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Symmetric semi-invariants for some Inonu-Wigner contractions-II-Case B even

Florence Fauquant-Millet

TL;DR

This paper develops a unified approach to the polynomiality of symmetric semi-invariants for Inönü-Wigner contractions of parabolic subalgebras, focusing on even maximal parabolics in type ${\rm B}_n$. By constructing canonical truncations, adapted pairs, and Weierstrass sections, it provides an upper bound that matches a previously known lower bound, hence proving that $Sy(\widetilde{\mathfrak p})=Y(\widetilde{\mathfrak p}_{\Lambda})$ is a polynomial algebra with an explicit affine slice. A detailed construction of sets $S$, Heisenberg sets, and $T$ yields a Weierstrass section for $\widetilde{\mathfrak p}$, and the derived subalgebra $\widetilde{\mathfrak p}'$ is shown to be nonsingular, reinforcing the structural clarity of these degenerate contractions. The results illuminate the invariant-theoretic geometry of IW contractions and supply explicit tools for studying degenerations of parabolic symmetries with potential applications in representation theory and algebraic geometry.

Abstract

We consider a proper parabolic subalgebra p of a simple Lie algebra g and the Inonu-Wigner contraction of p with respect to its decomposition into its standard Levi factor and its nilpotent radical : this is the Lie algebra which is isomorphic to p as a vector space, but where the nilpotent radical becomes an abelian ideal of this contraction. The study of the algebra of symmetric semi-invariants under the adjoint action associated with such a contraction was initiated in my paper entitled : Symmetric Semi-Invariants for some Inonu-Wigner Contractions-I, published in Transformation Groups, January 2025, wherein a lower bound for the formal character of this algebra was built, when the latter is well defined. Here in this paper we build an upper bound for this formal character, when p is a maximal parabolic subalgebra in a classical simple Lie algebra g in type B, when the Levi subalgebra of p is associated with the set of all simple roots without a simple root of even index with Bourbaki notation (we call this case the even case). We show that both bounds coincide. This provides a Weierstrass section for the algebra of symmetric semi-invariants associated with such a contraction and the polynomiality of such an algebra follows.

Symmetric semi-invariants for some Inonu-Wigner contractions-II-Case B even

TL;DR

This paper develops a unified approach to the polynomiality of symmetric semi-invariants for Inönü-Wigner contractions of parabolic subalgebras, focusing on even maximal parabolics in type . By constructing canonical truncations, adapted pairs, and Weierstrass sections, it provides an upper bound that matches a previously known lower bound, hence proving that is a polynomial algebra with an explicit affine slice. A detailed construction of sets , Heisenberg sets, and yields a Weierstrass section for , and the derived subalgebra is shown to be nonsingular, reinforcing the structural clarity of these degenerate contractions. The results illuminate the invariant-theoretic geometry of IW contractions and supply explicit tools for studying degenerations of parabolic symmetries with potential applications in representation theory and algebraic geometry.

Abstract

We consider a proper parabolic subalgebra p of a simple Lie algebra g and the Inonu-Wigner contraction of p with respect to its decomposition into its standard Levi factor and its nilpotent radical : this is the Lie algebra which is isomorphic to p as a vector space, but where the nilpotent radical becomes an abelian ideal of this contraction. The study of the algebra of symmetric semi-invariants under the adjoint action associated with such a contraction was initiated in my paper entitled : Symmetric Semi-Invariants for some Inonu-Wigner Contractions-I, published in Transformation Groups, January 2025, wherein a lower bound for the formal character of this algebra was built, when the latter is well defined. Here in this paper we build an upper bound for this formal character, when p is a maximal parabolic subalgebra in a classical simple Lie algebra g in type B, when the Levi subalgebra of p is associated with the set of all simple roots without a simple root of even index with Bourbaki notation (we call this case the even case). We show that both bounds coincide. This provides a Weierstrass section for the algebra of symmetric semi-invariants associated with such a contraction and the polynomiality of such an algebra follows.

Paper Structure

This paper contains 27 sections, 31 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Notation
  4. Inönü-Wigner contraction
  5. Symmetric semi-invariants
  6. Canonical truncation
  7. Centre and nilpotent radical of the contraction
  8. The contraction of the canonical truncation
  9. Adapted pairs and Weierstrass sections.
  10. The index
  11. Adapted pair
  12. Weierstrass section
  13. Coadjoint action
  14. The index of the canonical truncation of the contraction
  15. A Weierstrass section for the contraction
  16. ...and 12 more sections

Key Result

Lemma 2.5.1

Let $\mathfrak p=\mathfrak r\oplus\mathfrak m$ be a standard parabolic subalgebra in a simple Lie algebra $\mathfrak{g}$ with $\mathfrak r$ being the standard Levi factor of $\mathfrak p$ and $\mathfrak m$ the nilpotent radical of $\mathfrak p$. Let $\widetilde{\mathfrak p}$ be the Inönü-Wigner cont where $\mathfrak z\bigl(\widetilde{\mathfrak p}\bigr)$ denotes the centre of $\widetilde{\mathfrak

Figures (2)

  • Figure 1: The graph $C_{\alpha}\sqcup\widetilde{C_{\alpha}}$ for a stationary root $\alpha$ with the sequence $(\alpha^i)_{i\in\mathbb N}$ stationary at rank $1$ and the sequence $(\alpha^{(i)})_{i\in\mathbb N}$ stationary at rank $2$ and with $\alpha^1\in O_3$.
  • Figure 2: The graph $C_{\alpha}\sqcup\widetilde{C_{\alpha}}$ for a root $\alpha$ which is not stationary, with $\alpha^1\in O_3$.

Theorems & Definitions (76)

  • Remark 2.3.1
  • Remark 2.4.1
  • Lemma 2.5.1
  • proof
  • Proposition 2.5.2
  • proof
  • Corollary 2.5.3
  • proof
  • Corollary 2.5.4
  • proof
  • ...and 66 more