Complex Analysis and Operator Theory

, Volume 8, Issue 2, pp 513–528 | Cite as

Symplectic Twistor Operator on \(\mathbb R ^{2n}\) and the Segal–Shale–Weil Representation

  • Marie Dostálová
  • Petr Somberg


The aim of our article is the study of solution space of the symplectic twistor operator \(T_s\) in symplectic spin geometry on standard symplectic space \((\mathbb R ^{2n},\omega )\), which is the symplectic analogue of the twistor operator in (pseudo) Riemannian spin geometry. In particular, we observe a substantial difference between the case \(n=1\) of real dimension \(2\) and the case of \(\mathbb R ^{2n}, n>1\). For \(n>1\), the solution space of \(T_s\) is isomorphic to the Segal–Shale–Weil representation.


Symplectic twistor operator Symplectic Dirac operator  Metaplectic Howe duality 

Mathematics Subject Classification

53C27 53D05 81R25 



The authors gratefully acknowledge the support of the grant GA CR P201/12/G028 and SVV-2013-267317.


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Mathematical Institute of Charles UniversityPraha 8Czech Republic

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