The Signature of the Shapovalov Form on Irreducible Verma Modules

Author:
Wai Ling Yee

Journal:
Represent. Theory **9** (2005), 638-677

MSC (2000):
Primary 22E47; Secondary 20F55

DOI:
https://doi.org/10.1090/S1088-4165-05-00269-4

Published electronically:
December 8, 2005

MathSciNet review:
2183058

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A Verma module may admit an invariant Hermitian form, which is unique up to a real scalar when it exists. Suitably normalized, it is known as the Shapovalov form. The collection of highest weights decomposes under the affine Weyl group action into alcoves. The signature of the Shapovalov form for an irreducible Verma module depends only on the alcove in which the highest weight lies. We develop a formula for this signature, depending on the combinatorial structure of the affine Weyl group. Classifying the irreducible unitary representations of a real reductive group is equivalent to the algebraic problem of classifying the Harish-Chandra modules admitting a positive definite invariant Hermitian form. Finding a formula for the signature of the Shapovalov form is a related problem which may be a necessary first step in such a classification.

- Jacques Dixmier,
*Enveloping algebras*, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996. Revised reprint of the 1977 translation. MR**1393197** - T. J. Enright and N. R. Wallach,
*Notes on homological algebra and representations of Lie algebras*, Duke Math. J.**47**(1980), no. 1, 1–15. MR**563362** - James E. Humphreys,
*Introduction to Lie algebras and representation theory*, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR**0323842** - James E. Humphreys,
*Reflection groups and Coxeter groups*, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR**1066460** - Jens Carsten Jantzen,
*Moduln mit einem höchsten Gewicht*, Lecture Notes in Mathematics, vol. 750, Springer, Berlin, 1979 (German). MR**552943** - V. G. Kac and D. A. Kazhdan,
*Structure of representations with highest weight of infinite-dimensional Lie algebras*, Adv. in Math.**34**(1979), no. 1, 97–108. MR**547842**, DOI https://doi.org/10.1016/0001-8708%2879%2990066-5 - Anthony W. Knapp,
*Lie groups beyond an introduction*, Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 1996. MR**1399083** - Anthony W. Knapp and David A. Vogan Jr.,
*Cohomological induction and unitary representations*, Princeton Mathematical Series, vol. 45, Princeton University Press, Princeton, NJ, 1995. MR**1330919** - F. G. Malikov, B. L. Feĭgin, and D. B. Fuks,
*Singular vectors in Verma modules over Kac-Moody algebras*, Funktsional. Anal. i Prilozhen.**20**(1986), no. 2, 25–37, 96 (Russian). MR**847136** - David A. Vogan Jr.,
*Representations of real reductive Lie groups*, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR**632407** - David A. Vogan Jr.,
*Unitarizability of certain series of representations*, Ann. of Math. (2)**120**(1984), no. 1, 141–187. MR**750719**, DOI https://doi.org/10.2307/2007074 - Nolan R. Wallach,
*On the unitarizability of derived functor modules*, Invent. Math.**78**(1984), no. 1, 131–141. MR**762359**, DOI https://doi.org/10.1007/BF01388720

Retrieve articles in *Representation Theory of the American Mathematical Society*
with MSC (2000):
22E47,
20F55

Retrieve articles in all journals with MSC (2000): 22E47, 20F55

Additional Information

**Wai Ling Yee**

Affiliation:
Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada

Email:
wlyee@math.ualberta.ca

Received by editor(s):
January 18, 2005

Published electronically:
December 8, 2005

Additional Notes:
This research was supported in part by an NSERC postgraduate fellowship and by an NSF research assistantship.

Article copyright:
© Copyright 2005
American Mathematical Society