The Tits classes of simply connected simple real groups

Question

Let $G$ be a simply connected semisimple group over a perfect field $k$ (at the moment I am interested in the case $k=\mathbb R$). Then $G$ is an inner form of a quasi-split $k$-group $G_{\rm qs}$: there exists a quasi-split form $G_{\rm qs}$ of $G$ and a 1-cocycle $c\in Z^1(k,\overline{G}_{\rm qs})$ such that $G=\,_c G_{\rm qs}$, the inner twist of $G_{\rm qs}$ by the cocycle $c$, where $\overline{G}_{\rm qs}=G_{\rm qs}/Z_{\rm qs}$ and $Z_{\rm qs}=Z(G_{\rm qs})$. Let $\xi=[c]\in H^1(k,\overline{G}_{\rm qs})$, the cohomology class of the cocycle $c$. Let $$\Delta\colon H^1(k,\overline{G}_{\rm qs})\to H^2(k,Z_{\rm qs})$$ denote the connecting map from the cohomology exact sequence $$H^1(k,Z_{\rm qs})\to H^1(k,G_{\rm qs})\to H^1(k, \overline{G}_{\rm qs})\to H^2(k,Z_{\rm qs})$$ induced by the short exact sequence $$ 1\to Z_{\rm qs}\to G_{\rm qs}\to \overline{G}_{\rm qs}\to 1.$$ By definition, the Tits class of $G$ is $$t_G=\Delta(\xi)\in H^2(k,Z_{\rm qs}),$$ see The Book of Involutions, (31.7). Note that $Z_{\rm qs}=Z:=Z(G)$.

Question 1. Is there anywhere in the literature or in the Internet a table of $t_G$ for all simply connected absolutely simple ${\mathbb R}$-groups $G$?

For all connected Dynkin diagram except for ${\mathsf D}_{2m}$ (and also for ${\mathsf D}_{2m}$ when $G_{\rm qs}$ is not split) we have either $H^2({\mathbb R},Z_{\rm qs})=0$ or $H^2({\mathbb R},Z_{\rm qs})={\mathbb Z}/2{\mathbb Z}$. From the cohomology exact sequence we see that $t_G=0$ if and only if $G$ is a strong inner form of $G_{\rm qs}$, that is, $\xi$ comes from $H^1({\mathbb R}, G)$. I have a table of strong inner forms of $G_{\rm qs}$, and so I know when $t_G=0$; for non-strong inner forms we have $t_G\neq 0$, hence $t_G=1+2{\mathbb Z}\in {\mathbb Z}/2{\mathbb Z}$.

However, for ${\mathsf D}_{2m}$, in the case when $G_{\rm qs}$ is split, namely $G_{\rm qs}\simeq {\bf Spin}(2m,2m)$, we have $H^2({\mathbb R}, Z_{\rm qs})={\mathbb Z}/2{\mathbb Z}\times{\mathbb Z}/2{\mathbb Z}$. I know that $t_G=0$ if and only if $G$ is a strong inner form of $G_{\rm qs}$, that is, if $G\simeq{\bf Spin}(2m+4q, 2m-4q)$, but I need an explicit formula for the cases ${\bf Spin}(2m+4q+2, 2m-4q-2)$ and the quaternionic form ${\bf Spin}^*(4m)$. Thus Question 1 reduces to the following question:

Question 2. What are the Tits classes for the simple ${\mathbb R}$-groups ${\bf Spin}(2m+4q+2, 2m-4q-2)$ and ${\bf Spin}^*(4m)$ ?

In order to formulate an answer to Question 2, one needs an explicit description of the center of the split ${\mathbb R}$-group ${\bf Spin}(2m,2m)$. Such a description is given, for example, in Brian Conrad's cheat sheet.

Answer 1

Question 1: I haven't seen an explicit table in the literature of the Tits classes for simple $R$-groups.

That said, such a table can be constructed from tables in the literature. Specifically, the Tits class is determined by the Tits algebras corresponding to the minuscule dominant weights by Proposition 7 in my paper Outer automorphisms of algebraic groups and determining groups by their maximal tori (Michigan Mathematical Journal 61 #2 (2012), 227-237).

These Tits algebras are described for each simple type in general terms for any $k$ in section 27 of The Book of Involutions, and their precise values for $k = \mathbb{R}$ are available in several references, for example Tits's Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen.

Question 2: Victor answered this question in the language of Tits algebras, not the Tits class, so I just translate his answer. Identify the projections on the two components of $H^2(\mathbb{R}, Z_{qs}) \cong \mathbb{Z}/2 \times \mathbb{Z}/2$ with the highest weights of the half-spin representations as in section 4.3 of Tits's paper Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque; this agrees with the identification of $Z_{qs}$ with $\mu_2 \times \mu_2$ that you refer to. For $SO(2m + 4q + 2, 2m - 4q - 2)$, both half-spin representations are quaternionic, so the Tits class is $(1,1)$. For $SO^*(4m)$, one half-spin representation is quaternionic and the other is real, so the Tits class is $(0,1)$ or $(1,0)$.

Answer 2

To answer your second question you can use the fact that the Tits algebras of a group are the same as for its anisotropic kernel (J. Tits, Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque, 5.5), and for a quadratic form the Tits algebras are the components of the even Clifford algebra. So for Spin(2m+4q+2,2m-4q-2) the answer is the same as for the compact group Spin(8q+4), and by the Bott periodicity the same as for the compact group Spin(4), that are clearly just two quaternion algebras.