Theorem 1 Let $G$ be an absolutely almost simple algebraic group defined over $k$ that admits a Cartan-type automorphism. When the coefficient system is trivial, the cuspidal cohomology of $G$ over $S$ does not vanish, that is, every $S$-arithmetic subgroup of $G$ has a subgroup of finite index with non-zero cuspidal cohomology with respect to the trivial coefficient system.

Theorem 2 Let $k$ be a totally real number field and $G$ be an absolutely almost simple algebraic group defined over $k$. Consider the following cases:

1. (1) $G$ is $k$-split.

2. (2) $G = {\mathrm {Res}}_{k'/k}G'$ where $k'$ is a CM-field with $G'$ defined over $k$ totally real, satisfying one of the following hypotheses:

   1. (2a) $\mathbf{H}=G'(k\otimes \mathbb {R})$ has a compact Cartan subgroup;

   2. (2b) $G'$ is split over $k$ and simply connected.

Then the cuspidal cohomology of $G$ over $S$ with respect to the trivial coefficient system does not vanish.

Proof. In case (1), when $G$ is $k$-split, the proof is given in [Reference Borel, Labesse and SchwermerBLS96]: it relies on the first case of [Reference Borel, Labesse and SchwermerBLS96, Corollary 10.7] and Theorem 1. In case (2a) the result follows from Proposition 3 below and Theorem 1. In case (2b) the assertion is a particular case of [Reference LabesseLab99, Theorem 4.7.1], which in turn relies on case (1) above.

Proof. The automorphism $c$ is of Cartan type if, by definition, $\textbf {c}=\mathrm {Int}(x)\circ \boldsymbol {\theta }$ for some $x\in \textbf {G}$, that is, if and only if $U$ is an inner form of $G_0$ or equivalently of $G^*$. This proves the equivalence of (i) and (ii). Assume now that $U$ is an inner form of $G_0$. Up to conjugation under $\textbf {G}$, we may assume that $\boldsymbol {\theta }$ is of the form $\boldsymbol {\theta }=\mathrm {Int}(x)\circ \textbf {c}$ with $x$ in the normalizer of a maximal torus $T$ in $G_0$ defined over $\mathbb {R}$. In particular, $x$ is semisimple and its centralizer $\textbf {L}$ in $\textbf {G}$ is a complex reductive subgroup of maximal rank. The cocycle relation $\mathrm {Int}(x\,\textbf {c}(x))=1$ implies that $\textbf {L}$ is stable under $\textbf {c}$. Then $\textbf {c}$ induces an anti-holomorphic involution on $\textbf {L}$ whose fixed points $\textbf {M}=\textbf {L}\cap \textbf {H}=\textbf {U}\cap \textbf {H}$ are a compact real form of $\textbf {L}$. A Cartan subgroup $\textbf {C}$ of $\textbf {M}$ is a compact Cartan subgroup in $\textbf {H}$. Hence (ii) implies (iii). Now, consider a torus $T$ in $G_0$ such that $\textbf {C}=T(\mathbb {R})$ is compact. Then the complex conjugation $\textbf {c}$ acts by $-1$ on the weights of $T$. Hence there is $n\in \textbf {G}$ which belongs to the normalizer of $T^*\subset G^*$ such that $\mathrm {Int}(n)\circ \psi ^*$ acts as $-1$ on the root system of $T^*$. Now, since $\psi ^*$ preserves the set of positive roots, $w=\mathrm {Int}(n)|_{T^*}$ is the element of maximal length in the Weyl group. This shows that (iii) implies (v). The equivalence of (iii) and (iv) is a well-known theorem [Reference Harish-ChandraHar66, Theorem 13] due to Harish-Chandra. Finally, Lemma 4 below shows that (v) implies (ii).

Proof. Consider the complex Lie algebra $\mathfrak {g}=\mathrm {Lie}(\textbf {G})$. Let $\Sigma$ be the set of roots, $\Sigma ^+$ the set of positive roots and $\mathfrak {g}_\alpha$ the vector space attached to $\alpha \in \Sigma$ with respect to the torus $T^*(\mathbb {C})$. Following Weyl [Reference WeylWey26], Chevalley [Reference ChevalleyChe55] and Tits [Reference TitsTit66], one may choose elements $X_\alpha \in \mathfrak {g}_\alpha$ for ${\alpha \in \Sigma }$ such that, if we define $H_\alpha \in \mathrm {Lie}(T^*(\mathbb {C}))$ by $H_\alpha =[X_\alpha ,X_{-\alpha }]$, we have

\[ \alpha(H_\alpha)=2 \quad\mbox{and}\quad [X_\alpha,X_{\beta}]=N_{\alpha,\beta}X_{\alpha+\beta} \quad {\rm if }\ \alpha+\beta\in \Sigma\quad {\rm with}\ N_{\alpha,\beta}={-}N_{-\alpha,-\beta}\in\mathbb{Z}. \]

We assume the splitting compatible with this choice. Now let

\[ Y_\alpha=X_\alpha-X_{-\alpha},\quad Z_\alpha=i(X_\alpha+X_{-\alpha}), \quad W_\alpha=iH_\alpha. \]

The elements $Y_\alpha$ and $Z_\alpha$ for $\alpha \in \Sigma ^+$ together with the $W_\alpha$ for $\alpha \in \Delta$ build a basis for a real Lie algebra $\mathfrak {u}$. As in the proof of [Reference HelgasonHel62, Chapter III, Theorem 6.3], we see that the Killing form is negative definite on $\mathfrak {u}$ and hence the Lie subgroup $\textbf {U}\subset \textbf {G}$ with Lie algebra $\mathfrak {u}$ is compact. Since $\psi ^*$ preserves the splitting, $\psi ^*(X_\alpha )=X_{\psi ^*(\alpha )}$ for $\alpha \in \Delta$. Let $w$ be the element of maximal length in the Weyl group for $T^*$. There is an $n^*\in \textbf {G}$, uniquely determined modulo the center, such that the inner automorphism $w^*=\mathrm {Int}(n^*)$ acts as $w$ on $T^*$ and such that $w^*(X_\alpha )=-X_{w\alpha }$ for $\alpha \in \Delta$. This automorphism is of order 2 and commutes with $\psi ^*$. Now let $\phi =w^*\circ \psi ^*$. Since $w\circ \psi ^*$ acts by $-1$ on $\Sigma$ this implies $\phi (X_\alpha )=-X_{\phi (\alpha )}=-X_{-\alpha }$ for $\alpha \in \Delta$. It follows from the commutation relations and the relations $N_{\alpha ,\beta }=-N_{-\alpha ,-\beta }$ that $\phi (X_\alpha )=-X_{-\alpha }$ and $\phi (H_\alpha )=-H_{\alpha }$ for all $\alpha \in \Sigma$. Now $\phi$, which acts as an automorphism of the real Lie algebras $\mathfrak {g}^{**}$ generated by the $X_\alpha$ for $\alpha \in \Sigma$, can be extended to an antilinear involution of $\mathfrak {g}^{**}\otimes \mathbb {C}=\mathfrak {g}=\mathfrak {u}+i\mathfrak {u}$. This, in turn, induces a Cartan involution $\boldsymbol {\theta }$ on $\textbf {G}$: its fixed point set is the compact group $\textbf {U}=U(\mathbb {R})$ with Lie algebra $\mathfrak {u}$, and $U$ is the inner form of $G^*$ defined by the Galois cocycle $a_1=1$ and $a_\sigma =w^*$.