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Orbits of spherical representations and Pyasetskii duality

Daniil Shunin

TL;DR

This work classifies Pyasetskii duality for spherical representations of complex connected groups. By leveraging the commuting variety and conormal-bundle geometry, it reduces to indecomposable saturated modules and uses Knop’s spherical theory to build abstract orbit diagrams, then derives explicit orbit descriptions to realize dualities. For irreducible cases, it gives a precise duality criterion between orbit types O_{rs} and Q_{kt}, incorporating parity via floor(·)_2 and rank constraints; for reducible cases, it shows that apparent orbit duality is componentwise and treats non-apparent orbits through explicit matrix-conditions, yielding a complete duality description across all indecomposable saturated spherical modules. The results provide full orbit diagrams and duality maps, extending Pan’s abelian-action results to the broad landscape of spherical representations with structural tools from invariant theory and short gradings.

Abstract

Dual representations $V$ and $V^*$ of a complex connected algebraic group $G$ simultaneously have either infinitely or finitely many orbits. Whenever the latter holds, the orbits in $V$ and $V^*$ are in a bijective correspondence called Pyasetskii duality. We obtain a complete description of this duality in the case of spherical representations.

Orbits of spherical representations and Pyasetskii duality

TL;DR

This work classifies Pyasetskii duality for spherical representations of complex connected groups. By leveraging the commuting variety and conormal-bundle geometry, it reduces to indecomposable saturated modules and uses Knop’s spherical theory to build abstract orbit diagrams, then derives explicit orbit descriptions to realize dualities. For irreducible cases, it gives a precise duality criterion between orbit types O_{rs} and Q_{kt}, incorporating parity via floor(·)_2 and rank constraints; for reducible cases, it shows that apparent orbit duality is componentwise and treats non-apparent orbits through explicit matrix-conditions, yielding a complete duality description across all indecomposable saturated spherical modules. The results provide full orbit diagrams and duality maps, extending Pan’s abelian-action results to the broad landscape of spherical representations with structural tools from invariant theory and short gradings.

Abstract

Dual representations and of a complex connected algebraic group simultaneously have either infinitely or finitely many orbits. Whenever the latter holds, the orbits in and are in a bijective correspondence called Pyasetskii duality. We obtain a complete description of this duality in the case of spherical representations.

Paper Structure

This paper contains 7 sections, 16 theorems, 21 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Properties of dual orbits
  3. Orbits
  4. Spherical modules
  5. Explicit orbits description
  6. Proof of Theorem \ref{['thm: main_theorem']}a
  7. Proof of Theorem \ref{['thm: main_theorem']}b

Key Result

Theorem 1.1

Suppose $G$ has finitely many orbits in $V$ and $\mathfrak{C}=\cup_i \mathfrak{C}_i$ is the irreducible decomposition. Then 1. $\dim \mathfrak{C}_i = \dim V$ for each i; 2. the projection of $\mathfrak{C}$ onto the first (resp. second) factor sets up a one-to-one correspondence between the irreducib

Figures (2)

  • Figure 1: An abstract orbit diagram for the case \ref{['tablecase: irr_Sp_GL3']}, Table \ref{['table: reducible_modules']}
  • Figure 2: The algorithm applied to the representation \ref{['tablecase: red_SpxSLxGL']}, Table \ref{['table: reducible_modules']}

Theorems & Definitions (28)

  • Theorem 1.1: Py
  • Theorem 1.2: Pan
  • Remark 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Remark 1.7
  • proof : Proof of Lemma \ref{['lemma: same_orbits']}
  • proof : Proof of Lemma \ref{['lemma: duality_and_product']}
  • Lemma 2.1
  • ...and 18 more