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Reciprocity laws in the Verlinde formulae for the classical groups. (English) Zbl 0902.14031

For any smooth projective complex curve \(C\) of genus \(g\), and any reductive complex algebraic group \(G\), there exists a moduli space \({\mathcal M} (G)\) for algebraic principal \(G\)-bundles over \(C\), whose connected components are normal irreducible projective varieties. Moreover, any finite-dimensional representation \(G\to GL(V)\) gives rise to a natural line bundle \(\Theta (V)\) over \({\mathcal M} (G)\). This line bundle \({\mathcal L}: =\Theta (V)\) is usually called the associated determinant bundle (or theta bundle) on \({\mathcal M} (G)\). If \(G\) is simply-connected, then the so-called Verlinde number (associated to the given representation \(G \to GL(V))\) is defined to be the dimension of the space of sections (generalized theta functions on \({\mathcal M} (G))\), i.e. \(N_1(G,g): =\dim_C H^0({\mathcal M} (G), {\mathcal L})\).
More generally, to each such group with specified representation and each natural number \(k\) one associates the dimension number \(N_k(G): =N_k(G,g): =\dim_C H^0({\mathcal M} (G), {\mathcal L}^k)\), which is called the \(k\)-th Verlinde number of the given data. The famous Verlinde formula, which has recently been proved by several authors (in varying degrees of generality), provides the possibility to compute the Verlinde numbers \(N_k(G)\) in terms of certain trigonometric expressions. The purpose of the present paper is to show that the Verlinde formulae for each of the classical simple complex Lie groups \(G\) obey certain remarkable reciprocity laws with respect to the rank \(n\) (of the linear group \(G)\) and the level \(k\) (of the generalized theta functions on \({\mathcal M} (G))\). Such reciprocity laws have already been observed for the unitary groups by A. Beauville (1994), R. Donagi and L. Tu (1994), and D. Zagier (1994). In the present paper, the authors give a unified approach to verify analogous reciprocity formulae in the remaining cases of those groups, with a special emphasis on the cases \(G=Sp (n,\mathbb{C})\), \(G= \text{Spin} (m)\), and (again) \(SL(n, \mathbb{C})\). The corresponding reciprocity laws then state that
(i) \(k^g \cdot N_k(SL(n)) =n^g \cdot N_n(SL (k))\),
(ii) \(N_k(Sp(n)) =N_k(Sp(k))\), and
(iii) \(\dim_C H^0({\mathcal N} (m),L^k) =\dim_C H^0({\mathcal N} (k),L^m)\), for \(m,k\) both odd,
where \({\mathcal N} (r)\) denotes the two-component moduli space \({\mathcal M} (\text{Spin} (r)) \cup {\mathcal M}^- (\text{Spin} (r))\) in the case of \(r\) being odd. – The proofs of these (geometrical) reciprocity laws are based upon explicit computations of the Verlinde numbers for these classical Lie groups, which in turn are derived from some new, rather surprising trigonometric identities and their application to the general Verlinde formulae for arbitrary reductive complex algebraic groups [cf. G. Faltings, J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)]. This subtle analysis gives, in fact, much more numerical information than the above-mentioned reciprocity formulae incorporate (geometrically), whereas the latter ones lead to conjectural duality relations between the corresponding spaces of generalized theta functions. In the concluding section of this deep-going, excellently written paper, the authors discuss the duality conjecture for the spin reciprocity in the reciprocity in the odd-level case in greater detail, thereby referring to the earlier related works of W. M. Oxbury [cf. “Prym varieties and the moduli of spin bundles”, Annual Europroj Meeting (Barcelona 1994)] and S. Kumar, M. S. Narasimhan and A. Ramanathan [Math. Ann. 300, No. 1, 41-75 (1994; Zbl 0803.14012)]. The corresponding duality conjecture for the unitary case \((G=SL(n))\) is due to R. Donagi and L. W. Tu [cf. Math. Res. Lett. 1, No. 3, 345-357 (1994; Zbl 0847.14027)]. Altogether, the present work is an important contribution towards the geometric understanding of the Verlinde formulae.

MSC:

14K25 Theta functions and abelian varieties
14H15 Families, moduli of curves (analytic)
14D20 Algebraic moduli problems, moduli of vector bundles
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
14H30 Coverings of curves, fundamental group
22E30 Analysis on real and complex Lie groups
14H10 Families, moduli of curves (algebraic)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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References:

[1] A. Beauville, Vector bundles on curves and generalised theta functions: recent results and open problems, preprint, 1994.
[2] ——, Conformal blocks, fusion rules and the Verlinde formula, preprint, 1994.
[3] Ron Donagi and Loring W. Tu, Theta functions for \?\?(\?) versus \?\?(\?), Math. Res. Lett. 1 (1994), no. 3, 345 – 357. · Zbl 0847.14027
[4] Gerd Faltings, A proof for the Verlinde formula, J. Algebraic Geom. 3 (1994), no. 2, 347 – 374. · Zbl 0809.14009
[5] William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. · Zbl 0744.22001
[6] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. · Zbl 0716.17022
[7] Shrawan Kumar, M. S. Narasimhan, and A. Ramanathan, Infinite Grassmannians and moduli spaces of \?-bundles, Math. Ann. 300 (1994), no. 1, 41 – 75. · Zbl 0803.14012
[8] W.M. Oxbury, Prym varieties and the moduli of spin bundles, to appear in proceedings of the Annual Europroj Conference, Barcelona, 1994. · Zbl 0940.14020
[9] A. Szenes, The combinatorics of the Verlinde formulas , in Vector bundles in algebraic geometry, Cambridge, 1995. · Zbl 0823.14019
[10] D. Zagier, Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula, preprint, 1994. · Zbl 0854.14020
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