The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials

By O. Schiffmann, E. Vasserot. Compositio Mathematica, 2008

121 citations recorded by the source corpus.

Abstract We exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a finite field 𝔽l (or, more precisely, its spherical subalgebra U+X) and Cherednik’s double affine Hecke algebras $\ddot {\mathbf {H}}_n$ of type GLn, for all n. This allows us to obtain a geometric construction of the Macdonald polynomials Pλ(q,t−1) in terms of certain functions (Eisenstein series) on the moduli space of semistable vector bundles on the elliptic curve X.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ba2ec71d8f0d8d6006ed9fe3e84ebfdef37da508

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In Geometry and Topology

1. Peter Samuelson
2. H. Morton
3. O. Schiffmann
4. Boris Feigin
5. I. Cherednik
6. V. Toledano-Laredo
7. E. Mukhin
8. Michio Jimbo

In Mathematical Physics

1. Peter Samuelson
2. H. Morton
3. Boris Feigin
4. I. Cherednik
5. V. Toledano-Laredo
6. Tetsuji Miwa
7. Michio Jimbo
8. E. Mukhin

Citations in the corpus

• Quantum continuous $\mathfrak{gl}_{\infty}$: Semiinfinite construction of representations (2010, methodology): “n w= si1 ...sir. Here si denotes the transposition (i,i+1), l(w) is the length of wand [N]! = YN i=1 [i], [i] = v2i −1 v2 −1 . The algebra SH¨NSis called the spherical DAHA and is denoted by SH¨N. In [SV1] Schiffmann and Vasserot defined the elements PN 0,m,P N m,0 ∈ SH¨N (m∈ Z) by the formulas: PN 0,”
• Jones polynomials of torus knots via DAHA (2011, result): “The existence (a-stabilization) of the super-DAHA-polynomials (A-type) is confirmed now; it is actually similar to the approach from [37]; see Lemma 4.”
• Drinfeld realization of the elliptic Hall algebra (2010, background): “1 As it turns out, the (spherical) elliptic Hall algebra UX , or rather its rational form E appears in several other guises: as shown in [14] it projects to the spherical Cherednik algebra SḦn for any n ∈ Z and as such may be thought of as the stable limit SḦ∞ of these Cherednik algebras; as shown i”
• Yangians, quantum loop algebras and abelian difference equations (2013, background): “…relate the affine Yangian Y (cid:2) ( gl 1 ) appearing in the recent work of Maulik-Okounkov [42] to the quantum toro¨ıdal algebra U q ( gl 1 ) [21,43,46] (see also [49], where an isomorphism between appropriate completions of U q ( gl 1 ) and Y (cid:2) ( gl 1 ) is obtained in the formal setting by…”
• THE HOMFLYPT SKEIN ALGEBRA OF THE TORUS AND THE ELLIPTIC HALL ALGEBRA (2014, methodology): “ment is [SV11, Thm. 3.1]. The second statement follows from [SV13, Lemma 1.3] and the discussion directly preceding [SV13, Prop. 1.4]. (We remark that the definition of the PN m,n differs in [SV13] and [SV11] - we have chosen the latter because these make the map φN equivariant under the SL 2(Z) actio”
• Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on A2 (2012, unknown): “ = (q l−1)u(n) x , θ(u (n) x ) = t j(n−1)/2u(n) x −δi,0(t jn−1)/a j. We have the following formula [39, cor. 1.5] (7.19) θ(u(n) 0,l)·J (n) λ (q,t −1) = X s∈λ qlx(s)tly(s) J(n) λ (q,t −1), l> 1. By [38], [39, sec. 1.3] we have also (7.20) [u(n) 0,l,u (n) ±1,k] = ±sgn(l)u (n) ±1,k+l, where (7.21) s”