1 Notation and preliminaries

We fix the following notation.

• • $n \geq 3$ is an integer.Footnote ³

• • If k is a field, $\overline k$ denotes an algebraic closure of k.

• • When X is a square matrix, $\mathscr {EV}(X)$ denotes the multi-set of eigenvalues of X.

• • When A is a multi-set with elements in a ring R with $r\in R$ , write $r\cdot A$ for the multi-set formed by the elements $ra \in A$ as a ranges over A. For $n\in \mathbb Z_{>0}$ , write $A^{\oplus n}$ for the multi-set consisting of $a\in A$ whose multiplicity in $A^n$ is n times that in A.

• • F is a number field. (In the main text, F is a totally real field with a distinguished embedding into ${\mathbb {C}}$ .)

• • $\mathcal {O}_F$ is the ring of integers of F.

• • $\mathbb A_F$ is the ring of adèles of F, $\mathbb A_F := (F \otimes {\mathbb R}) \times (F \otimes \widehat {\mathbb {Z}})$ .

• • If S is a finite set of F-places, then $\mathbb A_F^S \subset \mathbb A_F$ is the ring of adèles with trivial components at the places in S, and $F_S := \prod _{v \in S} F_v$ ; $F_{\infty }:=F\otimes _{\mathbb {Q}} {\mathbb R}$ .

• • If ${{\mathfrak q}}$ is a finite F-place, we write $q_{{\mathfrak q}}$ for the cardinality of the residue field of ${{\mathfrak q}}$ .

• • $|\cdot |:\mathbb A_F^{\times } \rightarrow {\mathbb R}^{\times }_{>0}$ is the norm character on $\mathbb A_F^{\times }$ that is trivial on $F^{\times }$ . Denote by $|\cdot |_v:F_v^{\times } \rightarrow {\mathbb R}^{\times }_{>0}$ the restriction of $|\cdot |$ to the v-component. Our normalization is that $|\cdot |_{{\mathfrak q}}$ sends a uniformizer of $F_{\mathfrak {q}}$ to $q_{{\mathfrak q}}^{-1}$ , whereas $|\cdot |_v$ is the usual absolute value (resp. squared absolute value) when v is real (resp. complex).

• • If S is a set of prime numbers, we write $S^F$ for the set of F-places above S.

• • If p is a prime number, then $F_p := F \otimes _{\mathbb {Q}} {\mathbb {Q}_p}$ .

• • $\ell $ is a primenumber (typically different from p).

• • $\overline {\mathbb {Q}}_{\ell }$ is a fixed algebraic closure of $\mathbb {Q}_{\ell }$ , and $\iota \colon {\mathbb {C}} \overset \sim \to \overline {\mathbb {Q}}_{\ell }$ is an isomorphism.

• • For each prime number p, we fix the positive root $ p^{1/2} \in {\mathbb R}_{>0} \subset {\mathbb {C}}$ . From $\iota $ , we then obtain a choice for $p^{1/2} \in \overline {\mathbb {Q}}_{\ell }$ . If q is a power of p, we obtain similarly a preferred choice $q^{1/2}$ in $\overline {\mathbb {Q}}_{\ell }$ and in ${\mathbb {C}}$ .

• • $\Gamma =\Gamma _F:= {\mathrm {Gal}}(\overline {F}/F)$ is the absolute Galois group of F.

• • For a finite extension E of F in $\overline {F}$ , write $\Gamma _E:={\mathrm {Gal}}(\overline {F}/E)$ and $\Gamma _{E/F}:={\mathrm {Gal}}(E/F)$ .

• • $\Gamma _v = \Gamma _{F_v}:={\mathrm {Gal}}(\overline {F}_v/F_v)$ is (one of) the local Galois group(s) of F at the place v; $W_{F_v} \subset \Gamma _v$ is the corresponding Weil group.

• • For each F-place v, choose an embedding $\iota _v:\overline {F} \hookrightarrow \overline {F}_v$ , which induces $\Gamma _v\hookrightarrow \Gamma $ that is canonical up to conjugation.

• • $\mathcal V_{\infty }:= \mathrm {Hom}_{\mathbb {Q}}(F, {\mathbb R})$ is the set of infinite places of F.

• • $c_y\in \Gamma $ is the complex conjugation (well-defined as a conjugacy class) induced by any embedding $\overline {F}\hookrightarrow {\mathbb {C}}$ extending $y\in \mathcal V_{\infty }$ .

• • If S is a finite set of F-places, write $\Gamma _{F,S}$ for the Galois group ${\mathrm {Gal}}(F(S)/F)$ where $F(S) \subset \overline {F}$ is the maximal extension of F that is unramified away from S. If S is a set of rational places, we write $\Gamma _{F, S} := \Gamma _{F, S^F}$ .

• • $\mathrm {Frob}_{\mathfrak {q}}$ at a finite prime ${{\mathfrak q}}$ of F means the geometric Frobenius element in the quotient of $\Gamma _{\mathfrak {q}}$ by the inertia subgroup, or the image thereof in $\Gamma _{F,S}$ . (The image in $\Gamma _{F,S}$ depends on the choice of $\iota _{\mathfrak {q}}$ , but its conjugacy class is independent of the choice.)

• • When G is a connected reductive group over F, write $\widehat G$ and $^L G= \widehat G\rtimes \Gamma _F$ for the Langlands dual group and the L-group, respectively (with coefficients in ${\mathbb {C}}$ or $\overline {\mathbb {Q}}_{\ell }$ , depending on the context). If G splits over a finite extension $E/F$ in $\overline {F}$ , then $\widehat G\rtimes \Gamma _{E/F}$ denotes the L-group with respect to $E/F$ . (Namely, such a semi-direct product is always understood with the L-action of $\Gamma _{E/F}$ on $\widehat G$ .) Often we use $^L G$ to mean $\widehat G\rtimes \Gamma _{E/F}$ .Footnote ⁴

• • When H is a reductive group over $\overline {\mathbb {Q}}_{\ell }$ , we also use H to mean the topological group $H(\overline {\mathbb {Q}}_{\ell })$ by abuse of notation. This should be clear from the context and not leading to confusion.

• • When F is a p-adic field and G is the set of F-points of a reductive group over F, we write $\mathrm {St}_G$ for the Steinberg representation of G (defined in [Reference Borel and Wallach8, X.4.6], for instance). Moreover, we write $\textbf {1}_G$ for the trivial representation of G. In certain cases, when G is clear, we write $\mathrm {St} = \mathrm {St}_G$ or $\textbf {1} = \textbf {1}_G$ . We also sometimes write $\mathrm {St}_n$ for $\mathrm {St}_{{\mathrm {GL}}_n(F)}$ (in case F is clear from the context).

• • If G is an algebraic group, we write $Z(G)$ for its center.

• • An inner twist of a reductive group G over a perfect field k means a reductive group $G'$ over k together with an isomorphism $i:G_{\overline k}\rightarrow G^{\prime }_{\overline k}$ such that the automorphism $i^{-1}\sigma (i)$ of $G_{\overline k}$ is inner for every $\sigma \in {\mathrm {Gal}}(\overline k/k)$ . There is an obvious notion of isomorphism for inner twists $(G',i)$ , cf. [Reference Kaletha38, 2.2]. We often say $G'$ is an inner twist of G, keeping i implicit. If we forget i and only remember the k-group $G'$ and the existence of i, we refer to it as an inner form of G.

Fix G and $E/F$ as above. We introduce some notions on the Galois side. By an ( $\ell $ -adic) Galois representation of $\Gamma _F$ (with values in $\widehat G\rtimes \Gamma _{E/F}$ ), we mean a continuous morphism

$$ \begin{align*}\rho \colon \Gamma_F \rightarrow \widehat{G}(\overline{\mathbb{Q}}_{\ell})\rtimes \Gamma_{E/F} \end{align*} $$

which factors through $\Gamma _{F,S}$ for some finite set S and commutes with the obvious projections onto $\Gamma _{E/F}$ . Similarly, we define a Galois representation with the source $\Gamma _{\mathfrak {q}}$ or with values in $^L G(\overline {\mathbb {Q}}_{\ell })$ . Two Galois representations are considered isomorphic if they are conjugate by an element of $\widehat {G}(\overline {\mathbb {Q}}_{\ell })$ . We say that $\rho $ as above is (totally) odd if for every real place y of F, the following holds: writing $\mathrm {Ad}$ for the adjoint action of $^L G$ on $\mathrm {Lie}\, G(\overline {\mathbb {Q}}_{\ell })$ , which preserves the Lie algebra of the derived subgroup $G_{\mathrm {der}}$ , the image of $c_y$ under the composite

$$ \begin{align*}\Gamma_y \hookrightarrow \Gamma \stackrel{\rho}{\rightarrow} {}^L G(\overline{\mathbb{Q}}_{\ell}) \stackrel{\mathrm{Ad}}{\rightarrow} {\mathrm{GL}}(\mathrm{Lie}\, G_{\mathrm{der}}(\overline{\mathbb{Q}}_{\ell})) \end{align*} $$

has trace equal to the rank of $\widehat G_{\mathrm {der}}$ . (Compare with [Reference Gross27].)

An $^L G$ -valued Weil–Deligne representation of $W_{F_{\mathfrak {q}}}$ is a pair $(r,N)$ consisting of a morphism

$$ \begin{align*}r \colon W_{F_{\mathfrak{q}}} \rightarrow \widehat{G}(\overline{\mathbb{Q}}_{\ell})\rtimes \Gamma_{E_{\mathfrak p}/F_{\mathfrak{q}}}, \end{align*} $$

which has open kernel on the inertia subgroup and commutes with the canonical projections onto $ \Gamma _{E_{\mathfrak p}/F_{\mathfrak {q}}}$ , and a nilpotent operator $N\in \mathrm {Lie}\, \widehat {G}(\overline {\mathbb {Q}}_{\ell })$ such that $\mathrm {Ad} (r(w)) N= |w| N$ for $w\in W_{F_{\mathfrak {q}}}$ , where $|\cdot |:W_{F_{\mathfrak {q}}}\rightarrow \|{{\mathfrak q}}\|^{\mathbb Z}$ is the homomorphism sending a geometric Frobenius element to $\|{{\mathfrak q}}\|^{-1}$ ; here, $\|{{\mathfrak q}}\|\in \mathbb Z_{>0}$ denotes the norm of ${{\mathfrak q}}$ . The Frobenius-semisimplification $(r^{\mathrm {ss}},N)$ is obtained by replacing r with its semisimplification. We say $(r,N)$ is Frobenius-semisimple if $r=r^{\mathrm {ss}}$ .

Let $\rho \colon \Gamma _F \rightarrow \widehat {G}(\overline {\mathbb {Q}}_{\ell })\rtimes \Gamma _{E/F}$ be a Galois representation. Write ${\mathfrak p}$ for the prime of E induced by $\iota _{\mathfrak {q}}:\overline {F} \hookrightarrow \overline {F}_{\mathfrak {q}}$ . Then the restriction (via $\iota _{\mathfrak {q}}$ )

$$ \begin{align*}\rho|_{\Gamma_{\mathfrak{q}}} \colon \Gamma_{F_{\mathfrak{q}}} \rightarrow \widehat{G}(\overline{\mathbb{Q}}_{\ell})\rtimes \Gamma_{E_{\mathfrak p}/F_{\mathfrak{q}}} \end{align*} $$

gives rise to an $^L G$ -valued Weil–Deligne representation, to be denoted by $\mathrm {WD}(\rho |_{\Gamma _{F_{\mathfrak {q}}}})$ . The construction follows from the case of $G={\mathrm {GL}}_n$ by the Tannakian formalism via algebraic representations of $\widehat {G}(\overline {\mathbb {Q}}_{\ell })\rtimes \Gamma _{E_{\mathfrak p}/F_{\mathfrak {q}}}$ . (The case ${{\mathfrak q}}|\ell $ is more subtle than ${{\mathfrak q}}\nmid \ell $ . In the former case, a detailed explanation is given in the proof of [Reference Kret and Shin50, Lem. 3.2], where $\widehat G$ is denoted by H. In loc. cit. $\Gamma _{E_{\mathfrak p}/F_{\mathfrak {q}}}$ is trivial but the same argument extends.) When ${{\mathfrak q}}\nmid \ell $ , one can alternatively appeal to Grothendieck’s $\ell $ -adic monodromy theorem to construct $\mathrm {WD}(\rho |_{\Gamma _{F_{\mathfrak {q}}}})$ directly (without going through general linear groups).

A local L-parameter $\phi :W_{F_{\mathfrak {q}}} \times \mathrm {SL}(2)\rightarrow \widehat {G}(\overline {\mathbb {Q}}_{\ell })\rtimes \Gamma _{E_{\mathfrak p}/F_{\mathfrak {q}}}$ is associated with a Frobenius-semisimple $^L G$ -valued Weil–Deligne representation $(r,N)$ given by the following recipe:

$$ \begin{align*}r(w) = \phi\left( w, \begin{pmatrix} |w|^{1/2} & 0 \\ 0 & |w|^{-1/2} \end{pmatrix} \right), \qquad \mbox{and}\qquad N = \phi\left( 1,\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \right). \end{align*} $$

This induces a bijection on the sets of equivalence classes of such objects [Reference Gross and Reeder28, Prop. 2.2]. In practice (where only equivalence classes matter), we will use them interchangeably.

We introduce some further notation and conventions in representation theory. If $\pi $ is a representation on a complex vector space, then we set $\iota \pi :=\pi \otimes _{{\mathbb {C}},\iota } \overline {\mathbb {Q}}_{\ell }$ . Similarly, if $\phi $ is a local L-parameter of a connected reductive group G over a nonarchimedean local field so that $\phi $ maps into $^L G({\mathbb {C}})$ , then $\iota \phi $ is the parameter with values in $^L G(\overline {\mathbb {Q}}_{\ell })$ obtained from $\phi $ via $\iota $ . If G is a locally profinite group equipped with a Haar measure, then we write $\mathcal H(G)$ for the Hecke algebra of locally constant, complex valued functions with compact support. We write $\mathcal H_{\overline {\mathbb {Q}}_{\ell }}(G)$ for the same algebra but now consisting of $\overline {\mathbb {Q}}_{\ell }$ -valued functions. We normalize every parabolic induction by the half power of the modulus character as in [Reference Bernstein and Zelevinsky7, 1.8], so that it preserves unitarity.

Let G be a real reductive group, K a maximal compact subgroup of $G({\mathbb R})$ , and $\tilde K := K \cdot Z(G)({\mathbb R})$ . Let $\xi $ be an irreducible algebraic representation of G over ${\mathbb {C}}$ . An irreducible admissible representation $\pi $ of $G({\mathbb R})$ is said to be $\xi $ -cohomological if $H^i(\mathrm {Lie}\, G({\mathbb {C}}),\tilde K, \pi \otimes _{\mathbb {C}} \xi )\neq 0$ for some $i\ge 0$ . If this is the case, we assign a Hodge cocharacter over ${\mathbb {C}}$ (well-defined up to $\widehat G$ -conjugacy) as in [Reference Kret and Shin50, Def 1.14]:

$$ \begin{align*}\mu_{\mathrm{Hodge}}(\xi) \colon \mathbb{G}_{\mathrm{m}} \rightarrow \widehat G. \end{align*} $$

Let L be a finite extension of $\mathbb {Q}_{\ell }$ . Let H be a possibly disconnected reductive group over $\overline {\mathbb {Q}}_{\ell }$ (e.g., an L-group relative to a finite Galois extension), and $\rho \colon {\mathrm {Gal}}(\overline L/L)\rightarrow H(\overline {\mathbb {Q}}_{\ell })$ a continuous morphism. If $\rho $ is Hodge–Tate with respect to each $\mathbb {Q}_{\ell }$ -embedding $i \colon L \hookrightarrow \overline {\mathbb {Q}}_{\ell }$ , we define a Hodge–Tate cocharacter over $\overline {\mathbb {Q}}_{\ell }$ (well-defined up to H-conjugacy) as in [Reference Buzzard and Gee12, §2.4] (cf. [Reference Kret and Shin50, Def 1.10]):

$$ \begin{align*}\mu_{\mathrm{HT}}(\rho,i):\mathbb G_m \rightarrow H. \end{align*} $$

We recall the following lemma that can be easily deduced from the Chebotarev density theorem, as it will be needed in §10. Let F be a number field. The density of a set S consisting of primes of F is defined to be the limit $d(S) = \lim _{n \to \infty } a_n(S) / a_n(F)$ , where $a_n(F)$ is the number of primes ${{\mathfrak q}}$ with bounded norm $\|{{\mathfrak q}}\| < n$ and $a_n(S)$ is the number of ${{\mathfrak q}} \in S$ with $\|{{\mathfrak q}}\| < n$ [Reference Serre72, Sect. I.2.2]. Depending on S, the limit $d(S)$ may or may not exist — in the former case, we say S has density $d(S)$ , and otherwise we leave the density undefined.

Proof. Let $\mu $ be the Haar measure on $\Gamma _{S} = \Gamma _{F,S}$ with total volume $1$ . We write X to also mean $X(\overline {\mathbb {Q}}_{\ell })$ to simplify notation. Then $Y = r^{-1}(X)$ is a closed subset of $\Gamma _S$ (hence measurable) and stable under $\Gamma _S$ -conjugation. If we further have that $\mu (Y)=0$ , then the Chebotarev density theorem [Reference Serre72, I-8 Cor. 2b] implies that the set of places ${{\mathfrak q}} \notin S$ such that $\mathrm {Frob}_{{\mathfrak q}} \in Y$ has measure $0$ , so we will be done.

So it suffices to prove that $\mu (Y) = 0$ . We induct on $\dim (X)\in \{0,1,\ldots ,\dim (G)-1\}$ . We may assume that X is irreducible by induction. When $\dim X=0$ , then X is a point and the preimage Y of X is a torsor under $\ker (r)$ . We then have $\mathrm {vol}(Y) = \mathrm {vol}(\ker (r)) = 0$ , since $\ker (r)\subset \Gamma _S$ is a closed subgroup of infinite index by hypothesis. Now assume that the assertion is known whenever $\dim (X) < d$ and consider the case $\dim (X)=d< \dim (G)$ . There exists an infinite sequence $\gamma _1, \gamma _2, \ldots \in \Gamma _S$ such that the subset $r(\gamma _i) X$ are mutually distinct. (If the choice were impossible after $i=r$ , then multiplication by $r(g)$ preserves $\bigcup _{i<=r} r(\gamma _i) X$ for every $g\in \Gamma _S$ . This cannot happen because r has Zariski dense image, and the union has dimension $d < \dim (G)$ .) Consider

$$ \begin{align*}\Gamma_S \supset \bigcup_{i=1}^{\infty} \gamma_i r^{-1}(X). \end{align*} $$

The volume of $\Gamma _S$ is finite. Each term on the right-hand side is closed (so measurable), and the volumes of $\gamma _i r^{-1}(X)$ are all equal. We claim that their pairwise intersections have volume $0$ . If this is true, then we deduce that $\mathrm {vol} (\gamma _i r^{-1}(X)) = \mathrm {vol}( r^{-1}(X) ) =0$ , completing the proof.

It remains to verify the claim. Observe that the intersection

$$ \begin{align*}(*) \quad\quad \gamma_i r^{-1}(X) \cap \gamma_j r^{-1}(X), \quad\quad i \neq j \end{align*} $$

maps into the intersection $r(\gamma _i)X \cap r(\gamma _j)X$ in G, which has dimension less than d, so indeed (*) has measure $0$ by induction hypothesis. This completes the proof.

We also record a lemma on projective algebraic representations, which will be useful later on.

2 Root data of ${\mathrm {GSO}}_{2n}$ and $\mathrm {GSpin}_{2n}$

Let $\mathrm {GO}_{2n}/\mathbb {Q}$ be the algebraic group such that for all $\mathbb {Q}$ -algebras R, we have

(2.1) $$ \begin{align} \mathrm{GO}_{2n}(R) = \left\lbrace g \in {\mathrm{GL}}_{2n}(R)\ \left| \ \exists\, \mathrm{sim}(g) \in R^{\times} \ \colon\ g{}^{\mathrm{t}} \cdot \left(\begin{matrix} & {1_n} \cr {1_n} & \end{matrix}\right) \ \cdot g = \mathrm{sim}(g) \cdot \left(\begin{matrix} & {1_n} \cr {1_n} & \end{matrix}\right) \right. \ \right\rbrace. \end{align} $$

(In the above formula, $1_n$ is the $n\times n$ identity matrix.) The group $\mathrm {GO}_{2n}$ is disconnected; its neutral component ${\mathrm {GSO}}_{2n} \subset \mathrm {GO}_{2n}$ is defined by the condition $\det (g) = \mathrm {sim}(g)^n$ . The groups $\mathrm {GO}_{2n}$ , ${\mathrm {GSO}}_{2n}$ are split and defined by a quadratic form of signature $(n,n)$ . An element t of the diagonal torus $\mathrm {T}_{{\mathrm {GSO}}} \subset {\mathrm {GSO}}_{2n}$ is of the form

(2.2) $$ \begin{align} t = {\mathrm{diag}}(t_i)_{i=1}^{2n} = {\mathrm{diag}}(t_1, t_2, \ldots, t_n, t_0 t_1^{-1}, t_0 t_2^{-1}, \ldots, t_0 t_n^{-1}), \quad t_0:= \mathrm{sim}(t). \end{align} $$

Hence, $\mathrm {T}_{{\mathrm {GSO}}} \simeq \mathbb {G}_{\mathrm {m}}^{n+1}$ by sending t to $(t_0,t_1,\ldots ,t_n)$ . We identify $X^{*}(\mathrm {T}_{{\mathrm {GSO}}}) = \bigoplus _{i=0}^n \mathbb Z \cdot e_i$ and $X_*(\mathrm {T}_{{\mathrm {GSO}}}) = \bigoplus _{i=0}^n \mathbb Z \cdot e^{*}_i$ accordingly. We let $\mathrm {B}_{\mathrm {GSO}}$ be the Borel subgroup of ${\mathrm {GSO}}_{2n}$ of matrices of the form

(2.3) $$ \begin{align} g = \left(\begin{matrix} A & {AB} \cr 0 & {c A^{\mathrm{t}, -1}} \end{matrix}\right) , \quad A \in B_{{\mathrm{GL}}_n}, \ B \in \mathrm{M}_n, \ B^{\mathrm{t}} = -B \quad \mathrm{and} \quad c = \mathrm{sim}(g), \end{align} $$

where $B_{{\mathrm {GL}}_n} \subset {\mathrm {GL}}_n$ is the upper triangular Borel subgroup. (To see that $\mathrm {B}_{\mathrm {GSO}}$ is indeed a Borel subgroup, notice that any block matrix $g = {{\left ( \begin {smallmatrix} A & B \cr C & D \end {smallmatrix} \right )}}$ with $C = 0$ is of the above form if and only if $g \in {\mathrm {GSO}}_{2n}$ , and moreover, the displayed group is solvable of dimension $n^2 + 1$ ).

We realize the split forms of even (special) orthogonal groups in $\mathrm {GO}_{2n}/\mathbb {Q}$ . Namely, we write $\mathrm {O}_{2n}$ (resp. ${\mathrm {SO}}_{2n}$ ) for the subgroup of $\mathrm {GO}_{2n}$ (resp. ${\mathrm {GSO}}_{2n}$ ) where $\mathrm {sim}$ is trivial.

We define the following element (over any $\mathbb {Q}$ -algebra point of $\mathrm {O}_{2n}$ )Footnote ⁵

(2.4) $$ \begin{align} \vartheta^{\circ} := -\left( \begin{array}{cc;{1pt/1pt}cc}1_{n-1} & & & \\ & 0 & & 1 \\ \hdashline[1pt/1pt] & & 1_{n-1} & \\ & 1& & 0 \end{array}\right)\in \mathrm{O}_{2n}. \end{align} $$

Since $\det (\vartheta ^{\circ }) = -1$ , we have $\vartheta ^{\circ } \notin {\mathrm {SO}}_{2n}$ . We write $\theta ^{\circ } \in \mathrm {Aut}({\mathrm {GSO}}_{2n})$ for the automorphism given by $\vartheta ^{\circ }$ -conjugation.

Lemma 2.3. The automorphism $\theta ^{\circ }$ stabilizes $\mathrm {B}_{\mathrm {GSO}}$ and $\mathrm {T}_{{\mathrm {GSO}}}$ and acts on $\mathrm {T}_{{\mathrm {GSO}}}$ by

$$ \begin{align*}(t_0, t_1, \ldots, t_n) \mapsto (t_0, t_1, \ldots, t_{n-1}, t_0t_n^{-1}). \end{align*} $$

Furthermore, $\theta ^{\circ }(\alpha _i)= \alpha _i$ for $i < n-2$ , $\theta ^{\circ }(\alpha _{n-1}) = \alpha _n$ , and $\theta ^{\circ }(\alpha _n) = \alpha _{n-1}$ .

We define $\mathrm {GSpin}_{2n}$ to be the Langlands dual group $\widehat {{\mathrm {GSO}}_{2n}}$ over ${\mathbb {C}}$ (or later over $\overline {\mathbb {Q}}_{\ell }$ via $\iota :{\mathbb {C}}\simeq \overline {\mathbb {Q}}_{\ell }$ ). That is, $\mathrm {GSpin}_{2n}$ is the connected reductive group over ${\mathbb {C}}$ , equipped with a Borel subgroup $B_{\mathrm {GSpin}}$ and a maximal torus $\mathrm {T}_{\mathrm {GSpin}}$ , whose based root datum is dual to the one of ${\mathrm {GSO}}_{2n}$ that we described above. In particular,

$$ \begin{align*}X_*(\mathrm{T}_{\mathrm{GSpin}})=X^{*}(\mathrm{T}_{{\mathrm{GSO}}}) \quad \mathrm{and} \quad X^{*}(\mathrm{T}_{\mathrm{GSpin}})=X_*(\mathrm{T}_{{\mathrm{GSO}}}). \end{align*} $$

Via the identification $X^{*}(\mathrm {T}_{{\mathrm {GSO}}}) = \mathbb Z^{n+1}$ , we represent elements $s \in \mathrm {T}_{\mathrm {GSpin}}$ as $(s_0, s_1, \ldots , s_n)$ . In Section 3, we will also define an explicit model of $\mathrm {GSpin}_{2n}$ over $\mathbb {Q}$ using Clifford algebras.

Lemma 2.4. There is a unique $\theta \in \mathrm {Aut}(\mathrm {GSpin}_{2n})$ that fixes $\mathrm {T}_{\mathrm {GSpin}}$ and $\mathrm {B}_{\mathrm {GSpin}}$ , switches $\alpha _{n-1}^{\vee }$ and $\alpha _n^{\vee }$ , leaves the other $\alpha _i^{\vee }$ invariant, and induces the trivial automorphism of the cocenter of $\mathrm {GSpin}_{2n}$ . We have $\theta ^2 = 1$ , and on the torus $\mathrm {T}_{\mathrm {GSpin}}$ , the involution $\theta $ is given by

(2.5) $$ \begin{align} (s_0, s_1, \ldots, s_n) \mapsto (s_0s_n, s_1, \ldots, s_{n-1}, s_n^{-1}). \end{align} $$

Proof. We have $\theta (e_i^{*} - e_{i+1}^{*}) = e_i^{*} - e_{i+1}^{*}$ ( $1 \leq i < n$ ) and $\theta (e_{n-1}^{*} - e_n^{*}) = e_{n-1}^{*} + e_n^{*}$ . Thus,

(2.6) $$ \begin{align} \theta(e_{i}^{*}) = e_{i}^{*} \ \ (1 \leq i < n) \quad \mathrm{and} \quad \theta(e_n^{*}) = -e_n^{*}. \end{align} $$

The center of ${\mathrm {GSO}}_{2n}$ is the image of $ \mathbb {G}_{\mathrm {m}} \owns z \mapsto (z^2, z, \ldots , z) \in \mathrm {T}_{{\mathrm {GSO}}}. $ The dual map is

(2.7) $$ \begin{align} \mathrm{T}_{\mathrm{GSpin}} \to \mathbb{G}_{\mathrm{m}}, \quad (s_0, s_1, \ldots, s_n) \mapsto s_0^2 s_1 \cdots s_n. \end{align} $$

Thus, $\theta (2e_0^{*} + e_1^{*} + \cdots + e_n^{*}) = 2e_0^{*} + e_1^{*} + \cdots + e_n^{*}$ , so $\theta (2e_0^{*}) -e_n^{*} = 2e_0^{*} + e_n^{*}$ and $\theta (e_0^{*}) = e_0^{*} + e_n^{*}$ .

The Weyl group of ${\mathrm {GSO}}_{2n}$ (and $\mathrm {GSpin}_{2n}$ ) is equal to $\{\pm 1\}^{n, \prime } \rtimes \mathfrak S_n$ , where $\{\pm 1\}^{n,\prime }$ is the group of $a \in \{\pm 1\}^{n}$ such that $\prod _1^n a(i) = 1$ . The action of $\mathrm {W}_{{\mathrm {GSO}}}$ on $\mathrm {T}_{{\mathrm {GSO}}}$ is determined by

(2.8) $$ \begin{align} \begin{cases} \sigma \cdot (t_0, t_1, \ldots, t_n) = (t_0, t_{\sigma(1)}, \ldots t_{\sigma(n)}) & \sigma \in \mathfrak S_n \cr a \cdot (t_0, t_1, \ldots, t_n) = (t_0, t_0 t_1^{-1}, t_0t_2^{-1}, t_3, \ldots, t_n) & a = (-1, -1, 1\ldots, 1) \in \{\pm 1\}^{n,\prime}. \end{cases} \end{align} $$

We define, for $\varepsilon \in \{\pm 1\}$ , the following cocharacter:

(2.9) $$ \begin{align} \mu_{\varepsilon} := \begin{cases} (1, 1, \ldots, 1, 1) & \mathrm{if } \varepsilon = (-1)^n \cr (1, 1, \ldots, 1, 0) & \mathrm{if } \varepsilon = (-1)^{n+1} \end{cases} \quad \in \mathbb Z^{n+1} = X_*(\mathrm{T}_{{\mathrm{GSO}}}) = X^{*}(\mathrm{T}_{\mathrm{GSpin}}). \end{align} $$

Then $\mu _{\varepsilon }$ is a minuscule cocharacter of ${\mathrm {GSO}}_{2n}$ with $\langle \alpha _i, \mu _{\varepsilon } \rangle = 1$ if and only if $i = n$ (for $\varepsilon = (-1)^n$ ) and $i = n-1$ (for $\varepsilon = (-1)^{n+1}$ ).

These representations will be realized explicitly via Clifford algebras. Our sign convention is natural in that $\mathrm {spin}^+$ (resp. $\mathrm {spin}^-$ ) accounts for even (resp. odd) degree elements. See (4.2) and Lemma 4.1 below.

The minuscule $\mu _{\varepsilon }$ has $2^{n-1}$ translates under the Weyl group action. Thus, each half-spin representation has dimension $2^{n-1}$ . More precisely, the weights of $\mathrm {spin}^{(-1)^n}_{2n}$ are

(2.10) $$ \begin{align} \mathrm{T}_{\mathrm{GSpin}} \owns (s_0, s_1, \ldots, s_n) \mapsto \left( s_0 \prod_{i \in U} s_i \right)_{U \subset \{1, 2, \ldots, n\},\atop 2|\# U } \in \mathbb Z^{n+1} = X_*(\mathrm{T}_{{\mathrm{GSO}}}) = X^{*}(\mathrm{T}_{\mathrm{GSpin}}), \end{align} $$

and $\mathrm {spin}^{(-1)^{n+1}}_{2n}$ has similar weights, except that the cardinality of U is now required to be odd. By computing the $\theta $ -action on weights, we verify that (see Lemma 4.4 for an explicit intertwiner)

$$ \begin{align*}\mathrm{spin}^+ \circ \theta \simeq \mathrm{spin}^-\quad\mbox{and}\quad~\mathrm{spin}^- \circ \theta \simeq \mathrm{spin}^+. \end{align*} $$

Proof. Since $\mathrm {GSpin}_{2n}$ is simple modulo the center, the kernel $Z^{\varepsilon } \subset \mathrm {GSpin}_{2n}$ must be central. The central character is the restriction of $\mu _{\varepsilon } \colon \mathrm {T}_{\mathrm {GSpin}} \to \mathbb {G}_{\mathrm {m}}$ to the center $Z(\mathrm {GSpin}_{2n}) \subset \mathrm {T}_{\mathrm {GSpin}}$ . Let $s = (s_0, s_1, \ldots , s_n) = (a,b) \in Z(\mathrm {GSpin}_{2n}) \subset \mathrm {T}_{\mathrm {GSpin}}$ . Then (see proof of Lemma 2.5)

(2.11) $$ \begin{align} \mu_{\varepsilon}(s) = \begin{cases} s_0 s_1 \cdots s_n = ab^n& \mathrm{if } \varepsilon = (-1)^n \cr s_0 s_1 \cdots s_{n-1} = ab^{n-1}& \mathrm{if } \varepsilon = (-1)^{n+1}. \end{cases} \end{align} $$

The first assertion follows by considering the 4 different cases where n even or odd and $\varepsilon = \pm 1$ . For the second point, it suffices to observe that $Z^+ \cap Z^- = \{1\}$ .

Later on, the following fact on ${\mathrm {SO}}_{2n-1}$ will be needed, so we record it here.

3 Clifford algebras and Clifford groups

We recall how $\mathrm {GSpin}_{2n}$ is realized using the Clifford algebra, and we define a number of fundamental maps such as $i_{\mathrm {std}} \colon \mathrm {GSpin}_{2n-1} \hookrightarrow \mathrm {GSpin}_{2n}$ and the projections from $\mathrm {GSpin}_{2n}$ to ${\mathrm {GSO}}_{2n}$ and ${\mathrm {SO}}_{2n}$ . We also give a concrete definition of outer automorphisms $\theta $ of $\mathrm {GSpin}_{2n}$ and $\theta ^{\circ }$ of ${\mathrm {GSO}}_{2n}$ . Our main reference is [Reference Bass4], which introduces Clifford algebras over arbitrary commutative rings (with unity). Other useful references are [Reference Bourbaki9, §9] and [Reference Fulton and Harris25, §20].

Let V be a quadratic space over $\mathbb {Q}$ with quadratic form Q, giving rise to the groups $\mathrm {O}(V)$ , $\mathrm {GO}(V)$ , ${\mathrm {SO}}(V)$ and ${\mathrm {GSO}}(V)$ . The Clifford algebra $C(V)$ is a universal map $V \to C(V)$ which is initial in the category of $\mathbb {Q}$ -linear maps $f \colon V\to A$ into associative $\mathbb {Q}$ -algebras A with unity $1_A$ such that $f(v)^2=Q(v)\cdot 1_A$ for all $v\in V$ . (See [Reference Bass4, (2.3)] or [Reference Bourbaki9, §9.1].)

We define $\langle x, y \rangle := Q(x+y) - Q(x) - Q(y)$ for $x,y \in V$ , and similarly, $\langle x, y \rangle = (x+y)^2 - x^2 - y^2$ for $x,y \in C(V)$ . In particular, $\langle x, y \rangle $ measures if x and y anti-commute in $C(V)$ :

(3.1) $$ \begin{align} \langle x, y \rangle = (x+y)^2 - x^2 - y^2 = xy + yx \in C(V). \end{align} $$

The map $V\to C(V)$ induces a map $V \to C(V)^{\mathrm {opp}}$ (sending each $v\in V$ to the same element), where $C(V)^{\mathrm {opp}}$ is the opposite algebra. The latter factors through a unique $\mathbb {Q}$ -algebra map $\beta \colon C(V) \to C(V)^{\mathrm {opp}}$ . It is readily checked that $\beta ^2$ is the identity on $C(V)$ . By the universal property, $\beta $ is the unique involution of $C(V)$ that is the identity on V.

The universal property also yields a surjection from the tensor algebra

$$ \begin{align*}\bigoplus_{d \in \mathbb Z_{\geq 0}} V^{\otimes d} \twoheadrightarrow C(V). \end{align*} $$

Define $C^+ = C(V)^+$ (resp. $C^- = C(V)^-$ ) to be the image of $\oplus _{d \in \mathbb Z_{\geq 0}} V^{\otimes 2d}$ (resp. $\oplus _{d \in \mathbb Z_{\geq 0}} V^{\otimes 2d+1}$ ) so that $C(V) = C(V)^+ \oplus C(V)^-$ . In fact, the discussion of Clifford algebras so far works when V is replaced with a quadratic space on a module over an arbitrary commutative ring, in a way compatible with base change: in particular, if R is a (commutative) $\mathbb {Q}$ -algebra, then $C(V\otimes _{\mathbb {Q}} R)=C(V)\otimes _{\mathbb {Q}} R$ [Reference Bourbaki9, §9.1, Prop 2]. By scalars in $C(V\otimes _{\mathbb {Q}} R)$ , we mean R times the multiplicative unity. We keep using $\beta $ to denote the main involution of $C(V\otimes _{\mathbb {Q}} R)$ .

The Clifford group $\mathrm {GPin}(V)$ is the $\mathbb {Q}$ -group such that for every $\mathbb {Q}$ -algebra R,

$$ \begin{align*}\mathrm{GPin}(V)(R)=\{ x\in C(V\otimes_{\mathbb{Q}} R)^{\times} : x(V\otimes_{\mathbb{Q}} R) x^{-1} = V\otimes_{\mathbb{Q}} R, x \ \mathrm{is \ homogeneous}\}, \end{align*} $$

where homogeneity of x means that $x \in C(V\otimes _{\mathbb {Q}} R)^{\varepsilon }$ for some sign $\varepsilon $ . The special Clifford group $\mathrm {GSpin}(V)$ is defined similarly with $C^+$ in place of C. The embedding of invertible scalars in $C(V\otimes _{\mathbb {Q}} R)$ induces a central embedding

(3.2) $$ \begin{align} \mathbb G_m\rightarrow \mathrm{GSpin}(V). \end{align} $$

Since $x\beta (x)\in R$ for $x\in C(V\otimes _{\mathbb {Q}} R)$ by [Reference Bass4, Prop 3.2.1 (a)], we have the spinor norm morphism

$$ \begin{align*}\mathcal N \colon \mathrm{GPin}(V)\rightarrow \mathbb{G}_{\mathrm{m}},\qquad x\mapsto x\beta(x) \end{align*} $$

over $\mathbb {Q}$ . (The involution in loc. cit. differs from our $\beta $ by $C(-1_P)$ in their notation, so our $\mathcal N$ does not coincide with their N, but $\mathcal N$ and N have the same kernel.) Evidently, composing $\mathcal N$ with (3.2) yields the squaring map.

Define $\mathrm {Spin}(V)$ by the following exact sequence of algebraic groups:

$$ \begin{align*}1\rightarrow \mathrm{Spin}(V)\rightarrow \mathrm{GSpin}(V) \stackrel{\mathcal N}{\rightarrow} \mathbb G_m\rightarrow 1. \end{align*} $$

Proof. (i) The surjectivity can be checked on field-valued points. This is proved in [Reference Bourbaki9, §9.5, Thm. 4].

(ii) As $V\subset C(V)$ generates the Clifford algebra, the identity $xvx^{-1}=v$ implies $xyx^{-1}=y$ for all $y\in C(V)$ , and the analogue holds for $C(V\otimes _{\mathbb {Q}} R)$ for $\mathbb {Q}$ -algebras R. Thus, $\ker (\mathrm {pr}^{\circ })(R)$ consists of invertible elements in the center of $C(V\otimes _{\mathbb {Q}} R)$ . Let $W\subset V$ be an isotropic subspace. Then $C(V\otimes _{\mathbb {Q}} R)\simeq \mathrm {End}(\bigwedge (W\otimes _{\mathbb {Q}} R))$ as super R-algebras by [Reference Bass4, (2.4) Thm.], so the center of $C(V\otimes _{\mathbb {Q}} R)$ is R, implying that $\ker (\mathrm {pr}^{\circ }) = \mathbb {G}_{\mathrm {m}}$ .

(iii) We observe that $\mathrm {pr}(x)$ preserves V: as $x (V\otimes _{\mathbb {Q}} R) x^{-1} = V\otimes _{\mathbb {Q}} R$ and $x\beta (x)\in R^{\times }$ imply that $x(V\otimes _{\mathbb {Q}} R)\beta (x)=V\otimes _{\mathbb {Q}} R$ . Moreover, $\mathrm {pr}(x)\in \mathrm {GO}(V)$ as

(3.3) $$ \begin{align} Q(xv\beta(x))=xv\beta(x) xv\beta(x)=\mathcal N(x)^2 Q(v). \end{align} $$

Moreover, $\mathrm {pr}$ and $\mathrm {pr}^{\circ }$ coincide on $\mathrm {Pin}(V)$ , so $\mathrm {(S)O}(V)$ is in the image of $\mathrm {pr}$ . However, $\mathcal N$ is seen to be surjective by considering scalar elements, telling us that the image of $\mathrm {pr}$ also contains $\mathbb {G}_{\mathrm {m}}$ (scalar matrices in $\mathrm {GO}(V)$ ). Since $\mathbb {G}_{\mathrm {m}}$ and $\mathrm {(S)O}(V)$ generate $\mathrm {G(S)O}(V)$ , the surjectivity of $\mathrm {pr}$ follows. The equality $\mathrm {sim} \circ \mathrm {pr} = \mathcal N^2$ follows from (3.3).

(iv) The first part follows from $\mathrm {pr}(x)(v)=xv \beta (x)=xvx^{-1}x\beta (x)=\mathrm {pr}^{\circ }(x)(v) \mathcal N(x)$ when $x\in \mathrm {GPin}(V)$ and $v\in V$ . The second part is easily seen from (i) and (ii).

(v) This readily follows from the preceding points.

If V is odd dimensional, then ${\mathrm {SO}}(V) \times \{\pm 1\} = \mathrm {O}(V)$ , and the group $\mathrm {GO}(V)$ is connected. For convenience, we define ${\mathrm {GSO}}(V) := \mathrm {GO}(V)$ in this case. If $\dim (V)$ is even, then $\mathrm {O}(V)$ (resp. $\mathrm {GO}(V)$ ) has two connected components but does not admit a direct product decomposition into $\mathrm {O}(V)$ (resp. ${\mathrm {GSO}}(V)$ ) and $\{\pm 1\}$ .

Assume that we have an orthogonal sum decomposition $\varphi \colon W_1 \oplus W_2 \overset \sim \to V$ of nondegenerate quadratic spaces over $\mathbb {Q}$ . As super algebras, we have ([Reference Bass4, (2.3)] or [Reference Bourbaki9, §9.3, Cor. 3, Cor. 4])

$$ \begin{align*}C_{\varphi} \colon C(W_1) \widehat \otimes C(W_2) \overset \sim \to C(V), \quad w_1 \widehat \otimes w_2 \mapsto w_1 w_2. \end{align*} $$

By definition, the algebra given by $\widehat \otimes $ on the left side has underlying vector space $C(W_1) \otimes C(W_2)$ and product

$$ \begin{align*}(a \widehat \otimes b) \cdot (c \widehat \otimes d) := (-1)^{k_b \cdot k_c} ac \widehat \otimes bd, \end{align*} $$

if $a, c \in C(W_1)$ , $b, d \in C(W_2)$ are homogeneous elements of degree $k_a, k_b, k_c, k_d \in \mathbb Z/2\mathbb Z$ . The sign is there to make $C_{\varphi }$ compatible with products since $bc=(-1)^{k_b k_c}cb$ in $C(V)$ .

In fact, $C_{\varphi }$ intertwines the involution $\beta $ on $C(V)$ with the involution

$$ \begin{align*}\beta' \colon C(W_1) \widehat \otimes C(W_2) \to C(W_1) \widehat \otimes C(W_2),\quad \beta'(a \widehat \otimes b) = (-1)^{k_a k_b} \beta_1(a) \widehat \otimes \beta_2(b), \end{align*} $$

for homogeneous elements $a\in C(W_1)$ , $b\in C(W_2)$ of degree $k_a, k_b \in \mathbb Z/2\mathbb Z$ , where $\beta _1,\beta _2$ are the involutions of $C(W_1)$ and $C(W_2)$ (see below (3.1)). To verify that $\beta $ is compatible with $\beta '$ , observe that $\beta $ on $C(V)$ restricts to $\beta _1,\beta _2$ via the obvious inclusions $C(W_1)\hookrightarrow C(V)$ and $C(W_2)\hookrightarrow C(V)$ induced by $W_1\subset V$ and $W_2\subset V$ (since $\beta $ acts as the identity on both $W_1$ and $W_2$ ), and use the property that $\beta _1$ , $\beta _2$ , and $\beta $ are preserving degrees. It follows that

$$ \begin{align*}\beta(ab)= \beta(b)\beta(a)= (-1)^{k_a k_b}\beta(a)\beta(b)=(-1)^{k_a k_b}\beta_1(a)\beta_2(b). \end{align*} $$

Proof. We check that the image of $C_{\varphi }$ is in $\mathrm {GSpin}(V)$ . Let $g \in \mathrm {GSpin}(W_1)$ , $h \in \mathrm {GSpin}(W_2)$ . Note that $C_{\varphi }(g \widehat {\otimes }h)=gh \in C^+(V)$ . Let $w_1 + w_2 \in V$ with $w_i \in W_i$ , $i=1,2$ . To verify that $gh \in \mathrm {GSpin}(V)$ , since homogeneous elements of even degree commute with each other if they are perpendicular, we see that

$$ \begin{align*}gh(w_1+w_2)h^{-1}g^{-1} = gw_1g^{-1}+ hw_2h^{-1}\in V.\\[-42pt] \end{align*} $$

Lemma 3.3. The diagram

commutes, where $i_{W_1, W_2}$ is the block diagonal embedding.

In later chapters, we will carry out explicit computations. It will then be convenient to work with fixed bases and quadratic forms. For this reason, we now fix quadratic forms on the vector spaces $V_{2n}={\mathbb {C}}^{2n}$ and $V_{2n-1}={\mathbb {C}}^{2n-1}$ . We take the following quadratic forms:

(3.4) $$ \begin{align} Q_{2n} \colon x_1 x_{n+1} + x_2 x_{n+2} + \ldots + x_n x_{2n} & \mathrm{ on } {\mathbb{C}}^{2n} \cr Q_{2n-1} \colon y_1 y_{n+1} + \ldots + y_{n-2} y_{2n-2} + y_{2n-1}^2 & \mathrm{ on } {\mathbb{C}}^{2n-1}. \end{align} $$

Using them, we write ${\mathrm {SO}}_{m}={\mathrm {SO}}(V_{m})$ , ${\mathrm {GSO}}_{m}={\mathrm {GSO}}(V_{m})$ , and likewise for $\mathrm {O}_m$ , $\mathrm {GO}_m$ , for $m=2n$ and $m=2n-1$ . This is identical to the convention of §2 for m even. Similarly, we write $\mathrm {pr}^{\circ }_{2n-1}=\mathrm {pr}^{\circ }_{V_{2n-1}}$ and $\mathrm {pr}^{\circ }_{2n}=\mathrm {pr}^{\circ }_{V_{2n}}$ .

Now we claim that $\mathrm {GSpin}(V_{2n})$ is isomorphic to $\mathrm {GSpin}_{2n}$ of §8, that is, the Clifford algebra definition is compatible with the root-theoretic definition as the Langlands dual of ${\mathrm {GSO}}_{2n}$ . (An analogous argument shows that $\mathrm {GSpin}_{2n-1}$ is dual to $\mathrm {GSp}_{2n-2}$ .) As this is a routine exercise, we only sketch the argument. First, $\mathrm {pr}^{\circ }$ restricts to a connected double covering $\mathrm {Spin}(V_{m})\rightarrow {\mathrm {SO}}(V_{m})$ ([Reference Fulton and Harris25, Prop. 20.38]), which must then be the unique (up to isomorphism) simply connected covering. This determines the root datum of $\mathrm {Spin}(V_{m})$ . From this, we compute the root datum of $\mathrm {GSpin}(V_m)$ via the central isogeny $\mathrm {Spin}(V_{m})\times \mathbb G_m\rightarrow \mathrm {GSpin}(V_m)$ of Lemma 3.1. Finally, when $m=2n$ , we deduce that the outcome is dual to the root datum of ${\mathrm {GSO}}_{2n}$ in Lemma 2.1. Therefore, $\mathrm {GSpin}(V_{2n})$ is isomorphic to $\mathrm {GSpin}_{2n}$ of §8. Henceforth, we identify

(3.5) $$ \begin{align} \mathrm{GSpin}(V_{2n})=\mathrm{GSpin}_{2n}. \end{align} $$

In fact, we may and will choose $\mathrm {B}_{\mathrm {GSpin}}$ and $\mathrm {T}_{\mathrm {GSpin}}$ to be the preimages of $\mathrm {B}_{{\mathrm {SO}}}$ and $\mathrm {T}_{{\mathrm {SO}}}$ via $\mathrm {pr}^{\circ }: \mathrm {GSpin}_{2n}\rightarrow {\mathrm {SO}}_{2n}$ . We fix pinnings of $\mathrm {GSpin}_{2n}$ , ${\mathrm {GSO}}_{2n}$ , and ${\mathrm {SO}}_{2n}$ (which are $\Gamma _F$ -equivariant if $(V_{2n},Q_{2n})$ is defined over F) compatibly via $\mathrm {pr}$ and $\mathrm {pr}^{\circ }$ .

Proof. Write $Z^0$ for the identity component of the center of $\mathrm {GSpin}_{2n}$ , consisting of $(s_0,1,\ldots ,1)$ with $s_0\in {\mathbb G}_m$ in the notation of Lemma 2.5. The dual of $\mathrm {sim}:{\mathrm {GSO}}_{2n}\rightarrow {\mathbb G}_m$ is calculated as the central cocharacter $\mathbb G_m\rightarrow Z^0\subset \mathrm {GSpin}_{2n}$ , $z\mapsto (z,1,\ldots ,1)$ . The inclusion $\mathrm {cent}:\mathbb G_m\rightarrow \mathrm {GSpin}_{2n}$ identifies $\mathbb G_m$ with $Z^0$ . Thus, cent is dual to $\mathrm {sim}$ .

Both $\mathcal N\circ \mathrm {cent}$ and $\mathrm {sim}\circ \mathrm {cent}^{\circ }$ are the squaring map on $\mathbb G_m$ . Using the hat symbol to denote a dual morphism, we see that

$$ \begin{align*}\mathcal N\circ\mathrm{cent}=\widehat{\mathrm{cent}^{\circ}}\circ \widehat{\mathrm{sim}} = \widehat{\mathrm{cent}^{\circ}}\circ \mathrm{cent} \end{align*} $$

and that they are all equal to the squaring map. It follows that $\mathcal N$ is dual to $\mathrm {cent}^{\circ }$ .

We have the morphism of quadratic spaces

$$ \begin{align*}\varphi \colon ({\mathbb{C}}^{2n-1}, Q_{2n-1}) \to ({\mathbb{C}}^{2n}, Q_{2n}), \quad y \mapsto (y_1, y_2, \ldots, y_{n-1}, y_{2n-1}, y_n, y_{n+1}, \ldots, y_{2n-1}). \end{align*} $$

Indeed, $Q_{2n} \varphi = Q_{2n-1}$ as readily checked. We have the complementary embedding:

$$ \begin{align*}\varphi' \colon {\mathbb{C}} \to {\mathbb{C}}^{2n}, \quad u \mapsto x, \quad \mathrm{where} \quad \begin{cases} x_k = 0 & k \neq n,\ 2n \cr x_k = (-1)^{k/n} u & \mathrm{if}\ k = n\ \mathrm{or}\ k = 2n. \end{cases} \end{align*} $$

Write $U := \varphi '({\mathbb {C}})=(e_n-e_{2n})\cdot {\mathbb {C}}$ for the image. The induced quadratic form on U is then $a\cdot (e_n-e_{2n})\mapsto -a^2$ . This gives us an orthogonal decomposition of quadratic spaces ${\mathbb {C}}^{2n} = {\mathbb {C}}^{2n-1} \widehat {\oplus } U$ . Let $\mathrm {PO}_m$ denote the adjoint group of $\mathrm {O}_m$ . The decomposition induces morphisms (cf. Lemmas 3.2, 3.3)

(3.6) $$ \begin{align} i_{\mathrm{std}} &:= C_{\varphi, \varphi'} \colon \mathrm{GSpin}_{2n-1} \times \mathrm{GSpin}_1 \to \mathrm{GSpin}_{2n}, \cr i_{\mathrm{std}}^{\circ} &:= i_{{\mathbb{C}}^{2n-1}, {\mathbb{C}}} \colon \mathrm{O}_{2n-1} \times \mathrm{O}_1 \to \mathrm{O}_{2n},~~\mbox{and} \cr \overline {i_{\mathrm{std}}} &:= \mathrm{PO}_{2n-1} \to \mathrm{PO}_{2n}, \end{align} $$

where $\overline {i_{\mathrm {std}}}$ is induced from $i_{\mathrm {std}} \colon \mathrm {GSpin}_{2n-1} \times \mathrm {GSpin}_1 \to \mathrm {GSpin}_{2n} \twoheadrightarrow \mathrm {PSO}_{2n}\subset \mathrm {PO}_{2n}$ . By Lemma 3.3, we have $\mathrm {pr}^{\circ } \circ i_{\mathrm {std}}=i_{\mathrm {std}}^{\circ }\circ (\mathrm {pr}^{\circ }_{2n-1}\times \mathrm {pr}^{\circ }_{U})$ .

Let $1_{2n-1}, 1_U$ denote the identity map on ${\mathbb {C}}^{2n-1},U$ . Then (cf. (2.4))

$$ \begin{align*} i_{\mathrm{std}}^{\circ}(-1_{2n-1}, 1_U) =-\left( \begin{array}{cc;{1pt/1pt}cc}1_{n-1} & & & \\ & 0 & & 1 \\ \hdashline[1pt/1pt] & & 1_{n-1} & \\ & 1& & 0 \end{array}\right) = \vartheta^{\circ} \in \mathrm{O}_{2n}. \end{align*} $$

Fix $\sqrt {-1}\in \mathbb G_m=Z(\mathrm {GPin}_{2n})$ . Define

(3.7) $$ \begin{align} \vartheta := \sqrt{-1}\cdot i_{\mathrm{std}}(1_{C({\mathbb{C}}^{2n-1})} \widehat \otimes (e_n-e_{2n})) = \sqrt{-1}(e_n - e_{2n}) \in \mathrm{GPin}_{2n}\backslash \mathrm{GSpin}_{2n}. \end{align} $$

We have fixed pinnings of $\mathrm {GSpin}_{2n}$ , ${\mathrm {GSO}}_{2n}$ and ${\mathrm {SO}}_{2n}$ compatibly via $\mathrm {pr}$ . They are fixed by $\theta \in \mathrm {Aut}(\mathrm {GSpin}_{2n})$ and $\theta ^{\circ }\in \mathrm {Aut}({\mathrm {GSO}}_{2n})$ . It is easy to see that $\theta $ and $\theta ^{\circ }$ induce automorphisms of based root data, which correspond to each other via duality of the two based root data. Thus, letting $E/F$ be a quadratic extension of fields of characteristic 0, and ${\mathrm {GSO}}_{2n}^{E/F}$ an outer form of ${\mathrm {GSO}}_{2n}$ over F with respect to the Galois action $\Gamma _{E/F}=\{1,c\}\stackrel {\sim }{\to } \{1,\theta \}$ , we can identify

$$ \begin{align*}^L ({\mathrm{GSO}}_{2n}^{E/F}) = \mathrm{GSpin}_{2n}\rtimes \{1,c\} = \mathrm{GPin}_{2n}, \end{align*} $$

where the semi-direct product is given by $c g c^{-1} = \theta (g)$ . (Of course, $c=c^{-1}$ .) The second identification above is via $c\mapsto \vartheta $ . Similarly, for ${\mathrm {SO}}_{2n}^{E/F}$ an outer form of ${\mathrm {SO}}_{2n}$ with respect to $\Gamma _{E/F}=\{1,c\}\stackrel {\sim }{\to } \{1,\theta ^{\circ }\}$ , we have

$$ \begin{align*}^L ({\mathrm{SO}}_{2n}^{E/F}) = {\mathrm{SO}}_{2n}\rtimes \{1,c\} = \mathrm{O}_{2n}\qquad\mbox{via}\quad c\mapsto \vartheta^{\circ}. \end{align*} $$

Let us describe the center $Z(\mathrm {Spin}_{2n})$ of $\mathrm {Spin}_{2n}=\mathrm {Spin}(V_{2n})$ explicitly as this is going to be useful for classifying inner twists of (quasi-split forms of) ${\mathrm {SO}}_{2n}$ and ${\mathrm {GSO}}_{2n}$ in §8. In what follows, we identify $Z(\mathrm {GSpin}_{2n})=\{(s_0,s_1)\,:\,s_0\in \mathbb G_m,~s_1\in \{\pm 1\}\}$ as in Lemma 2.5 and write $1,-1$ for $(1,1),(1,-1)\in Z(\mathrm {GSpin}_{2n})$ .

Proof. We have $Z(\mathrm {Spin}_{2n})=Z(\mathrm {GSpin}_{2n}) \cap \mathrm {Spin}_{2n}= \{z\in Z(\mathrm {GSpin}_{2n}): \mathcal N(z)=1\}$ , where $\mathcal N$ is described by (2.7) (Remark 3.5). It follows from Lemma 2.5 that

$$ \begin{align*}Z(\mathrm{Spin}_{2n})=\{(s_0,s_1): s_0^2 = s_1^n\},\end{align*} $$

which is alternatively described as in (i). Assertion (ii) is also clear from that lemma.

4 The spin representations

We recollect how to construct the spin representations via Clifford algebras, and we show that they coincide with the highest weight representations in Section 2. We also check some compatibility of maps that will become handy.

Consider the quadratic space $V_{2n} := {\mathbb {C}}^{2n}$ from (3.4) with standard basis $\{e_1,..,e_{2n}\}$ and quadratic form $Q_{2n}$ . Define $W_{2n} := \oplus _{i=1}^n {\mathbb {C}} e_i$ and $W^{\prime }_{2n} := \oplus _{i=n+1}^{2n} {\mathbb {C}} e_i$ . We often omit the subscript $2n$ to lighten notation, when there is no danger of confusion. Since W is isotropic, we obtain a morphism $\bigwedge W \overset \sim \to C(W) \hookrightarrow C(V)$ . Through this injection, we view $\bigwedge W$ as a subspace of $C(V)$ . The space $\bigwedge W$ carries an $C(V)$ -module structure

$$ \begin{align*}\mathrm{spin} \colon C(V) \to \mathrm{End}(\bigwedge W) \end{align*} $$

that is uniquely characterized by the following:

• • $w \in W \subset V$ acts through left multiplication,

• • and $w' \in W' \subset V$ acts as

  (4.1) $$ \begin{align} w' (w_1 \wedge w_2 \wedge \cdots \wedge w_r) = \sum_{i=1}^r (-1)^{i+1} \langle w', w_i \rangle (w_1 \wedge w_2 \wedge \cdots \wedge \widehat {w_i} \wedge \cdots \wedge w_r) \end{align} $$
  on $w_1 \wedge \cdots \wedge w_r \in \bigwedge ^r W \subset \bigwedge W$ .

The subspaces ${\bigwedge }^+ W := \bigwedge _{i \in 2\mathbb Z_{\geq 0}} W$ and ${\bigwedge }^- W := \bigwedge _{i \in 1+2\mathbb Z_{\geq 0}} W$ of $\bigwedge W$ are stable under $C^+(V)$ . By restriction, we obtain the spin representations

(4.2) $$ \begin{align} \mathrm{spin} \colon \mathrm{GPin}_{2n} \to {\mathrm{GL}} \left( \bigwedge W \right) \quad \mathrm{and} \quad \mathrm{spin}^{\pm} \colon \mathrm{GSpin}_{2n} \to {\mathrm{GL}} \left( {\bigwedge}^{\pm} W \right). \end{align} $$

We recall that the representations $\mathrm {spin}^{\pm }$ are irreducible. In (4.5) and (4.6) below, we will choose (ordered) bases for $ \bigwedge W$ and $ \bigwedge ^{\pm } W$ coming from $\{e_1,\ldots ,e_n\}$ to view $\mathrm {spin}$ and $\mathrm {spin}^{\pm }$ as ${\mathrm {GL}}_{2^n}$ and ${\mathrm {GL}}_{2^{n-1}}$ -valued representations, respectively. We had another definition of $\mathrm {spin}^{\varepsilon }$ as the representation with highest weight $\mu _{\varepsilon }$ (Definition 2.6), $\varepsilon \in \{+,-\}$ . Let us check that the two definitions coincide via (3.5).

Let us introduce a bilinear pairing on $\bigwedge W$ which is invariant under the spin representation up to scalars. Let $\mathrm {pr}_n: \bigwedge W\rightarrow {\mathbb {C}}$ denote the projection onto $\bigwedge ^n W$ , identified with ${\mathbb {C}}$ via $e_1\wedge \cdots \wedge e_n \mapsto 1$ . Write $\tau : \bigwedge W\stackrel {\sim }{\to } \bigwedge W$ for the ${\mathbb {C}}$ -linear anti-automorphism $w_1\wedge \cdots \wedge w_r \mapsto w_r \wedge \cdots \wedge w_1$ for $r\ge 1$ and $w_1,\ldots ,w_r\in W$ . Define

$$ \begin{align*}(\!( \dot w_1,\dot w_2 )\!):=\mathrm{pr}_n(\tau(\dot w_1) \wedge \dot w_2),\qquad \dot w_1,\dot w_2\in \bigwedge W. \end{align*} $$

We write $\mathrm {spin}^{\vee }$ and $\mathrm {spin}^{\varepsilon ,\vee }$ for the dual representations of $\mathrm {spin}$ and $\mathrm {spin}^{\varepsilon }$ . By the preceding lemma, the highest weight of $\mathrm {spin}^{\varepsilon ,\vee }$ is in the Weyl group orbit of $(\mu _{\varepsilon })^{-1}$ .

Lemma 4.2. The pairing $(\!(~,~)\!)$ is nondegenerate; it is alternating if $n\equiv 2,3~(\mathrm {mod}~4)$ and symmetric if $n\equiv 0,1~(\mathrm {mod}~4)$ . The restriction of $(\!(~,~)\!)$ to each of $\bigwedge ^+ W$ and $\bigwedge ^- W$ is nondegenerate if n is even, and identically zero if n is odd. We have

(4.3) $$ \begin{align} (\!( \mathrm{spin}(g)\dot w_1, \mathrm{spin}(g)\dot w_2 )\!)=\mathcal N(g) (\!( \dot w_1,\dot w_2 )\!),\qquad g\in \mathrm{GPin}_{2n}({\mathbb{C}}),~~\dot w_1,\dot w_2\in \bigwedge W. \end{align} $$

In particular, we have $\mathrm {spin}^{\varepsilon } \simeq \mathrm {spin}^{(-1)^n \varepsilon ,\vee } \otimes \mathcal N$ .

Proof. The first two assertions are elementary and left to the reader. The last assertion follows from the rest. For the equality (4.3), we claim that

(4.4) $$ \begin{align} (\!( c \dot w_1, \dot w_2 )\!)= (\!( \dot w_1,\beta(c) \dot w_2 )\!),\qquad c\in C(V),~\dot w_1,\dot w_2\in \bigwedge W. \end{align} $$

Since $\mathrm {GPin}_{2n}\subset C(V)$ , this implies (4.3) as

$$ \begin{align*}(\!( \mathrm{spin}(g)\dot w_1, \mathrm{spin}(g)\dot w_2 )\!)=(\!( \dot w_1, \mathrm{spin}(\beta(g)g)\dot w_2 )\!)=\beta(g)g(\!(\dot w_1, \dot w_2 )\!). \end{align*} $$

It remains to prove the claim. The proof of (4.4) reduces to the case $c\in V$ , then to the two cases $c\in W$ and $c\in W'$ by linearity. In both cases, (4.4) follows from the explicit description of the $C(V)$ -action as in (4.1). Indeed, (4.4) is obvious if $c\in W$ . When $c\in W'$ , it is enough to show that for $0\le r,s\le n$ , $1\le i_1<\cdots <i_r\le n$ , $1\le j_1<\cdots <j_s\le n$ , and $1\le k\le n$ ,

$$ \begin{align*}\tau(e_{n+k}(e_{i_1}\wedge \cdots \wedge e_{i_r})) \wedge (e_{j_1}\wedge \cdots \wedge e_{j_s}) = \tau(e_{i_1}\wedge \cdots \wedge e_{i_r})\wedge \left( e_{n+k}(e_{j_1}\wedge \cdots \wedge e_{j_s})\right). \end{align*} $$

(This implies (4.4) by taking $\mathrm {pr}_n$ .) The equality is simply $0=0$ unless $k=r_0=s_0$ for some $1\le r_0\le r$ and $1\le s_0\le s$ . In the latter case, the equality boils down to

$$ \begin{align*}&(-1)^{r_0+1} e_{i_r}\wedge \cdots \wedge \widehat{e}_{i_{r_0}}\wedge \cdots \wedge e_{i_1}\wedge e_{j_1}\wedge \cdots \wedge e_{j_s} \\&\quad= (-1)^{s_0+1} e_{i_r}\wedge \cdots \wedge e_{i_1}\wedge e_{j_1}\wedge \cdots \wedge \widehat{e}_{j_{s_0}} \wedge \cdots \wedge e_{j_s},\end{align*} $$

which is clear. The proof is complete.

We also discuss the odd case. Equip $V_{2n-1}={\mathbb {C}}^{2n-1}$ with standard basis $\{f_1,\ldots ,f_{2n-1}\}$ and quadratic form $Q_{2n-1}$ of (3.4). As in [Reference Fulton and Harris25, p.306], we decompose

$$ \begin{align*}V_{2n-1} := {\mathbb{C}}^{2n-1} = W_{2n-1} \oplus W_{2n-1}' \oplus U_{2n-1}, \end{align*} $$

where $W_{2n-1}:=\oplus _{i=1}^{n-1} {\mathbb {C}} f_i$ , $W_{2n-1}':=\oplus _{i=n}^{2n-2} {\mathbb {C}} f_i$ , and $U_{2n-1}:={\mathbb {C}} f_{2n-1}$ . Again, we omit the subscript $2n-1$ when it is clear from the context. Then W and $W'$ are $(n-1)$ -dimensional isotropic subspaces, and U is a line perpendicular to them. As in the even case, each of $\bigwedge W $ and $\bigwedge ^{\pm } W$ can be viewed as a subspace of $C(V)$ and has a unique structure of left $C(V)$ -module where

• • $w \in W \subset V$ acts on $\bigwedge W$ through left multiplication,

• • $w' \in W' \subset V$ acts as in (4.1) (cf. [Reference Fulton and Harris25, 20.16]),

• • $f_{2n-1}$ acts trivially on $\bigwedge ^+ W$ and as $-1$ on $\bigwedge ^- W$ .

Consider the bijection

$$ \begin{align*}\psi \colon \bigwedge W_{2n-1} \overset \sim \to {\bigwedge}^+ W_{2n}, \quad w_1 \wedge \cdots \wedge w_r \mapsto \begin{cases} w_1 \wedge \cdots \wedge w_r \wedge e_n, & r \mathrm{ \ odd} \cr w_1 \wedge \cdots \wedge w_r, & r \mathrm{ \ even}. \end{cases} \end{align*} $$