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Lagrangian subvarieties of hyperspherical varieties

Michael Finkelberg, Victor Ginzburg, Roman Travkin

TL;DR

The article develops and tests conjectures about Lagrangian subvarieties Λ_X arising from hyperspherical G-varieties and a dual correspondence with Λ_{X^∨}, tying geometric, algebraic, and combinatorial data through moment maps and flag-variety geometry. It recasts the problem in terms of coisotropic-to-Lagrangian fiber criteria, proposes a Koszul-type duality between appropriate B-equivariant and B^∨-monodromic categories, and leverages loop-rotation localization for a graded perspective. The bulk of the work verifies these ideas for key families: the F4 example, GL(N|M) (with rooks/mazes indexing), and orthosymplectic cases, where the irreducible Lagrangian components are counted by combinatorial models of non-attacking rooks and centrally symmetric mazes. The results provide a concrete, checkable bridge between geometric representation theory, orbit combinatorics, and dualities, with potential implications for Weyl-group representations and microlocal/Kazhdan-Lusztig-type decompositions in these settings.

Abstract

Given a hyperspherical $G$-variety $\mathscr X$ we consider the zero moment level $Λ_{\mathscr X}\subset{\mathscr X}$ of the action of a Borel subgroup $B\subset G$. We conjecture that $Λ_{\mathscr X}$ is Lagrangian. For the dual $G^\vee$-variety ${\mathscr X}^\vee$, we conjecture that that there is a bijection between the sets of irreducible components $\mathrm{Irr}Λ_{\mathscr X}$ and $\mathrm{Irr}Λ_{{\mathscr X}^\vee}$. We check this conjecture for all the hyperspherical equivariant slices, and for all the basic classical Lie superalgebras.

Lagrangian subvarieties of hyperspherical varieties

TL;DR

The article develops and tests conjectures about Lagrangian subvarieties Λ_X arising from hyperspherical G-varieties and a dual correspondence with Λ_{X^∨}, tying geometric, algebraic, and combinatorial data through moment maps and flag-variety geometry. It recasts the problem in terms of coisotropic-to-Lagrangian fiber criteria, proposes a Koszul-type duality between appropriate B-equivariant and B^∨-monodromic categories, and leverages loop-rotation localization for a graded perspective. The bulk of the work verifies these ideas for key families: the F4 example, GL(N|M) (with rooks/mazes indexing), and orthosymplectic cases, where the irreducible Lagrangian components are counted by combinatorial models of non-attacking rooks and centrally symmetric mazes. The results provide a concrete, checkable bridge between geometric representation theory, orbit combinatorics, and dualities, with potential implications for Weyl-group representations and microlocal/Kazhdan-Lusztig-type decompositions in these settings.

Abstract

Given a hyperspherical -variety we consider the zero moment level of the action of a Borel subgroup . We conjecture that is Lagrangian. For the dual -variety , we conjecture that that there is a bijection between the sets of irreducible components and . We check this conjecture for all the hyperspherical equivariant slices, and for all the basic classical Lie superalgebras.
Paper Structure (27 sections, 20 theorems, 6 equations, 2 figures, 1 table)

This paper contains 27 sections, 20 theorems, 6 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Dual hyperspherical varieties
  3. Basic classical Lie superalgebras and equivariant slices
  4. Acknowledgments
  5. Generalities
  6. Coisotropic varieties and moment maps
  7. F(4)
  8. Setup
  9. Equivariant slice
  10. Linear side
  11. ${\mathop{\operatorname{\rm GL}}}(M|N)$
  12. Setup
  13. Rooks, colored permutations and mazes
  14. Linear side
  15. Equivariant slice, I
  16. ...and 12 more sections

Key Result

Proposition 2.1.1

Let ${\mathscr X}$ be a smooth affine Hamiltonian $G$-variety, and $Z\subset{\boldsymbol{\mu}}({\mathscr X})$ a locally closed $G$-stable subset. Let $\bar{\lambda}\in{\mathfrak{b}}^*$ be a character of ${\mathfrak{b}}$, so that its preimage in ${\mathfrak{g}}^*$ is $\bar{\lambda}+{\mathfrak{b}}^\pe (2) The variety ${\mathscr X}$ is coisotropic if and only if the equivalent conditions in (1) hold

Figures (2)

  • Figure 1: A $9\times10$-maze
  • Figure 2: ${\mathfrak{M}}_{4,4}^\iota\mathbin{\newline{\overset{\sim}{\newline{\longrightarrow}}}}{\mathfrak{M}}_{4,5}^\iota$

Theorems & Definitions (43)

  • Conjecture 1.1.1
  • Conjecture 1.1.2
  • Conjecture 1.1.3
  • Remark 1.1.4
  • Proposition 2.1.1
  • proof
  • Remark 2.1.2
  • Proposition 3.1.1
  • Lemma 3.3.1
  • proof
  • ...and 33 more