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An analogue of Weyl’s law for quantized irreducible generalized flag manifolds. (English) Zbl 1357.17017

The paper under review presents an analogue of Weyl’s law for quantized irreducible generalized flag manifolds. The class of irreducible generalized flag manifolds coincides with the class of compact irreducible Hermitian symmetric spaces. The latter are Riemannian symmetric spaces admitting a complex structure, which furthermore cannot be written as a product of spaces of the same type. By quantized irreducible generalized flag manifolds the paper means the following: Let \(g\) be a complex semisimple Lie algebra. Denote by \(G_C\) the corresponding simply connected Lie group. Let \(p\) be a parabolic subalgebra, that is any subalgebra that contains a Borel subalgebra. Denote by \(P\) the corresponding subgroup of \(G_C\). Then a generalized flag manifold is defined to be the complex manifold \(\frac{G_C}{P}\). As a real manifold it is diffeomorphic to \(\frac{G}{K}\), where \(G\) is the compact real form of \(G_C\) and \(K\); the intersection of \(L\) and \(G\); is the real form of the Levi factor \(L\). The case when the adjoint action of \(p\) on \(\frac{g}{p}\) is irreducible corresponds to \(\frac{G_C}{P}\) being a symmetric space. If in addition \(g\) is simple, then the generalized flag manifold is called irreducible. The aim of the paper under review is to investigate whether an analogue of the residue formula holds in the case of compact quantum groups and their homogeneous spaces. There are two properties of this formula that the paper maintains: (1) the proportionality between the (residue of the) trace of an operator and the integral, (2) the appearance of the dimension of the space as the first singularity of the zeta function. In the quantum setting some of the classical ingredients have to be replaced by their appropriate counterparts. For example, the Haar integral of a compact Lie group has to be replaced by the Haar state, which satisfies analogues of the classical invariance conditions. In this regard the replacement in this paper is done by defining a zeta function, similarly to the classical setting, and showing that it satisfies the following two properties: as a functional on the quantized algebra it is proportional to the Haar state; its first singularity coincides with the classical dimension. The relevant formulae are given for the more general case of compact quantum groups.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
58B32 Geometry of quantum groups
58B34 Noncommutative geometry (à la Connes)
46N50 Applications of functional analysis in quantum physics
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations

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