What does a maximal torus in GSpin$_{2n}$ look like?

Question

I am interested in spin groups for a project at my university, and I was wondering: what would a maximal torus in GSpin$_{2n}$ look like, and how does one come to it? Does anyone maybe have a reference that explains? I have tried about 5 books already on clifford algebra's en Lie groups, but unfortunately nothing! Any help would be greatly appreciated!

Gspin$_k$, or the general spin group, is defined as all invertible even elements $x$ in the Clifford algebra Cliff$(\mathbb{C}^k,Q)$ with the property that for all $v \in \mathbb{C}^n$ it holds that $xvx^{-1} \in \mathbb{C}^k$. Here $Q$ is the quadratic form defined by $Q(x) = x_1x_k + x_2x_{k-1} + \dots+ x_kx_1$.

Kind regards and many thanks!

PS: I am also interested in the Gspin$_{2n+1}$ case! Any thoughts on that hopefully might help me figure out the even-dimensional case.

Answer 1

I will first work everything out for the hyperbolic plane $\mathbb H=\mathbb Ce_0+\mathbb Ce_1$, whose inner product is: $q(e_0,e_0)=q(e_1,e_1)=0$ and $q(e_0,e_1)=1$.

Then, $\mathrm{Spin}(\mathbb H)$ is abelian, and it consists of elements of the form $$(ae_0+be_1)(ce_0+de_1)=ade_0e_1+bce_1e_0,$$ where $ab=cd=\frac12$, i.e., $$ \mathrm{Spin}(\mathbb H)=\{\frac12xe_0e_1+\frac12ye_1e_0:x,y\in\mathbb C,xy=1\}. $$ It is easy to check that the element $\frac12xe_0e_1+\frac12ye_1e_0\in\mathrm{Spin}(\mathbb H)$ acts on $\mathbb H$ as $e_0\mapsto x^2e_0,e_1\mapsto y^2e_1$. Moreover, the product on $\mathrm{Spin}(\mathbb H)$ can be explicitly described: $$(\frac12xe_0e_1+\frac12ye_1e_0)(\frac12x'e_0e_1+\frac12y'e_1e_0)=\frac12xx'e_0e_1+\frac12yy'e_1e_0.$$ Now, $\mathrm{GSpin}(\mathbb H)$ is just $$\mathrm{GSpin}(\mathbb H)=\{\frac12xe_0e_1+\frac12ye_1e_0:x,y\in\mathbb C^\times\},$$ which simply acts on $\mathbb H$ as $e_0\mapsto xy^{-1}e_0,e_1\mapsto x^{-1}ye_1$.

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Now, an even orthogonal space is isomorphic to $V=\mathbb H^n$, say with basis $e_0^{(1)},e_1^{(1)},\dots,e_0^{(n)},e_1^{(n)}$. A maximal torus of $\mathrm{GSpin}(V)$ is then $$T=\{\prod_{i=1}^n(\frac12x^{(i)}e_0^{(i)}e_1^{(i)}+\frac12y^{(i)}e_1^{(i)}e_0^{(i)}):x^{(i)},y^{(i)}\in\mathbb C^\times\}.$$ Maximal tori of $\mathrm{GSpin}_{2n+1}$ admit a similar description.