ind}_B^G({}^\iota \chi _1 \otimes {}^\iota \chi _2)^{K(m)}$ and $\tilde {\mathfrak {I}} = {}^{\mathrm {a}}\mathrm {Ind}_B^G({}^\iota \chi _2(1) \otimes {}^\iota \chi _1(-1))^{K(m)}$ ; these induced representations admit a natural E-structures; define

$$ \begin{align*}\mathfrak{I}_0 \ := \ {}^{\mathrm{a}}\mathrm{Ind}_B^G(\chi_1 \otimes \chi_2)^{K(m)}, \quad \tilde{\mathfrak{I}}_0 \ := \ {}^{\mathrm{a}}\mathrm{Ind}_B^G(\chi_2(1) \otimes \chi_1(-1))^{K(m)}. \end{align*} $$

If $\chi : F^\times \to E^\times $ is a locally constant homomorphism then for any integer n, we denote $\chi (n) = \chi \otimes |\ |^n$ the E-valued character: $u \mapsto \chi (u)$ for all $u \in \mathcal O^\times $ and $\varpi \mapsto q^{-n}\chi (\varpi )$ . It is clear then that $\mathfrak {I} = \mathfrak {I}_0 \otimes _{E, \iota } \mathbb {C}$ and $\tilde {\mathfrak {I}} = \tilde {\mathfrak {I}}_0 \otimes _{E, \iota } \mathbb {C}.$ Note that $\mathfrak {I}_0$ consists of all E-valued functions in $\mathfrak {I}$ ; similarly, $\tilde {\mathfrak {I}}_0.$ The local L-value that appears in (5.4.3) is E-rational, i.e., $L(-1, {}^\iota \chi _1 \otimes {}^\iota \chi _2^{-1}) = \left (1 - {}^\iota \chi _1(\varpi ){}^\iota \chi _2(\varpi ^{-1}) q\right )^{-1} \in \iota (E),$ and furthermore, if $L_0(-1, \chi _1 \otimes \chi _2^{-1}) = \left (1 - \chi _1(\varpi )\chi _2(\varpi ^{-1}) q\right )^{-1} \in E$ then $\iota (L_0(-1, \chi _1 \otimes \chi _2^{-1})) = L(-1, {}^\iota \chi _1 \otimes {}^\iota \chi _2^{-1}).$ It is clear now from (5.4.3) that $T(\mathfrak {I}_0) \subset \tilde {\mathfrak {I}}_0;$ also that if $T_0 = T|_{\mathfrak {I}_0}$ then $T = T_0 \otimes _{E, \iota } \mathbb {C}.$