The Chern character of the symmetric space \(SU(2n)/SO(2n)\) (Q1804696)

For \(n \geq 1\), let \(t : \text{SU} (n) \to \text{SU}(n)\) be the map defined by \(t(x) = \overline x\) for \(x \in \text{SU} (n)\), where \(\overline x\) is the complex conjugate of a unitary matrix \(x\). The natural inclusion \(\mathbb{R} \subset \mathbb{C}\) yields a monomorphism \(i_1 : \text{SO} (n) \to \text{SU} (n)\) of topological groups. Clearly \(i_1 (\text{SO} (n)) = \{x \in \text{SU} (n) |t(x) = x\}\). So the quotient space \(\text{SU} (n)/i_1 (\text{SO} (n))\), which we abbreviate to \(\text{SU} (n)/ \text{SO} (n)\), forms a compact symmetric space. It is denoted by \(AI\) (of rank \(n - 1)\) in É. Cartan's notation. In this paper we compute its Chern character \[ ch : K^* \bigl( \text{SU} (2n)/ \text{SO} (2n) \bigr) \to H^{**} \bigl( \text{SU} (2n)/ \text{SO} (2n); \mathbb{Q} \bigr), \] while that of \(\text{SU} (2n + 1)/ \text{SO} (2n + 1)\) has been described in [the author, J. Math. Kyoto Univ. 34, No. 1, 149-169 (1994; Zbl 0818.22003)].