Kirillov-Kostant theory and Feynman path integrals on coadjoint orbits of a certain real semisimple Lie group (Q1321767)

\textit{A. Alekseev}, \textit{L. D. Faddeev} and \textit{S. Shatashvili} showed in [J. Geom. Phys. 5, 391-406 (1989)] that any irreducible unitary representation of compact groups can be obtained by path integrals. They computed characters of the representations. We showed in [the author, \textit{K. Ogura}, \textit{K.Okamoto}, \textit{R. Sawae} and \textit{H. Yasunaga}, Hokkaido J. Math. 20, 353-405 (1991; Zbl 0733.58007)] that path integrals give unitary operators of the representation which is constructed by Kirillov-Kostant theory for the Heisenberg group, the affine transformation group on the real line, \(\text{SL}(2, {\mathbf R})\) \((\cong \text{SU}(1, 1))\) and \(\text{SU}(2)\). For the affine transformation group, we took a real polarization, for \(\text{SU}(2)\) a complex polarization (but computed without Hamiltonians), and for the Heisenberg group and \(\text{SL}(2,{\mathbf R})\) both a real polarization and a complex polarization. (For a complex polarization of \(\text{SU}(2, {\mathbf R})\), we realized it as \(\text{SU}(1,1)\) and computed it without Hamiltonians.) In [the authors, Kirillov-Kostant theory and Feynman path integrals on coadjoint orbits of \(\text{SU}(2)\) and \(\text{SU}(1,1)\). Proc. RIMS Res. Project 91 on Infinite Analysis (to appear)] we found that, in order to compute the path integrals with non-trivial Hamiltonians for \(\text{SU}(1, 1)\) and \(\text{SU}(2)\) to obtain unitary operators realized by Borel-Weil theory, we have to regularize the Hamiltonian functions, and in [the author, \textit{K. Ogura}, \textit{K. Okamoto} and \textit{R. Sawae}, Hiroshima Math. J. 23, 234-247 (1993; Zbl 0838.22009)] we extended the results to the case that the maximal compact subgroup \(K\) of a connected semisimple Lie group \(G\) has equal rank to the complex rank of \(G\). In this paper, we work with a linear connected noncompact semisimple Lie group \(G\) and consider real polarizations. Let \(\mathfrak g\) be the Lie algebra of \(G\). We fix a Cartan involution \(\theta\) of \(\mathfrak g\) and let the corresponding Cartan decomposition [cf. \textit{S. Helgason}, Differential geometry and symmetric spaces, New York-London (1962; Zbl 0111.181)] be \({\mathfrak g}= {\mathfrak k}\oplus {\mathfrak p}\). Let \(\mathfrak a\) be a maximal Abelian subalgebra of \(\mathfrak p\) and \(\mathfrak m\) the centralizer of \(\mathfrak a\) in \(\mathfrak k\). If we fix a notion of positivity for \({\mathfrak a}\)-roots, we can let \(\mathfrak n\) be the nilpotent subalgebra given as the sum of the root spaces for the positive roots. In this paper, we explicitly compute the path integrals with Hamiltonians for \(Y\in {\mathfrak m}\oplus {\mathfrak a}\) or \(\mathfrak n\), to give unitary operators of the representation which is constructed by Kirillov-Kostant theory. When we compute the path integral with the Hamiltonian for \(Y\in {\mathfrak n}\), we make the following assumption. Put \({\mathcal C}^0{\mathfrak n}= {\mathfrak n}\) and \({\mathcal C}^{i+ 1}{\mathfrak n}= [{\mathfrak n}, {\mathcal C}^i{\mathfrak n}]\). Then \[ \text{Assumption:}\qquad {\mathcal C}^i{\mathfrak n}= \{0\}\qquad\text{if} \quad i\geq 3. \] Lie groups which satisfy the above assumption include \(\text{SL}(n, k)\) \((n= 2, 3, 4, k=\mathbb{R},\mathbb{C})\) and linear connected semisimple Lie groups of real rank one etc. For \(Y\in \overline{{\mathfrak n}}= \theta{\mathfrak n}\), we have not yet succeeded in computing the path integral explicitly, even under the above assumption. We also show that one can obtain the formal intertwining operator between the representations which are constructed by Kirillov-Kostant theory with two polarizations, by the path integral. This is a generalization of the results in the paper mentioned in the first quotation to our \(G\), which we showed for the Heisenberg group and \(\text{SL}(2,\mathbb{R})\).