# The Tits classes of simply connected simple real groups

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.

• What is the oldest or most convenient published reference for the explicit description of the center of the split group in terms of co-roots for which you refer to Brian's cheat sheet? Of course one can just work out the exercise, but for writing a paper it is convenient to be able to refer to something. Some of it is in section 17 of Springer's Linear algebraic groups, 2nd edition. It is also in my paper with Anne Queguiner Restricting the Rost invariant to the Center ( dx.doi.org/10.1090/S1061-0022-08-00993-X ), where $Spin(2m,2m)$ is Example 8.6. – Skip Jun 19 '18 at 15:25
• @Skip: The most convenient published reference? One can find a description of $P^\vee/Q^\vee$ in the book "Lie Groups and Algebraic Groups" by Onishchik and Vinberg, Springer-Verlag 1990. See Table 3 on page 298. – Mikhail Borovoi Jun 20 '18 at 14:02
• @Skip: The oldest published reference? The book by Onishchik and Vinberg of 1969 ("Seminar on algebraic groups and Lie groups", in Russian, mimeographed notes, 200 copies). – Mikhail Borovoi Jun 20 '18 at 14:10

Question 1: I haven't seen an explicit table in the literature of the Tits classes for simple $R$-groups.
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 $(1,0)$.