Reciprocity Laws in the Verlinde Formulae

for the Classical Groups

Authors:
W. M. Oxbury and S. M. J. Wilson

Journal:
Trans. Amer. Math. Soc. **348** (1996), 2689-2710

MSC (1991):
Primary 14D20, 14H15

DOI:
https://doi.org/10.1090/S0002-9947-96-01563-2

MathSciNet review:
1340183

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Abstract | References | Similar Articles | Additional Information

Abstract: The Verlinde formula is computed for each of the simply-connected classical Lie groups, and it is shown that the resulting formula obeys certain reciprocity laws with respect to the exchange of the rank and the level. Some corresponding dualities between spaces of sections of theta line bundles over moduli spaces of -bundles on curves are conjectured but not proved.

**[B1]**A. Beauville,*Vector bundles on curves and generalised theta functions: recent results and open problems*, preprint, 1994.**[B2]**------,*Conformal blocks, fusion rules and the Verlinde formula*, preprint, 1994.**[DT]**R. Donagi, L.W. Tu,*Theta functions for versus*, Math. Research Letters 1 (1994), 345--357. MR**95j:14012****[F]**G. Faltings,*A proof for the Verlinde formula*, J. Alg. Geometry**3**(1994), 347--374. MR**95j:14013****[FH]**W. Fulton, J. Harris,*Representation theory*, Springer, 1991. MR**93a:20069****[K]**V.G. Kac,*Infinite dimensional Lie algebras*, 3rd edition, Cambridge, 1990. MR**92k:17038****[KNR]**S. Kumar, M.S. Narasimhan, A. Ramanathan,*Infinite Grassmannian and moduli space of -bundles*, Math. Annalen**300**(1994), 41--75. MR**96e:14011****[O]**W.M. Oxbury,*Prym varieties and the moduli of spin bundles*, to appear in proceedings of the Annual Europroj Conference, Barcelona, 1994.**[Sz]**A. Szenes,*The combinatorics of the Verlinde formulas*(N.J. Hitchin et al., ed.), in Vector bundles in algebraic geometry, Cambridge, 1995.**[Z]**D. Zagier,*Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula*, preprint, 1994.

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Additional Information

**W. M. Oxbury**

Affiliation:
Department of Mathematical Sciences, Science Laboratories, South Road, Durham DH1 3LE, U.K.

Email:
w.m.oxbury@durham.ac.uk

**S. M. J. Wilson**

Affiliation:
Department of Mathematical Sciences, Science Laboratories, South Road, Durham DH1 3LE, U.K.

Email:
s.m.j.wilson@durham.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-96-01563-2

Received by editor(s):
March 6, 1995

Received by editor(s) in revised form:
May 21, 1995

Article copyright:
© Copyright 1996
American Mathematical Society