Exceptional isomorphisms for low-rank general spin groups

In this post, we define general spin groups via Clifford algebra and give exceptional isomorphisms with other groups for low rank cases. Main references are the chapter 2 of M. Emory’s “On the global Gan-Gross-Prasad conjecture for general spin groups”, and T. Y. Lam’s “Introduction to Quadratic Forms over Fields”, but with a bit of more details.

Clifford algebra and general spin groups

Let $F$ be a field of characteristic $\neq 2$ and $(V, q)$ be a quadratic space over $F$ of dimension $n$. It means that $q: V \to F$ is a map such that $q(av) = a^2 q(v)$ for all $a \in F$ and $v \in V$, and $q(v + w) - q(v) - q(w)$ is a bilinear map. We often associate a bilinear map $B = B_q: V \times V \to F$ as

\[B(v, w) = \frac{1}{2} (q(v+w) - q(v) - q(w))\]

which recovers $q$ by $q(v) = B(v, v)$. We say that $v$ and $w$ are orthogonal if $B(v, w) = 0$. Also, we say that $V$ is regular if $B$ defines a non-degenerate bilinear paring on $V$, i.e. $v \mapsto B(v, -)$ is an isomorphism between $V$ and $V^{\ast}$. For regular quadratic spaces, we define their discriminant (signed determinant) as

\[\mathrm{disc}(V) = (-1)^{\frac{n(n-1)}{2}} \det(V) \in F^\times / (F^\times)^2\]

where $\det(V) = \det(Q)$ is the determinant of a matrix $Q = Q_q$ representing the quadratic form $q$. For two isomorphic quadratic spaces $(V, q) \simeq (V, q’)$, there exists $C \in \mathrm{GL}(V)$ such that $Q_{q’} = C^t Q_{q} C$ hence $\det(Q_{q’}) = \det(Q_q) \det(C)^2$ and $\det(Q_q)$ are in the same equivalence class of $F^\times / (F^\times)^2$.

Clifford algebra $C(V)$ associated to the quadratic space $(V, q)$ is defined as a universal unital associative algebra $C(V) = C(V, q)$ over $F$ with inclusion $i: V \to C(V)$ satisfying $i(v)^2 = q(v)\cdot 1$ for all $v \in V$. It satisfies the following universal property: given any unital associative algebra $A$ over $F$ with a linear map $j: V \to A$ satisfying $j(v)^2 = q(v) \cdot 1_A$ for all $v \in V$, it factors through $i$. Explicitly, it can be constructed as follows. Let $T(V)$ be a tensor algebra associated to $V$, which is

\[T(V) = F \oplus V \oplus V^{\otimes 2} \oplus V^{\otimes 3} \oplus \cdots.\]

Let $I(V)$ be an ideal of $T(V)$ generated by the elements of the form

\[v \otimes v - q(v)\cdot 1\]

for $v \in V$. Then we can define $C(V)$ as $C(V):= T(V) / I(V)$, which satisfies the above universal property. We also write $v_1 \otimes v_2 \otimes \cdots \otimes v_n = v_1 v_2 \cdots v_n$, and especially we have $v^2 = q(v)$. One can easily check that $vw = -wv$ if $v$ and $w$ are orthogonal. From this, we can construct a basis of $C(V)$ as follows: for an orthogonal basis $\{v_1, \dots, v_n\}$ of $V$, each element of $C(V)$ is a linear combination of elements of the form $v_{i_1}v_{i_2} \cdots v_{i_k}$ with $i_1 < i_2 < \cdots < i_k$, so if we define

\[C^k(V) = \left\{ \sum_i c_i v_{i_1} \cdots v_{i_k} : i_1 < i_2 < \cdots < i_k\right\}\]

then we have

\[C(V) = C^0(V) \oplus C^1(V) \oplus \cdots \oplus C^k(V)\]

and its dimension (as an $F$-vector space) becomes

\[\sum_{k=0}^{n} \binom{n}{k} = 2^n.\]

From this viewpoint, $C(V)$ has a natural $\mathbb{Z} / 2$-grading (induced from that of $T(V)$) as

\[\begin{align*} C^+(V) &= C^0(V) \oplus C^2(V) \oplus C^4(V) \oplus \cdots \\ C^-(V) &= C^1(V) \oplus C^3(V) \oplus C^5(V) \oplus \cdots \\ \end{align*}\]

which we call the even and odd Clifford algebra, with $\dim C^+(V) = \dim C^-(V) = 2^{n-1}$.

From now, we will assume that $(V, q)$ is a regular quadratic space. Define the general spin group $\mathrm{GSpin}(V) = \mathrm{GSpin}(V, q)$ associated to $(V, q)$ as

\[\mathrm{GSpin}(V, q):= \{ g\in C^+(V) : g^{-1} \text{ exists and } gVg^{-1} = V\}.\]

One of the most important property of $\mathrm{GSpin}(V)$ is that it becomes a $\mathrm{GL}_1$-cover of the special orthogonal group $\mathrm{SO}(V) = \mathrm{SO}(V, q)$.

Theorem. There exists a short exact sequence of (algebraic) groups

\[1 \to F^\times \xrightarrow{\iota} \mathrm{GSpin}(V) \xrightarrow{p} \mathrm{SO}(V) \to 1\]

where $\mathrm{SO}(V)$ is the special orthogonal group associated to the quadratic space $(V, q)$ defined as

\[\mathrm{SO}(V):= \{g \in \mathrm{GL}(V): q(gv) = q(v)\text{ for all }v \in V, \det(g) = 1\}.\]

Proof.

First, we need to defined the maps occur in the exact sequence. The first map $\iota: F^\times \to \mathrm{GSpin}(V)$ is given by the natural inclusion $F^\times \hookrightarrow F \hookrightarrow C^0(V) \hookrightarrow C^+(V)$, whose image is $C^0(V)^\times \simeq F^\times$ (and injective). For the second map, we define $p: \mathrm{GSpin}(V) \to \mathrm{SO}(V)$ as $$ p(g): v \mapsto gvg^{-1}. $$ This is well-defined as a may $\mathrm{GSpin}(V) \to \mathrm{GL}(V)$ by definition of $\mathrm{GSpin}(V)$ (we have $p(g)^{-1} = p(g^{-1})$), and it also preserves the quadratic form since $$ q(gvg^{-1}) = (gvg^{-1})^{2} = gv^{2}g^{-1} = q(v) \cdot gg^{-1} = q(v). $$ Also, it has determinant 1 since it is a conjugate of the identity map $v \mapsto v$. Hence its image is in $\mathrm{SO}(V)$. To prove surjectivity of $p$, we use the following theorem:

Theorem (Cartan-Dieudonné). Let $(V, q)$ be a regular quadratic space of dimension $n$. Then every $\sigma \in \mathrm{O}(V, q)$ is a product of at most $n$ reflections. In other words, there exists anistropic vectors $w_1, \dots, w_r \in V$ with $r \leq n$ such that $\sigma = \tau_{w_1} \cdots \tau_{w_r}$, where $$ \tau_w(v):= v - \frac{2B_q(v, w)}{q(w)}w. $$

Proof of the theorem can be found in [Lam, page 20]. Now, if $\sigma = \tau_{w_1} \cdots \tau_{w_r} \in \mathrm{SO}(V)$, then $$ \det(\sigma) = 1 = \det(\tau_{w_1}) \cdots \det(\tau_{w_r}) = (-1)^r $$ and $r$ should be even. So it is enough to show that every product of two reflections, $\tau_{w} \tau_{w'}$, is in the image of $p$. In fact, a direct computation gives that in $C(V)$, we have $$ \tau_w(x) = x - \frac{2B_q(x, w)}{q(w)} w = x - \frac{(xw + wx)}{q(w)} w = -wxw^{-1} $$ hence $$ \tau_w(\tau_{w'}(x)) = -w(-w'xw'^{-1})w^{-1} = (ww')x(ww')^{-1} $$ and $\tau_w \tau_w' = p(ww')$. Obviously, we have $\mathrm{img}(\iota) \subset \ker(p)$ since $\mathrm{img}(\iota)$ lies in the center of $\mathrm{GSpin}(V)$. For the reverse inclusion, it is equivalent to $Z(C(V)) = C^0(V) \simeq F$, which is true because Clifford algebras are central simple.

Note that the connected component of the center $Z^\circ$ of $\mathrm{GSpin}(V)$ is isomorphic to $F^\times$ for all $n \neq 2$.

There is a natural involution on the Clifford algebra, given by

\[(v_1 v_2 \cdots v_k)^* := v_k v_{k-1} \cdots v_1\]

where each $v_i$ is in $V$. This gives a map

\[N: C(V) \to C(V), \quad N(x) = xx^*\]

for $x \in C(V)$, which we call spinor norm. From anti-commutativity of orthogonal elements in $C(V)$, it follows that $C^+(V)$ is closed under the involution. In fact, we have the following.

Proposition. For $g \in \mathrm{GSpin}(V)$, $g^* \in \mathrm{GSpin}(V)$ and $N(g) \in C^0(V)^\times = F^\times$.

Proof [Scharlau, Lemma 3.2, page 335].

Let $g \in \mathrm{GSpin}(V)$. By definition, we have $gV = Vg$, and taking involution on the both sides gives $g^* V = V g^*$, so $g^* \in \mathrm{GSpin}(V)$. To show $gg^* \in C^0(V)^\times$, it is enough to show that $p(gg^*) = 1$ (where $p: \mathrm{GSpin}(V) \to \mathrm{SO}(V)$ is defined above). Since $v^* = v$ for all $v \in V$, we have $$ g^* v (g^{*})^{-1} = (g^* v (g^{-1})^{*})^* = g^{-1} v^* g = g^{-1} v g $$ so $$ gg^* v (gg^*)^{-1} = g(g^* v (g^*)^{-1})g^{-1} = g(g^{-1}vg)g^{-1} =v $$ for all $v \in V$. Hence $p(gg^*) = p(N(g)) = 1$ and $N(g) \in F^\times$. Note that Scharlau worked with slightly larger group called Clifford group.

Structures of Clifford algebra

To compute (low-rank) general spin groups, one needs to understand the structure of (even) Clifford algebras. They are so-called central simple graded algebras (CSGAs), and we will see the structure theorems of general CSGAs and how they applies to the Clifford algebras. This mostly follows the Lam’s book, chapter IV and V.

CSGAs are simply central simple algebras (CSAs) with $\mathbb{Z}/2$-grading. An $F$-algebra $A$ is a ($\mathbb{Z}/2$-)graded if it has a decomposition $A = A^+ \oplus A^-$ as even and odd parts, so that $F = F \cdot 1 \subset A^+$ and multiplication respects grading (e.g. if $a, b \in A^-$, then $ab \in A^+$). We call $a$ is homogeneous if $a \in h(A):= A^+ \cup A^-$, and for such $a$, we denote $|a|$ for its degree: $|a| = 0$ (resp. $1$) if $a \in A^+$ (resp. $a \in A^-$). $A$ is called simply graded algebra (SGA) if it has no proper graded ideal, and we say $A$ is CSGA if it is both SGA and CSA. For two graded algebras $A$ and $B$, we can take its graded tensor product $A \hat{\otimes} B$ whose grading is given by

\[\begin{align*} (A \hat{\otimes} B)^+ &= (A^+ \otimes B^+) \oplus (A^- \otimes B^-) \\ (A \hat{\otimes} B)^- &= (A^+ \otimes B^-) \oplus (A^- \otimes B^+) \end{align*}\]

and the multiplication is induced by

\[(a \otimes b) (a' \otimes b') = (-1)^{|b||a'|} (aa') \otimes (bb')\]

for $a, a’ \in h(A)$ and $b, b’ \in h(B)$. If $A$ and $B$ are CSGA, it can be shown that $A \hat{\otimes} B$ is also a CSGA [Lam, Theorem 2.3 (3), page 85].

Examples

(1) A quadratic extension $A = F(\sqrt{a})$ of $F$ ($a\in F^\times$) can be graded as $A^+ = F$ and $A^- = F \cdot \sqrt{a}$, which we denote as $F\langle \sqrt{a} \rangle$ to indicate the fact that $A$ is made into a graded algebra in this way. It can be shown that $F \langle \sqrt{a} \rangle$ is a CSGA, and it is even true when $a \in (F^\times)^2$ and $A \simeq F \oplus F$ (so is not a field).

(2) A quaternion algebra $C = \left(\frac{a, b}{F}\right)$, which can be graded as

\[C^+ = F \oplus F \cdot k, \quad C^- = F\cdot i \oplus F \cdot j.\]

As a previous example, we will denote the quaternion algebra with the grading as $\langle \frac{a, b}{F} \rangle$. In fact, we can check that

\[\left\langle \frac{a, b}{F} \right\rangle \simeq (F \oplus F \cdot i) \hat{\otimes} (F \oplus F \cdot j) = F\langle \sqrt{a} \rangle \hat{\otimes} F\langle \sqrt{b} \rangle\]

so is also CSGA.

(3) For a graded algebra $A$, there’s a natural grading on the matrix algebra $M_r(A)$ over $A$ (which we denote as $\tilde{M}_r(A)$):

\[\tilde{M}_r(A)^+ := M_r(A^+), \quad \tilde{M}_r(A)^- := M_r(A^-)\]

However, there’s one more interesting grading, so-called checkerboard grading (denote as $\hat{M}_r(A)$):

\[\begin{align*} \hat{M}_r(A)^+ := \begin{pmatrix} A^+ & A^- & \cdots \\ A^- & A^+ & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix} \\ \hat{M}_r(A)^- := \begin{pmatrix} A^- & A^+ & \cdots \\ A^+ & A^- & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix} \end{align*}\]

This may looks weird at first glance. However, when $A = A^+$ (i.e. $A$ is concentrated at even degree), such a grading coincides with the natural grading on $\mathrm{End}(V)$ of a graded vector space $V = V^+ \oplus V^-$ induced from it.

For a CSGA $A$ over $F$, we write $Z = Z(A) = Z^+ \oplus Z^- = F \oplus Z^-$ for the center of $A$ ($Z^- \subseteq A^-$). If $Z^- = 0$, we say that $A$ is of even type, and if $Z^- \neq 0$, then we say that $A$ is of odd type. One can check that $A$ is of even type iff $A$ is a CSA as an ungraded algebra. Also, $A$ is of odd type iff $A^- \neq 0$ and $A^+$ is a CSA over $F$, in which case $A$ is not a CSA.

Now, the following two theorems describe structures of odd and even type CSGAs. You can find the proof in [Lam].

Theorem (odd type, [Lam, Theorem 3.6, page 92]). Let $A$ be a CSGA of odd type. Then

(1) $Z(A) = F\langle \sqrt{a} \rangle$ for some $a \in F^\times$. The square class of $a$ is uniquely determined.

(2) There are graded algebra isomorphisms

\[A \simeq A^+ \hat{\otimes} F\langle \sqrt{a} \rangle \simeq A^+ \otimes F\langle \sqrt{a} \rangle\]

(3) If $a \not\in (F^\times)^2$, then $A$ is a CSA over $Z(A) \simeq F(\sqrt{a})$. If $a \in (F^\times)^2$, then $Z(A) \simeq F \times F$, and $A \simeq A^+ \times A^+$.

Theorem (even type [Lam, Theorem 3.8, page 94]). Let $A$ be a CSGA of even type, so is a CSA over $F$. Assume that $A \simeq M_n(D)$ for some central division algebra $D$ over $F$ (Wedderburn’s theorem). Then

(1) $Z(A^+) = F\langle \sqrt{a}\rangle$ for some $a \in F^\times$, whose square class is uniquely determined.

(2) Suppose $a \in (F^\times)^2$. Then $Z(A^+) \simeq F \times F$ and there exists a graded $F$-vector space $V = V^+ \oplus V^-$, such that $A \simeq \mathrm{End}(V) \hat{\otimes} D$ as graded algebras. Furthermore, $A^+ \simeq M_r(D) \times M_s(D)$ where $r =\dim V^+$ and $s = \dim V^-$.

(3) Suppose $a \not \in (F^\times)^2$ and $F(\sqrt{a})$ can be embedded into $D$. Then there exists a grading on $D$ such that $A \simeq \tilde{M}(D)$. In this case, $A^+ \simeq M_n(D^+)$ is a CSA over $F(\sqrt{a})$.

(4) Suppose $a \not \in (F^\times)^2$ and $F(\sqrt{a})$ cannot be embedded into $D$. Then $n = 2m$ is even, and $A \simeq (M_m(D)) \hat{\otimes} \langle \frac{-a, 1}{F} \rangle$ as graded algebras. In this case, $A^+ \simeq M_m(D) \otimes F(\sqrt{a})$ is a CSA over $F(\sqrt{a})$.

Now, let’s get back to the Clifford algebras. For two quadratic spaces $(V, q)$ and $(V’, q’)$, we have an isomorphism

\[C(V \oplus V) \simeq C(V) \hat{\otimes} C(V')\]

induced from

\[(x, x') \mapsto x \otimes 1 + 1 \otimes x'\]

for $x \in V$ and $x’ \in V$. From this and the previous results, by decomposing $V$ into orthogonal sum of 1-dimensional quadratic spaces, one can see that $C(V)$ is a CSGA and we can apply the above theorems. One needs to know the type of $C(V)$ (even or odd), and this is precisely determined by the dimension $n = \dim_F V$ of $V$.

Lemma. The type of $C(V)$ is same as $\dim_F V\,(\mathrm{mod}\,2)$.

Proof [Lam, Theorem 2.2, page 109].

Choose an orthogonal basis $e_1, \dots, e_n$ of $V$ and set $z = e_1 e_2 \cdots e_n$. Write $Z = Z(C(V)) = F \oplus Z^-$ ($Z^- \subseteq C^-(V)$). If $n$ is odd, then $ze_i = e_i z$ for all $i$, since $e_i e_j = -e_j e_i$ for $i \neq j$ and there are $(n-1)$ swaps from $ze_i$ to $e_i z$, which is even. Hence $z \in Z^-$ defines a nonzero element and $C(V)$ is of odd type. If $n$ is even, we have $z e_i = -e_i z$ for all $i$, so $z e_i e_j = e_i e_j z$ for all $i, j$. In particular, $z \in Z(C^+(V))$ and so $C^+(V)$ is not a CSA, which implies that $C(V)$ is of even type.

Now we can state the structure theorem for Clifford algebras, which follows from the above structure theorems for CSGAs.

Theorem. Let $n = \dim(V)$, $d = \mathrm{disc}(V)$, and $E = F(\sqrt{d})$. Then we have

\[C^+(V) = \begin{cases} A & 2 \nmid n \\ A \times A & 2 \mid n \text{ and } d = 1 \\ A_E & 2 \mid n \text{ and } d \neq 1 \end{cases}\]

where $A$ is a central simple algebra over $F$ and $A_E$ is a central simple algebra over $E$. Note that $\dim_F A = \dim_F C^+(V) = 2^{n-1}$ if $n$ is odd and $\dim_F A = \dim_E A_E = 2^{n-2}$ if $n$ is even. Also, when $n$ is even, $d \in (F^\times)^2$, and $C(V) \simeq M_t(D)$ for some central division algebra $D$ over $F$, then $t$ is a power of 2 and

\[C^+(V) \simeq M_{t/2}(D) \times M_{t/2}(D)\]

and $C^+(V) \subset C(V)$ corresponds to the diagonal embedding $M_{t/2}(D) \times M_{t/2}(D) \hookrightarrow M_t(D)$.

Proof [Lam, Theorem 2.4, Theorem 2.5, page 110].

When $n$ is odd, $C(V)$ is off odd type (by Lemma) and so $C^+(V)$ is a CSA over $F$. When $n$ is even, $C(V)$ is a CSA over $F$ and so $C(V) \simeq M_t(D)$ for some central division algebra $D$ over $F$. By observing the dimensions, we have $$ \dim_F M_t(D) = t^2 (\dim_F D) = \dim_F C(V) = 2^n, $$ so $t$ and $\dim_F D$ should be powers of 2. If $d \not \in (F^\times)^2$. Then $C^+(V)$ is a CSA over $E = F(\sqrt{d})$. (We can give a finer classification depending on whether $E$ embeds into $D$ or not, but we may not need this.) When $d \in (F^\times)^2$, choose an orthogonal basis $\\{ e_1, \dots, e_n\\}$ and let $z = e_1 e_2 \cdots e_n$. Then $z^2 = \mathrm{disc}(V)$ and we may assume $z^2 = 1$ by multiplying a scalar if necessary. Then $$ e = \frac{1 + z}{2}, \quad f = \frac{1 - z}{2} $$ are two elements in $C^+(V)$ that are orthogonal idempotents and $e + f = 1$. Hence the two simple factors of $C^+(V)$ are just $C^+(V)\cdot e$ and $C^+(V) \cdot f$. Now the reflection map $\tau = \tau_{e_1}: V \to V$ extends to an isomorphism $C(\tau): C(V) \to C(V)$ with $C(\tau)(z) = -e_1 e_2 \cdots e_n = -z$. Hence it swaps $e$ and $f$ and so $C^+(V)\cdot e \simeq C^+(V) \cdot f$. This implies $r = s = t / 2$.

We also needs to discuss about the types of involutions on $C(V)$. First, let $W$ be a quadratic space with the associated non-degenerate bilinear form $B:W \times W \to F$. It induces an isomorphism $\hat{B}: W \to W^\ast$. Using this, we may then define a map $\sigma_B: \mathrm{End}_F(W) \to \mathrm{End}_F(W)$ as

\[\sigma_B(f) := \hat{B}^{-1} \circ f^* \circ \hat{B}\]

where $f^\ast: W^\ast \to W^\ast$ is a dual (transpose) map of $f:W \to W$. Then $\sigma_B$ is a $F$-linear anti-automorphism of $\mathrm{End}_F(W)$, and one has the following theorem that allows us to define the types of ($F$-linear) involutions.

Theorem ([Knus et al., Theorem, page 1]). The above map induces a bijection between equivalence classes of non-degenerate bilinear forms on $W$ up to a nonzero constant and linear anti-automorphisms of $\mathrm{End}_F(W)$. Under this bijection, $F$-linear involutions on $\mathrm{End}_F(W)$ correspond to non-degenerate bilinear forms which are either symmetric or skew-symmetric.

There are two kinds of involutions, depending on whether the involution restricted to the center ($\simeq F$) fixes each element (first kind) or not (second kind). For the involutions of the first kind, we call it orthogonal if a corresponding bilinear form (under the above theorem) is symmetric, and symplectic otherwise (anti-symmetric). For the involutions of the second kind, the center is a quadratic étale algebra, i.e. either $F \times F$ or a quadratic field extension $E$ of $F$. Such involutions are called unitary, and it restricts to the canonical (non-identity) involution on the center: $(x, y) \mapsto (y, x)$ (when $F \times F$) or $x \mapsto \bar{x}$ (when $E/F$ is a quadratic extension and $x \mapsto \bar{x}$ is the Galois conjugation).

The following theorem determines the types of the canonical involution on $C^+(V)$ in terms of dimensions of $V$.

Theorem ([Knus et al., (8.4) Proposition, page 89]). The involution $*$ on $C^+(V)$ is

\[\begin{cases} \text{unitary} & n \equiv 2, 6\,(\mathrm{mod}\,8) \\ \text{symplectic} & n \equiv 3, 4, 5\, (\mathrm{mod}\,8) \\ \text{orthogonal} & n \equiv 0, 1, 7\, (\mathrm{mod}\,8) \end{cases}\]

and furthermore if $n\equiv 0, 4\,(\mathrm{mod}\,8)$ and $C^+(V) = A\times A$ then $*$ is orthogonal or symplectic type on each factor of $C^+(V)$.

In case of a quaternion algebra, the only symplectic involution is the canonical involution, i.e. the quaternion conjugation.

Proposition ([Knus et al. (2.21) Proposition, page 26]) The quaternion conjugation on a quaternion algebra $D$ is the unique symplectic involution on $D$.

Proof [Knus et al., (2.21) Proposition, page 26].

Let $\sigma$ be an arbitrary symplectic involution on $D$ and $\sigma_D$ be the canonical involution (quaternion conjugation) on $D$. Since $\sigma$ is an involution of the first kind (fixes every elements in the center), $\sigma \sigma_D = \sigma \sigma_D^{-1}$ is a $F$-linear automorphism. By Skolem-Noether theorem, it should be an inner automorphism $\mathrm{Inn}(u)$ by $u \in D^\times$, so $\sigma = \mathrm{Inn}(u) \circ \sigma_D$ ($\mathrm{Inn}(u)(x):= uxu^{-1}$). By a direct computation, we have $1 = \sigma^2 = \mathrm{Inn}(u \sigma_D(u)^{-1})$, so $\sigma_D(u) = \lambda u$ for some $\lambda \in F$. Taking $\sigma_D$ on both sides gives $\lambda^2 = 1 \Leftrightarrow \lambda = \pm 1$. One can check that $\lambda = 1$ because $\sigma$ and $\sigma_D$ has a same type ([Knus et al., (2.7) Proposition, page 17]).

Exceptional isomorphisms, for $n \leq 6$

Now let’s compute general spin groups for $n \leq 6$. To do this, we consider the similarity group $\mathrm{Sim}(V)$ defined as

\[\mathrm{Sim}(V):= \{g \in C^+(V): N(g) \in C^0(V)^\times = F^\times\}\]

which, by definition, contains $\mathrm{GSpin}(V)$. Both $\mathrm{Sim}(V)$ and $\mathrm{GSpin}(V)$ are connected as algebraic groups, so they should be the same if they have the same dimension. This happens when $n$ is small, as we will see.

First of all, using the short exact sequence above, we can compute the dimension of $\mathrm{GSpin}(V)$ as

\[\dim \mathrm{GSpin}(V) = \dim \mathrm{SO}(V) + \dim \mathrm{GL}_1 = \frac{n(n-1)}{2} + 1\]

where $n = \dim_F V$. Now, let’s compute $\mathrm{Sim}(V)$ for small $n$’s.

$n = 1$

By the structure theorem, $C^+(V)$ is a central simple algebra of dimension 1, hence $F$. Then

\[\begin{align*} \mathrm{Sim}(V) &= \{ g \in F: N(g) = g^2 \in F^\times \} = F^\times \end{align*}\]

and $\dim \mathrm{Sim}(V) = 1 = \dim \mathrm{GSpin}(V)$, hence

\[\mathrm{GSpin}(V) = \mathrm{Sim}(V) \simeq \mathrm{GL}_{1}.\]

$n = 2$

Since $n$ is even, there are two cases to consider.

Case 1. Assume $d = \mathrm{disc}(V) = 1$, so that $V = \mathbb{H}$ is a hyperbolic plane, i.e. the unique regular isotropic quadratic space (up to isomorphism), which can be also thought as $V = F \oplus F$ with $q(x, y) = xy$. Then $C^+(V) = A \times A$ for a central simple algebra $A$ over $F$ with $\dim_F A = 1$, so $A = F$ and $C^+(V) \simeq F \times F$. In this case, the unitary involution is the exchange involution $(a, b)^* := (b, a)$, and

\[\begin{align*} \mathrm{Sim}(V) &\simeq \{(a, b) \in F \times F: N(a, b)=(ab, ab) \in \Delta F^\times\} \\ & = F^\times \times F^\times \end{align*}\]

whose $F$-dimension is $2 = \frac{2(2-1)}{2} + 1$. Hence we have

\[\mathrm{GSpin}(V) = \mathrm{Sim}(V) \simeq \mathrm{GL}_1 \times \mathrm{GL}_1.\]

Case 2. If $d \neq 1$, then $V = E$ with norm form $q(x) = N_{E/F}(x) = xx^* = x\bar{x}$ ($\bar{x}$ is the Galois conjugate of $x$). By the structure theorem again, $C^+(V)$ is a central simple algebra over $E$ with $\dim_E C^+(V) = 1$, hence $C^+(V) = E$. This gives

\[\mathrm{Sim}(V) \simeq \{g \in E: gg^* \in F^\times\} = E^\times \simeq \mathrm{Res}_{E/F}\mathrm{GL}_{1, E}\]

whose dimension is $2 = \dim \mathrm{GSpin}(V)$. Hence

\[\mathrm{GSpin}(V) = \mathrm{Sim}(V) \simeq \mathrm{Res}_{E/F} \mathrm{GL}_{1, E}.\]

$n = 3$

Since $n$ is odd, $C^+(V)$ is a central simple algebra over $F$ of dimension $2^{3-1} = 4$, hence a quaternion algebra $D$ over $F$. If $d \neq 1$, then $C(V) \simeq C^+(V) \otimes_F E = D_E$ is a quaternion algebra over $E = F(\sqrt{d})$, and if $d = 1$, then $C(V) = C^+(V) \times C^+(V) = D \times D$. For both cases, the symplectic involution $*$ on $C(V)$ restricts to the quaternion conjugation on $D$ so

\[\mathrm{Sim}(V) = \{ g \in D : g\bar{g} \in F^\times \} = D^\times\]

which has dimension $4 = \frac{3(3-1)}{2} + 1$. Hence

\[\mathrm{GSpin}(V) = \mathrm{Sim}(V) \simeq D^\times.\]

Especially, $\mathrm{GSpin}(V) \simeq \mathrm{GL}_2$ when $D$ splits ($D \simeq M_2(F)$).

$n = 4$

As in the case of $n = 2$, there are two cases to consider.

Case 1. Assume $d = 1$, so $C^+(V) = A \times A$ for a central simple algebra $A$ over $F$ of dimension $\dim_F A = 2^{4-2} = 4$. Then $C^+(V) = D \times D$ for a quaternion algebra $D$ over $F$, and since $n \equiv 4\,(\mathrm{mod}\,8)$, the involution is symplectic on each factor $D$. This means that the involution is given by

\[(x, y)^* = (\bar{x}, \bar{y})\]

for $(x, y) \in D$, for the quaternion conjugation $x \mapsto \bar{x}$, and

\[\begin{align*} \mathrm{Sim}(V) &= \{(x, y) \in D \times D: (x, y)(\bar{x}, \bar{y}) = (x\bar{x}, y\bar{y}) \in \Delta F^\times \} \\ &= \{ (x, y) \in D^\times \times D^\times: N_D(x) = N_D(y)\} \\ &= D^\times \times_{\mathrm{GL}_1} D^\times \end{align*}\]

where $N_D: D \to \mathrm{GL}_1$ is a reduced norm on $D$. Then dimension of $\mathrm{Sim}(V)$ is $2 \dim_F D - 1 = 8 - 1 = 7$ (the dimension of $D^\times \times D^\times$ is $2 \dim_F D = 8$, and it is cut out by a single equation $N_D(x) = N_D(y)$ which decreases a dimension by $1$.) Since this equals to the dimension of $\mathrm{GSpin}(V)$, we have

\[\mathrm{GSpin}(V) = \mathrm{Sim}(V) \simeq D^\times \times_{\mathrm{GL}_1} D^\times.\]

When $D = M_2(F)$ splits, then $\bar{x} = \mathrm{adj}(x)$ is the adjoint matrix of $x$ and $N_D(x) = \det(x)$, so

\[\mathrm{GSpin}(V) \simeq \mathrm{GL}_2 \times_{\mathrm{GL}_1} \mathrm{GL}_2.\]

Case 2. Assume $d \neq 1$, so $C^+(V) = A_E$ is a central simple algebra over $E = F(\sqrt{d})$ with $\dim_E A_E = 2^{4-2} = 4$. Thus $A_E = D_E$ is a quaternion algebra over $E$ and the involution $*$ is the quaternion conjugation on $D_E$ (since it is symplectic). Then

\[\mathrm{Sim}(V) = \{ g \in D_E^\times: N_{D_E}(g) = g\bar{g} \in \mathrm{GL}_{1, F} \}\]

whose dimension over $F$ is $8 - 1 = 7 = \frac{3(3-1)}{2} + 1$ (the dimension of $D_E^\times$ over $F$ is $4 \cdot 2 = 8$, and the the condition $N_{D_E}(g) \in F^\times$ is equivalent to $N_{D_E}(g) = \overline{N_{D_E}(g)}$ for a Galois conjugation $x \mapsto \bar{x}$, which gives a single equation over $F$ and gives a codimension 1 subvariety). Hence

\[\mathrm{GSpin}(V) = \mathrm{Sim}(V) \simeq \{ g \in D_E: N_{D_E}(g) \in \mathrm{GL}_{1, F} \}.\]

Especially, when $D_E = M_2(E)$ (splits over $E$), then $N_{D_E}(g) = \det(g)$ and

\[\mathrm{GSpin}(V) = \{ g\in \mathrm{GL}_2(E): \det(g) \in \mathrm{GL}_{1, F} \}.\]

$n = 5$

Since $n$ is odd, $C^+(V)$ is a CSA over $F$ of dimension $2^{5-1} = 16$. Then there are three possible cases to consider:

\[C^+(V) \simeq \begin{cases} M_4(F) \\ M_2(D) & D \text{ is a quaternion algebra} \\ A & A\text{ is a degree }4 \text{ central division algebra} \end{cases}\]

where the first case happens when $d = \mathrm{disc}(V) = 1$.

Case 1. Assume $C^+(V) \simeq M_4(F)$. Then involution $*$ on $C+(V)$ is symplectic, hence there exists a 4-dimensional vector space $V_0$ with a symplectic bilinear form $B_0: V_0 \times V_0 \to F$ such that $(M_4(F), \ast) \simeq (\mathrm{End}(V_0), \ast_{0})$ as involution algebras, where $\ast_{0}$ is the adjoint involution for $B_0$. Then we have

\[\mathrm{Sim}(V) \simeq \{ g \in \mathrm{End}(V): gg^{\ast_0} \in F^\times\}.\]

Since $\ast_0$ is the adjoint involution for $B_0$,

\[B_0(gv, gv') = B_0(v, g^{\ast_0} g v') = N(g) B_0(v, v')\]

for all $v, v’$, which implies $\mathrm{Sim}(V) \simeq \mathrm{GSp}(V_0, B_0)$. In general, the dimension of the general simplectic group $\mathrm{GSp}_{2n}$ is

\[\begin{align*} \dim \mathrm{GSp}_{2n} &= 1 + \dim \mathrm{Sp}_{2n} \\ &= 1 + \dim \mathfrak{sp}_{2n} \\ &= 1 + n^2 + \frac{n(n+1)}{2} + \frac{n(n+1)}{2} \\ &= 2n^2 + 2n + 1 \end{align*}\]

and for $n = 2$, the dimension is $11 = \frac{5(5-1)}{2} + 1$, that coincides with the dimension of $\mathrm{GSpin}(V)$. Here we use the standard form of symplectic Lie algebra to compute the dimension. Its elements are block matrices $\left(\begin{smallmatrix} X & Y \\ Z & W\end{smallmatrix}\right)$ where $X, Y, Z, W$ are $n \times n$ matrices satisfying

\[W = -X^T, \quad Y = Y^T, \quad Z = Z^T.\]

So we have

\[\mathrm{GSpin}(V) = \mathrm{Sim}(V) \simeq \mathrm{GSp}(V_0), \quad \dim_F V_0 = 4.\]

Case 2. Assume $C^+(V) \simeq M_2(D)$ for a quaternion algebra $D$ over $F$. In this case, we say that $C^+(V)$ is Brauer equivalent to $D$ (i.e. defines a same element in a Brauer group). Becher proved that, $(C^+(V), \ast)$ is isomorphic to $(D, \ast_D) \otimes (M_2(F), \ast’)$ where $\ast_D: D \to D$ is the quaternion conjugation (i.e. the unique symplectic involution on $D$), and $\ast’: M_2(F) \to M_2(F)$ is an orthogonal involution of the form

\[g \mapsto \begin{pmatrix} 1 & \\ & -a \end{pmatrix}^{-1} g^T \begin{pmatrix}1 & \\ & -a \end{pmatrix}\]

for some $a \in F^\times$. Then the induced involution on $M_2(D) \simeq D \otimes M_2(F)$ is

\[\begin{pmatrix} x & y \\ z & w \end{pmatrix} \mapsto \begin{pmatrix} \bar{x} & -a \bar{z} \\ -\frac{\bar{y}}{a} & \bar{w} \end{pmatrix}\]

where $x \mapsto \bar{x}$ is the quaternion conjugation on $D$. Then the similarity group is isomorphic to

\[\begin{align*} &\{g \in \mathrm{GL}_2(D): gg^* \in F^\times \cdot I_2\} \\ &= \left\{ \begin{pmatrix} x & y \\ z & w\end{pmatrix} \in \mathrm{GL}_2(D) : \begin{pmatrix} N_D(x) - \frac{N_D(y)}{a} & -ax\bar{z} + y \bar{w} \\ \bar{x}z - \frac{\bar{y}w}{a} & -aN_D(z) + N_D(w) \end{pmatrix}\in F^\times \cdot I_2\right\} \\ &=\left\{\begin{pmatrix} x&y \\ z&w \end{pmatrix} \in \mathrm{GL}_2(D): y\bar{w} = ax\bar{z},\,\, N_D(x) - \frac{N_D(y)}{a} = -aN_D(z) + N_D(w) \ne 0 \right\}. \end{align*}\]

The set elements with $y \neq 0$ is a Zariski open subset of the group. and on the set, $w$ is simply determined as $w = a\bar{x}z / \bar{y}$. Hence we have three free variables $x, y, z \in D$ with a single equation over $F$, and the dimension becomes $3 \cdot 4 - 1 = 11$. Hence $\mathrm{GSpin}(V)$ is isomorphic to the above group.

Case 3. Let $A$ be a central division algebra over $F$ of degree $4$ (dimension $16$). Note that it is equipped with a symplectic involution $\ast$. In [Rowen, Theorem B, page 296], the author proved that $A$ is always a tensor product of two $\ast$-invariant quaternion subalgebras, say $A \simeq D_1 \otimes D_2$. Also, it contains a maximal subfield $E$ that is a bi-quadratic extension of $F$, i.e. $E = F(\sqrt{d_1}, \sqrt{d_2})$ for some $d_1, d_2 \in F^\times - (F^\times)^2$ that are not in the same square classes (so $\mathrm{Gal}(E/F) \simeq \mathbb{Z}/2 \times \mathbb{Z}/2$) [Rowen, Theorem 4.5, page 293].

In honest, I don’t know how proceed from here, especially how to describe the (symplectic) involution in this case. I believe that there may exists a Becher-type theorem for such degree 4 central division algebras (with no split factors). I’ll add the details later once I figure it out.

$n = 6$

For sanity check, we will try to compute $\mathrm{GSpin}(V)$ when $\dim V = 6$ using the similarity group, which would fail because their dimension won’t be the same anymore (and for higher $n$, there will be the same issue). Let’s consider the simplest $6$-dimensional quadratic space, which is $V = \mathbb{H} \oplus \mathbb{H} \oplus \mathbb{H}$ (so that $\mathrm{disc}(V) = 1$). Then the structure theorem gives

\[C(V) \simeq M_{2^3}(F) = M_8(F), \quad C^+(V) \simeq M_4(F) \times M_4(F)\]

where $M_4(F) \times M_4(F)$ embeds into $M_8(F)$ diagonally. The involution $*$ on $C^+(V)$ is unitary, and by [Knus et al. (2.20) Proposition, page 24], any such involution has a form of

\[(X, Y) \mapsto (uY^T u^{-1}, uX^T u^{-1}), \quad u \in \mathrm{GL}_4(F)\]

where $u(u^T)^{-1} \in F^\times \cdot I_4$. If we assume $u = 1$, then it is simply given by the exchange involution $(X, Y) \mapsto (Y^T, X^T)$. It acts as an identity on the center $Z(C(V)) \simeq { a I_8: a \in F^\times } \simeq F^\times$ where $I_8$ is the $8 \times 8$ identity matrix. Then the similarity group becomes

\[\begin{align*} \mathrm{Sim}(V) &\simeq \{(X, Y) \in \mathrm{GL}_4 \times \mathrm{GL}_4: N(X, Y) = (XY^T, YX^T) \in Z(C(V))\} \\ & = \{ (X, aX^{-T}) : X \in \mathrm{GL}_4, a \in \mathrm{GL}_1\} \simeq \mathrm{GL}_4 \times \mathrm{GL}_1 \end{align*}\]

(here $X^{-T}:= (X^T)^{-1}$) which has the dimension $\dim \mathrm{GL}_4 + \dim \mathrm{GL}_1 = 16 + 1 = 17$. Unfortunately, the dimension of the corresponding general spin group is $\frac{6(6-1)}{2} + 1 = 16$, which is smaller by $1$.

However, it may possible to investigate the isomorphisms further and find an appropriate codimension 1 subgroup of the above $\mathrm{Sim}(V) \simeq \mathrm{GL}_4 \times \mathrm{GL}_1$ that is isomorphic to $\mathrm{GSpin}(V)$ for $n = 6$. At least, it is known that the spin group $\mathrm{Spin}_6(\mathbb{C})$ is isomorphic to $\mathrm{SL}_4(\mathbb{C})$, and we should have $\mathrm{GSpin}(V) \simeq \mathrm{GL}_4$ over a general field when $V \simeq \mathbb{H}\oplus \mathbb{H} \oplus \mathbb{H}$. (I’ll add this later once I figure it out.)

References.

[Albert] Albert, Structure of Algebras, Amer. Math. Soc. Colloq. Publ. no. 24, Amer. Math. Soc , Providence, R.I., 1961.

[Becher] Becher, A proof of the Pfister Factor Conjecture, Invent. Math. 173 (2008), no. 1, 1-6

[Emory] Emory, On the Global Gan-Gross-Prasad conjecture for general spin groups. Pacific J. Math. 306 (2020), no. 1, 115-151.

[Knus et al.] Knus et al. The Book of Involutions. Americal Mathematical Society Colloquium Publications, Volume 44, 1998

[Lam] Lam, Introduction to Quadratic Forms over fields. Graduate Studies in Mathematics, Volume 67, 2004

[Rowen] Rowen, Central Simple Algebras, Israel J. Math. 29 (1978), 285-301.

[Scharlau] Scharlau, Quadratic and Hermitian Forms. A Series of Comprehensive Studies in Mathematics, 1985

Tags: math