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Relative Lie algebra cohomology of SU(2,1) and Eisenstein classes on Picard surfaces

Jitendra Bajpai, Mattia Cavicchi

Abstract

We consider Picard surfaces, locally symmetric varieties $S_Γ$ attached to the Lie group SU(2,1), and we construct explicit differential forms on $S_Γ$ representing Eisenstein classes, i.e. cohomology classes restricting non-trivially to the boundary of the Borel-Serre compactification. This is needed for the computation of the class of the extensions of the Hodge structure that we have constructed in [2] according to the predictions of the Bloch-Beilinson conjectures. The tool for the construction of the differential forms is an analysis of relative Lie algebra cohomology of the principal series of SU(2,1) using recent methods of Buttcane and Miller.

Relative Lie algebra cohomology of SU(2,1) and Eisenstein classes on Picard surfaces

Abstract

We consider Picard surfaces, locally symmetric varieties attached to the Lie group SU(2,1), and we construct explicit differential forms on representing Eisenstein classes, i.e. cohomology classes restricting non-trivially to the boundary of the Borel-Serre compactification. This is needed for the computation of the class of the extensions of the Hodge structure that we have constructed in [2] according to the predictions of the Bloch-Beilinson conjectures. The tool for the construction of the differential forms is an analysis of relative Lie algebra cohomology of the principal series of SU(2,1) using recent methods of Buttcane and Miller.
Paper Structure (15 sections, 13 theorems, 99 equations, 1 table)

This paper contains 15 sections, 13 theorems, 99 equations, 1 table.

Table of Contents

  1. Introduction
  2. Outline
  3. Notations
  4. Structure of $\mathop{\rm SU} \nolimits(2,1)$ and $\mathop{\rm U} \nolimits(2)$ and of their Lie algebras
  5. The group $G$ and its maximal compact subgroups
  6. Lie algebras
  7. Principal series representations of SU$(2,1)$
  8. Relative Lie algebra cohomology
  9. Finite dimensional $(\mathfrak{g}, K_{\infty})$-modules
  10. Relative Lie algebra cohomology and the Delorme isomorphism
  11. Explicit representatives of a generator
  12. Eisenstein classes on Picard surfaces
  13. Picard surfaces
  14. Cohomology of local systems on Picard surfaces and Lie algebra cohomology
  15. Eisenstein cohomology of Picard surfaces

Key Result

Lemma 3.3

The $K_{\infty}$-finite vectors in $I_{\phi_{\infty}}$ are finite linear combinations of Wigner $D$-functions $W^{j,n}_{m_1,m_2}$ whose parameters $j,n,m_1,m_2$ with $j \in \frac{1}{2}{\mathbb{Z}}_{\geq0}, n, m_1, m_2 \in \frac{1}{2}{\mathbb{Z}}$ satisfy and

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • Lemma 3.5
  • Remark 3.6
  • Theorem 3.7
  • ...and 27 more