Drinfeld realization of the elliptic Hall algebra
Olivier Schiffmann
DOI: 10.1007/s10801-011-0302-8
Abstract
We give a new presentation of the Drinfeld double E \boldsymbol{\mathcal {E}} of the (spherical) elliptic Hall algebra E + \boldsymbol{\mathcal{E}}^{+} introduced in our previous work (Burban and Schiffmann in Duke Math. J. preprint math.AG/0505148, 2005). This presentation is similar in spirit to Drinfeld's `new realization' of quantum affine algebras. This answers, in the case of elliptic curves, a question of Kapranov concerning functional relations satisfied by (principal, unramified) Eisenstein series for GL( n) over a function field. It also provides proofs of some recent conjectures of Feigin, Feigin, Jimbo, Miwa and Mukhin ( arXiv:1002.3100, 2010).
Pages: 237–262
Keywords: keywords Hall algebras; Cherednik algebras; shuffle algebras; Drinfeld new realization
Full Text: PDF
References
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2. Burban, I., Schiffmann, O.: On the Hall algebra of an elliptic curve, I. Preprint (2005), to appear in Duke Math. J.
3. Drinfeld, V.: A new realization of Yangians and of quantum affine algebras. Sov. Math. Dokl. 36(2), 212-216 (1988)
4. Feigin, B., Odesskii, A.: Vector bundles on an elliptic curve and Sklyanin algebras. In: Topics in Quantum Groups and Finite-Type Invariants. Am. Math. Soc. Transl. Ser. 2, vol. 185, pp. 65-84. Am. Math. Soc., Providence (1998)
5. Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Mukhin, E.: Quantum continuous gl\infty : semi-infinite construction of representations. Preprint (2010)
6. Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Mukhin, E.: Quantum continuous gl\infty : tensor products of Fock modules and Wn characters. Preprint (2010)
7. Feigin, B., Hoshino, A., Shibahara, J., Shiraishi, J., Yanagida, S.: Kernel function and quantum algebras. Preprint (2010)
8. Feigin, B., Hashizume, K., Hoshino, A., Shiraishi, J., Yanagida, S.: A commutative algebra on degenerate CP 1 and Macdonald polynomials. J. Math. Phys. 50, 095215 (2009)
9. Feigin, B., Tsymbaliuk, A.: Heisenberg action in the equivariant K-theory of Hilbert schemes via Shuffle Algebra. Preprint (2009)
10. Fratila, D.: Cusp eigenforms for elliptic curves and Hall algebras. Preprint (2011)
11. Kapranov, M.: Eisenstein series and quantum affine algebras. J. Math. Sci. 84, 1311-1360 (1997)
12. Ringel, C.: Hall algebras and quantum groups. Invent. Math. 101(3), 583-591 (1990)
13. Schiffmann, O.: Lectures on Hall algebras. Preprint (2006), to appear in Sém. & Cong. (SMF)
14. Schiffmann, O., Vasserot, E.: The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials. Compos. Math. 147, 188-234 (2011)
15. Schiffmann, O., Vasserot, E.: The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of A2. Preprint (2009)
16. Schiffmann, O., Vasserot, E.: Hall algebras of curves, commuting varieties and Langlands duality.
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