Definition of $\textrm{GSpin}_{2n}$ and its root datum

Question

I'm trying to get my hands on the general spin group $G = \textrm{GSpin}_{2n}$. It have seen it mentioned as a connected, reductive group whose derived group is $\textrm{Spin}_{2n}$, which is the unique semisimple, simply connected group having the same root system as $\textrm{SO}_{2n}$.

The general spin group is for example mentioned in a paper here https://math.okstate.edu/people/asgari/Files/gspin.pdf .

I have read that it is difficult to view $\textrm{Spin}_{2n}$ as a closed subgroup of some general linear group. According to (9.16) in Linear Algebraic Groups and Finite Groups of Lie Type, the smallest $m$ for which $\textrm{Spin}_{2n}$ can be embedded in $\textrm{GL}_{m}$ is $2^{[\frac{2n-1}{2}]}$.

As for $G$, I am not really sure how that should be defined. Since its derived group is $\textrm{Spin}_{2n}$, it should be quotient of $\textrm{Spin}_{2n} \times S$ by a finite normal subgroup, where $S$ is some torus.

So my questions are:

1 . How should the general spin group be defined?

2 . What is the most straightforward way to compute the roots, coroots, and root datum of the general spin group as well as its derived group?

For the group $\textrm{SO}_{2n}$, I did (2) by finding a maximal torus and computing the Lie algebra. I imagine there must be a different approach when one is not working with an explicit embedding into $\textrm{GL}_{m}$.

Answer 1

A good concise reference is Deligne's article on the Weil conjectures for K3 surfaces. See "La conjecture de Weil pour les surfaces K3" in Inventiones 15 (1971/72): 206-226. An English version is easy to Google too. In Deligne, and in other sources, the group GSpin is called CSpin. Here are the pertinent details.

Let $(V,q)$ be a finite-dimensional vector space with nondegenerate quadratic form, over any field (any base scheme is fine, adapting definitions appropriately... see the comment by nfdc23 below). Let $C(V,q)$ be the Clifford algebra, and $C^+$ the even part of the Clifford algebra. Embed $V$ in $C(V,q)$ as the degree 1 part; thus for $v \in V$, we view $v \in C(V,q)$ and $v \cdot v = q(v)$.

Then the group $CSpin$ is defined: $$CSpin(V,q) = \{ g \in C^+ : g V g^{-1} = V \}.$$ From above, we have a natural homomorphism $CSpin(V,q) \rightarrow SO(V,q)$ sending $g \in CSpin(V,q)$ to the map $(v \mapsto g v g^{-1})$. The kernel consists of scalars, giving the short exact sequence $$1 \rightarrow G_m \rightarrow CSpin(V,q) \rightarrow SO(V,q) \rightarrow 1.$$

The group $Spin(V,q)$ is the kernel of the spinor norm $CSpin(V,q) \rightarrow G_m$. Altogether, this gives the diagram used by Deligne: $$ \begin{array}{ccccc} & & Spin(V,q) & & \\ & & \downarrow & \searrow & \\ G_m & \rightarrow & CSpin(V,q) & \rightarrow & SO(V,q) \\ & \searrow & \downarrow & & \\ & & G_m & & \\ \end{array} $$ (Excuse the pooly formatted diagram!) The bottom-left diagonal arrow is the map $x \mapsto x^{-2}$.

This pretty quickly gives an identification $$CSpin(V,q) \cong \frac{Spin(V,q) \times G_m}{\mu_2}.$$ Here $\mu_2$ is identified with the kernel of $Spin(V,q) \rightarrow SO(V,q)$.

If you want to describe this $\mu_2$ using coroots, the only subtlety is in type $D_n$ with $n$ even, i.e., in $Spin_{4n}$. (Thanks to Mikhail Borovoi for the correction and nfdc23 for the suggestion.) Outside of $n=4$, the central $\mu_2$ in $Spin(V,q)$ should be the only one fixed by the Dynkin diagram automorphism. In type $D_4$, i.e., for $Spin_8$, the representation $Spin(V,q) \rightarrow SO(V,q)$ determines a node on the Dynkin diagram. The central $\mu_2$ in $Spin(V,q)$ is the only one fixed by the Dynkin diagram automorphism fixing that node.

This makes it pretty easy to relate the root datum of a (split) $CSpin$ group to that of the simply-connected $Spin$ group (which is its derived subgroup).

For simply-connected groups, one can look up the full root datum in any good reference (e.g., Bourbaki).

Answer 2

If you'd like to be completely explicit, the Atlas of Lie Groups and Representations software can do this. See www.liegroups.org. Comments are in braces { }.

atlas> {start with simply connected group Spin(8)xGL(1),
of type D4.T1, then mod out by specified element of order 2
so G=Spin(8)xGL(1)/Z_2
Z_2 given by the vector [1/2,1/2,1/2]: element of center
of Spin(8)xGL(1)=Z/2 x Z/2 x C^x, i.e. the element (-1,-1,-1)}
atlas> set rd=root_datum(Lie_type ("D4.T1"),[1/2,1/2,1/2])
Variable rd: RootDatum
atlas> rd
Value: simply connected root datum of Lie type 'D4.T1'
{derived group is simply connected}
atlas> set roots=simple_roots (rd)  
{simple roots as columns of a matrix}
Variable roots: mat
{The software chooses a basis so that X^*(T)=Z^5, X_*(T)=Z^5, 
with the dot product pairing, and then gives roots and corooots 
as integral vectors}
atlas> set coroots=simple_coroots (rd)  {simple coroots as columns}
Variable coroots: mat
atlas> roots
Value:  {roots are columns, 4 vectors of size 5} 
|  2, -1,  0,  0 |
|  3,  0, -1, -1 |
|  1,  0,  0,  0 |
| -2,  0,  0,  2 |
|  0,  0,  0,  0 |

atlas> coroots
Value: {coroots are columns}
| 1, -2,  1, 1 |
| 0,  1, -2, 0 |
| 0,  0,  2, 0 |
| 0,  0, -1, 1 | 
| 0,  0, -1, 0 |

atlas> ^roots*coroots  {compute the pairing of roots and coroots}
Value: 
|  2, -1,  0,  0 |
| -1,  2, -1, -1 |
|  0, -1,  2,  0 |
|  0, -1,  0,  2 |

atlas> ^roots*coroots=Cartan_matrix(rd)
Value: true  {this confirms that these are valid roots and 
coroots of the root system}