JOURNAL BROWSE
Search
Advanced SearchSearch Tips
SPHERICAL HALL ALGEBRAS OF CURVES AND HARDER-NARASIMHAN STRATAS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
SPHERICAL HALL ALGEBRAS OF CURVES AND HARDER-NARASIMHAN STRATAS
Schiffmann, Olivier;
  PDF(new window)
 Abstract
We show that the characteristic function of any Harder-Narasimhan strata belongs to the spherical Hall algebra of a smooth projective curve X (defined over a finite field ). We prove a similar result in the geometric setting: the intersection cohomology complex IC() of any Harder-Narasimhan strata belongs to the category of spherical Eisenstein sheaves of X. We show by a simple example how a complete description of all spherical Eisenstein sheaves would necessarily involve the Brill-Noether stratas of .
 Keywords
Hall algebras;Harder-Narasimhan stratas;Eisenstein sheaves;
 Language
English
 Cited by
1.
Elliptic Springer theory, Compositio Mathematica, 2015, 151, 08, 1568  crossref(new windwow)
2.
Indecomposable vector bundles and stable Higgs bundles over smooth projective curves, Annals of Mathematics, 2016, 297  crossref(new windwow)
 References
1.
E. Arbarello, M. Cornalba, P. Griffiths, and J. Harris, Geometry of algebraic curves. Volume I, Grundlehren Math. Wiss. 267, Springer-Verlag, 1985.

2.
P. Baumann and C. Kassel, The Hall algebra of the category of coherent sheaves on the projective line, J. Reine Angew. Math. 533 (2001), 207-233.

3.
A. Beilinson, I. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), 5-171, Asterisque, 100, Soc. Math. France, Paris, 1982.

4.
I. Burban and O. Schiffmann, On the Hall algebra of an elliptic curve, I., preprint math.AG/0505148, to appear in Duke Math. Jour. crossref(new window)

5.
G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75), 215-248. crossref(new window)

6.
M. Kapranov, Eisenstein series and quantum affine algebras, Algebraic geometry 7, J. Math. Sci. (New York) 84 (1997), no. 5, 1311-1360. crossref(new window)

7.
M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J. 89 (1997), no. 1, 9-36. crossref(new window)

8.
A. King, A survey of Brill-Noether theory on algebraic curves, preprint (1993), available at http://www.maths.bath.ac.uk/ masadk/papers/.

9.
G. Laumon, Faisceaux automorphes lies aux series d'Eisenstein, Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), 227-281, Perspect. Math., 10, Academic Press, Boston, MA, 1990.

10.
G. Lusztig, Introduction to Quantum Groups, Birkhauser, 1994.

11.
H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), no. 3, 515-560. crossref(new window)

12.
M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2003), no. 2, 349-368. crossref(new window)

13.
C. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583-591. crossref(new window)

14.
O. Schiffmann, Noncommutative projective curves and quantum loop algebras, Duke Math. J. 121 (2004), no. 1, 113-168. crossref(new window)

15.
O. Schiffmann, Lectures on Hall algebras, preprint arXiv:math/0611617 (2006), to appear.

16.
O. Schiffmann, Canonical bases and moduli spaces of sheaves on curves, Invent. Math. 165 (2006), no. 3, 453-524. crossref(new window)

17.
O. Schiffmann, On the Hall algebra of an elliptic curve, II, preprint arXiv:math/0508553 (2005).

18.
O. Schiffmann and E. Vasserot, The elliptic Hall algebra, Cherednick Hecke algebras and Macdonald polynomials, Compos. Math. 147 (2011), no. 1, 188-234. crossref(new window)

19.
O. Schiffmann and E. Vasserot, Hall algebras of curves, quiver varieties and Langlands duality, arXiv:1009.0678 (2010).

20.
S. Shatz, The decomposition and specialization of algebraic families of vector bundles, Compositio Math. 35 (1977), no. 2, 163-187.

21.
M. Varagnolo and E. Vasserot, On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), no. 2, 267-297. crossref(new window)