Compositio Mathematica

Self-extensions of Verma modules and differential forms on opers

Edward Frenkel a1 and Constantin Teleman a2
a1 Department of Mathematics, University of California, Berkeley, CA 94720, USA
a2 DPMMS, Centre for Mathematical Sciences, Wilberforce Road,Cambridge CB3 0WB, UK

Article author query
frenkel e   [Google Scholar] 
teleman c   [Google Scholar] 


We compute the algebras of self-extensions of the vacuum module and the Verma modules over an affine Kac–Moody algebra $\hat{\mathfrak g}$ in suitable categories of Harish-Chandra modules. We show that at the critical level these algebras are isomorphic to the algebras of differential forms on various spaces of opers associated to the Langlands dual Lie algebra of ${\mathfrak g}$, whereas away from the critical level they become trivial. These results rely on and generalize the description of the corresponding algebras of endomorphisms obtained by Feigin and Frenkel and the description of the corresponding graded versions due to Fishel, Grojnowski and Teleman.

(Published Online March 13 2006)
(Received May 7 2004)
(Accepted September 16 2005)

Key Words: affine Kac–Moody algebra; critical level; Verma module; oper.

Maths Classification

17B67 (primary); 17B56 (secondary).