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Representation of complex semi-simple Lie groups and Lie algebras. (English) Zbl 0177.18004
Let $$\mathfrak g$$ 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 $$O$$.

##### MSC:
 22E46 Semisimple Lie groups and their representations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B20 Simple, semisimple, reductive (super)algebras
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