Elliptic Springer theory

David Ben-Zvi; David Nadler

doi:10.1112/s0010437x14008021

Elliptic Springer theory

Compositio Mathematica ◽

10.1112/s0010437x14008021 ◽

2015 ◽

Vol 151 (8) ◽

pp. 1568-1584 ◽

Cited By ~ 1

Author(s):

David Ben-Zvi ◽

David Nadler

Keyword(s):

Elliptic Curve ◽

Weyl Group ◽

Principal Series ◽

Degree Zero ◽

Loop Groups ◽

Characteristic Variety ◽

Affine Weyl Group ◽

Springer Theory ◽

Character Sheaves

We introduce an elliptic version of the Grothendieck–Springer sheaf and establish elliptic analogues of the basic results of Springer theory. From a geometric perspective, our constructions specialize geometric Eisenstein series to the resolution of degree-zero, semistable $G$-bundles by degree-zero $B$-bundles over an elliptic curve $E$. From a representation theory perspective, they produce a full embedding of representations of the elliptic or double affine Weyl group into perverse sheaves with nilpotent characteristic variety on the moduli of $G$-bundles over $E$. The resulting objects are principal series examples of elliptic character sheaves, objects expected to play the role of character sheaves for loop groups.

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Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

Forum of Mathematics Sigma ◽

10.1017/fms.2020.60 ◽

2021 ◽

Vol 9 ◽

Author(s):

Alex Chirvasitu ◽

Ryo Kanda ◽

S. Paul Smith

Keyword(s):

Elliptic Curve ◽

Symmetric Power ◽

Canonical Homomorphism ◽

Coordinate Ring ◽

Characteristic Variety ◽

Closed Subvariety ◽

Coordinate Rings ◽

Cubic Hypersurfaces ◽

Elliptic Algebras ◽

Twisted Homogeneous Coordinate Ring

Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

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