Representations of the $p$ -adic $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$ and the Adjoint $L$ -Function

Mahdi Asgari and Kwangho Choiy Mahdi Asgari
Department of Mathematics, Oklahoma State University, Stillwater, OK 74078-1058, U.S.A. [email protected] Kwangho Choiy
School of Mathematical and Statistical Sciences, Southern Illinois University, Carbondale, IL 62901-4408, U.S.A. [email protected]

Abstract.

We prove a conjecture of B. Gross and D. Prasad about determination of generic $L$ -packets in terms of the analytic properties of the adjoint $L$ -function for $p$ -adic general even spin groups of semi-simple ranks 2 and 3. We also explicitly write the adjoint $L$ -function for each $L$ -packet in terms of the local Langlands $L$ -functions for the general linear groups.

1. Introduction

In this article, we provide further details on the local $L$ -packets for the non-Archimedean split general spin groups $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$ , following our earlier work [AC17]. We then use our explicit description of these $L$ -packets to prove a conjecture of B. Gross and D. Prasad [Gr22, GP92] determining which of the $L$ -packets are “generic” (i.e., contain an irreducible representation with a Whittaker model) in terms of the analytic properties at $s=1$ of the adjoint $L$ -function of the packet. We also write the adjoint $L$ -function for each $L$ -packet in terms of the local Langlands $L$ -functions of the general linear groups. In addition to details about the representations that our results provide, given that the adjoint $L$ -functions have a significant role in the Gan-Gross-Prasad conjectures, we expect that our results in this paper would be helpful in that direction as well. Particularly striking is the generalization of the Gan-Gross-Prasad to the non-tempered case [GGP20] where the relevant adjoint $L$ -function does have a pole at $s=1$ .

Let $F$ be a $p$ -adic field of characteristic zero. Denote by $W_{F}$ the Weil group of $F$ and let $W^{\prime}_{F}=W_{F}\times\mathrm{SL}_{2}(\mathbb{C})$ be the Weil-Deligne group of $F$ . Let $G$ be a connected, reductive, linear algebraic group over $F$ . The local Langlands Conjecture (LLC) predicts a surjective, finite-to-one map $\mathcal{L}$ from the set $\operatorname{Irr}(G)$ of equivalence classes of irreducible, smooth, complex representations of $G(F)$ to the set $\Phi(G)$ of $\widehat{G}$ -conjugacy classes of $L$ -parameters of $G(F)$ , i.e., admissible homomorphisms $\phi:W^{\prime}_{F}\longrightarrow{}^{L}G$ . Here, ${}^{L}G$ denotes the $L$ -group of $G$ with $\widehat{G}={}^{L}G^{0}$ its connected component, i.e., the complex dual of $G$ [Bor79]. Among other properties, the map $\mathcal{L}$ is supposed to preserve the local $L$ -, $\epsilon$ -, and $\gamma$ -factors. Moreover, the (finite) fibers $\Pi_{\phi}$ , for $\phi\in\Phi(G)$ , of the map $\mathcal{L}$ are called the $L$ -packets of $G$ and their structures are expected to be controlled by certain finite subgroups of $\widehat{G}$ .

Consider the split general spin groups $G=\mathrm{GSpin}_{4}$ and $G=\mathrm{GSpin}_{6}$ , of type $D_{2}=A_{1}\times A_{2}$ and $D_{3}=A_{3}$ respectively, whose algebraic structure we review in Section 2.3. We constructed most of the $L$ -packets for these two groups in [AC17] and proved that they satisfy the expected properties of preservation of the local factors and their internal structure. We review and complete the construction of these $L$ -packets. In particular, using the classification of representations of $GL_{n},$ we give more explicit descriptions of the $L$ -packets for $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$ in terms of given representations of $\mathrm{GL}_{2}\times\mathrm{GL}_{2}$ and $\mathrm{GL}_{4}\times GL_{1},$ respectively. As a byproduct, we are able to give the criteria for determining the size of the $L$ -packets for $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$ (see Sections 4 and 5).

The known cases of the LLC for the $p$ -adic groups include $\mathrm{GL}_{n}$ [HT01, Hen00, Sch13]; $\mathrm{SL}_{n}$ [GK82]; non-quasi-split $F$ -inner forms of $\mathrm{GL}_{n}$ and $\mathrm{SL}_{n}$ [HS12, ABPS16]; $\operatorname*{GSp}_{4}$ and $\operatorname*{Sp}_{4}$ [GT11, GT10]; non-quasi-split $F$ -inner form $\operatorname*{GSp}_{1,1}$ of $\operatorname*{GSp}_{4}$ [GT14]; $\operatorname*{Sp}_{2n},\operatorname*{SO}_{n},$ and quasi-split $\operatorname*{SO}^{*}_{2n}$ [Art13]; $\operatorname*{U}_{n}$ [Rog90, Mok15]; non quasi-split $F$ -inner forms of $\operatorname*{U}_{n}$ [Rog90, KMSW14]; non-quasi-split $F$ -inner form $\operatorname*{Sp}_{1,1}$ of $\operatorname*{Sp}_{4}$ [Cho17]; $\mathrm{GSpin}_{4},\mathrm{GSpin}_{6}$ and their inner forms [AC17]; $\operatorname*{GSp}_{2n}$ and $\mathrm{GO}_{2n}$ [Xu18].

Going back to the case of general $G$ , assume that $\rho$ is a finite-dimensional complex representation of ${}^{L}G$ . When LLC is known, one can define the local Langlands $L$ -functions

L(s,\pi,\rho)=L(s,\rho\circ\phi)

for each $\pi\in\Pi_{\phi}$ . Here, the $L$ -factors on the right hand side are the Artin local factors associated to the given representation of $W^{\prime}_{F}$ .

B. Gross and D. Prasad conjectured (in the generality of quasi-split groups) that the local $L$ -packet $\Pi_{\phi}(G)$ is generic if and only if the adjoint $L$ -function $L(s,\mathrm{Ad}\circ\phi)$ is regular at $s=1$ [GP92, Conj. 2.6]. Here, $\mathrm{Ad}$ denotes the adjoint representation of ${}^{L}G$ on the dual Lie algebra $\widehat{\mathfrak{g}}$ of $\widehat{G}$ . (Note that in the body of this paper we use $\mathrm{Ad}$ exclusively for the restriction of the adjoint representation to the derived group of $\widehat{\mathfrak{g}}$ to distinguish it from the full adjoint $L$ -function, which would have an extra factor of the $L$ -function for the trivial character when $\widehat{\mathfrak{g}}$ has a one-dimensional center.)

We prove the above conjecture for the groups $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$ as a consequence of our construction of the $L$ -packets for these groups. In fact, we prove the conjecture for a larger class of groups $G=G_{m,n}^{r,s}$ , which are given as subgroups of $\mathrm{GL}_{m}\times\mathrm{GL}_{n}$ satisfying a certain determinant equality (2.6). We are able to work in the slightly larger generality because, as in the construction of the $L$ -packets, we use the approach of restricting representations from $\mathrm{GL}_{m}(F)\times\mathrm{GL}_{n}(F)$ to the subgroup $G$ .

Moreover, we also give the adjoint $L$ -function in all cases explicitly in terms of local Langlands $L$ -functions of the general linear groups. While we are able to prove the Gross-Prasad conjecture already without the explicit knowledge of the adjoint $L$ -function, the explicit description of the adjoint $L$ -function certainly also verifies the conjecture and we include it here since it may lead to other number theoretic or representation theoretic results.

Finally, we take this opportunity to correct a few inaccuracies in [AC17]. They do not affect the main results in that paper and fix some errors in our description of the $L$ -packets. The details are given in Section 6.

Acknowledgements

We are grateful to Behrang Noohi and Ralf Schmidt for helpful discussions. We also thank B. Gross for his interest in this paper and clarifying the history of his conjecture and the context in which it was made.

K. Choiy was supported by a gift from the Simons Foundation (#840755).

2. Preliminaries

2.1. Local Langlands Correspondence (LLC)

Let $p$ be a prime number and let $F$ be a $p$ -adic field of characteristic zero, i.e., a finite extension of $\mathbb{Q}_{p}$ . We fix an algebraic closure $\bar{F}$ of $F.$ Denote the ring of integers of $F$ by $\mathcal{O}_{F}$ and its unique maximal ideal by $\mathcal{P}_{F}$ . Moreover, let $q$ denote the cardinality of the residue field $\mathcal{O}_{F}/\mathcal{P}_{F}$ and fix a uniformizer $\varpi$ with $|\varpi|_{F}=q^{-1}$ . Also, let $W_{F}$ denote the Weil group of $F$ , $W^{\prime}_{F}$ the Weil-Deligne group of $F$ , and $\Gamma$ the absolute Galois group $\operatorname*{Gal}(\bar{F}/F)$ . Throughout the paper, we will use the notation $\nu(\cdot)=|\cdot|_{F}$ .

Let $G$ be a connected, reductive, linear algebraic group over $F$ . Fixing $\Gamma$ -invariant splitting data we define the $L$ -group of $G$ as a semi-direct product ${}^{L}G:=\widehat{G}\rtimes\Gamma$ , where $\widehat{G}={}^{L}G^{0}$ denotes the connected component of the $L$ -group of $G,$ i.e., the complex dual of $G$ (see [Bor79, §2]).

LLC (still conjectural in this generality) asserts that there is a surjective, finite-to-one map from the set $\operatorname*{Irr}(G)$ of isomorphism classes of irreducible smooth complex representations of $G(F)$ to the set $\Phi(G)$ of $\widehat{G}$ -conjugacy classes of $L$ -parameters, i.e., admissible homomorphisms $\varphi:W^{\prime}_{F}\longrightarrow{}^{L}G$ .

Given $\varphi\in\Phi(G),$ its fiber $\Pi_{\varphi}(G)$ , which is called an $L$ -packet for $G,$ is expected to be controlled by a certain finite group living in the complex dual group $\widehat{G}.$ Furthermore, for $\pi\in\Pi_{\varphi}(G)$ and $\rho$ a finite dimensional algebraic representation of ${}^{L}G$ one defines the local factors

$\displaystyle L(s,\pi,\rho)$	$\displaystyle=$	$\displaystyle L(s,\rho\circ\phi),$	(2.1)
$\displaystyle\epsilon(s,\pi,\rho,\psi)$	$\displaystyle=$	$\displaystyle\epsilon(s,\rho\circ\phi,\psi),$	(2.2)
$\displaystyle\gamma(s,\pi,\rho,\psi)$	$\displaystyle=$	$\displaystyle\gamma(s,\rho\circ\phi,\psi).$	(2.3)

provided that LLC is known for the case in question. Here, the factors on the right are Artin factors.

2.2. The Adjoint $L$ -Function

What we recall in this subsection holds for $G$ quasi-split ([GP92, §2]). However, for simplicity we will take $G$ to be split over $F$ since the groups we are working with in this article are split. When $G$ is split over $F$ , we may replace the $L$ -group ${}^{L}G$ by its connected component $\widehat{G}={}^{L}G^{0}$ . Take $\rho$ to be the adjoint action of $\widehat{G}$ on its Lie algebra. Then we obtain the adjoint $L$ -function $L(s,\pi,\mathrm{Ad}_{\widehat{G}})=L(s,\mathrm{Ad}_{\widehat{G}}\circ\phi)$ for all $\pi\in\Pi_{\varphi}(G)$ . The following is a conjecture of D. Gross and D. Prasad (see [GP92, Conj. 2.6]).

Conjecture 2.1.

$\Pi_{\varphi}(G)$ contains a generic member if and only if $L(s,\mathrm{Ad}_{\widehat{G}}\circ\phi)$ is regular at $s=1$ . (Equivalently, $\pi$ is generic if and only if $L(s,\pi,\mathrm{Ad}_{\widehat{G}})$ is regular at $s=1$ .)

The conjecture is known in many cases in which the LLC is known. To mention a few, it was verified for $\mathrm{GL}_{n}$ by B. Gross and D. Prasad [GP92], for $\operatorname*{GSp}_{4}$ in [GT11] and, for non-supercuspidals, in [AS08], and for $\operatorname*{SO}$ and $\operatorname*{Sp}$ groups, it follows from the work of Arthur on endoscopic classification [Art13]. We will verify this conjecture for the small rank split groups $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$ .

2.3. The Groups $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$

We gave detailed information about the structure of these two groups (as well as their inner forms) in [AC17, §2.2]. For now we just recall the incidental isomorphisms

	$\displaystyle{\mathrm{GSpin}}_{4}$	$\displaystyle\cong$	$\displaystyle\left\{(g_{1},g_{2})\in{\mathrm{GL}}_{2}\times{\mathrm{GL}}_{2}:\det g_{1}=\det g_{2}\right\}$		(2.4)
	$\displaystyle{\mathrm{GSpin}}_{6}$	$\displaystyle\cong$	$\displaystyle\left\{(g_{1},g_{2})\in{\mathrm{GL}}_{1}\times{\mathrm{GL}}_{4}:g_{1}^{2}=\det g_{2}\right\}.$		(2.5)

While our main interests in this article are the split general spin groups $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$ , for the purposes of Conjecture 2.1 it is no more difficult, and perhaps also more natural, to consider a slightly more general setup as follows.

Fix integers $m,n\geq 1$ and $r,s\geq 1$ and assume that $\operatorname{gcd}(r,s)=1$ . Define

G=G_{m,n}^{r,s}:=\left\{(g,h)\in\mathrm{GL}_{m}\times\mathrm{GL}_{n}\mid(\det g)^{r}=(\det h)^{s}\right\}

(2.6)

Proposition 2.2.

The group $G_{m,n}^{r,s}$ is a split, connected, reductive, linear algebraic group over $F$ .

Proof.

Let $X=(X_{ij})$ and $Y=(Y_{kl})$ be $m\times m$ and $n\times n$ matrices, respectively. It is clear that $G_{m,n}^{r,s}$ , being an almost direct product of $SL_{m}\times\mathrm{SL}_{n}$ and a torus, is reductive. The only issue that requires justification is that the polynomial $f(X,Y)=(\det X)^{r}-(\det Y)^{s}$ is irreducible in $F[X_{ij},Y_{kl}]$ if and only if $d=\gcd(r,s)=1$ . It is clear that if $d>1$ , then $f$ is reducible since it would be divisible by $(\det X)^{(r/d)}-(\det Y)^{(s/d)}$ . It remains to show that if $d=1$ , then $f(X,Y)$ is irreducible. This assertion should be easy to see via elementary arguments considering the polynomials in a possible factorization of $f$ . However, we prove it below as a special case of a more general fact.

Assume that $f(x,y)$ is an (arbitrary) irreducible polynomial in $F[x,y]$ . Let

p(x_{1},x_{2},\dots,x_{a})\in F[x_{1},x_{2},\dots,x_{a}]\quad\mbox{ and }\quad p(y_{1},y_{2},\dots,y_{b})\in F[y_{1},y_{2},\dots,y_{b}]

be two polynomials such that $p-\alpha$ and $q-\alpha$ are irreducible for all constants $\alpha$ . Then, $f(p,q)$ is irreducible in $F[x_{1},x_{2},\dots,x_{a},y_{1},y_{2},\dots,y_{b}]$ .

Our Proposition would clearly follow from the above assertion since $(\det-\alpha)$ is always an irreducible polynomial and it is well-known that the two-variable polynomial $x^{r}-y^{s}$ is irreducible in $F[x,y]$ provided that $d=\gcd(r,s)=1$ .

To prove the assertion above, we proceed as follows. By base extension to an algebraic closure we may assume, without loss of generality, that $F$ is algebraically closed.

Let $A$ be the subscheme of $\operatorname{Spec}F[x_{1},x_{2},\dots,x_{a},y_{1},y_{2},\dots,y_{b}]$ defined by $f(p,q)$ , and let $B$ be the subscheme of $\operatorname{Spec}F[x,y]$ defined by $x^{r}-y^{s}$ . The latter is irreducible since $x^{r}-y^{s}$ is an irreducible polynomial by our assumption that $d=1$ . There is a natural map $A\to B$ which has irreducible (geometric) fibers. The result now follows from the following claim.

Claim: Let $g:A\to B$ be an open morphism of schemes of finite type over an algebraically closed field $F$ such that the (geometric) fibers of $g$ are irreducible and $B$ is irreducible. Then $A$ is irreducible.

To see the claim let $U$ be an open in $A$ . We want to show that for any other open $V$ , we have that $U\cap V$ is nonempty. Since $B$ is irreducible and $g$ is open, we have that $g(U)\cap g(V)$ is nonempty so there is a fiber $F_{0}$ of $g$ such that $F_{0}\cap U$ and $F_{0}\cap V$ are nonempty. Hence, by irreducibility of $F_{0}$ , they have a nonempty intersection in $F_{0}$ . In particular, $U\cap V$ is nonempty, which gives the claim.

It only remains to check that the map $A\to B$ above is open. In fact, it is flat since it is a base extension of the cartesian product of two flat morphisms $p:\operatorname{Spec}F[x_{1},...,x_{a}]\to\operatorname{Spec}F[x]$ and $q:\operatorname{Spec}F[y_{1},...,y_{b}]\to\operatorname{Spec}F[y]$ . (Here, we are using the fact that $\operatorname{Spec}F[x]$ is a curve.) This finishes the proof. ∎

Of particular interest to us in this paper are the cases

•

$m=n=2$ and $r=s=1$ , when $G=\mathrm{GSpin}_{4}$ , and
•

$m=1$ , $n=4$ and $r=2$ , $s=1$ , when $G=\mathrm{GSpin}_{6}$ .

The (connected) $L$ -group of $G$ is

{{}^{L}{G}_{m,n}^{{r,s}\,0}}=\widehat{G}\cong({\mathrm{GL}}_{m}(\mathbb{C})\times{\mathrm{GL}}_{n}(\mathbb{C}))/\{(z^{-r}I_{m},z^{s}I_{n}):z\in\mathbb{C}^{\times}\}

(2.7)

and we have the exact sequence

1\longrightarrow\{(z^{-r}I_{m},z^{s}I_{n}):z\in\mathbb{C}^{\times}\}\cong\mathbb{C}^{\times}\longrightarrow{\mathrm{GL}}_{m}(\mathbb{C})\times{\mathrm{GL}}_{n}(\mathbb{C})\xrightarrow{pr_{m,n}^{r,s}}\widehat{G_{m,n}^{r,s}}\longrightarrow 1.

(2.8)

2.4. Computation of the Adjoint $L$ -Function for $G$

Let $\pi$ be an irreducible admissible representation of $G(F)$ . There exist irreducible admissible representations $\pi_{m}$ and $\pi_{n}$ of $\mathrm{GL}_{m}(F)$ and $\mathrm{GL}_{n}(F)$ , respectively, such that

\pi\hookrightarrow{\operatorname*{Res}}_{G(F)}^{\mathrm{GL}_{m}(F)\times\mathrm{GL}_{n}(F)}\left(\pi_{m}\otimes\pi_{n}\right).

(2.9)

Let $\mathrm{Ad}_{\widehat{G}}$ denote the adjoint action of $\widehat{G}$ on its Lie algebra

\widehat{\mathfrak{g}}=\left\{(X,Y)\in\mathfrak{gl}_{m}(\mathbb{C})\times\mathfrak{gl}_{n}(\mathbb{C})\mid r\operatorname{tr}(X)=s\operatorname{tr}(Y)\right\}.

(2.10)

In what follows, let us write

\mathrm{Ad}_{\widehat{G}}=\operatorname{triv}\oplus\mathrm{Ad}

(2.11)

and for $i\in\{m,n\}$ we similarly write $\mathrm{Ad}_{i}=\mathrm{Ad}_{\widehat{GL}_{i}}=\operatorname{triv}\oplus\mathrm{Ad},$ where $\mathrm{Ad}$ here denotes the action of $\mathrm{GL}_{i}(\mathbb{C})$ on the space of traceless $i\times i$ complex matrices ${\mathfrak{s}l}_{i}(\mathbb{C})$ .

Let $\phi_{\pi}:W_{F}\times\mathrm{SL}_{2}(\mathbb{C})\to\widehat{G}$ be the $L$ -parameter of $\pi$ and let $\phi_{i}:W_{F}\times\mathrm{SL}_{2}(\mathbb{C})\to\mathrm{GL}_{i}(\mathbb{C})$ , $i=m,n$ , be the $L$ -parameter of $\pi_{i}$ . Recall by (2.8) that we have a natural map

pr={pr_{m,n}^{r,s}}:\mathrm{GL}_{m}(\mathbb{C})\times\mathrm{GL}_{n}(\mathbb{C})\longrightarrow\widehat{G}.

(2.12)

Then we have

\phi_{\pi}=pr\circ(\phi_{m}\otimes\phi_{n}).

(2.13)

Since the subgroup $\{(z^{-r}I_{m},z^{s}I_{n}):z\in\mathbb{C}^{\times}\}$ is central in $\mathrm{GL}_{m}(\mathbb{C})\times\mathrm{GL}_{n}(\mathbb{C})$ the following diagram commutes.

Note that the adjoint action $\mathrm{Ad}_{m}$ of $\mathrm{GL}_{m}(\mathbb{C})$ on ${\mathfrak{g}l}_{m}(\mathbb{C})$ preserves the trace, and similarly for $n$ , so we obtain a right downward arrow by simply restricting any automorphism to the set of those pairs satisfying the trace equality in (2.10). We have

$\displaystyle L(s,1_{F^{\times}})L(s,\pi,\mathrm{Ad})\cdot L(s,1_{F^{\times}})$	$\displaystyle=$	$\displaystyle L(s,\pi,\mathrm{Ad}_{\widehat{G}})\cdot L(s,1_{F^{\times}})$	(2.14)
	$\displaystyle=$	$\displaystyle L(s,\mathrm{Ad}_{\widehat{G}}\circ\phi_{\pi})\cdot L(s,1_{F^{\times}})$
	$\displaystyle=$	$\displaystyle L\left(s,(\mathrm{Ad}_{m}\otimes\mathrm{Ad}_{n})\circ(\phi_{m}\otimes\phi_{n})\right)$
	$\displaystyle=$	$\displaystyle L(s,\mathrm{Ad}_{m}\circ\phi_{m})L(s,\mathrm{Ad}_{n}\circ\phi_{n})$
	$\displaystyle=$	$\displaystyle L(s,\pi_{m},\mathrm{Ad}_{m})L(s,\pi_{n},\mathrm{Ad}_{n})$
	$\displaystyle=$	$\displaystyle L(s,1_{F^{\times}})^{2}L(s,\pi_{m},\mathrm{Ad})L(s,\pi_{n},\mathrm{Ad}).$

Therefore, we obtain the more convenient equality

L(s,\pi,\mathrm{Ad})=L(s,\pi_{m},\mathrm{Ad})L(s,\pi_{n},\mathrm{Ad}),

(2.15)

which holds thanks to our choice of the notation $\mathrm{Ad}$ . In Section 3.2 this relation helps verify Conjecture 2.1 for the groups of interest to us.

3. Genericity and The Conjecture of B. Gross and D. Prasad

3.1. Restriction of Generic Representations

Let us write $\square^{D}$ for the group $\operatorname*{Hom}(\square,\mathbb{C}^{\times})$ of all continuous characters on a topological group $\square$ . Dente by $\square_{\operatorname*{der}}$ the derived group of $\square.$ Let $G$ and $\widetilde{G}$ be connected, reductive, linear, algebraic groups over $F$ satisfying the property that

G_{\operatorname*{der}}=\widetilde{G}_{\operatorname*{der}}\subseteq G\subseteq\widetilde{G}.

(3.1)

For any connected, reductive, linear, algebraic group $\square$ over $F,$ we write $\operatorname*{Irr}_{\rm sc}(\square)$ and $\operatorname*{Irr}_{\rm esq}(\square)$ for the set of equivalence classes of supercuspidal and essentially square-integrable representations of $\square(F),$ respectively.

Assume $\widetilde{G}$ and $G$ to be $F$ -split. Let $\widetilde{B}$ be a Borel subgroup of $\widetilde{G}$ with Levi decomposition $\widetilde{B}=\widetilde{T}\widetilde{U}.$ Then $B=\widetilde{B}\cap G$ is a Borel subgroup of $G$ with $B=TU$ . Note that $T=\widetilde{T}\cap G$ and $\widetilde{U}=U.$ Let $\psi$ be a generic character of $U(F)$ . From [Tad92, Proposition 2.8] we know that given a $\psi$ -generic irreducible representation $\widetilde{\sigma}$ of $\widetilde{G}(F)$ we have a unique $\psi$ -generic $\sigma$ of $G(F)$ such that

\sigma\hookrightarrow{\operatorname*{Res}}^{\widetilde{G}}_{G}(\widetilde{\sigma}).

The generic character associated with $\sigma$ is not unique though.

Proposition 3.1.

Each generic character associated with $\sigma$ is determined up to the action of $\widetilde{T}(F)/T(F).$

Proof.

We let $\widetilde{\sigma}\in\operatorname*{Irr}(\widetilde{G})$ be $\psi$ -generic. Then there is a unique $\psi$ -generic $\sigma_{\psi}\in\Pi_{\widetilde{\sigma}}(G)$ . On the other hand, for each $\sigma\in\Pi_{\widetilde{\sigma}}(G)$ there exists $t\in\widetilde{T}(F)/T(F)\cong\widetilde{G}/G(F)$ such that $\sigma={{}^{t}}\sigma_{\psi},$ where ${}^{t}\sigma_{\psi}(g)=\sigma(t^{-1}gt).$ This implies that $\sigma$ is ${}^{t}\psi$ -generic. Here ${}^{t}\psi$ is defined as ${}^{t}\psi(u)=\psi(t^{-1}ut).$ ∎

Remark 3.2.

We say $\sigma\in\operatorname*{Irr}(G)$ , resp. $\widetilde{\sigma}\in\operatorname*{Irr}(\widetilde{G})$ , is generic if it is $\psi$ -generic with respect to some generic character $\psi$ . With this notation, $\sigma\in\operatorname*{Irr}(G)$ is generic if and only if is $\widetilde{\sigma}\in\operatorname*{Irr}(\widetilde{G}).$

3.2. Criterion for Genericity

In this section we verify Conjecture 2.1 for the small rank general spin groups we are considering in this article.

Theorem 3.3.

Let $G=G_{m,n}^{r,s}$ be the group defined in (2.6). Let $\pi$ be an irreducible admissible representation of $G(F)$ . Then $\pi$ is generic if and only if $L(s,\pi,\mathrm{Ad})$ is regular at $s=1$ .

Proof.

Given $\pi$ there exist irreducible admissible representations $\pi_{m}$ of $\mathrm{GL}_{m}(F)$ and $\pi_{n}$ of $\mathrm{GL}_{n}(F)$ such that $\pi$ is a subrepresentation of the restriction to $G(F)$ of $\pi_{m}\otimes\pi_{n}$ as in (2.9). Now, $\pi$ is generic if and only if both $\pi_{m}$ and $\pi_{n}$ are generic. By the truth of Conjecture 2.1 for the general linear groups, the latter is equivalent to both $L(s,\pi_{m},\mathrm{Ad})$ and $L(s,\pi_{n},\mathrm{Ad})$ being regular at $s=1$ . Hence, by (2.15) and the fact that neither of the $L$ -functions can have a zero at $s=1$ , we have that $\pi$ is generic if and only if $L(s,\pi,\mathrm{Ad})$ is regular at $s=1$ . This proves the theorem. ∎

As we observed in Section 2.3, the split groups $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$ are special cases of $G_{m,n}^{r,s}$ . Therefore, we have the following.

Corollary 3.4.

Conjecture 2.1 holds for the groups $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$ .

4. Representations of $\mathrm{GSpin}_{4}$

In this section we list all the irreducible representations of $\mathrm{GSpin}_{4}(F)$ and then calculate their associated adjoint $L$ -function explicitly. To this end, we give the nilpotent matrix associated to their parameter in each case.

4.1. The Reprsentations

4.1.1. Classification of representations of $\mathrm{GSpin}_{4}$

Following [AC17], we have

1\longrightarrow{\mathrm{GSpin}}_{4}(F)\longrightarrow{\mathrm{GL}}_{2}(F)\times{\mathrm{GL}}_{2}(F)\longrightarrow F^{\times}\longrightarrow 1.

(4.1)

Recall that

{\mathrm{GSpin}}_{4}(F)\cong\{(g_{1},g_{2})\in{\mathrm{GL}}_{2}(F)\times{\mathrm{GL}}_{2}(F):\det g_{1}=\det g_{2}\},

(4.2)

{{}^{L}{\mathrm{GSpin}}_{4}}=\widehat{{\mathrm{GSpin}}_{4}}={\mathrm{GSO}}_{4}(\mathbb{C})\cong({\mathrm{GL}}_{2}(\mathbb{C})\times{\mathrm{GL}}_{2}(\mathbb{C}))/\{(z^{-1},z):z\in\mathbb{C}^{\times}\},

(4.3)

and

1\longrightarrow\mathbb{C}^{\times}\longrightarrow{\mathrm{GL}}_{2}(\mathbb{C})\times{\mathrm{GL}}_{2}(\mathbb{C})\overset{pr_{4}}{\longrightarrow}\widehat{{\mathrm{GSpin}}_{4}}\longrightarrow 1.

(4.4)

When convenient, we view $\mathrm{GSO}_{4}$ as the group similitude orthogonal $4\times 4$ matrices with respect to the anti-diagonal matrix

J=J_{4}=\begin{bmatrix}0&0&0&1\\ 0&0&1&0\\ 0&1&0&0\\ 1&0&0&0\end{bmatrix}.

(4.5)

The Lie algebra of this group is also defined with respect to $J$ and an element $X$ in this Lie algebra satisfies

{}^{t}XJ+JX=0.

4.1.2. Construction of the $L$ -packets of $\mathrm{GSpin}_{4}$ (recalled from [AC17])

Given $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{4})$ we have a lift $\widetilde{\sigma}\in\operatorname*{Irr}({\mathrm{GL}}_{2}\times{\mathrm{GL}}_{2})$ such that

\sigma\hookrightarrow{\operatorname*{Res}}_{{\mathrm{GSpin}}_{4}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}(\widetilde{\sigma}).

It follows form the LLC for $GL_{n}$ [HT01, Hen00, Sch13] that there is a unique $\widetilde{\varphi}_{\widetilde{\sigma}}\in\Phi({\mathrm{GL}}_{2}\times{\mathrm{GL}}_{2})$ corresponding to the representation $\widetilde{\sigma}.$ We now have a surjective, finite-to-one map

	$\displaystyle{\mathcal{L}}_{4}:{\operatorname*{Irr}}({\mathrm{GSpin}}_{4})$	$\displaystyle\longrightarrow$	$\displaystyle\Phi({\mathrm{GSpin}}_{4})$		(4.6)
	$\displaystyle\sigma$	$\displaystyle\longmapsto$	$\displaystyle pr_{4}\circ\widetilde{\varphi}_{\widetilde{\sigma}},$

which does not depend on the choice of the lifting $\widetilde{\sigma}.$ Then, for each $\varphi\in\Phi(\mathrm{GSpin}_{4}),$ all inequivalent irreducible constituents of $\widetilde{\sigma}$ constitutes the $L$ -packet

\Pi_{\varphi}({\mathrm{GSpin}}_{4}):=\Pi_{\widetilde{\sigma}}({\mathrm{GSpin}}_{4})=\left\{\sigma\,\middle|\,\sigma\hookrightarrow{\operatorname*{Res}}_{{\mathrm{GSpin}}_{4}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}(\widetilde{\sigma})\right\}\Big{/}\cong.

(4.7)

Here, $\widetilde{\sigma}$ is the member in the singleton $\Pi_{\widetilde{\varphi}}(\mathrm{GL}_{2}\times\mathrm{GL}_{2})$ and $\widetilde{\varphi}\in\Phi(\mathrm{GL}_{2}\times\mathrm{GL}_{2})$ is such that $pr_{4}\circ\widetilde{\varphi}=\varphi.$ We note that the construction does not depends on the choice of $\widetilde{\varphi},$ due to the LLC for $\mathrm{GL}_{2}$ , [GK82, Lemma 2.4], [Tad92, Corollary 2.5], and [HS12, Lemma 2.2]. Further details can be found in [AC17, Section 5.1].

4.1.3. The $L$ -parameters of ${\mathrm{GL}}_{2}$

We recall the generic representations of $\mathrm{GL}_{2}(F)$ in this paragraph. We refer to [Wed08, Kud94, GR10] for details. Let $\chi:F^{\times}\rightarrow\mathbb{C}^{\times}$ denote a continuous quasi-character of $F^{\times}$ . By Zelevinski ([Zel80, Theorem 9.7] or [Kud94, Theorem 2.3.1]) we know that the generic representations of ${\mathrm{GL}}_{2}$ are: the supercuspidals, $\mathsf{St}\otimes(\chi\circ\det)$ where $\mathsf{St}$ denotes the Steinberg representation, and normally induced representations $i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\otimes\chi_{2})$ with $\chi_{1}\neq\chi_{2}\nu^{\pm 1}.$ The only non-generic representation is $\chi\circ\det.$

4.2. Generic Representations of $\mathrm{GSpin}_{4}$

Following [AC17, Section 5.3], given $\varphi\in\Phi(\mathrm{GSpin}_{4}),$ fix the lift

\widetilde{\varphi}=\widetilde{\varphi}_{1}\otimes\widetilde{\varphi}_{2}\in\Phi({\mathrm{GL}}_{2}\times{\mathrm{GL}}_{2})

with $\widetilde{\varphi}_{i}\in\Phi({\mathrm{GL}}_{2})$ such that $\varphi=pr_{4}\circ\widetilde{\varphi}$ . Let

\widetilde{\sigma}=\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2}\in\Pi_{\widetilde{\varphi}}({\mathrm{GL}}_{2}\times{\mathrm{GL}}_{2})

be the unique member such that $\{\widetilde{\sigma}_{i}\}=\Pi_{\widetilde{\varphi}_{i}}({\mathrm{GL}}_{2}).$

Recall the notation

I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma}):=\left\{\chi\in\left({\mathrm{GL}}_{2}(F)\times{\mathrm{GL}}_{2}(F)/{\mathrm{GSpin}}_{4}(F)\right)^{D}\,\middle|\,\widetilde{\sigma}\otimes\chi\cong\widetilde{\sigma}\right\}.

Then we have

\Pi_{\varphi}({\mathrm{GSpin}}_{4})\,\overset{1-1}{\longleftrightarrow}\,I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma}),

(4.8)

and we recall that, by [AC17, Proposition 5.7], we have

I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma})=\left\{\begin{array}[]{lll}I^{{\mathrm{SL}}_{2}}(\widetilde{\sigma}_{1}),&\mbox{if $\widetilde{\sigma}_{2}\cong\widetilde{\sigma}_{1}\widetilde{\eta}$ for some $\widetilde{\eta}\in(F^{\times})^{D}$};\\ I^{{\mathrm{SL}}_{2}}(\widetilde{\sigma}_{1})\cap I^{{\mathrm{SL}}_{2}}(\widetilde{\sigma}_{2}),&\mbox{if $\widetilde{\sigma}_{2}\not\cong\widetilde{\sigma}_{1}\widetilde{\eta}$ for any $\widetilde{\eta}\in(F^{\times})^{D}$}.\end{array}\right.

(4.9)

4.2.1. Irreducible Parameters

Let $\varphi\in\Phi(\mathrm{GSpin}_{4})$ be irreducible. Then $\widetilde{\varphi},$ $\widetilde{\varphi}_{1},$ and $\widetilde{\varphi}_{2}$ are all irreducible. By Section 3.1, we have the following.

Proposition 4.1.

Let $\varphi\in\Phi(\mathrm{GSpin}_{4})$ be irreducible. Then every member in $\Pi_{\varphi}(\mathrm{GSpin}_{4})$ is supercuspidal and generic.

To study the internal structure of $\Pi_{\varphi}(\mathrm{GSpin}_{4})$ , by (4.8), we need to know the structure of $I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma})$ , as we now recall from [AC17].

\mathfrak{gnr}

-(a)

When $\widetilde{\sigma}_{2}\cong\widetilde{\sigma}_{1}\widetilde{\eta}$ for some $\widetilde{\eta}\in(F^{\times})^{D},$ we have

I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma})\cong\left\{\begin{array}[]{llll}\{1\},&\mbox{if $\widetilde{\varphi}_{1}$ (and hence also $\widetilde{\varphi}_{2}$) is primitive or non-trivial on $\mathrm{SL}_{2}(\mathbb{C})$};\\ \mathbb{Z}/2\mathbb{Z},&\mbox{if $\widetilde{\varphi}_{1}$ (and hence also $\widetilde{\varphi}_{2}$) is dihedral w.r.t. one quadratic extension};\\ (\mathbb{Z}/2\mathbb{Z})^{2},&\mbox{if $\widetilde{\varphi}_{1}$ (and hence also $\widetilde{\varphi}_{2}$) is dihedral w.r.t. three quadratic extensions}.\end{array}\right.

$\mathfrak{gnr}$ -(b)
When $\widetilde{\sigma}_{2}\not\cong\widetilde{\sigma}_{1}\widetilde{\eta}$ for any $\widetilde{\eta}\in(F^{\times})^{D}$ , then by (4.9) we have

$I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma})\cong\{1\}~{}\text{or }\mathbb{Z}/2\mathbb{Z}.$

Since $\widetilde{\sigma}_{2}\not\cong\widetilde{\sigma}_{1}\widetilde{\eta}$ for any $\widetilde{\eta}\in(F^{\times})^{D}$ , the case of both $\widetilde{\varphi}_{1}$ and $\widetilde{\varphi}_{2}$ being diredral w.r.t. three quadratic extensions is excluded. Thus, we have the following list:
- •
  
  If at least one of $\widetilde{\varphi}_{i}$ is primitive, then $I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma})\cong\{1\}.$
- •
  
  If both are dihedral, then $I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma})\cong\mathbb{Z}/2\mathbb{Z}.$

From [AC17, Proposition 2.1], we recall the identification

\Delta^{\vee}=\left\{\beta^{\vee}_{1}=f^{*}_{11}-f^{*}_{12},\beta^{\vee}_{2}=f^{*}_{21}-f^{*}_{22}\right\},

(4.10)

using the notation $f_{ij}$ and $f^{*}_{ij},$ $1\leq i,j\leq 2,$ for the usual $\mathbb{Z}$ -basis of characters and cocharacters of $\mathrm{GL}_{2}\times\mathrm{GL}_{2}$ and $\beta_{1},\beta_{2}$ denote the simple roots of $\mathrm{GSpin}_{4}$ . We can use this identification to relate the nilpotent matrices associated to the parameters of $\mathrm{GL}_{2}\times\mathrm{GL}_{2}$ and $\mathrm{GSpin}_{4}$ , respectively.

For both (a) and (b) above, we have

N_{{\mathrm{GL}}_{2}(\mathbb{C})\times{\mathrm{GL}}_{2}(\mathbb{C})}=\left(\begin{bmatrix}0&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&0\end{bmatrix}\right)\overset{\eqref{indentity}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{4}(\mathbb{C})}=0_{4\times 4}.

Remark 4.2.

We note that case (b) above was mentioned, less precisely, in [AC17, Remark 5.10].

4.2.2. Reducible Parameters

If $\varphi\in\Phi(\mathrm{GSpin}_{4})$ is reducible, then at least one $\widetilde{\varphi}_{i}$ must be reducible. Since the number of irreducible constituents in $\operatorname*{Res}_{\mathrm{SL}_{2}}^{{\mathrm{GL}}_{2}}(\widetilde{\sigma}_{i})$ is at most 2, we have $I^{\mathrm{SL}_{2}}(\widetilde{\sigma}_{i})\cong\{1\},~{}\text{or }\mathbb{Z}/2\mathbb{Z}.$ This implies that

I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma})\cong\{1\},~{}\text{or }\mathbb{Z}/2\mathbb{Z}.

If $\widetilde{\varphi}_{i}$ is reducible and generic, then $\widetilde{\sigma}_{i}$ is either the Steinberg representation twisted by a character or an irreducibly induced representation from the Borel subgroup of ${\mathrm{GL}}_{2}.$ We make case-by-case arguments as follows.

\mathfrak{gnr}

-(i)

Note that the Steinberg representation of $\mathrm{GL}_{2}\times\mathrm{GL}_{2}$ is of the form ${\mathsf{St}}_{\mathrm{GL}_{2}}\boxtimes{\mathsf{St}}_{\mathrm{GL}_{2}}.$ We have

{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}({\mathsf{St}}_{\mathrm{GL}_{2}}\boxtimes{\mathsf{St}}_{\mathrm{GL}_{2}})={\mathsf{St}}_{\mathrm{GSpin}_{4}}

(4.11)

and

{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}\left({\mathsf{St}}_{\mathrm{GL}_{2}}\otimes\chi_{1}\boxtimes{\mathsf{St}}_{\mathrm{GL}_{2}}\otimes\chi_{2}\right)={\mathsf{St}}_{\mathrm{GSpin}_{4}}\otimes\chi

for some $\chi.$ We have $I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma})\cong\{1\}$ as $I^{G}(\mathsf{St}_{G})\cong\{1\}.$ Thus, by (4.9), the $L$ -packet remains a singleton and the restriction is irreducible.

•

To determine $\chi,$ we use the required properties of $\chi_{1},\chi_{2}$ . Using

$T=\left\{\left(\begin{bmatrix}a&0\\ 0&b\end{bmatrix},\begin{bmatrix}c&0\\ 0&d\end{bmatrix}\right)\,\middle|\,ab=cd\right\},$ (4.12)

we have $\chi_{1}(ab)=\chi_{2}(cd)~{}~{}\Leftrightarrow~{}~{}\chi_{1}=\chi_{2}.$ Denote $\chi_{1}=\chi_{2}$ by $\chi$ .

For (4.11), we have

N_{{\mathrm{GL}}_{2}(\mathbb{C})\times{\mathrm{GL}}_{2}(\mathbb{C})}=\left(\begin{bmatrix}0&1\\ 0&0\end{bmatrix},\begin{bmatrix}0&1\\ 0&0\end{bmatrix}\right)\overset{\eqref{indentity}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{4}(\mathbb{C})}=\begin{bmatrix}0&1&1&0\\ 0&0&0&-1\\ 0&0&0&-1\\ 0&0&0&0\end{bmatrix}

\mathfrak{gnr}

-(ii)

Next we consider

{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}\left(i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\otimes\chi_{2})\boxtimes{\mathsf{St}}_{\mathrm{GL}_{2}}\otimes\chi\right).

(4.13)

By (4.9), the fact that $\widetilde{\sigma}_{2}\not\cong\widetilde{\sigma}_{1}\widetilde{\eta}$ for any $\widetilde{\eta}\in(F^{\times})^{D}$ , and since $I^{G}(\mathsf{St}_{G})\cong\{1\}$ , it follows that

I^{\mathrm{GSpin}_{4}}(\widetilde{\sigma})\cong\{1\}.

Thus, the $L$ -packet remains a singleton and the restriction (4.13) is irreducible.

•

To describe the restriction (4.13), we proceed similarly as above. We have

\chi_{1}(a)\chi_{2}(b)=\chi(cd)=\chi(ab)~{}~{}\Leftrightarrow~{}~{}\chi_{1}\chi^{-1}(a)=\chi_{2}^{-1}\chi(b)

Specializing to $a=b$ and $c=d$ in the center, we have

\chi_{1}\chi_{2}\chi^{-2}=1

For (4.13) , we have

N_{{\mathrm{GL}}_{2}(\mathbb{C})\times{\mathrm{GL}}_{2}(\mathbb{C})}=\left(\begin{bmatrix}0&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&1\\ 0&0\end{bmatrix}\right)\overset{\eqref{indentity}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{4}(\mathbb{C})}=\begin{bmatrix}0&0&1&0\\ 0&0&0&-1\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}.

\mathfrak{gnr}

-(iii)

We consider

{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}\left(i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\otimes\chi_{2})\boxtimes i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{3}\otimes\chi_{4})\right)=i_{T}^{\mathrm{GSpin}_{4}}\left(\chi_{1}\otimes\chi_{2},\chi_{3}\otimes\chi_{1}\chi_{2}\chi_{3}^{-1}\right).

Here, $\chi_{1}\neq\chi_{2}\nu^{\pm 1}$ and $\chi_{3}\neq\chi_{4}\nu^{\pm 1}.$ Note that by (4.9) this induced representation may be irreducible or consist of two irreducible inequivalent constituents. We have

N_{{\mathrm{GL}}_{2}(\mathbb{C})\times{\mathrm{GL}}_{2}(\mathbb{C})}=\left(\begin{bmatrix}0&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&0\end{bmatrix}\right)\overset{\eqref{indentity}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{4}(\mathbb{C})}=\begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}.

\mathfrak{gnr}

-(iv)

Given a supercuspidal $\widetilde{\sigma}\in\operatorname*{Irr}(\mathrm{GL}_{2})$ , we consider

{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}\left(\widetilde{\sigma}\boxtimes{\mathsf{St}}_{\mathrm{GL}_{2}}\otimes\chi\right).

(4.14)

Since $I^{G}(\mathsf{St}_{G})\cong\{1\},$ due to (4.9), the restriction (4.14) is irreducible. We then have

N_{\mathrm{GL}_{2}(\mathbb{C})\times\mathrm{GL}_{2}(\mathbb{C})}=\left(\begin{bmatrix}0&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&1\\ 0&0\end{bmatrix}\right)\overset{\eqref{indentity}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{4}(\mathbb{C})}=\begin{bmatrix}0&0&1&0\\ 0&0&0&-1\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}.

\mathfrak{gnr}

-(v)

Given supercuspidal $\widetilde{\sigma}\in\operatorname*{Irr}(\mathrm{GL}_{2}),$ we next consider

{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}\left(\widetilde{\sigma}\boxtimes i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\otimes\chi_{2})\right).

Note from (4.9) that this may be irreducible or consist of two irreducible inequivalent constituents. We have

N_{{\mathrm{GL}}_{2}(\mathbb{C})\times{\mathrm{GL}}_{2}(\mathbb{C})}=\left(\begin{bmatrix}0&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&0\end{bmatrix}\right)\overset{\eqref{indentity}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{4}(\mathbb{C})}=0_{4\times 4}.

4.3. Non-Generic Representations of $\mathrm{GSpin}_{4}$

If $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{4})$ is non-generic, then $\sigma$ is of the form

{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}\left((\chi\circ\det)\boxtimes\widetilde{\sigma}\right),

(4.15)

with $\widetilde{\sigma}\in\operatorname*{Irr}(\mathrm{GL}_{2}).$ Note this restriction is irreducible due to (4.9), and that as $\chi\circ\det$ is non-generic, so is the restriction $\sigma$ for any $\widetilde{\sigma}\in\operatorname*{Irr}(\mathrm{GL}_{2}).$

For $\widetilde{\sigma}=\mathsf{St}\in\operatorname*{Irr}(\mathrm{GL}_{2}),$ we have

N_{{\mathrm{GL}}_{2}(\mathbb{C})\times{\mathrm{GL}}_{2}(\mathbb{C})}=\left(\begin{bmatrix}0&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&1\\ 0&0\end{bmatrix}\right)\overset{\eqref{indentity}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{4}(\mathbb{C})}=\begin{bmatrix}0&0&1&0\\ 0&0&0&-1\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},

and otherwise we have

N_{{\mathrm{GL}}_{2}(\mathbb{C})\times{\mathrm{GL}}_{2}(\mathbb{C})}=\left(\begin{bmatrix}0&0\\ 0&0\end{bmatrix},\begin{bmatrix}0&0\\ 0&0\end{bmatrix}\right)\overset{\eqref{indentity}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{4}(\mathbb{C})}=0_{4\times 4}.

We summarize the above information about the representations of $\mathrm{GSpin}_{4}$ in Table 1.

4.4. Computation of the Adjoint $L$ -function for $\mathrm{GSpin}_{4}$

We now give explicit expressions for the adjoint $L$ -function for each of the representations of $\mathrm{GSpin}_{4}(F)$ . We start by recalling that the adjoint $L$ -functions of the representations $\widetilde{\sigma}\in\operatorname*{Irr}(\mathrm{GL}_{2})$ are as follows.

L(s,\widetilde{\sigma},\mathrm{Ad}_{2})=\begin{cases}L(s)^{2}L(s,\chi_{1}\chi_{2}^{-1})L(s,\chi_{1}^{-1}\chi_{2}),&\mbox{ if }\widetilde{\sigma}=i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\boxtimes\chi_{2})\mbox{ with }\chi_{1}\chi_{2}^{-1}\neq\nu^{\pm 1};\\ L(s)L(s+1),&~{}~{}\text{if }\widetilde{\sigma}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\chi;\\ L(s)L(s,\widetilde{\sigma},\operatorname{Sym}^{2}\otimes\omega_{\widetilde{\sigma}}^{-1}),&\mbox{ if }\widetilde{\sigma}\mbox{ is supercuspidal};\\ L(s)^{2}L(s-1)L(s+1),&\mbox{ if }\widetilde{\sigma}=\chi\circ\det.\end{cases}

Here, $L(s)=L(s,1_{F^{\times}})$ . Recall our choice of notation

L(s,\widetilde{\sigma},\mathrm{Ad}_{2})=L(s)L(s,\widetilde{\sigma},\mathrm{Ad}).

Combining with (2.14), Sections 4.2.1 and 4.2.2, we have the following.

\mathfrak{gnr}

-(a)&(b)

Given a supercuspidal $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{4}),$ we recall that

\sigma\subset{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}(\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2})

for some supercuspidal $\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2}\in\operatorname*{Irr}(\mathrm{GL}_{2}\times\mathrm{GL}_{2}).$ By (2.15) we have

L(s,\sigma,\mathrm{Ad})=L(s,\widetilde{\sigma}_{1},\operatorname{Sym}^{2}\otimes\omega_{\widetilde{\sigma}_{1}}^{-1})L(s,\widetilde{\sigma}_{2},\operatorname{Sym}^{2}\otimes\omega_{\widetilde{\sigma}_{2}}^{-1}).

$\mathfrak{gnr}$ -(i)

Given

$\sigma=\mathsf{St}_{\mathrm{GSpin}_{4}}\otimes\chi\in\operatorname*{Irr}(\mathrm{GSpin}_{4}),$

by (2.15) we have

$L(s,\sigma,\mathrm{Ad})=L(s+1)^{2}.$

\mathfrak{gnr}

-(ii)

Given $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{4})$ such that

\sigma={\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}\left(i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\otimes\chi_{2})\boxtimes{\mathsf{St}}_{\mathrm{GL}_{2}}\otimes\chi\right),

by (2.15) we have

L(s,\sigma,\mathrm{Ad})=L(s)L(s,\chi_{1}\chi_{2}^{-1})L(s,\chi_{1}^{-1}\chi_{2})L(s+1).

\mathfrak{gnr}

-(iii)

Given $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{4})$ such that

\sigma\subset{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}\left(i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\otimes\chi_{2})\boxtimes i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{3}\otimes\chi_{4})\right)

by (2.15) we have

L(s,\sigma,\mathrm{Ad})=L(s)^{2}L(s,\chi_{1}\chi_{2}^{-1})L(s,\chi_{1}^{-1}\chi_{2})L(s,\chi_{3}\chi_{4}^{-1})L(s,\chi_{3}^{-1}\chi_{4}).

\mathfrak{gnr}

-(iv)

Given $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{4})$ such that

\sigma={\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}\left(\widetilde{\sigma}\boxtimes{\mathsf{St}}_{\mathrm{GL}_{2}}\otimes\chi\right)

by (2.15) we have

L(s,\sigma,\mathrm{Ad})=L(s,\widetilde{\sigma}_{2},\operatorname{Sym}^{2}\otimes\omega_{\widetilde{\sigma}_{2}}^{-1})L(s+1).

\mathfrak{gnr}

-(v)

Given $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{4})$ such that

\sigma\subset{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}\left(\widetilde{\sigma}\boxtimes i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\otimes\chi_{2})\right)

by (2.15) we have

L(s,\sigma,\mathrm{Ad})=L(s)L(s,\widetilde{\sigma}_{2},\operatorname{Sym}^{2}\otimes\omega_{\widetilde{\sigma}_{2}}^{-1})L(s,\chi_{1}\chi_{2}^{-1})L(s,\chi_{1}^{-1}\chi_{2}).

$\mathfrak{nongnr}$

Given a non-generic $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{4}),$ from (4.15), we recall that

$\sigma={\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}(\chi\circ\det\,\boxtimes\,\widetilde{\sigma})$

and by (2.15) we have

$L(s,\sigma,\mathrm{Ad})=L(s)L(s-1)L(s+1)L(s,\widetilde{\sigma},\mathrm{Ad}).$

We summarize the explicit computations above in Table 2.

5. Representations of $\mathrm{GSpin}_{6}$

We now list all the representations of $\mathrm{GSpin}_{6}(F)$ and then calculate their associated adjoint $L$ -function explicitly. Again, we do this explicit calculation by finding the $6\times 6$ nilpotent matrix in the complex dual group $\mathrm{GSO}_{6}(\mathbb{C})$ in each case that is associated with the parameter of the representation.

5.1. The Represenations

5.1.1. Classification of representations of $\mathrm{GSpin}_{6}$

Again, following [AC17], we have

1\longrightarrow{\mathrm{GSpin}}_{6}(F)\longrightarrow{\mathrm{GL}}_{1}(F)\times{\mathrm{GL}}_{4}(F)\longrightarrow F^{\times}\longrightarrow 1.

(5.1)

Recall that

{\mathrm{GSpin}}_{6}(F)\cong\left\{(g_{1},g_{2})\in{\mathrm{GL}}_{1}(F)\times{\mathrm{GL}}_{4}(F):g_{1}^{2}=\det g_{2}\right\},

(5.2)

{{}^{L}{\mathrm{GSpin}}_{6}}=\widehat{{\mathrm{GSpin}}_{6}}={\mathrm{GSO}}_{6}(\mathbb{C})\cong({\mathrm{GL}}_{1}(\mathbb{C})\times{\mathrm{GL}}_{4}(\mathbb{C}))/\{(z^{-2},z):z\in\mathbb{C}^{\times}\},

(5.3)

and

1\longrightarrow\mathbb{C}^{\times}\longrightarrow{\mathrm{GL}}_{1}(\mathbb{C})\times{\mathrm{GL}}_{4}(\mathbb{C})\overset{pr_{6}}{\longrightarrow}\widehat{{\mathrm{GSpin}}_{6}}\longrightarrow 1.

(5.4)

Just as the rank two case, here too we view $\mathrm{GSO}_{6}$ as the group similitude orthogonal $6\times 6$ matrices with respect to the analogous $6\times 6$ , anti-diagonal, matrix $J=J_{6}$ as in (4.5), and similarly define its Lie algebra with respect to $J$ .

5.1.2. Construction of the $L$ -packets of $\mathrm{GSpin}_{6}$ (recalled from [AC17])

Given $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{6})$ we have a lift $\widetilde{\sigma}\in\operatorname*{Irr}({\mathrm{GL}}_{1}\times{\mathrm{GL}}_{4})$ such that

\sigma\hookrightarrow{\operatorname*{Res}}_{{\mathrm{GSpin}}_{6}}^{\mathrm{GL}_{1}\times\mathrm{GL}_{4}}(\widetilde{\sigma}).

It follows from the LLC for $GL_{n}$ [HT01, Hen00, Sch13] that there is a unique $\widetilde{\varphi}_{\widetilde{\sigma}}\in\Phi({\mathrm{GL}}_{1}\times{\mathrm{GL}}_{4})$ corresponding to the representation $\widetilde{\sigma}.$ We now have a surjective, finite-to-one map

	$\displaystyle{\mathcal{L}}_{6}:{\operatorname*{Irr}}({\mathrm{GSpin}}_{6})$	$\displaystyle\longrightarrow$	$\displaystyle\Phi({\mathrm{GSpin}}_{6})$		(5.5)
	$\displaystyle\sigma$	$\displaystyle\longmapsto$	$\displaystyle pr_{6}\circ\widetilde{\varphi}_{\widetilde{\sigma}},$

which does not depend on the choice of the lifting $\widetilde{\sigma}.$ Then, for each $\varphi\in\Phi({\mathrm{GSpin}}_{6}),$ all inequivalent irreducible constituents of $\widetilde{\sigma}$ constitutes the $L$ -packet

\Pi_{\varphi}({\mathrm{GSpin}}_{6}):=\Pi_{\widetilde{\sigma}}({\mathrm{GSpin}}_{6})=\left\{\sigma:\sigma\hookrightarrow{\operatorname*{Res}}_{{\mathrm{GSpin}}_{6}}^{\mathrm{GL}_{1}\times\mathrm{GL}_{4}}(\widetilde{\sigma})\right\}\Big{/}\cong,

(5.6)

where $\widetilde{\sigma}$ is the unique member of $\Pi_{\widetilde{\varphi}}(\mathrm{GL}_{1}\times\mathrm{GL}_{4})$ and $\widetilde{\varphi}\in\Phi(\mathrm{GL}_{1}\times\mathrm{GL}_{4})$ is such that $pr_{6}\circ\widetilde{\varphi}=\varphi.$ We note that the construction does not depends on the choice of $\widetilde{\varphi}$ . Further details can be found in [AC17, Section 6.1].

Following [AC17, Section 6.3], given $\varphi\in\Phi(\mathrm{GSpin}_{6}),$ fix the lift

\widetilde{\varphi}=\widetilde{\eta}\otimes\widetilde{\varphi}_{0}\in\Phi({\mathrm{GL}}_{1}\times{\mathrm{GL}}_{4})

with $\widetilde{\varphi}_{0}\in\Phi({\mathrm{GL}}_{4})$ such that $\varphi=pr_{6}\circ\widetilde{\varphi}$ . Let

\widetilde{\sigma}=\widetilde{\eta}\boxtimes\widetilde{\sigma}_{0}\in\Pi_{\widetilde{\varphi}}({\mathrm{GL}}_{1}\times{\mathrm{GL}}_{4})

be the unique member such that $\{\widetilde{\sigma}_{0}\}=\Pi_{\widetilde{\varphi}_{0}}({\mathrm{GL}}_{4}).$

Recall that

I^{\mathrm{GSpin}_{6}}(\widetilde{\sigma}):=\left\{\widetilde{\chi}\in\Big{(}{\mathrm{GL}}_{1}(F)\times{\mathrm{GL}}_{4}(F)/{\mathrm{GSpin}}_{6}(F)\Big{)}^{D}:\widetilde{\sigma}\otimes\widetilde{\chi}\cong\widetilde{\sigma}\right\}.

Then we have

\Pi_{\varphi}({\mathrm{GSpin}}_{6})\,\overset{1-1}{\longleftrightarrow}\,I^{\mathrm{GSpin}_{6}}(\widetilde{\sigma}),

(5.7)

and by [AC17, Lemma 6.5 and Proposition 6.6] we have

I^{\mathrm{GSpin}_{6}}(\widetilde{\sigma})\cong\{\widetilde{\chi}\in I^{\mathrm{SL}_{4}}(\widetilde{\sigma}_{0}):\widetilde{\chi}^{2}=1_{F^{\times}}\}

(5.8)

and any $\widetilde{\chi}\in I^{\mathrm{GSpin}_{6}}(\widetilde{\sigma})$ is of the form

\widetilde{\chi}=(\widetilde{\chi}^{\prime})^{-2}\boxtimes\widetilde{\chi}^{\prime},

for some $\widetilde{\chi}^{\prime}\in(F^{\times})^{D}.$

5.2. Generic Representations of $\mathrm{GSpin}_{6}$

Thanks to the group structure (5.2) and the relation of generic representations in Section 3.1, in order to classify the generic representations of $\mathrm{GSpin}_{6},$ it suffices to classify the generic representations of $GL_{4}$ .

Here are two key facts from the $\mathrm{GL}$ theory.

•

Recall from [Zel80, Theorem 9.7] and [Kud94, Theorem 2.3.1] that a generic representation of $\mathrm{GL}_{4}$ is of the form

$i_{M_{\flat}}^{GL_{4}}(\sigma_{\flat})$

where $M_{\flat}$ runs through any $F$ -Levi subgroup of $\mathrm{GL}_{4}$ (including $\mathrm{GL}_{4}$ itself) and $\sigma_{\flat}$ is any essentially square-integrable representation of $M_{\flat}.$
•

For their $L$ -parameters, we note from [Kud94, §5.2] that the generic representations of $GL_{4}$ have Langlands parameters (i.e., 4-dimensional Weil-Deligne representations $(\rho,N)$ ) of the form

$(\rho_{1}\otimes sp(r_{1}))\otimes..\otimes(\rho_{t}\otimes sp(r_{t}))$

with $t\leq 4,$ where $\rho_{i}$ ’s are irreducible and no two segments are linked.

5.2.1. Irreducible Parameters

Let $\varphi\in\Phi(\mathrm{GSpin}_{6})$ be irreducible. Then $\widetilde{\varphi}$ and $\widetilde{\varphi}_{0}$ are also irreducible. By Section 3.1, we have the following.

Proposition 5.1.

Let $\varphi\in\Phi(\mathrm{GSpin}_{6})$ be irreducible. Every member in $\Pi_{\varphi}(\mathrm{GSpin}_{6})$ is supercuspidal and generic.

To see the internal structure of $\Pi_{\varphi}(\mathrm{GSpin}_{6}),$ we need, by (5.7), to know the detailed structure of $I^{\mathrm{GSpin}_{6}}(\widetilde{\sigma})$ as follows.

\mathfrak{gnr}

-(a)

Given $\sigma\in\operatorname*{Irr}_{\rm sc}(\mathrm{GSpin}_{6}),$ we have

\widetilde{\sigma}=\widetilde{\sigma}_{0}\boxtimes\widetilde{\eta}\in{\operatorname*{Irr}}_{\rm sc}(\mathrm{GL}_{4}\times\mathrm{GL}_{1}).

(5.9)

From [AC17, Proposition 2.1], we recall the identification:

\Delta^{\vee}=\left\{\beta^{\vee}_{1}=f^{*}_{2}-f^{*}_{3},\beta^{\vee}_{2}=f^{*}_{1}-f^{*}_{2},\beta^{\vee}_{3}=f^{*}_{3}-f^{*}_{4}\right\}.

(5.10)

using the notation $f_{ij}$ and $f^{*}_{ij},$ $1\leq i,j\leq 4,$ for the usual $\mathbb{Z}$ -basis of characters and cocharacters of $\mathrm{GL}_{4}$ . Also, $\{\beta_{1},\beta_{2},\beta_{3}\}$ are the simple roots of $\mathrm{GSpin}_{6}$ .

We have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(0_{4\times 4},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=0_{6\times 6}.

5.2.2. Reducible Parameters

When $\widetilde{\varphi}_{0}$ is not irreducible, we have proper parabolic inductions. An exhaustive list of $F$ -Levi subgroups $\mathbf{M}$ of $\mathrm{GSpin}_{6}$ (up to isomorphism) is as follows.

•

$\mathbf{M}\cong\mathrm{GL}_{1}\times GL_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}=\widetilde{\mathbb{M}}\cap\mathrm{GSpin}_{6}$ , where $\widetilde{\mathbb{M}}=(\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}.$
•

$\mathbf{M}\cong\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}=\widetilde{\mathbb{M}}\cap\mathrm{GSpin}_{6}$ , where $\widetilde{\mathbb{M}}=(\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}.$
•

$\mathbf{M}\cong\mathrm{GL}_{3}\times\mathrm{GL}_{1}=\widetilde{\mathbb{M}}\cap\mathrm{GSpin}_{6}$ , where $\widetilde{\mathbb{M}}=(\mathrm{GL}_{3}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}.$ (Note: The factor $\mathrm{GL}_{1}$ of $\mathbf{M}$ is $\mathrm{GSpin}_{0}$ by convention.)
•

$\mathbf{M}\cong\mathrm{GL}_{1}\times\mathrm{GSpin}_{4}=\widetilde{\mathbb{M}}\cap\mathrm{GSpin}_{6}$ , where $\widetilde{\mathbb{M}}=(\mathrm{GL}_{2}\times\mathrm{GL}_{2})\times\mathrm{GL}_{1}.$
•

$\mathbf{M}\cong\mathrm{GSpin}_{6}=\widetilde{\mathbb{M}}\cap\mathrm{GSpin}_{6},$ where $\widetilde{\mathbb{M}}=\mathrm{GL}_{4}\times\mathrm{GL}_{1}.$

(Note that $\mathbf{M}\cong\mathrm{GL}_{2}\times\mathrm{GL}_{2}$ does not occur on this list.) We now consider each case and, by abuse of notation, conflate algebraic groups and their $F$ -points.

\mathfrak{gnr}

-(I)

$\mathbf{M}\cong\mathrm{GL}_{1}\times GL_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}$ .

Given $\chi_{i}\in(F^{\times})^{D}$ we consider

i_{M}^{\mathrm{GSpin}_{6}}(\chi_{1}\boxtimes\chi_{2}\boxtimes\chi_{3}\boxtimes\chi_{4}).

(5.11)

Write $\chi_{1}\boxtimes\chi_{2}\boxtimes\chi_{3}\boxtimes\chi_{4}=(\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}\boxtimes\widetilde{\eta})|_{M}$ with $\widetilde{\chi}_{i},\widetilde{\eta}\in(F^{\times})^{D}$ so that

\widetilde{\chi}_{1}\widetilde{\chi}_{2}\widetilde{\chi}_{3}\widetilde{\chi}_{4}=\widetilde{\eta}^{2}.

Then we have the following relations

\chi_{1}=\widetilde{\chi}_{1},~{}\chi_{2}=\widetilde{\chi}_{2},~{}\chi_{3}=\widetilde{\chi}_{3},~{}\chi_{4}=\widetilde{\eta}^{2}(\widetilde{\chi}_{2}\widetilde{\chi}_{3}\widetilde{\chi}_{4})^{-1}.

(5.12)

By Section 3.1, we know that the representation (5.11) is generic if and only if its lift

i_{\widetilde{M}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}\boxtimes\widetilde{\eta})

(5.13)

is generic if and only if

i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}}(\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4})

(5.14)

is generic. By the classification of the generic representations of $\mathrm{GL}_{n}$ ([Zel80, Theorem 9.7] and [Kud94, Theorem 2.3.1]), this amounts to (5.14) being irreducible. By [Kud94, Theorem 2.1.1] and [BZ77, Zel80], the necessary and sufficient condition for this to occur is that there is no pair $i,j$ with $i\neq j$ such that

\widetilde{\chi}_{i}=\nu\widetilde{\chi}_{j}.

We have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(0_{4\times 4},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=0_{6\times 6}

\mathfrak{gnr}

-(II)

$\mathbf{M}\cong\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}$ .

Given $\sigma_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{2})$ and $\chi_{1},\chi_{2}\in(F^{\times})^{D}$ , we consider

i_{M}^{\mathrm{GSpin}_{6}}(\sigma_{0}\boxtimes\chi_{1}\boxtimes\chi_{2}).

(5.15)

Write $\sigma_{0}\boxtimes\chi_{1}\boxtimes\chi_{2}=(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\eta})|_{M}$ with $\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{2}),\widetilde{\chi}_{i},\widetilde{\eta}\in(F^{\times})^{D}$ .

Given $(g,h_{1},h_{2},h_{3})\in\widetilde{M}$ with $\det(gh_{1}h_{2})=h_{3}^{2},$

•

if we set $(g,h_{1},h_{3})\in M$ , we have

$\displaystyle\widetilde{\sigma}_{0}(g)\widetilde{\chi}_{1}(h_{1})\widetilde{\chi}_{2}(h_{2})\widetilde{\eta}(h_{3})$	$\displaystyle=$	$\displaystyle\widetilde{\sigma}_{0}(g)\widetilde{\chi}_{1}(h_{1})\widetilde{\chi}_{2}(\det g^{-1}h_{1}^{-1}h_{3}^{2})\widetilde{\eta}(h_{3})$
	$\displaystyle=$	$\displaystyle(\widetilde{\sigma}_{0}\widetilde{\chi}^{-1}_{2}\circ\det)(g)(\widetilde{\chi}_{1}\widetilde{\chi}_{2}^{-1})(h_{1})(\widetilde{\chi}_{2}^{2}\widetilde{\eta})(h_{3})$
	$\displaystyle=$	$\displaystyle\sigma(g)\chi_{1}(h_{1})\chi_{2}(h_{3}).$

Then we have

\widetilde{\sigma}_{0}=\sigma_{0}\widetilde{\chi}_{2},~{}~{}\widetilde{\chi}_{1}=\chi_{1}\widetilde{\chi}_{2},~{}~{}\widetilde{\eta}=\chi_{2}\widetilde{\chi}_{2}^{-2}.

•

If we set $(g,h_{2},h_{3})\in M$ , we have

$\displaystyle\widetilde{\sigma}_{0}(g)\widetilde{\chi}_{1}(h_{1})\widetilde{\chi}_{2}(h_{2})\widetilde{\eta}(h_{3})$	$\displaystyle=$	$\displaystyle\widetilde{\sigma}_{0}(g)\widetilde{\chi}_{1}(\det g^{-1}h_{2}^{-1}h_{3}^{2})\widetilde{\chi}_{2}(h_{2})\widetilde{\eta}(h_{3})$
	$\displaystyle=$	$\displaystyle(\widetilde{\sigma}_{0}\widetilde{\chi}^{-1}_{1}\circ\det)(g)(\widetilde{\chi}_{2}\widetilde{\chi}_{1}^{-1})(h_{2})(\widetilde{\chi}_{1}^{2}\widetilde{\eta})(h_{3})$
	$\displaystyle=$	$\displaystyle\sigma(g)\chi_{1}(h_{2})\chi_{2}(h_{3}).$

Then we have

\widetilde{\sigma}_{0}=\sigma_{0}\widetilde{\chi}_{1},~{}~{}\widetilde{\chi}_{2}=\chi_{2}\widetilde{\chi}_{1},~{}~{}\widetilde{\eta}=\chi_{1}\widetilde{\chi}_{1}^{-2}.

(5.16)

As before, the representation (5.15) is generic if and only if its lift

i_{\widetilde{M}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\eta})

(5.17)

is generic if and only if

i_{\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2})

(5.18)

is generic. Again by the classification of the generic representations of $\mathrm{GL}_{n}$ this amounts to (5.18) being irreducible. Hence, we must have

\widetilde{\chi}_{1}\neq\nu^{\pm 1}\widetilde{\chi}_{2}.

In other words, given $(g,h_{1},h_{2},h_{3})\in\widetilde{M}$ with $\det(gh_{1}h_{2})=h_{3}^{2},$

•

if we set $(g,h_{1},h_{3})\in M,$ then

$\chi_{1}\neq\nu^{\pm 1};$
•

if we set $(g,h_{2},h_{3})\in M,$ then

$\chi_{2}\neq\nu^{\pm 1}.$

We have the following two cases. If $\sigma_{0}$ is supercuspidal, then

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(0_{4\times 4},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=0_{6\times 6}.

If $\sigma_{0}$ is non-supercuspidal, then

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix}.

\mathfrak{gnr}

-(III)

$\mathbf{M}\cong\mathrm{GL}_{3}\times\mathrm{GL}_{1}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{3}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}.$

Given $\sigma_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{3})$ and $\chi\in(F^{\times})^{D}$ , we consider

i_{M}^{\mathrm{GSpin}_{6}}(\sigma_{0}\boxtimes\chi).

(5.19)

Write $\sigma_{0}\boxtimes\chi=(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}\boxtimes\widetilde{\eta})|_{M}$ with $\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{3}),\widetilde{\chi},\widetilde{\eta}\in(F^{\times})^{D}.$

Given $(g,h_{1},h_{2})\in\widetilde{M}$ with $\det(gh_{1})=h_{2}^{2}$ , if we set $(g,h_{2})\in M$ , then we have

$\displaystyle\widetilde{\sigma}_{0}(g)\widetilde{\chi}(h_{1})\widetilde{\eta}(h_{2})$	$\displaystyle=$	$\displaystyle\widetilde{\sigma}_{0}(g)\widetilde{\chi}(\det g^{-1}h_{2}^{2})\widetilde{\eta}(h_{2})$
	$\displaystyle=$	$\displaystyle(\widetilde{\sigma}_{0}\widetilde{\chi}^{-1}\circ\det)(g)(\widetilde{\chi}^{2}\widetilde{\eta})(h_{2})$
	$\displaystyle=$	$\displaystyle\sigma(g)\chi(h_{2}).$

Then, we have

\widetilde{\sigma}_{0}=\sigma_{0}\widetilde{\chi}\quad\mbox{ and }\quad\widetilde{\eta}=\chi_{2}\widetilde{\chi}^{-2}.

As before, (5.19) is generic if and only if its lift

i_{\widetilde{M}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}\boxtimes\widetilde{\eta})

(5.21)

is generic if and only if

i_{\mathrm{GL}_{3}\times\mathrm{GL}_{1}}^{GL_{4}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi})

(5.22)

is generic. This amounts to (5.22) being irreducible as before, which is always true since $\widetilde{\sigma}_{0}$ is an essentially square integrable representation of $\mathrm{GL}_{3}$ . Note that by the classification of essentially square-integrable representations of $\mathrm{GL}_{3}$ ([Kud94, Proposition 1.1.2]), $\widetilde{\sigma}_{0}$ must be either supercuspidal or the unique subrepresentation of

i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{3}}\left(\nu\chi\boxtimes\chi\boxtimes\nu^{-1}\chi\right)

(5.23)

with any $\chi\in(F^{\times})^{D}.$

We have the following two cases. If $\sigma_{0}$ is supercuspidal, then

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(0_{4\times 4},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=0_{6\times 6}.

If $\sigma_{0}$ is the non-supercuspidal, unique, subrepresentation of (5.23), then

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&-1\\ 0&0&0&0&0&0\end{bmatrix}.

\mathfrak{gnr}

-(IV)

$\mathbf{M}\cong\mathrm{GL}_{1}\times\mathrm{GSpin}_{4}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{2}\times\mathrm{GL}_{2})\times\mathrm{GL}_{1}.$

Given $\sigma_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GSpin}_{4})$ and $\chi\in(F^{\times})^{D}$ we consider

i_{M}^{\mathrm{GSpin}_{6}}(\chi\boxtimes\sigma_{0}).

(5.24)

Write $\chi\boxtimes\sigma_{0}\subset(\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2}\boxtimes\widetilde{\eta})|_{M}$ with $\widetilde{\sigma}_{i}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{2}),\widetilde{\eta}\in(F^{\times})^{D}.$

As before, (5.24) is generic if and only if its lift

i_{\widetilde{M}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2}\boxtimes\widetilde{\eta})

(5.25)

is generic if and only if

i_{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}^{GL_{4}}(\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2})

(5.26)

is generic. This amounts to (5.26) being irreducible. Thus, we must have

\widetilde{\sigma}_{1}\neq\nu^{\pm 1}\widetilde{\sigma}_{2}.

We have several cases to consider. If $\sigma_{0}$ is supercuspidal (so are $\widetilde{\sigma}_{1}$ and $\widetilde{\sigma}_{2}$ ), then

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(0_{4\times 4}0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=0_{6\times 6}.

If $\sigma_{0}$ is non-supercuspidal, then for supercuspidal $\widetilde{\sigma}_{1}$ and non-supercuspidal $\widetilde{\sigma}_{2}$ we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&0&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix};

for non-supercuspidal $\widetilde{\sigma}_{1}$ and supercuspidal $\widetilde{\sigma}_{2}$ we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix};

and for non-supercuspidal $\widetilde{\sigma}_{1}$ and $\widetilde{\sigma}_{2}$ we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&0&0&0&0&0\\ 0&0&1&1&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix}.

\mathfrak{gnr}

-(V)

$\mathbf{M}\cong\mathrm{GSpin}_{6}$ and $\widetilde{\mathbb{M}}=\mathrm{GL}_{4}\times\mathrm{GL}_{1}.$

Given $\sigma\in\operatorname*{Irr}_{\rm esq}(\mathrm{GSpin}_{6})\setminus\operatorname*{Irr}_{\rm sc}(\mathrm{GSpin}_{6})$ , we consider

\sigma\subset(\widetilde{\sigma}\boxtimes\widetilde{\eta})|_{M}

with $\widetilde{\sigma}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{4})\setminus\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{4}),\widetilde{\eta}\in(F^{\times})^{D}.$ Here, we note that $\varphi\in\Phi(\mathrm{GSpin}_{6})$ is not irreducible and neither $\widetilde{\sigma}$ nor $\sigma$ is supercuspidal. It is clear that $\sigma$ is generic as $\widetilde{\sigma}\boxtimes\widetilde{\eta}$ is. By the classification of essentially square-integrable representations of $\mathrm{GL}_{4}$ ([Kud94, Proposition 1.1.2]), $\widetilde{\sigma}$ must be the unique subrepresentation of either

i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{4}}\left(\nu^{3/2}\widetilde{\chi}\boxtimes\nu^{1/2}\widetilde{\chi}\boxtimes\nu^{-1/2}\widetilde{\chi}\boxtimes\nu^{-3/2}\widetilde{\chi}\right)

(5.27)

with any $\widetilde{\chi}\in(F^{\times})^{D}$ (i.e., $\widetilde{\sigma}=\mathsf{St}_{\mathrm{GL}_{4}}\otimes\widetilde{\chi}$ ), or of

i_{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}^{\mathrm{GL}_{4}}\left(\nu^{1/2}\widetilde{\tau}\boxtimes\nu^{-1/2}\widetilde{\tau}\right)

(5.28)

with any $\widetilde{\tau}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2})$ .

Now, for (5.27) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&1&0&0&0&0\\ 0&0&1&1&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&-1\\ 0&0&0&0&0&0\end{bmatrix};

and for (5.28) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&0&1&1&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&-1\\ 0&0&0&0&0&-1\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix}.

(We note, cf. [Tat79, (4.1.5)], that $N_{{\mathrm{GL}}_{4}(\mathbb{C})}$ is of the form $O_{2\times 2}\otimes I_{2\times 2}+\begin{bmatrix}0&1\\ 0&0\end{bmatrix}\otimes I_{2\times 2}.$ )

5.3. Non-Generic Representaions of $\mathrm{GSpin}_{6}$

Using the transitivity of the parabolic induction and the classification of generic representations of $\mathrm{GL}_{n}$ , ([Zel80, Theorem 9.7] and [Kud94, Theorem 2.3.1]), the non-generic representations of $\mathrm{GSpin}_{6}$ are as follows.

\mathfrak{nongnr}

-(A)

Given $\chi_{i}\in(F^{\times})^{D}$ , by Section 3.1 and using (5.12), the representation (5.11) contains a non-generic constituent if and only if the same is true for

i_{\widetilde{M}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}\boxtimes\widetilde{\eta})

(5.29)

if and only if

i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}}(\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4})

(5.30)

contains a non-generic constituent. This amounts to (5.30) being reducible. As before, the necessary and sufficient condition for this to occur is that there is some pair $i,j$ with $i\neq j$ such that $\widetilde{\chi}_{i}=\nu\widetilde{\chi}_{j}$ .

By the Langlands classification and the description of constituents of the parabolic induction (see [Zel80, Theorem 7.1], [Rod82, Theorem 7.1], and [Kud94, Theorems 2.1.1 §5.1.1]), each constituent can be described as a Langlands quotient, denoted by $Q(...)$ , as follows.

The first case is when there is only one pair, say $\widetilde{\chi}_{1}=\nu^{1/2}\widetilde{\chi}$ and $\widetilde{\chi}_{2}=\nu^{-1/2}\widetilde{\chi}$ for some $\widetilde{\chi}\in(F^{\times})^{D}$ while $\widetilde{\chi}_{3}\neq\nu^{\pm 1}\widetilde{\chi}_{j}$ for $j\neq 3$ and $\widetilde{\chi}_{4}\neq\nu^{\pm 1}\widetilde{\chi}_{j}$ for $j\neq 4.$ Then we have the non-generic constituent

Q\left([\nu^{1/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}],[\widetilde{\chi}_{3}],[\widetilde{\chi}_{4}]\right),

(5.31)

which is the Langlands quotient of

i_{\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}}\left(Q\left([\nu^{1/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}]\right)\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}\right)=i_{\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}}\left(\left(\widetilde{\chi}\circ\det\right)\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}\right).

We have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(0_{4\times 4},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=0_{6\times 6}.

Note that the other constituent of this induced representation, which is generic, is

	$\displaystyle Q\left([\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi}],[\widetilde{\chi}_{3}],[\widetilde{\chi}_{4}]\right)$	$\displaystyle=$	$\displaystyle i_{\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}}\left(Q\left([\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi}]\right)\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}\right)$
		$\displaystyle=$	$\displaystyle i_{\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}}\left((\mathsf{St}\otimes\widetilde{\chi})\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}\right).$

The next case is when there are two pairs, say $\widetilde{\chi}_{1}=\nu\widetilde{\chi}$ , $\widetilde{\chi}_{2}=\widetilde{\chi}$ , and $\widetilde{\chi}_{3}=\nu^{-1}\widetilde{\chi}$ for some $\widetilde{\chi}\in(F^{\times})^{D}$ and $\widetilde{\chi}_{4}\neq\nu^{\pm 1}\widetilde{\chi}_{i}$ for $i=1,2,3$ . Then we have the following three non-generic constituents:

$\displaystyle Q\left([\nu\widetilde{\chi}],[\widetilde{\chi}],[\nu^{-1}\widetilde{\chi}],[\widetilde{\chi}_{4}]\right)$	$\displaystyle=$	$\displaystyle i_{\mathrm{GL}_{3}\times\mathrm{GL}_{1}}^{GL_{4}}((\widetilde{\chi}\circ\det)\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4});$	(5.32)
$\displaystyle Q\left([\widetilde{\chi},\nu\widetilde{\chi}],[\nu^{-1}\widetilde{\chi}],[\widetilde{\chi}_{4}]\right);$			(5.33)
$\displaystyle Q\left([\nu\widetilde{\chi}],[\widetilde{\chi},\nu^{-1}\widetilde{\chi}],[\widetilde{\chi}_{4}]\right).$			(5.34)

For (5.32) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(0_{4\times 4},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=0_{6\times 6},

for (5.33) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix},

and for (5.34) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&0&0&0\\ 0&0&1&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&1&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&-1\\ 0&0&0&0&0&0\end{bmatrix}.

Finally, in the case where we have three pairs we are in the situation of (5.27). Then we have the following seven non-generic constituents:

$\displaystyle Q\left([\nu^{3/2}\widetilde{\chi}],[\nu^{1/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}]\right)$	$\displaystyle=\widetilde{\chi}\circ\det;$	(5.35)
$\displaystyle Q\left([\nu^{1/2}\widetilde{\chi},\nu^{3/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}]\right);$		(5.36)
$\displaystyle Q\left([\nu^{3/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}]\right);$		(5.37)
$\displaystyle Q\left([\nu^{3/2}\widetilde{\chi}],[\nu^{1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi},\nu^{-1/2}\widetilde{\chi}]\right);$		(5.38)
$\displaystyle Q\left([\nu^{1/2}\widetilde{\chi},\nu^{3/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi},\nu^{-1/2}\widetilde{\chi}]\right);$		(5.39)
$\displaystyle Q\left([\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi},\nu^{3/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}]\right);$		(5.40)
$\displaystyle Q\left([\nu^{3/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi},\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi}]\right).$		(5.41)

For (5.35) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(0_{4\times 4},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=0_{6\times 6},

for (5.36) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&0&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix},

for (5.37) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&0&0&0\\ 0&0&1&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&1&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&-1\\ 0&0&0&0&0&0\end{bmatrix},

for (5.38) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&0&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix},

for (5.39) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&0&0&0&0&0\\ 0&0&1&1&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix},

for (5.40) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&-1\\ 0&0&0&0&0&0\end{bmatrix},

and for (5.41) we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&1&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&-1\\ 0&0&0&0&0&0\end{bmatrix}.

\mathfrak{nongnr}

-(B)

$\mathbf{M}\cong\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}.$

Given $\sigma_{0}\in\operatorname*{Irr}(\mathrm{GL}_{2})$ and $\chi_{1},\chi_{2}\in(F^{\times})^{D}$ , we consider

i_{M}^{\mathrm{GSpin}_{6}}(\sigma_{0}\boxtimes\chi_{1}\boxtimes\chi_{2}).

(5.42)

Write

\sigma_{0}\boxtimes\chi_{1}\boxtimes\chi_{2}=(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\eta})|_{M}

with $\widetilde{\sigma}_{0}\in\operatorname*{Irr}(\mathrm{GL}_{2})$ and $\widetilde{\chi}_{i},\widetilde{\eta}\in(F^{\times})^{D}$ . By (5.16), it follows that (5.42) contains a non-generic constituent if and only if its lift

i_{\widetilde{M}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\eta})

(5.43)

contains a non-generic constituent if and only if

i_{\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2})

(5.44)

does. Recalling $\mathfrak{nongnr}$ -(A), it is sufficient to consider the case of $\widetilde{\sigma}_{0}\in\operatorname*{Irr}(\mathrm{GL}_{2})$ , $\widetilde{\chi}_{1}=\nu^{1/2}\widetilde{\chi}$ , and $\widetilde{\chi}_{2}=\nu^{-1/2}\widetilde{\chi}$ for $\widetilde{\chi}\in(F^{\times})^{D}$ , where the segment $\Delta_{\widetilde{\sigma}_{0}}$ of $\widetilde{\sigma}_{0}$ does not precede either $\widetilde{\chi}_{1}$ or $\widetilde{\chi}_{2}$ . We then have the following sole non-generic constituent:

Q([\Delta_{\widetilde{\sigma}_{0}}],[\nu^{1/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}]).

(5.45)

We have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(0_{4\times 4},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=0_{6\times 6}.

$\mathfrak{nongnr}$ -(C)

$\mathbf{M}\cong\mathrm{GL}_{3}\times\mathrm{GL}_{1}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{3}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}.$

Given a non-generic $\sigma_{0}\in\operatorname*{Irr}(\mathrm{GL}_{3})$ and any $\chi\in(F^{\times})^{D}$ , we consider

$i_{M}^{\mathrm{GSpin}_{6}}(\sigma_{0}\boxtimes\chi).$ (5.46)

Write

$\sigma_{0}\boxtimes\chi=(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}\boxtimes\widetilde{\eta})|_{M}$

with non-generic $\widetilde{\sigma}_{0}\in\operatorname*{Irr}(\mathrm{GL}_{3})$ and $\widetilde{\chi},\widetilde{\eta}\in(F^{\times})^{D}.$ As in (III) we have

$\widetilde{\sigma}_{0}=\sigma_{0}\widetilde{\chi},\quad\mbox{ and }\quad\widetilde{\eta}=\chi_{2}\widetilde{\chi}^{-2}.$

As before, (5.46) contains a non-generic constituent if and only if its lift

$i_{\widetilde{M}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}\boxtimes\widetilde{\eta})$ (5.47)

also contains one if and only if

$i_{\mathrm{GL}_{3}\times\mathrm{GL}_{1}}^{GL_{4}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi})$ (5.48)

does. To have a non-generic $\widetilde{\sigma}_{0}$ of $\mathrm{GL}_{3}(F)$ , the irreducible representation $\widetilde{\sigma}_{0}$ must be some constituent in a reducible induction. This case has been covered in $\mathfrak{nongnr}$ -(A) and (B) above.

\mathfrak{nongnr}

-(D)

$\mathbf{M}\cong\mathrm{GL}_{1}\times\mathrm{GSpin}_{4}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{2}\times\mathrm{GL}_{2})\times\mathrm{GL}_{1}.$

Given a non-generic $\sigma_{0}\in\operatorname*{Irr}(\mathrm{GSpin}_{4})$ , by Section 4.3, we know that it must be of the form

{\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}((\chi\circ\det)\boxtimes\widetilde{\sigma})

for $\widetilde{\sigma}\in\operatorname*{Irr}(\mathrm{GL}_{2}).$ For $\eta\in(F^{\times})^{D},$ the induced representation

i_{M}^{\mathrm{GSpin}_{6}}((\chi\circ\det)\boxtimes\widetilde{\sigma}\boxtimes\eta)

(5.49)

contains a non-generic constituent if and only if so does

i_{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}^{\mathrm{GL}_{4}}((\chi\circ\det)\boxtimes\widetilde{\sigma}),

which is always the case. Therefore, if $\widetilde{\sigma}$ is supercuspidal, then

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(0_{4\times 4},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=0_{6\times 6}.

If $\widetilde{\sigma}$ is non-supercuspidal, then it suffices to consider the case $\widetilde{\sigma}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\eta$ with $\eta\in(F^{\times})^{D}$ since the other case has been covered in $\mathfrak{nongnr}$ -(A). Thus, we have

N_{{\mathrm{GL}}_{4}(\mathbb{C})\times{\mathrm{GL}}_{1}(\mathbb{C})}=\left(\begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix},0\right)\overset{\eqref{indentitygspin6}}{\Longleftrightarrow}N_{{\mathrm{GSO}}_{6}(\mathbb{C})}=\begin{bmatrix}0&0&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{bmatrix}.

\mathfrak{nongnr}

-(E)

$\mathbf{M}\cong\mathrm{GSpin}_{6}$ and $\widetilde{\mathbb{M}}=\mathrm{GL}_{4}\times\mathrm{GL}_{1}.$

Given a non-generic $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{6}),$ it must be of the form

{\operatorname*{Res}}^{\mathrm{GL}_{4}\times\mathrm{GL}_{1}}_{\mathrm{GSpin}_{6}}\left(\widetilde{\chi}\circ\det\boxtimes\widetilde{\eta}\right)=\chi\circ\det,

(5.50)

for some $\widetilde{\chi},\widetilde{\eta}\in(F^{\times})^{D}.$ This is the case $Q([\nu^{3/2}\widetilde{\chi}],[\nu^{1/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}])$ in $\mathfrak{nongnr}$ -(A).

5.4. Computation of the Adjoint L-function for $\mathrm{GSpin}_{6}$

We now give explicit expressions for the adjoint $L$ -function of each of the representations of $\mathrm{GSpin}_{6}(F)$ . Recall that if we have a parameter $(\phi,N)$ with $N$ a nilpotent matrix on the vector space $V$ , then its adjoint $L$ -function is

L(s,\phi,\mathrm{Ad})=\det\left(1-q^{-s}\mathrm{Ad}(\phi)|V_{N}^{I}\right)^{-1},

where $V_{N}=\ker(N)$ , $V^{I}$ the vectors fixed by the inertia group, and $V_{N}^{I}=V^{I}\cap V_{N}$ . Below for the cases where $N$ is non-zero, we write $\ker(\mathrm{Ad}(N))$ and we use $L_{\alpha}$ to denote the root group associated with the root $\alpha$ .

We now consider each case. Using (2.14) and Sections 5.2, and 5.3, we have the following.

$\mathfrak{gnr}$ -(a)

Given $\sigma\in\operatorname*{Irr}_{\rm sc}(\mathrm{GSpin}_{6})$ , we have $\widetilde{\sigma}=\widetilde{\sigma}_{0}\boxtimes\widetilde{\eta}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{4}\times\mathrm{GL}_{1}).$ Then

$L(s,1_{F^{\times}})L(s,\sigma,\mathrm{Ad})=L(s,\widetilde{\sigma}_{0},\mathrm{Ad}_{\widehat{GL}_{4}})$

or

$L(s,\sigma,\mathrm{Ad})=L(s,\widetilde{\sigma}_{0},\mathrm{Ad}).$

\mathfrak{gnr}

-(I)

Given $\mathbf{M}\cong\mathrm{GL}_{1}\times GL_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}$ , we recall

i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}}(\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4})

must be irreducible. Thus, given $\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{6})$ such that

\sigma=i_{M}^{\mathrm{GSpin}_{6}}(\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}),

we have

L(s,\sigma,\mathrm{Ad})=L(s)^{3}\prod_{i\neq j}L(s,\widetilde{\chi}_{i}\widetilde{\chi}_{j}^{-1}).

\mathfrak{gnr}

-(II)

Given $\mathbf{M}\cong\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}$ , for $\sigma_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{2})$ and $\chi_{1},\chi_{2}\in(F^{\times})^{D}$ , we have an irreducible induced representation

\sigma=i_{M}^{\mathrm{GSpin}_{6}}(\sigma_{0}\boxtimes\chi_{1}\boxtimes\chi_{2})={\operatorname*{Res}}_{\mathrm{GSpin}_{6}}^{\mathrm{GL}_{4}\times\mathrm{GL}_{1}}\left(i_{\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\eta})\right),

for some $\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{2})$ , and $\widetilde{\chi}_{i},\widetilde{\eta}\in(F^{\times})^{D}$ . For supercuspidal $\widetilde{\sigma}_{0}$ we have

	$\displaystyle L(s,\sigma,\mathrm{Ad})$	$\displaystyle=$	$\displaystyle L(s)^{2}L(s,\widetilde{\sigma}_{0},\mathrm{Ad})L(s,\widetilde{\sigma}_{0}\times\widetilde{\chi}_{1}^{-1})L(s,\widetilde{\sigma}_{0}^{\vee}\times\widetilde{\chi}_{1})$
			$\displaystyle L(s,\widetilde{\sigma}_{0}\times\widetilde{\chi}_{2}^{-1})L(s,\widetilde{\sigma}_{0}^{\vee}\times\widetilde{\chi}_{2})L(s,\widetilde{\chi}_{1}\widetilde{\chi}_{2}^{-1})L(s,\widetilde{\chi}_{2}\widetilde{\chi}_{1}^{-1}).$

For non-supercuspidal $\widetilde{\sigma}_{0}\in\operatorname*{Irr}(\mathrm{GL}_{2})$ , i.e., $\sigma_{0}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\widetilde{\chi}$ for some $\widetilde{\chi}\in(F^{\times})^{D}$ , we have

\ker\left(\mathrm{ad}\begin{bmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}\right)=\left\langle\begin{bmatrix}a&0&0&0\\ 0&a&0&0\\ 0&0&b&0\\ 0&0&0&c\end{bmatrix},L_{f_{1}-f_{2}},L_{f_{1}-f_{3}},L_{f_{1}-f_{4}},L_{f_{3}-f_{2}},L_{f_{3}-f_{4}},L_{f_{4}-f_{2}},L_{f_{4}-f_{3}}\right\rangle.

(5.51)

It follows that

	$\displaystyle L(s,\sigma,\mathrm{Ad})$	$\displaystyle=$	$\displaystyle L(s)^{2}L(s+1)L(s+1,\widetilde{\chi}\widetilde{\chi}_{1}^{-1})L(s+1,\widetilde{\chi}\widetilde{\chi}_{2}^{-1})$
			$\displaystyle\cdot L(s,\widetilde{\chi}^{-1}\widetilde{\chi}_{1})L(s,\widetilde{\chi}^{-1}\widetilde{\chi}_{2})L(s,\widetilde{\chi}_{1}\widetilde{\chi}_{2}^{-1})L(s,\widetilde{\chi}_{2}\widetilde{\chi}_{1}^{-1}).$

\mathfrak{gnr}

-(III)

Given $\mathbf{M}\cong\mathrm{GL}_{3}\times\mathrm{GL}_{1}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{3}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}$ , for $\sigma_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{3})$ and $\chi\in(F^{\times})^{D}$ , we have an irreducible induced representation

\sigma=i_{M}^{\mathrm{GSpin}_{6}}(\sigma_{0}\boxtimes\chi)={\operatorname*{Res}}_{\mathrm{GSpin}_{6}}^{\mathrm{GL}_{4}\times\mathrm{GL}_{1}}\left(i_{\mathrm{GL}_{3}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{GL_{4}\times\mathrm{GL}_{1}}\left(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}\boxtimes\widetilde{\eta}\right)\right),

for $\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{3})$ and $\widetilde{\chi},\widetilde{\eta}\in(F^{\times})^{D}$ . If $\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{3})$ is supercuspidal, then we have

L(s,\sigma,\mathrm{Ad})=L(s)L(s,\widetilde{\sigma}_{0},\mathrm{Ad})L(s,\widetilde{\sigma}_{0}\times\widetilde{\chi}^{-1})L(s,\widetilde{\sigma}_{0}^{\vee}\times\widetilde{\chi}).

For non-supercuspidal $\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{3}),$ i.e., $\sigma_{0}=\mathsf{St}_{\mathrm{GL}_{3}}\otimes\widetilde{\chi}_{0}$ for some $\widetilde{\chi}_{0}\in(F^{\times})^{D},$ we have

\ker\left(\mathrm{ad}\begin{bmatrix}0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}\right)=\left\langle\begin{bmatrix}a&c&0&0\\ 0&a&c&0\\ 0&0&a&0\\ 0&0&0&b\end{bmatrix},L_{f_{1}-f_{3}},L_{f_{1}-f_{4}},L_{f_{4}-f_{3}}\right\rangle.

(5.52)

It follows that

L(s,\sigma,\mathrm{Ad})=L(s)L(s+1)L(s+2)L(s+1,\widetilde{\chi}\widetilde{\chi}_{0}^{-1})L(s+1,\widetilde{\chi}^{-1}\widetilde{\chi}_{0}).

\mathfrak{gnr}

-(IV)

Given $\mathbf{M}\cong\mathrm{GL}_{1}\times\mathrm{GSpin}_{4}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{2}\times\mathrm{GL}_{2})\times\mathrm{GL}_{1}$ , we have the representation (5.24)

\sigma=i_{M}^{\mathrm{GSpin}_{6}}(\chi\boxtimes\sigma_{0})

with $\sigma_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GSpin}_{4})$ , and $\chi\in(F^{\times})^{D}$ . We have the irreducible $i_{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}^{GL_{4}}(\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2})$ as in (5.26), where $\chi\boxtimes\sigma_{0}\subset(\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2}\boxtimes\widetilde{\eta})|_{M}$ with $\widetilde{\sigma}_{i}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{2}),\widetilde{\eta}\in(F^{\times})^{D}.$ Thus, if $\sigma_{0}$ is supercuspidal (and hence so are $\widetilde{\sigma}_{1}$ and $\widetilde{\sigma}_{2}$ ) we have

L(s,\sigma,\mathrm{Ad})=L(s)L(s,\widetilde{\sigma}_{1},\mathrm{Ad})L(s,\widetilde{\sigma}_{2},\mathrm{Ad})L(s,\widetilde{\sigma}_{1}\times\widetilde{\sigma}_{2}^{\vee})L(s,\widetilde{\sigma}_{1}^{\vee}\times\widetilde{\sigma}_{1}).

If $\sigma_{0}$ is non-supercuspidal, with $\widetilde{\sigma}_{1}$ supercuspidal and $\widetilde{\sigma}_{2}$ non-supercuspidal, i.e., $\widetilde{\sigma}_{2}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\widetilde{\chi}$ for some $\widetilde{\chi}\in(F^{\times})^{D}$ , we have

\ker\left(\mathrm{ad}\begin{bmatrix}0&0&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix}\right)=\left\langle\begin{bmatrix}a&0&0&0\\ 0&b&0&0\\ 0&0&c&0\\ 0&0&0&c\end{bmatrix},L_{f_{1}-f_{2}},L_{f_{1}-f_{4}},L_{f_{2}-f_{1}},L_{f_{2}-f_{4}},L_{f_{3}-f_{1}},L_{f_{3}-f_{2}},L_{f_{3}-f_{4}}\right\rangle,

(5.53)

and it then follows that

L(s,\sigma,\mathrm{Ad})=L(s)L(s+1)L(s,\widetilde{\sigma}_{1},\mathrm{Ad})L(s+\frac{1}{2},\widetilde{\sigma}_{1}^{\vee}\times\widetilde{\chi})L(s+\frac{1}{2},\widetilde{\sigma}_{1}\times\widetilde{\chi}^{-1}).

If $\sigma_{0}$ is non-supercuspidal, with $\widetilde{\sigma}_{1}$ non-supercuspidal and $\widetilde{\sigma}_{2}$ supercuspidal, i.e., $\widetilde{\sigma}_{1}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\widetilde{\chi}$ for some $\widetilde{\chi}\in(F^{\times})^{D}$ , then $\ker(\mathrm{ad}(N))$ is as in (5.51) and we have

L(s,\sigma,\mathrm{Ad})=L(s)L(s+1)L(s,\widetilde{\sigma}_{2},\mathrm{Ad})L(s+\frac{1}{2},\widetilde{\sigma}_{2}^{\vee}\times\widetilde{\chi})L(s+\frac{1}{2},\widetilde{\sigma}_{2}\times\widetilde{\chi}^{-1}).

If both $\widetilde{\sigma}_{1}$ and $\widetilde{\sigma}_{2}$ are non-supercuspidal, i.e., $\widetilde{\sigma}_{i}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\widetilde{\chi}_{i}$ with $\widetilde{\chi}_{1},\widetilde{\chi}_{2}\in(F^{\times})^{D}$ satisfying $\widetilde{\chi}_{1}\neq\widetilde{\chi}_{2}\nu^{\pm 1}$ , we have

\ker\left(\mathrm{ad}\begin{bmatrix}0&1&0&0\\ 0&0&0&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix}\right)=\left\langle\begin{bmatrix}a&0&c&0\\ 0&a&0&c\\ d&0&b&0\\ 0&d&0&b\end{bmatrix},L_{f_{1}-f_{2}},L_{f_{1}-f_{4}},L_{f_{3}-f_{2}},L_{f_{3}-f_{4}}\right\rangle,

(5.54)

and it follows that

L(s,\sigma,\mathrm{Ad})=L(s)L(s+1)^{2}L(s+1,\widetilde{\chi}_{1}\widetilde{\chi}_{2}^{-1})L(s+1,\widetilde{\chi}_{1}^{-1}\widetilde{\chi}_{2})L(s,\widetilde{\chi}_{1}^{-1}\widetilde{\chi}_{2})L(s,\widetilde{\chi}_{1}\widetilde{\chi}_{2}^{-1}).

\mathfrak{gnr}

-(V)

Given $\mathbf{M}\cong\mathrm{GL}_{1}\times\mathrm{GSpin}_{4}$ and $\widetilde{\mathbb{M}}=(\mathrm{GL}_{2}\times\mathrm{GL}_{2})\times\mathrm{GL}_{1}$ , we consider $\sigma\in\operatorname*{Irr}_{\rm esq}(\mathrm{GSpin}_{6})$ and $\widetilde{\sigma}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{4})$ and $\widetilde{\eta}\in(F^{\times})^{D}$ such that $\sigma\subset(\widetilde{\sigma}\boxtimes\widetilde{\eta})|_{M}.$ Then, $\widetilde{\sigma}$ must be either (5.27) or (5.28).

For (5.27) (i.e., $\widetilde{\sigma}=\mathsf{St}_{\mathrm{GL}_{4}}\otimes\widetilde{\chi}$ ), we have

\ker\left(\mathrm{ad}\begin{bmatrix}0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix}\right)=\left\langle\begin{bmatrix}a&b&c&0\\ 0&a&b&c\\ 0&0&a&b\\ 0&0&0&a\end{bmatrix},L_{f_{1}-f_{4}}\right\rangle,

(5.55)

and it follows that

L(s,\sigma,\mathrm{Ad})=L(s+3)L(s+2)L(s+1).

For (5.28) (i.e., $\widetilde{\tau}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2})$ ), we have

\ker\left(\mathrm{ad}\begin{bmatrix}0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}\right)=\left\langle\begin{bmatrix}a&c&0&0\\ d&b&0&0\\ 0&0&a&c\\ 0&0&d&b\end{bmatrix},L_{f_{1}-f_{3}},L_{f_{1}-f_{4}},L_{f_{2}-f_{3}},L_{f_{2}-f_{4}}\right\rangle,

(5.56)

and it follows that

L(s,\sigma,\mathrm{Ad})=L(s,\widetilde{\tau},\mathrm{Ad})L(s,\widetilde{\tau}\times\widetilde{\tau}^{\vee}).

\mathfrak{nongnr}

-(A)

For $Q([\nu^{1/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}],[\widetilde{\chi}_{3}],[\widetilde{\chi}_{4}])$ (5.31), we have

	$\displaystyle L(s,\sigma,\mathrm{Ad})$	$\displaystyle=$	$\displaystyle L(s)^{3}L(s+1)L(s-1)L(s,\widetilde{\chi}_{3}\widetilde{\chi}_{4}^{-1})L(s,\widetilde{\chi}_{3}^{-1}\widetilde{\chi}_{4})$
			$\displaystyle\prod_{i=3,4}\left(L(s+\frac{1}{2},\widetilde{\chi}\widetilde{\chi}_{i}^{-1})L(s-\frac{1}{2},\widetilde{\chi}^{-1}\widetilde{\chi}_{i})L(s-\frac{1}{2},\widetilde{\chi}\widetilde{\chi}_{i}^{-1})L(s+\frac{1}{2},\widetilde{\chi}^{-1}\widetilde{\chi}_{i})\right)$

For $Q\left([\nu\widetilde{\chi}],[\widetilde{\chi}],[\nu^{-1}\widetilde{\chi}],[\widetilde{\chi}_{4}]\right)$ (5.32), we have

L(s,\sigma,\mathrm{Ad})=L(s)^{3}L(s+1)^{2}L(s-1)^{2}L(s+2)L(s-2)\prod_{t=0,1,-1}\left(L(s+t,\widetilde{\chi}\widetilde{\chi}_{4}^{-1})L(s+t,\widetilde{\chi}^{-1}\widetilde{\chi}_{4})\right),

For $Q([\widetilde{\chi},\nu\widetilde{\chi}],[\nu^{-1}\widetilde{\chi}],[\widetilde{\chi}_{4}])$ (5.33), we have $\ker(\mathrm{ad}(N))$ as in (5.51) and

L(s,\sigma,\mathrm{Ad})=L(s)^{2}L(s-1)^{2}L(s-2)\prod\limits_{t=-1,0}L(s+t,\widetilde{\chi}\widetilde{\chi}_{4}^{-1})\prod\limits_{t=\pm 1}L(s+t,\widetilde{\chi}^{-1}\widetilde{\chi}_{4}).

For $Q\left([\nu\widetilde{\chi}],[\widetilde{\chi},\nu^{-1}\widetilde{\chi}],[\widetilde{\chi}_{4}]\right)$ (5.34), since

\ker\left(\mathrm{ad}\begin{bmatrix}0&0&0&0\\ 0&0&1&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}\right)=\left\langle\begin{bmatrix}a&0&0&0\\ 0&b&0&0\\ 0&0&b&0\\ 0&0&0&c\end{bmatrix},L_{f_{1}-f_{3}},L_{f_{1}-f_{4}},L_{f_{2}-f_{1}},L_{f_{2}-f_{3}},L_{f_{2}-f_{4}},L_{f_{4}-f_{1}},L_{f_{4}-f_{3}}\right\rangle,

(5.57)

we have

L(s,\sigma,\mathrm{Ad})=L(s)^{2}L(s+2)L(s-1)L(s+1)\prod\limits_{t=0,1}L(s+t,\widetilde{\chi}\widetilde{\chi}_{4}^{-1})\prod_{t=\pm 1}L(s+t,\widetilde{\chi}^{-1}\widetilde{\chi}_{4}).

For $Q([\nu^{3/2}\widetilde{\chi}],[\nu^{1/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}])$ (5.35), we have

L(s,\sigma,\mathrm{Ad})=L(s)^{3}L(s+1)^{3}L(s-1)^{3}L(s+2)^{2}L(s-2)^{2}L(s+3)L(s-3).

For $Q\left([\nu^{1/2}\widetilde{\chi},\nu^{3/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}]\right)$ (5.36), we have $\ker(\mathrm{ad}(N))$ is as in (5.51) and

L(s,\sigma,\mathrm{Ad})=L(s)^{2}L(s-1)^{2}L(s+1)^{2}L(s-2)L(s+2)L(s-3).

For $Q([\nu^{3/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}])$ (5.37), we have $\ker(\mathrm{ad}(N))$ is as in (5.57) and

L(s,\sigma,\mathrm{Ad})=L(s)^{2}L(s+1)^{2}L(s+2)L(s-1)^{2}L(s-3)L(s-2).

For $Q([\nu^{3/2}\widetilde{\chi}],[\nu^{1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi},\nu^{-1/2}\widetilde{\chi}])$ (5.38), we have $\ker(\mathrm{ad}(N))$ is as in (5.53) and

L(s,\sigma,\mathrm{Ad})=L(s)^{2}L(s+1)^{2}L(s-1)^{2}L(s-2)L(s+2)L(s-3).

For $Q([\nu^{1/2}\widetilde{\chi},\nu^{3/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi},\nu^{-1/2}\widetilde{\chi}])$ (5.39), we have $\ker(\mathrm{ad}(N))$ is as in (5.54) and

L(s,\sigma,\mathrm{Ad})=L(s)L(s-1)^{2}L(s+1)L(s+2)L(s-2)L(s-3).

For $Q([\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi},\nu^{3/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}])$ (5.40), we have $\ker(\mathrm{ad}(N))$ is as in (5.52) and

L(s,\sigma,\mathrm{Ad})=L(s)L(s-1)L(s-2)L(s+1)L(s-3).

Finally, for $Q\left([\nu^{3/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi},\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi}]\right)$ (5.41), since

\ker\left(\mathrm{ad}\begin{bmatrix}0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix}\right)=\left\langle\begin{bmatrix}a&0&0&0\\ 0&b&c&0\\ 0&0&b&c\\ 0&0&0&b\end{bmatrix},L_{f_{1}-f_{4}},L_{f_{2}-f_{1}},L_{f_{2}-f_{4}}\right\rangle,

(5.58)

we have

L(s,\sigma,\mathrm{Ad})=L(s)L(s+1)L(s-1)L(s-2)L(s-3).

\mathfrak{nongnr}

-(B)

For $Q([\Delta_{\widetilde{\sigma}_{0}}],[\nu^{1/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}])$ (5.45), with say $[\Delta_{\widetilde{\sigma}_{0}}]=i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\widetilde{\eta}_{1}\boxtimes\widetilde{\eta}_{2}),~{}\widetilde{\eta}_{1}\widetilde{\eta}_{2}^{-1}\neq\nu^{\pm 1}$ we have

	$\displaystyle L(s,\sigma,\mathrm{Ad})$	$\displaystyle=$	$\displaystyle L(s)^{3}L(s+1)L(s-1)L(s,\widetilde{\eta}_{1}\widetilde{\eta}_{2}^{-1})L(s,\widetilde{\eta}_{1}^{-1}\widetilde{\eta}_{2})$
			$\displaystyle\prod_{i=1,2}\left(L(s-\frac{1}{2},\widetilde{\eta}_{i}\widetilde{\chi}^{-1})L(s+\frac{1}{2},\widetilde{\eta}_{i}\widetilde{\chi}^{-1})L(s+\frac{1}{2},\widetilde{\eta}_{i}^{-1}\widetilde{\chi})L(s-\frac{1}{2},\widetilde{\eta}_{i}^{-1}\widetilde{\chi})\right).$

$\mathfrak{nongnr}$ -(C)

As mentioned before, all the possibilities in this case were covered in (A) and (B) above.

\mathfrak{nongnr}

-(D)

For (5.49) with $\widetilde{\sigma}$ supercuspidal, we have

	$\displaystyle L(s,\sigma,\mathrm{Ad})$	$\displaystyle=$	$\displaystyle L(s)^{2}L(s+1)L(s-1)L(s,\sigma,\mathrm{Ad})$
			$\displaystyle L(s-\frac{1}{2},\sigma\times\chi^{-1})L(s+\frac{1}{2},\sigma\times\chi^{-1})L(s-\frac{1}{2},\sigma^{\vee}\times\chi)L(s+\frac{1}{2},\sigma^{\vee}\times\chi),$

For (5.49) with non-supercuspidal $\widetilde{\sigma}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\eta,~{}\eta\in(F^{\times})^{D}$ we have $\ker(\mathrm{ad}(N))$ as in (5.53) and

\displaystyle L(s,\sigma,\mathrm{Ad})

\displaystyle=

\displaystyle L(s)^{2}L(s+1)^{2}L(s-1)L(s,\chi\eta^{-1})L(s+1,\chi\eta^{-1})L(s+1,\chi^{-1}\eta)L(s,\chi^{-1}\eta).

Recall that the remaining possibilities in this case were already covered in (A) above.

$\mathfrak{nongnr}$ -(E)

Finally, as mentioned before, all the possibilities in this case we also covered in (A).

6. Correction to [AC17]

We take this opportunity to correct the following errors in our earlier work [AC17]. They do not affect the main results in that paper.

6.1. Proposition 5.5 and 6.4

•

Change “1,2,4,8, if $p=2$ ” to “1,2,4,8,…, $2^{[F:\mathbb{Q}_{2}]+2},$ if $p=2.$ ” We have $2^{[F:\mathbb{Q}_{p}]+2}$ due to the fact that $\left|F^{\times}/(F^{\times})^{2}\right|=2^{[F:\mathbb{Q}_{2}]+2}$ .
•

For Proposition 5.5, using [GP92, Corollary 7.7], it follows that the case of $p=2$ is bounded by $|(\mathbb{Z}/2\mathbb{Z})^{4-1}|=8.$ Here $4$ is coming from $\widehat{\mathrm{GSpin}_{4}}=\mathrm{GSO}(4,\mathbb{C}).$
•

For Proposition 6.4, using [GP92, Corollary 7.7], it follows that the case of $p=2$ is bounded by $|(\mathbb{Z}/2\mathbb{Z})^{6-1}|=32.$ Here $6$ is coming from $\widehat{\mathrm{GSpin}_{6}}=\mathrm{GSO}(6,\mathbb{C}).$

6.2. Remark 5.11

•

The formula (5.13) should read as follows:

\Big{|}\,\Pi_{\varphi}\left(\mathrm{GSpin}_{4}\right)\Big{|}=\Big{|}\,\Pi_{\varphi}(\mathrm{GSpin}_{4}^{1,1})\Big{|}=4,\quad\quad\left|\,\Pi_{\varphi}\left(\mathrm{GSpin}_{4}^{2,1}\right)\right|=1.

(5.13)

Also, in the following sentence change “in which case the multiplicity is 2” to “in which case the multiplicity 2 could also occur”. We thank Hengfei Lu [Lu20] for bringing this error to our attention.

•

In addition, it is more accurate that we use ‘not irreducible’ rather than ‘reducible’ in this Remark since one may have indecomposable parameters. Alternatively, we may write $\widetilde{\varphi}_{i}|_{W_{F}}$ is reducible. Thus, at the beginning the Remark, change “When $\widetilde{\varphi}_{i}$ is reducible,” to “When $\widetilde{\varphi}_{i}$ is not irreducible,”.

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Table 1. Representations of

\mathrm{GSpin}_{4}(F)

\begin{array}[]{|c|l|l|c|}\hline\cr&\mbox{${\operatorname*{Res}}^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\mathrm{GSpin}_{4}}$ of}&\mbox{$L$-packet Structure}&\mbox{generic}\\ \hline\cr\hline\cr\mbox{(a)}&(\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2}),\quad\widetilde{\sigma}_{2}\cong\widetilde{\sigma}_{1}\widetilde{\eta},\widetilde{\sigma}_{i}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2})&\{1\},\mathbb{Z}/2\mathbb{Z},(\mathbb{Z}/2\mathbb{Z})^{2}&\bullet\\ \hline\cr\mbox{(b)}&(\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2}),\quad\widetilde{\sigma}_{2}\not\cong\widetilde{\sigma}_{1}\widetilde{\eta},\widetilde{\sigma}_{i}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2})&\{1\},\mathbb{Z}/2\mathbb{Z}&\bullet\\ \hline\cr\mbox{(i)}&({\mathsf{St}}_{\mathrm{GL}_{2}}\boxtimes{\mathsf{St}}_{\mathrm{GL}_{2}})={\mathsf{St}}_{\mathrm{GSpin}_{4}}\quad\mbox{(irreducible)}&\{1\}&\bullet\\ \hline\cr\mbox{(ii)}&(i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\otimes\chi_{2})\boxtimes{\mathsf{St}}_{\mathrm{GL}_{2}}\otimes\chi)\quad\mbox{(irreducible)}&\{1\}&\bullet\\ \hline\cr\mbox{(iii)}&(i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\otimes\chi_{2})\boxtimes i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{3}\otimes\chi_{4})),\chi_{1}\neq\nu^{\pm 1\chi_{2}},\chi_{3}\neq\nu^{\pm 1}\chi_{4}&\{1\},\mathbb{Z}/2\mathbb{Z}&\bullet\\ \hline\cr\mbox{(iv)}&(\widetilde{\sigma}\boxtimes{\mathsf{St}}_{\mathrm{GL}_{2}}\otimes\chi),\quad\widetilde{\sigma}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2})\quad\mbox{(irreducible)}&\{1\}&\bullet\\ \hline\cr\mbox{(v)}&(\widetilde{\sigma}\boxtimes i_{\mathrm{GL}_{1}\times\mathrm{GL}_{1}}^{\mathrm{GL}_{2}}(\chi_{1}\otimes\chi_{2})),\quad\widetilde{\sigma}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2})&\{1\},\mathbb{Z}/2\mathbb{Z}&\bullet\\ \hline\cr\mathfrak{nongnr}&(\chi\circ\det\boxtimes\widetilde{\sigma}),\quad\widetilde{\sigma}\in\operatorname*{Irr}(\mathrm{GL}_{2})\quad\mbox{(irreducible)}\par&\{1\}&\\ \hline\cr\end{array}

Table 2. The adjoint

L

-function

L(s,\sigma,\mathrm{Ad})

for

\mathrm{GSpin}_{4}

\begin{array}[]{|c|l|c|}\hline\cr&L(s,\sigma,\mathrm{Ad})&\operatorname{ord}_{s=1}\\ \hline\cr\hline\cr\mbox{(a)\&(b)}&L(s,\widetilde{\sigma}_{1},\operatorname{Sym}^{2}\otimes\omega_{\widetilde{\sigma}_{1}}^{-1})L(s,\widetilde{\sigma}_{2},\operatorname{Sym}^{2}\otimes\omega_{\widetilde{\sigma}_{2}}^{-1})&0\\ \hline\cr\mbox{(i)}&L(s+1)^{2}&0\\ \hline\cr\mbox{(ii)}&L(s)L(s+1)L(s,\chi_{1}\chi_{2}^{-1})L(s,\chi_{1}^{-1}\chi_{2})&0\\ \hline\cr\mbox{(iii)}&L(s)^{2}L(s,\chi_{1}\chi_{2}^{-1})L(s,\chi_{1}^{-1}\chi_{2})L(s,\chi_{3}\chi_{4}^{-1})L(s,\chi_{3}^{-1}\chi_{4})&0\\ \hline\cr\mbox{(iv)}&L(s+1)L(s,\widetilde{\sigma}_{2},\operatorname{Sym}^{2}\otimes\omega_{\widetilde{\sigma}_{2}}^{-1})&0\\ \hline\cr\mbox{(v)}&L(s)L(s,\chi_{1}\chi_{2}^{-1})L(s,\chi_{1}^{-1}\chi_{2})L(s,\widetilde{\sigma}_{2},\operatorname{Sym}^{2}\otimes\omega_{\widetilde{\sigma}_{2}}^{-1})&0\\ \hline\cr\mathfrak{nongnr}&L(s-1)L(s)L(s+1)L(s,\widetilde{\sigma},\mathrm{Ad})&1+\operatorname{ord}_{s=1}L(s,\widetilde{\sigma},\mathrm{Ad})\\ \hline\cr\end{array}

Table 3. Representations of

\mathrm{GSpin}_{6}(F)

\begin{array}[]{|c|l|c|}\hline\cr&\mbox{${\operatorname*{Res}}^{\mathrm{GL}_{4}\times\mathrm{GL}_{1}}_{\mathrm{GSpin}_{6}}$ of}&\mbox{generic}\\ \hline\cr\hline\cr\mbox{(a)}&(\widetilde{\sigma}_{0}\boxtimes\widetilde{\eta}),\quad\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{4})&\bullet\\ \hline\cr\mbox{(I)}&i_{(\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}\boxtimes\widetilde{\eta}),\quad\widetilde{\chi}_{i}\neq\nu\widetilde{\chi}_{j}&\bullet\\ \hline\cr\mbox{(II)}&i_{(\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\eta}),\quad\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{2}),\widetilde{\chi}_{1}\neq\nu^{\pm 1}\widetilde{\chi}_{2}&\bullet\\ \hline\cr\mbox{(III)}&i_{(\mathrm{GL}_{3}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}\boxtimes\widetilde{\eta}),\quad\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{3})&\bullet\\ \hline\cr\mbox{(IV)}&i_{(\mathrm{GL}_{2}\times\mathrm{GL}_{2})\times\mathrm{GL}_{1}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\sigma}_{1}\boxtimes\widetilde{\sigma}_{2}\boxtimes\widetilde{\eta}),\quad\widetilde{\sigma}_{i}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{2}),\widetilde{\sigma}_{1}\neq\nu^{\pm 1}\widetilde{\sigma}_{2}&\bullet\\ \hline\cr\mbox{(V)}&(\widetilde{\sigma}\boxtimes\widetilde{\eta}),\quad\widetilde{\sigma}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{4})\setminus\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{4})&\bullet\\ \hline\cr\mbox{(A)}&i_{(\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}\boxtimes\widetilde{\eta}),\quad\widetilde{\chi}_{i}=\nu\widetilde{\chi}_{j}&\\ \hline\cr\mbox{(B)}&i_{(\mathrm{GL}_{2}\times\mathrm{GL}_{1}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\eta}),\quad\widetilde{\sigma}_{0}\not\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{2}),\mbox{ or }\widetilde{\chi}_{1}=\nu^{\pm 1}\widetilde{\chi}_{2}&\\ \hline\cr\mbox{(C)}&i_{(\mathrm{GL}_{3}\times\mathrm{GL}_{1})\times\mathrm{GL}_{1}}^{\mathrm{GL}_{4}\times GL_{1}}(\widetilde{\sigma}_{0}\boxtimes\widetilde{\chi}\boxtimes\widetilde{\eta}),\quad\mbox{non-generic }\widetilde{\sigma}_{0}\in\operatorname*{Irr}(\mathrm{GL}_{3})&\\ \hline\cr\mbox{(D)}&i_{(\mathrm{GL}_{2}\times\mathrm{GL}_{2})\times\mathrm{GL}_{1}}^{\mathrm{GL}_{4}\times GL_{1}}((\chi\circ\det)\boxtimes\widetilde{\sigma}\boxtimes\widetilde{\eta}),\quad\widetilde{\sigma}\in\operatorname*{Irr}(\mathrm{GL}_{2})&\\ \hline\cr\mbox{(E)}&(\widetilde{\chi}\circ\det\boxtimes\widetilde{\eta}),\quad\widetilde{\sigma}\in\operatorname*{Irr}_{\rm esq}(\mathrm{GL}_{4})\setminus\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{4})&\\ \hline\cr\end{array}

Table 4. The adjoint

L

-function

L(s,\sigma,\mathrm{Ad})

for

\mathrm{GSpin}_{6}

\begin{array}[]{|c|l|l|c|}\hline\cr&\sigma\in\operatorname*{Irr}(\mathrm{GSpin}_{6}(F))\mbox{ determined by }&\begin{array}[]{l}L(s,\sigma,\mathrm{Ad})\end{array}&\mbox{ord}_{s=1}\\ \hline\cr\hline\cr\mbox{(a)}&\eqref{sc gl4}\,\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{4})&\begin{array}[]{l}L(s,\widetilde{\sigma}_{0},\mathrm{Ad})\end{array}&0\\ \hline\cr\mbox{(I)}&\eqref{ps gl4}\,\widetilde{\chi}_{1}\boxtimes\widetilde{\chi}_{2}\boxtimes\widetilde{\chi}_{3}\boxtimes\widetilde{\chi}_{4}\boxtimes\widetilde{\eta}&\begin{array}[]{l}L(s)^{3}\prod_{i\neq j}L(s,\widetilde{\chi}_{i}\widetilde{\chi}_{j}^{-1})\end{array}&0\\ \hline\cr\mbox{(II)}&\eqref{ps gl4 II}\,\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2})&\begin{array}[]{l}L(s)^{2}L(s,\widetilde{\sigma}_{0},\mathrm{Ad})L(s,\widetilde{\sigma}_{0}\times\widetilde{\chi}_{1}^{-1})L(s,\widetilde{\sigma}_{0}^{\vee}\times\widetilde{\chi}_{1})\\ L(s,\widetilde{\sigma}_{0}\times\widetilde{\chi}_{2}^{-1})L(s,\widetilde{\sigma}_{0}^{\vee}\times\widetilde{\chi}_{2})L(s,\widetilde{\chi}_{1}\widetilde{\chi}_{2}^{-1})L(s,\widetilde{\chi}_{2}\widetilde{\chi}_{1}^{-1})\end{array}&0\\ \hline\cr\mbox{(II)}&\eqref{ps gl4 II}\,\widetilde{\sigma}_{0}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\widetilde{\chi}&\begin{array}[]{l}L(s)^{2}L(s+1)L(s+1,\widetilde{\chi}\widetilde{\chi}_{1}^{-1})L(s+1,\widetilde{\chi}\widetilde{\chi}_{2}^{-1})\\ L(s,\widetilde{\chi}^{-1}\widetilde{\chi}_{1})L(s,\widetilde{\chi}^{-1}\widetilde{\chi}_{2})L(s,\widetilde{\chi}_{1}\widetilde{\chi}_{2}^{-1})L(s,\widetilde{\chi}_{2}\widetilde{\chi}_{1}^{-1})\end{array}&0\\ \hline\cr\mbox{(III)}&\eqref{ps gl4 III}\,\widetilde{\sigma}_{0}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{3})&\begin{array}[]{l}L(s)L(s,\widetilde{\sigma}_{0},\mathrm{Ad})L(s,\widetilde{\sigma}_{0}\times\widetilde{\chi}^{-1})L(s,\widetilde{\sigma}_{0}^{\vee}\times\widetilde{\chi})\end{array}&0\\ \hline\cr\mbox{(III)}&\eqref{ps gl4 III}\,\widetilde{\sigma}_{0}=\mathsf{St}_{\mathrm{GL}_{3}}\otimes\widetilde{\chi}_{0}&\begin{array}[]{l}L(s)L(s+1)L(s+2)L(s+1,\widetilde{\chi}\widetilde{\chi}_{0}^{-1})L(s+1,\widetilde{\chi}^{-1}\widetilde{\chi}_{0})\end{array}&0\\ \hline\cr\mbox{(IV)}&\eqref{ps gl4 IV}\,\widetilde{\sigma}_{i}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2})&\begin{array}[]{l}L(s)L(s,\widetilde{\sigma}_{1},\mathrm{Ad})L(s,\widetilde{\sigma}_{2},\mathrm{Ad})\\ L(s,\widetilde{\sigma}_{1}\times\widetilde{\sigma}_{2}^{\vee})L(s,\widetilde{\sigma}_{1}^{\vee}\times\widetilde{\sigma}_{1})\end{array}&0\\ \hline\cr\mbox{(IV)}&\eqref{ps gl4 IV}\,\widetilde{\sigma}_{1}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2}),\widetilde{\sigma}_{2}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\widetilde{\chi}&\begin{array}[]{l}L(s)L(s+1)L(s,\widetilde{\sigma}_{1},\mathrm{Ad})\\ L(s+\frac{1}{2},\widetilde{\sigma}_{1}^{\vee}\times\widetilde{\chi})L(s+\frac{1}{2},\widetilde{\sigma}_{1}\times\widetilde{\chi}^{-1})\end{array}&0\\ \hline\cr\mbox{(IV)}&\eqref{ps gl4 IV}\,\widetilde{\sigma}_{2}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2}),\widetilde{\sigma}_{1}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\widetilde{\chi}&\begin{array}[]{l}L(s)L(s+1)L(s,\widetilde{\sigma}_{2},\mathrm{Ad})\\ L(s+\frac{1}{2},\widetilde{\sigma}_{2}^{\vee}\times\widetilde{\chi})L(s+\frac{1}{2},\widetilde{\sigma}_{2}\times\widetilde{\chi}^{-1})\end{array}&0\\ \hline\cr\mbox{(IV)}&\eqref{ps gl4 IV}\,\widetilde{\sigma}_{1}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\widetilde{\chi}_{1}\widetilde{\sigma}_{2}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\widetilde{\chi}_{2}&\begin{array}[]{l}L(s)L(s+1)^{2}L(s,\widetilde{\chi}_{1}^{-1}\widetilde{\chi}_{2})L(s,\widetilde{\chi}_{1}\widetilde{\chi}_{2}^{-1})\\ L(s+1,\widetilde{\chi}_{1}\widetilde{\chi}_{2}^{-1})L(s+1,\widetilde{\chi}_{1}^{-1}\widetilde{\chi}_{2})\end{array}&0\\ \hline\cr\mbox{(V)}&\eqref{4gl1}\,\widetilde{\sigma}=\mathsf{St}_{\mathrm{GL}_{4}}\otimes\widetilde{\chi}&\begin{array}[]{l}L(s+1)L(s+2)L(s+3)\end{array}&0\\ \hline\cr\mbox{(V)}&\eqref{2gl2}\,\widetilde{\sigma}=\Delta[\nu^{1/2},\nu^{-1/2}],\widetilde{\tau}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2})&\begin{array}[]{l}L(s,\widetilde{\tau},\mathrm{Ad})L(s,\widetilde{\tau}\times\widetilde{\tau}^{\vee})\end{array}&0\\ \hline\cr\mbox{(A)}&\eqref{nongenericA1}\,Q\left([\nu^{1/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}],[\widetilde{\chi}_{3}],[\widetilde{\chi}_{4}]\right)&\begin{array}[]{l}L(s-1)L(s)^{3}L(s+1)L(s,\widetilde{\chi}_{3}\widetilde{\chi}_{4}^{-1})L(s,\widetilde{\chi}_{3}^{-1}\widetilde{\chi}_{4})\\ \prod\limits_{i=3,4}\left(\begin{array}[]{l}L(s+\frac{1}{2},\widetilde{\chi}\widetilde{\chi}_{i}^{-1})L(s-\frac{1}{2},\widetilde{\chi}^{-1}\widetilde{\chi}_{i})\\ L(s-\frac{1}{2},\widetilde{\chi}\widetilde{\chi}_{i}^{-1})L(s+\frac{1}{2},\widetilde{\chi}^{-1}\widetilde{\chi}_{i})\end{array}\right)\end{array}&\geq 1\\ \hline\cr\mbox{(A)}&\eqref{nongenericA2}\,Q\left([\nu\widetilde{\chi}],[\widetilde{\chi}],[\nu^{-1}\widetilde{\chi}],[\widetilde{\chi}_{4}]\right)&\begin{array}[]{l}L(s-2)L(s-1)^{2}L(s)^{3}L(s+1)^{2}L(s+2)\\ \prod\limits_{t=-1,0,1}\left(L(s+t,\widetilde{\chi}\widetilde{\chi}_{4}^{-1})L(s+t,\widetilde{\chi}^{-1}\widetilde{\chi}_{4})\right)\end{array}&\geq 2\\ \hline\cr\mbox{(A)}&\eqref{nongenericA3}\,Q\left([\widetilde{\chi},\nu\widetilde{\chi}],[\nu^{-1}\widetilde{\chi}],[\widetilde{\chi}_{4}]\right)&\begin{array}[]{l}L(s-2)L(s-1)^{2}L(s)^{2}\\ \prod\limits_{t=-1,0}L(s+t,\widetilde{\chi}\widetilde{\chi}_{4}^{-1})\prod\limits_{t=-1,1}L(s+t,\widetilde{\chi}^{-1}\widetilde{\chi}_{4})\end{array}&\geq 2\\ \hline\cr\mbox{(A)}&\eqref{nongenericA4}\,Q\left([\nu\widetilde{\chi}],[\widetilde{\chi},\nu^{-1}\widetilde{\chi}],[\widetilde{\chi}_{4}]\right)&\begin{array}[]{l}L(s-1)L(s)^{2}L(s+1)L(s+2)\\ \prod\limits_{t=0,1}L(s+t,\widetilde{\chi}\widetilde{\chi}_{4}^{-1})\prod\limits_{t=-1,1}L(s+t,\widetilde{\chi}^{-1}\widetilde{\chi}_{4})\end{array}&\geq 1\\ \hline\cr\mbox{(A)}&\eqref{nongenericA5}\,Q\left([\nu^{3/2}\widetilde{\chi}],[\nu^{1/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}]\right)&\begin{array}[]{l}L(s-3)L(s-2)^{2}L(s-1)^{3}L(s)^{3}\\ L(s+1)^{3}L(s+2)^{2}L(s+3)\end{array}&3\\ \hline\cr\mbox{(A)}&\eqref{nongenericA6}\,Q\left([\nu^{1/2}\widetilde{\chi},\nu^{3/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}]\right)&L(s-3)L(s-2)L(s-1)^{2}L(s)^{2}L(s+1)^{2}L(s+2)&2\\ \hline\cr\mbox{(A)}&\eqref{nongenericA7}\,Q\left([\nu^{3/2}\widetilde{\chi}],[\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}]\right)&L(s-3)L(s-2)L(s-1)^{2}L(s)^{2}L(s+1)^{2}L(s+2)&2\\ \hline\cr\mbox{(A)}&\eqref{nongenericA8}\,Q\left([\nu^{3/2}\widetilde{\chi}],[\nu^{1/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi},\nu^{-1/2}\widetilde{\chi}]\right)&L(s-3)L(s-2)L(s-1)^{2}L(s)^{2}L(s+1)^{2}L(s+2)&2\\ \hline\cr\mbox{(A)}&\eqref{nongenericA9}\,Q\left([\nu^{1/2}\widetilde{\chi},\nu^{3/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi},\nu^{-1/2}\widetilde{\chi}]\right)&L(s-3)L(s-2)L(s-1)^{2}L(s)L(s+1)L(s+2)&2\\ \hline\cr\mbox{(A)}&\eqref{nongenericA10}\,Q\left([\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi},\nu^{3/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi}]\right)&L(s-3)L(s-2)L(s-1)L(s)L(s+1)&1\\ \hline\cr\mbox{(A)}&\eqref{nongenericA11}\,Q\left([\nu^{3/2}\widetilde{\chi}],[\nu^{-3/2}\widetilde{\chi},\nu^{-1/2}\widetilde{\chi},\nu^{1/2}\widetilde{\chi}]\right)&L(s-3)L(s-2)L(s-1)L(s)L(s+1)&1\\ \hline\cr\mbox{(B)}&\eqref{nongenericB}\begin{array}[]{c}Q\left([i_{B}^{\mathrm{GL}_{2}}(\widetilde{\eta}_{1}\boxtimes\widetilde{\eta}_{2})],[\widetilde{\chi}\nu^{1/2}],[\widetilde{\chi}\nu^{-1/2}]\right)\!,\\ \widetilde{\eta}_{1}\widetilde{\eta}_{2}^{-1}\neq\nu^{\pm 1}\end{array}&\begin{array}[]{l}L(s-1)L(s)^{3}L(s+1)L(s,\widetilde{\eta}_{1}\widetilde{\eta}_{2}^{-1})L(s,\widetilde{\eta}_{1}^{-1}\widetilde{\eta}_{2})\\ \prod\limits_{t=\pm\frac{1}{2}}\prod\limits_{i=1,2}\left(L(s+t,\widetilde{\eta}_{i}\widetilde{\chi}^{-1})L(s+t,\widetilde{\eta}_{i}^{-1}\widetilde{\chi})\right)\end{array}&\geq 1\\ \hline\cr\mbox{(B)}&\eqref{nongenericB}\,\mbox{ (others covered in (A)) }&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\ \hline\cr\mbox{(C)}&\eqref{ps gl4 III non}\,\mbox{ (covered in (A) and (B)) }&\vrule\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol\\ \hline\cr\mbox{(D)}&\eqref{ps gspin6 IV non}\,\mbox{ with }\widetilde{\sigma}\in\operatorname*{Irr}_{\rm sc}(\mathrm{GL}_{2})&\begin{array}[]{l}L(s-1)L(s)^{2}L(s+1)L(s,\sigma,\mathrm{Ad})\\ \prod\limits_{t=\pm\frac{1}{2}}\left(L(s+t,\sigma\times\chi^{-1})L(s+t,\sigma^{\vee}\times\chi)\right)\\ \end{array}&1\\ \hline\cr\mbox{(D)}&\eqref{ps gspin6 IV non}\,\mbox{ with }\widetilde{\sigma}=\mathsf{St}_{\mathrm{GL}_{2}}\otimes\eta&\begin{array}[]{l}L(s-1)L(s)^{2}L(s+1)^{2}\\ L(s,\chi\eta^{-1})L(s+1,\chi\eta^{-1})L(s+1,\chi^{-1}\eta)L(s,\chi^{-1}\eta)\end{array}&\geq 1\\ \hline\cr\mbox{(D)}&\eqref{ps gspin6 IV non}\,\mbox{ (others covered in (A)) }&\vrule\lx@intercol\hfil\lx@intercol\vrule\lx@intercol\\ \hline\cr\mbox{(E)}&\eqref{ps gspin6 E non}\,\mbox{ (covered in (A)) }&\vrule\lx@intercol\hfil\lx@intercol\vrule\lx@intercol\\ \hline\cr\end{array}

Representations of the pp-adic GSpin4\mathrm{GSpin}_{4} and GSpin6\mathrm{GSpin}_{6} and the Adjoint LL-Function

Abstract.

1. Introduction

Acknowledgements

2. Preliminaries

2.1. Local Langlands Correspondence (LLC)

2.2. The Adjoint LL-Function

Conjecture 2.1.

2.3. The Groups GSpin4\mathrm{GSpin}_{4} and GSpin6\mathrm{GSpin}_{6}

Proposition 2.2.

Proof.

2.4. Computation of the Adjoint LL-Function for GG

3. Genericity and The Conjecture of B. Gross and D. Prasad

3.1. Restriction of Generic Representations

Proposition 3.1.

Proof.

Remark 3.2.

3.2. Criterion for Genericity

Theorem 3.3.

Proof.

Corollary 3.4.

4. Representations of GSpin4\mathrm{GSpin}_{4}

4.1. The Reprsentations

4.1.1. Classification of representations of GSpin4\mathrm{GSpin}_{4}

4.1.2. Construction of the LL-packets of GSpin4\mathrm{GSpin}_{4} (recalled from [AC17])

4.1.3. The LL-parameters of GL2{\mathrm{GL}}_{2}

4.2. Generic Representations of GSpin4\mathrm{GSpin}_{4}

4.2.1. Irreducible Parameters

Proposition 4.1.

Remark 4.2.

4.2.2. Reducible Parameters

4.3. Non-Generic Representations of GSpin4\mathrm{GSpin}_{4}

4.4. Computation of the Adjoint LL-function for GSpin4\mathrm{GSpin}_{4}

5. Representations of GSpin6\mathrm{GSpin}_{6}

5.1. The Represenations

5.1.1. Classification of representations of GSpin6\mathrm{GSpin}_{6}

5.1.2. Construction of the LL-packets of GSpin6\mathrm{GSpin}_{6} (recalled from [AC17])

5.2. Generic Representations of GSpin6\mathrm{GSpin}_{6}

5.2.1. Irreducible Parameters

Proposition 5.1.

5.2.2. Reducible Parameters

5.3. Non-Generic Representaions of GSpin6\mathrm{GSpin}_{6}

5.4. Computation of the Adjoint L-function for GSpin6\mathrm{GSpin}_{6}

6. Correction to [AC17]

6.1. Proposition 5.5 and 6.4

6.2. Remark 5.11

References

Representations of the $p$ -adic $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$ and the Adjoint $L$ -Function

2.2. The Adjoint $L$ -Function

2.3. The Groups $\mathrm{GSpin}_{4}$ and $\mathrm{GSpin}_{6}$

2.4. Computation of the Adjoint $L$ -Function for $G$

4. Representations of $\mathrm{GSpin}_{4}$

4.1.1. Classification of representations of $\mathrm{GSpin}_{4}$

4.1.2. Construction of the $L$ -packets of $\mathrm{GSpin}_{4}$ (recalled from [AC17])

4.1.3. The $L$ -parameters of ${\mathrm{GL}}_{2}$

4.2. Generic Representations of $\mathrm{GSpin}_{4}$

4.3. Non-Generic Representations of $\mathrm{GSpin}_{4}$

4.4. Computation of the Adjoint $L$ -function for $\mathrm{GSpin}_{4}$

5. Representations of $\mathrm{GSpin}_{6}$

5.1.1. Classification of representations of $\mathrm{GSpin}_{6}$

5.1.2. Construction of the $L$ -packets of $\mathrm{GSpin}_{6}$ (recalled from [AC17])

5.2. Generic Representations of $\mathrm{GSpin}_{6}$

5.3. Non-Generic Representaions of $\mathrm{GSpin}_{6}$

5.4. Computation of the Adjoint L-function for $\mathrm{GSpin}_{6}$