The FQS criterion for the Virasoro algebra was discovered by Friedan, Qiu and Shenker ([1]), but the mathematicians found their proof insufficient, so that, FQS ([2]) and Langlands ([3]), published in the same time a complete proof.   

There is an error in the paper of Langlands : ([3]) lemma 7b page 148 (see also [here][5] page 7) :     
 
 - $p=2$,  $q=1$,  $m=2$,  $h_{p,q}(m)= \frac{5}{8}$,  $M=4$  
 - $p=4$, $q=1$, $m=3$, $h_{p,q}(m)= \frac{7}{2}$,  $M=13$  
 - ...  

yield case $(B)$, but $(p,q) \ne (1,1)$ and $m \ngtr q+p-1$ (in fact, we need to distinguish between $q \ne 1$ and $q=1$, not between
$(p,q)  \ne (1,1)$ and $q=(1,1)$).

This lemma is used in the rest of the paper.  
>**Question**: Is there a way to fix the rest of the paper ?

 **Remark** : This way was used by Sauvageot (([4]) lemma 2 (ii) p 648), without fixing.

**References** :    
  ([1]) D. Friedan, Z. Qiu, S. Shenker, *Conformal invariance, unitarity, and critical exponents in two dimensions.*  Phys. Rev. Lett.  52  (1984),  no. 18, 1575--1578.   

  ([2]) D. Friedan, Z. Qiu, S. Shenker, *Details of the nonunitarity proof for highest weight representations of the Virasoro algebra.* Comm. Math. Phys.  107  (1986),  no. 4, 535--542.   

>    ([3]) R. P. Langlands, *On unitary representations of the Virasoro
> algebra.* Infinite-dimensional Lie algebras and their applications
> (Montreal, PQ, 1986),  141--159, World Sci. Publ., Teaneck, NJ, 1988.

   ([4]) F. Sauvageot, *Représentations unitaires des super-algèbres de Ramond et de Neveu-Schwarz.*   Comm. Math. Phys.  121  (1989),  no. 4, 639--657.   

  
  [1]: http://prl.aps.org/abstract/PRL/v52/i18/p1575_1
  [2]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104116227
  [3]: http://books.google.fr/books/about/Infinite_Dimensional_Lie_Algebras_and_th.html?id=PI7gmgEACAAJ&redir_esc=y
  [4]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104178251
  [5]: http://sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/VirasoroAlg-ps.pdf