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Two families of Dirac-like operators for Drinfeld's Hecke algebra

Kieran Calvert

Abstract

In this paper, we define two generalisations of Dirac operators for Drinfeld's Hecke algebra. One generalisation, Parthasarathy operators inherit the notion of the Dirac inequality. The second generalisation, warped Dirac operators are such that every unitary module must have a non-zero warped Dirac cohomology. An open question is whether non-zero warped Dirac cohomology can determine the infinitesimal character akin to the fact that non-zero Dirac cohomology does. For a type $A$ Hecke algebra we give a family of operators in each class.

Two families of Dirac-like operators for Drinfeld's Hecke algebra

Abstract

In this paper, we define two generalisations of Dirac operators for Drinfeld's Hecke algebra. One generalisation, Parthasarathy operators inherit the notion of the Dirac inequality. The second generalisation, warped Dirac operators are such that every unitary module must have a non-zero warped Dirac cohomology. An open question is whether non-zero warped Dirac cohomology can determine the infinitesimal character akin to the fact that non-zero Dirac cohomology does. For a type Hecke algebra we give a family of operators in each class.
Paper Structure (19 sections, 27 theorems, 58 equations)

This paper contains 19 sections, 27 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Drinfeld's Hecke algebra
  4. Clifford Algebra
  5. Hermitian forms
  6. The original Dirac element
  7. Parthasarathy Operators
  8. A family of Parthasarathy operators for $\mathbb{H}$
  9. $P$-admissible elements
  10. A family of Dirac inequalities
  11. Warped Dirac operators
  12. A family of warped Dirac operators for $\mathbb{H}$
  13. A formula for $\mathcal{D}_\omega^2$
  14. $wD$-Admissible elements
  15. Vogan's Dirac morphism
  16. ...and 4 more sections

Key Result

Theorem 2.2

D86RS03 The algebra $\mathbb{H}$ is a Drinfeld algebra if and only if for every $g,h \in G$ and $u,v \in V$, $h' \in Z_G(g)$, $g' \in G(b) \setminus \ker \pi_V$, where $V^{\pi_V(g)^\perp }= \{ v - \pi_V(g)(v): v\in V\}$.

Theorems & Definitions (64)

  • Definition 2.1
  • Theorem 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 54 more