Invariant eigendistributions on \(\text{SU}_{p,q}\)
Let \((\rho_j,\mathcal H_j)\) \((0 \leq j \leq \min \{p,q\})\) be an irreducible (strongly continuous) unitary representation of \(\text{SU}_{p,q}\) (on a complex Hilbert space \(\mathcal H_j\)) associated with a Cartan subgroup \(H_j\), where \(H_j\) and \(H_{j'}\) are non-conjugate if \(j' \neq j\) [the authors, Int. J. Math. 12, 1--36 (2001; Zbl 1107.22300) and ibid. 12, 37--47 (2001; Zbl 1108.22300)]. Then, for each \(f \in C^\infty_\roman{c}(\text{SU}_{p,q})\) satisfying \[ \int_{\text{SU}_{p,q}}|f(g)|^2\,d\mu(g) < \infty, \] where \(\mu\) is a Haar measure on \(\text{SU}_{p,q}\), the bounded linear operator \(\rho_{j(f)} = \int_{\text{SU}_{p,q}}f(g)\rho_j(g)\, d\mu(g)\) in \(\mathcal H_j\) has a trace and is a Hilbert-Schmidt operator. The linear functional \(T_{\omega_j}\) on \(C^\infty_\roman{c}(\text{SU}_{p,q})\) defined by \[ T_{\omega_j}(f) = \text{tr}(\rho_{j(f)}) = \sum_{i \geq 1}\int_{\text{SU}_{p,q}}f(g)(\rho_j(g) \psi_i,\psi_i)\,d\mu(g) \quad (\rho_j \in \omega_j), \] where \((\psi_i)_{i \geq 1}\) is an orthonormal basis of \(\mathcal H_j\) and \(\omega_j\) is the unitary equivalence class of the representation \((\rho_j,\mathcal H_j)\), does not vary within \(\omega_j\) and is called the \textit{global character} of \((\rho_j,\mathcal H_j)\). This global character is an invariant eigendistribution on \(\text{SU}_{p,q}\), since it is conjugation-invariant under \(\text{SU}_{p,q}\) and the center of the universal enveloping algebra of \(\mathfrak{sl}_n(\mathbb C)\) acts on it via scalars. In this paper, the authors determine the global characters of all irreducible unitary representations of \(\text{SU}_{p,q}\) associated with the \(\min \{p,q\} + 1\) non-conjugate Cartan subgroups \(H_j\) and show that they are tempered distributions. They determine also the global characters of the contragredient representations of \((\rho_j,\mathcal H_j)\) and some properties of the adjoints of the global characters \(T_{\omega_j}\) \((0 \leq j \leq \min\{p,q\})\).
scientific article; zbMATH DE number 1758077
scientific article