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Register a new userSingular fibers of the Gelfand--Cetlin system on $mathfrak{u}(n)^*$
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In this paper, we show that every singular fiber of the Gelfand--Cetlin system on coadjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a $2$-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibers can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibers, and give a detailed description of these singular fibers in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibers are degenerate for the Gelfand--Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids, and also general ideas coming from the theory of singularities of integrable Hamiltonian systems.
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We show that the Ginzburg-Weinstein diffeomorphism $mathfrak{u}(n)^* to U(n)^*$ of Alekseev-Meinrenken admits a scaling tropical limit on an open dense subset of $mathfrak{u}(n)^*$. The target of the limit map is a product $mathcal{C} times T$, where $mathcal{C}$ is the interior of a cone, $T$ is a torus, and $mathcal{C} times T$ carries an integrable system with natural action-angle coordinates. The pull-back of these coordinates to $mathfrak{u}(n)^*$ recovers the Gelfand-Zeitlin integrable system of Guillemin-Sternberg. As a by-product of our proof, we show that the Lagrangian tori of the Flaschka-Ratiu integrable system on the set of upper triangular matrices meet the set of totally positive matrices for sufficiently large action coordinates.
We classify, up to symplectomorphisms, a neighborhood of a singular fiber of an integrable system (which is proper and has connected fibers) containing $k > 1$ focus-focus critical points. Our result shows that there is a one-to-one correspondence between such neighborhoods and $k$ formal power series, up to a $(mathbb{Z}_2 times D_k)$-action, where $D_k$ is the $k$-th dihedral group. The $k$ formal power series determine the dynamical behavior of the Hamiltonian vector fields $X_{f_1}, X_{f_2}$ associated to the components $f_1, f_2 colon (M, omega) to mathbb{R}$ of the integrable system on the symplectic manifold $(M,omega)$ via the differential equation $omega(X_{f_i}, cdot) = mathop{}!mathrm{d} f_i$, near the singular fiber containing the $k$ focus-focus critical points. This proves a conjecture of San Vu Ngoc from 2002.
We determine the center of a localization of ${mathcal U}_q({mathfrak n}_omega)subseteq {mathcal U}^+_q({mathfrak g})$ by the covariant elements (non-mutable elements) by means of constructions and results from quantum cluster algebras. In our set-up, ${mathfrak g}$ is any finite-dimensional complex Lie algebra and $omega$ is any element in the Weyl group $W$. The non-zero complex parameter $q$ is mostly assumed not to be a root of unity, but our method also gives many details in case $q$ is a primitive root of unity. We point to a new and very useful direction of approach to a general set of problems which we exemplify here by obtaining the result that the center is determined by the null space of $1+omega$. Further, we use this to give a generalization to double Schubert Cell algebras where the center is proved to be given by $omega^{mathfrak a}+omega^{mathfrak c}$. Another family of quadratic algebras is also considered and the centers determined.
Actions of $U(n)$ on $U(n+1)$ coadjoint orbits via embeddings of $U(n)$ into $U(n+1)$ are an important family of examples of multiplicity free spaces. They are related to Gelfand-Zeitlin completely integrable systems and multiplicity free branching rules in representation theory. This paper computes the Hamiltonian local normal forms of all such actions, at arbitrary points, in arbitrary $U(n+1)$ coadjoint orbits. The results are described using combinatorics of interlacing patterns; gadgets that describe the associated Kirwan polytopes.
For the quantum algebra U_q(gl(n+1)) in its reduction on the subalgebra U_q(gl(n)) an explicit description of a Mickelsson-Zhelobenko reduction Z-algebra Z_q(gl(n+1),gl(n)) is given in terms of the generators and their defining relations. Using this Z-algebra we describe Hermitian irreducible representations of a discrete series for the noncompact quantum algebra U_q(u(n,1)) which is a real form of U_q(gl(n+1)), namely, an orthonormal Gelfand-Graev basis is constructed in an explicit form.
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