[Paper Review] Eigenvalue Coincidences and Multiplicity Free Spherical Pairs

Mark Colarusso, Sam Evens|arXiv (Cornell University)|Oct 15, 2014

Advanced Algebra and Geometry18 references3 citations

TL;DR

This paper extends eigenvalue coincidence varieties from gl(n,C) to so(n,C), establishing a geometric and invariant-theoretic framework for K-orbits on the orthogonal Lie algebra. It proves that maximal-dimensional K-orbits correspond to points where the differentials of K-invariant generators are linearly independent, generalizing Kostant’s theorem and enabling a new characterization of strongly regular elements crucial for the Gelfand-Zeitlin integrable system on so(n,C).

ABSTRACT

In recent work, we related the structure of subvarieties of $n imes n$ complex matrices defined by eigenvalue coincidences to $GL(n-1,\mathbb{C})$-orbits on the flag variety of $\mathfrak{gl}(n,\mathbb{C})$. In the first part of this paper, we extend these results to the complex orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n,\mathbb{C})$. In the second part of the paper, we use these results to study the geometry and invariant theory of the $K$-action on $\mathfrak{g}$, in the cases where $(\mathfrak{g}, K)$ is $(\mathfrak{gl}(n,\mathbb{C}), GL(n-1,\mathbb{C}))$ or $(\mathfrak{so}(n,\mathbb{C}), SO(n-1,\mathbb{C}))$. We study the geometric quotient $\mathfrak{g} o \mathfrak{g}//K$ and describe the closed $K$-orbits on $\mathfrak{g}$ and the structure of the zero fibre. We also prove that for $x\in \mathfrak{g}$, the $K$-orbit $Ad(K)\cdot x$ has maximal dimension if and only if the algebraically independent generators of the invariant ring $\mathbb{C}[\mathfrak{g}]^{K}$ are linearly independent at $x$, which extends a theorem of Kostant. We give applications of our results to the Gelfand-Zeitlin system.

Motivation & Objective

• To extend the study of eigenvalue coincidence varieties from gl(n,C) to the complex orthogonal Lie algebra so(n,C).
• To analyze the geometry and invariant theory of the K-action on g = so(n,C), where K = SO(n-1,C), via the quotient map g → g//K.
• To characterize closed K-orbits and describe the structure of the zero fibre of the Kostant-Wallach morphism Φn.
• To generalize Kostant’s theorem by showing that maximal-dimensional K-orbits correspond to linear independence of differentials of invariant generators.
• To apply these results to the Gelfand-Zeitlin integrable system on so(n,C), providing a simpler definition of strongly regular elements.

Proposed method

• Define eigenvalue coincidence varieties g(≥i) in so(n,C) as the set of x such that the spectra of x and its (n−1)×(n−1) upper-left submatrix x_k share at least 2i eigenvalues (counted with multiplicity, accounting for ± symmetry).
• Use the Kostant-Wallach morphism Φn: g → C^k, factoring through the quotient g//K, to study the geometric quotient and the zero fibre Φn⁻¹(0).
• Apply Knop’s theorem to show that C[g]^K is a polynomial algebra, so the quotient map is identified with Φn.
• Relate K-orbits on the flag variety B_g to the structure of the eigenvalue coincidence varieties via the K-saturation of Borel subalgebras.
• Prove flatness of the morphism Ψ: h^⊥ → h^⊥//H under a spherical pair condition, using dimension bounds and properness of the moment map.
• Use the fact that f|_h^⊥ is a polynomial in the generators g₁,…,g_k to show that elements in Ψ⁻¹(0) are nilpotent and lie in the image of the moment map.

Experimental results

Research questions

• RQ1 How do eigenvalue coincidence varieties in so(n,C) generalize those in gl(n,C), and what is their geometric structure?
• RQ2 What is the structure of the closed K-orbits on g = so(n,C) under the action of K = SO(n−1,C)?
• RQ3 How does the zero fibre of the Kostant-Wallach morphism Φn relate to the geometry of g and the invariant ring C[g]^K?
• RQ4 Can the condition for maximal-dimensional K-orbits be characterized by linear independence of the differentials of the K-invariant generators?
• RQ5 How can the new characterization of strongly regular elements simplify the construction and analysis of the Gelfand-Zeitlin integrable system on so(n,C)?

Key findings

• The irreducible components of the eigenvalue coincidence variety g(≥i) in so(n,C) are indexed by K-orbits on the flag variety B_g.
• The zero fibre Φn⁻¹(0) is equidimensional of dimension dim g − dim g//K, and its irreducible components are related to K-orbits on B_g.
• The Kostant-Wallach morphism Φn is flat, as shown via dimension bounds and properness of the moment map.
• A K-orbit Ad(K)·x has maximal dimension if and only if the differentials of the algebraically independent generators of C[g]^K are linearly independent at x, extending Kostant’s theorem.
• The n-strongly regular set in so(n,C) is nonempty only when n = 3, and the nilfibre Φ⁻¹(0) contains no strongly regular elements.
• The new definition of n-strongly regular elements via linear independence of differentials allows for a direct construction of the Gelfand-Zeitlin system on so(n,C) without using slices.