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Extensions of the constant family of Harish-Chandra pairs of $SL_2(\mathbb{R})$

Eyal Subag

TL;DR

This work classifies algebraic extensions of the constant Harish-Chandra pair $(\mathfrak{sl}_2(\mathbb{C}),SO(2,\mathbb{C}))/\mathbb{C}^{\times}$ over both the affine and projective complex lines. It constructs a family \( (\boldsymbol{\mathfrak{g}}(n),SO(2,\mathbb{C})) \) for each \(n\in\mathbb{N}_0\), with special cases: \(n=0\) giving the constant family, \(n=1\) yielding a contraction, and \(n=2\) a deformation; crucially, the contraction \(n=1\) is universal in the sense that every extension is a pullback of it. The paper develops canonical realizations inside the constant family, analyzes morphisms between extensions and their localizations, and describes the centers of the corresponding universal enveloping algebras via Casimir-type elements. Over \(\mathbb{P}^1_{\mathbb{C}}\) it provides a complete classification of extensions and shows that they correspond to vector bundle decompositions \(\mathcal{O}(0)\oplus\mathcal{O}(-k)\oplus\mathcal{O}(k+n-m)\) with explicit gluing maps. These results give a robust algebraic framework for understanding contractions and deformations of representations in the Harish-Chandra setting and illuminate their geometric realization on affine and projective bases.

Abstract

We study and classify algebraic families of Harish-Chandra pairs over the complex affine line and over the complex projective line with generic fiber that is isomorphic to the Harish-Chandra pair of $SL_2(\mathbb{R})$.

Extensions of the constant family of Harish-Chandra pairs of $SL_2(\mathbb{R})$

TL;DR

This work classifies algebraic extensions of the constant Harish-Chandra pair over both the affine and projective complex lines. It constructs a family \( (\boldsymbol{\mathfrak{g}}(n),SO(2,\mathbb{C})) \) for each , with special cases: giving the constant family, yielding a contraction, and a deformation; crucially, the contraction is universal in the sense that every extension is a pullback of it. The paper develops canonical realizations inside the constant family, analyzes morphisms between extensions and their localizations, and describes the centers of the corresponding universal enveloping algebras via Casimir-type elements. Over it provides a complete classification of extensions and shows that they correspond to vector bundle decompositions \(\mathcal{O}(0)\oplus\mathcal{O}(-k)\oplus\mathcal{O}(k+n-m)\) with explicit gluing maps. These results give a robust algebraic framework for understanding contractions and deformations of representations in the Harish-Chandra setting and illuminate their geometric realization on affine and projective bases.

Abstract

We study and classify algebraic families of Harish-Chandra pairs over the complex affine line and over the complex projective line with generic fiber that is isomorphic to the Harish-Chandra pair of .

Paper Structure

This paper contains 17 sections, 17 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Algebraic families of Harish-Chandra pairs
  3. Families of Harish-Chandra pairs over affine base
  4. Extensions of $(\mathfrak{sl}_2(\mathbb{C}),SO(2,\mathbb C))/\mathbb C^{\times}$ over $\mathbb{A}^1_{\mathbb C}$
  5. Classification of extensions
  6. The universality of $(\boldsymbol{\mathfrak{g}}(1),SO(2,\mathbb C))$
  7. Some properties of extensions
  8. Morphisms between extensions
  9. Morphisms between localizations of extensions
  10. Canonical realization of $(\boldsymbol{\mathfrak{g}}(n),SO(2,\mathbb C))$ inside the constant family
  11. The even case
  12. The general case
  13. The center of $\mathcal{U}(\boldsymbol{\mathfrak{g}}(n))$
  14. Extensions of $(\mathfrak{sl}_2(\mathbb{C}),SO(2,\mathbb C))/\mathbb C^{\times}$ over $\mathbb{P}_{\mathbb C}^1$
  15. Extensions over $\mathbb{P}_{\mathbb C}^1$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(\boldsymbol{\mathfrak{g}},\boldsymbol{K})$ be an extension of $(\mathfrak{sl}_2(\mathbb{C}),SO(2,\mathbb C))/\mathbb C^{\times}$. Then $\boldsymbol{K}$ is isomorphic to $\mathbb{A}^1(\mathbb C)\times_{\operatorname{Spec}(\mathbb C)}SO(2,\mathbb C)$ and there exists a basis $\{{X},{Y},{H}\}$ of Moreover, the section ${H}$ is unique and the sections ${X}$ and ${Y}$ are unique up to simultaneou

Theorems & Definitions (36)

  • Theorem 1.1: Sec. \ref{['sec3']}, Thm. \ref{["th1'"]}
  • Theorem 1.2: Sec. \ref{['sec3']}, Thm. \ref{['th2']}
  • Theorem 1.3: Sec. \ref{['sec3.5']}, Thm. \ref{['th6']}
  • Example 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • ...and 26 more