Representation of complex semi-simple Lie groups and Lie algebras.

*(English)*Zbl 0177.18004Let \(\mathfrak g\) be a complex semisimple Lie algebra, \(\mathfrak h\) a Cartan subalgebra of \(\mathfrak g\) and \(\Delta\) the set of roots of \(\mathfrak g\) with respect to \(\mathfrak h\). For any positive system \(P\) of roots we denote by \(D_P\) the set of all \(\lambda\in\mathfrak h^*\) which are integral and dominant relative to \(P\). Let \(G = K\cdot A_+\cdot N\) be the Iwasawa decomposition of the simply connected Lie group \(G\) whose Lie algebra is \(\mathfrak g\). Let \(\mathfrak H\) be the Hilbert space of \(L^2\)-functions on the compact group \(K\) with the normalized Haar measure \(dk\). For \(x\in G\) and \(k\in K\), let

\[ xk = \sigma_x(k) \cdot a_+(x,k) \cdot n(x,k), \]

where \(\sigma_x(k)\in K\), \(a_+(x,k)\in A_+\), \(n(x,k)\in N\). Let \(\xi\in\mathfrak h^*\). For \(x\in G\) and \(f\in \mathfrak H\), let

\[ (n_\xi(x)f)(k) = \exp((\xi+2\delta)(\log a_+(x^{-1},k))) \cdot f(\sigma_x^{-1}(k)) \]

for \(k\in K\), where \(2\delta = \displaystyle\sum_{\alpha\in P} \alpha\). The map \(\pi_\xi\) is a representation of \(G\) in \(\mathfrak H\).

For any integral \(\nu\in\mathfrak h^*\), let \(\sigma_\nu(\exp\sqrt{-1}H) = \exp(\sqrt{-1} \nu(H))\) for \(H\in \mathfrak h_0\), \(\mathfrak h_0\) being the Lie algebra of \(A_+\). Then \(\sigma_\nu\) is a character of the group \(M = \exp\sqrt{-1}\mathfrak h_0\). For any integral \(\nu\in\mathfrak h^*\), we denote by \(\mathfrak H(\nu)\) the set of all \(f\in\mathfrak H\) such that \(R_r(m)f = \sigma_{-\nu}(m)f\) for all \(m\in M\), where \(R_r\) denotes the right regular representation of \(K\) in \(\mathfrak H\). We know that \(\mathfrak H\) is the orthogonal sum of the subspaces \(\mathfrak H(\nu)\) and each \(\mathfrak H(\nu)\) is invariant under \(\pi_\xi\). We denote by \(\pi_{\xi,\nu}\) the representation of \(G\) induced on \(\mathfrak H(\nu)\) by \(\pi_\xi\). Let \(\nu_0\) be the unique element of \(D_P\) in the orbit \(W\cdot \nu\), where \(W\) is the Weyl group of \((\mathfrak g, \mathfrak h)\) and let \(\pi_{\nu_0}\) be the irreducible representation of \(\mathfrak g\) with \(\nu_0\) as its highest weight relative to \(P\).

The authors construct an irreducible infinite dimensional representation \(\pi_{\lambda,\nu}\) of \(G\) with a certain quotient space of a subspace of \(\mathfrak H(\nu)\) as its representation space by making use of \(\pi_{\xi,\nu}\) and \(\pi_{\nu_0}\), where \(\lambda\) is defined by \(\lambda = (1/2)(\xi + \nu) - \delta\).

The authors study the properties of this class \(\left\{\pi_{\lambda,\nu}\right\}\) of irreducible representations of \(G\). For that purpose they consider the enveloping algebra \(\hat{\mathcal G}\) of the complexification \(\hat{\mathfrak g}\) of \(\mathfrak g\) and the centralizer \(\Omega\) in \(\hat{\mathcal G}\) of \(\mathfrak g_u\) where \(\mathfrak g_u\) is a compact form of \(\mathfrak g\) and then they construct, for each positive system \(Q\) of roots a homomorphism \(h^Q: \omega\to h^Q(\omega;\cdot,\cdot)\) of \(\Omega\) into the algebra \(P(\mathfrak h^*\times \mathfrak h^*)\) of all complex valued polynomial functions on \(\mathfrak h^*\times \mathfrak h^*\).

The authors consider also the ring \(R_\nu\) of all polynomials \(h^Q(\omega;\cdot,\nu)\). Using the results and ideas about the rings \(R_\nu\) the authors study the irreducible representations of \(G\) which are of class \(O\).

\[ xk = \sigma_x(k) \cdot a_+(x,k) \cdot n(x,k), \]

where \(\sigma_x(k)\in K\), \(a_+(x,k)\in A_+\), \(n(x,k)\in N\). Let \(\xi\in\mathfrak h^*\). For \(x\in G\) and \(f\in \mathfrak H\), let

\[ (n_\xi(x)f)(k) = \exp((\xi+2\delta)(\log a_+(x^{-1},k))) \cdot f(\sigma_x^{-1}(k)) \]

for \(k\in K\), where \(2\delta = \displaystyle\sum_{\alpha\in P} \alpha\). The map \(\pi_\xi\) is a representation of \(G\) in \(\mathfrak H\).

For any integral \(\nu\in\mathfrak h^*\), let \(\sigma_\nu(\exp\sqrt{-1}H) = \exp(\sqrt{-1} \nu(H))\) for \(H\in \mathfrak h_0\), \(\mathfrak h_0\) being the Lie algebra of \(A_+\). Then \(\sigma_\nu\) is a character of the group \(M = \exp\sqrt{-1}\mathfrak h_0\). For any integral \(\nu\in\mathfrak h^*\), we denote by \(\mathfrak H(\nu)\) the set of all \(f\in\mathfrak H\) such that \(R_r(m)f = \sigma_{-\nu}(m)f\) for all \(m\in M\), where \(R_r\) denotes the right regular representation of \(K\) in \(\mathfrak H\). We know that \(\mathfrak H\) is the orthogonal sum of the subspaces \(\mathfrak H(\nu)\) and each \(\mathfrak H(\nu)\) is invariant under \(\pi_\xi\). We denote by \(\pi_{\xi,\nu}\) the representation of \(G\) induced on \(\mathfrak H(\nu)\) by \(\pi_\xi\). Let \(\nu_0\) be the unique element of \(D_P\) in the orbit \(W\cdot \nu\), where \(W\) is the Weyl group of \((\mathfrak g, \mathfrak h)\) and let \(\pi_{\nu_0}\) be the irreducible representation of \(\mathfrak g\) with \(\nu_0\) as its highest weight relative to \(P\).

The authors construct an irreducible infinite dimensional representation \(\pi_{\lambda,\nu}\) of \(G\) with a certain quotient space of a subspace of \(\mathfrak H(\nu)\) as its representation space by making use of \(\pi_{\xi,\nu}\) and \(\pi_{\nu_0}\), where \(\lambda\) is defined by \(\lambda = (1/2)(\xi + \nu) - \delta\).

The authors study the properties of this class \(\left\{\pi_{\lambda,\nu}\right\}\) of irreducible representations of \(G\). For that purpose they consider the enveloping algebra \(\hat{\mathcal G}\) of the complexification \(\hat{\mathfrak g}\) of \(\mathfrak g\) and the centralizer \(\Omega\) in \(\hat{\mathcal G}\) of \(\mathfrak g_u\) where \(\mathfrak g_u\) is a compact form of \(\mathfrak g\) and then they construct, for each positive system \(Q\) of roots a homomorphism \(h^Q: \omega\to h^Q(\omega;\cdot,\cdot)\) of \(\Omega\) into the algebra \(P(\mathfrak h^*\times \mathfrak h^*)\) of all complex valued polynomial functions on \(\mathfrak h^*\times \mathfrak h^*\).

The authors consider also the ring \(R_\nu\) of all polynomials \(h^Q(\omega;\cdot,\nu)\). Using the results and ideas about the rings \(R_\nu\) the authors study the irreducible representations of \(G\) which are of class \(O\).

Reviewer: Akihiko Morimoto (Nagoya)