Homological algebra/Topology
Two functions on Sp(g,R)
[Deux fonctions sur Sp(g,R)]

Présenté par : Michèle Vergne

Gilmer, Patrick M.1
1 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA
Comptes Rendus. Mathématique, Tome 353 (2015) no. 6, pp. 477-481.

Nous considérons deux fonctions sur Sp(g,R) à valeurs dans le groupe cyclique d'ordre quatre {±1,±i}. L'une a été définie par Lion et Vergne. L'autre est −i élevé à la puissance donnée par une fonction à valeurs entières définie par Masbaum et l'auteur (initialement sur le groupe modulaire d'une surface). Nous montrons que ces deux fonctions coïncident sur Sp(g,Z). Nous conjecturons qu'elles coïncident sur Sp(g,R).

We consider two functions on Sp(g,R) with values in the cyclic group of order four {±1,±i}. One was defined by Lion and Vergne. The other is −i raised to the power given by an integer valued function defined by Masbaum and the author (initially on the mapping class group of a surface). We identify these functions when restricted to Sp(g,Z). We conjecture the identity of these functions on Sp(g,R).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.03.006
Gilmer, Patrick M. 1

1 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA 
@article{CRMATH_2015__353_6_477_0,
     author = {Gilmer, Patrick M.},
     title = {Two functions on $ \mathrm{Sp}(g,\mathbb{R})$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {477--481},
     publisher = {Elsevier},
     volume = {353},
     number = {6},
     year = {2015},
     doi = {10.1016/j.crma.2015.03.006},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.03.006/}
}
TY  - JOUR
AU  - Gilmer, Patrick M.
TI  - Two functions on $ \mathrm{Sp}(g,\mathbb{R})$
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 477
EP  - 481
VL  - 353
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2015.03.006/
DO  - 10.1016/j.crma.2015.03.006
LA  - en
ID  - CRMATH_2015__353_6_477_0
ER  - 
%0 Journal Article
%A Gilmer, Patrick M.
%T Two functions on $ \mathrm{Sp}(g,\mathbb{R})$
%J Comptes Rendus. Mathématique
%D 2015
%P 477-481
%V 353
%N 6
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2015.03.006/
%R 10.1016/j.crma.2015.03.006
%G en
%F CRMATH_2015__353_6_477_0
Gilmer, Patrick M. Two functions on $ \mathrm{Sp}(g,\mathbb{R})$. Comptes Rendus. Mathématique, Tome 353 (2015) no. 6, pp. 477-481. doi : 10.1016/j.crma.2015.03.006. http://archive.numdam.org/articles/10.1016/j.crma.2015.03.006/

[1] Cappell, S.; Lee, R.; Miller, E. On the Maslov index, Commun. Pure Appl. Math., Volume 47 (1994)

[2] Gilmer, P. Integrality for TQFTs, Duke Math. J., Volume 125 (2004), pp. 389-413

[3] Gilmer, P.; Masbaum, G. Maslov index, Lagrangians, mapping class groups and TQFT, Forum Math., Volume 25 (2013) no. 5, pp. 1067-1106

[4] Lion, G.; Vergne, M. The Weil Representation, Maslov Index and Theta Series, Progress in Mathematics, vol. 6, Birkhauser, Basel, Switzerland, 1980

[5] Masbaum, G.; Roberts, J. On central extensions of mapping class groups, Math. Ann. (1995), pp. 131-150

[6] Putman, A. The Picard group of the moduli space of curves with level structures, Duke Math. J., Volume 161 (2012), pp. 623-674

[7] Turaev, V. A cocycle of the symplectic first Chern class and Maslov indices, Funkc. Anal. Prilozh., Volume 18 (1984) no. 1, pp. 43-48

[8] Turaev, V. The first symplectic Chern class and Maslov indices, J. Sov. Math., Volume 37 (1987), pp. 1115-1127

[9] Turaev, V. Quantum Invariants of Knots and 3-Manifolds, De Gruyter Studies in Mathematics, vol. 18, Walter de Gruyter & Co., Berlin, 2010

[10] Walker, K. On Witten's 3-manifold invariants, 1991 http://canyon23.net/math/ (Preliminary version)

[11] Wang, X. Extra structures on three-dimensional cobordisms, 2013 http://etd.lsu.edu/docs/available/etd-07032013-142703// (LSU thesis)

Cité par Sources :