Twisted zastava and q-Whittaker functions
01 Oct 2017-Journal of The London Mathematical Society-second Series (Oxford University Press (OUP))-Vol. 96, Iss: 2, pp 309-325
TL;DR: In this article, the twisted zastava spaces were introduced and studied for solving q-difference Toda equations from the geometry of quasimaps' spaces, and the twisted Zastava space was extended to the case of nonsimply laced simple Lie algebras.
Abstract: We implement the program outlined in Section 7 of our earlier paper [J. Amer. Math. Soc. 27 (2014) 1147–1168] extending to the case of nonsimply laced simple Lie algebras the construction of solutions of q-difference Toda equations from geometry of quasimaps' spaces. To this end we introduce and study the twisted zastava spaces.
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TL;DR: In this paper, the Coulomb branches of unframed and framed quiver gauge theories of type $ADE were studied and shown to be isomorphic to the moduli space of based rational maps from the flag variety.
Abstract: This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type $ADE$. In the unframed case they are isomorphic to the moduli space of based rational maps from ${\\mathbb C}P^1$ to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.
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TL;DR: In this paper, the Pieri-Chevalley formula is used to describe the structure sheaf of a semi-infinite Schubert variety with a line bundle associated to a dominant integral weight.
Abstract: We propose a definition of equivariant (with respect to an Iwahori subgroup) K -theory of the formal power series model Q G of semi-infinite flag manifolds, and we prove the Pieri–Chevalley formula, which describes the product, in the K -theory of Q G , of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over Q G . In order to achieve this, we provide a number of fundamental results on Q G and its Schubert subvarieties including the Borel–Weil–Bott theory, whose precise shape was conjectured by Braverman and Finkelberg in 2014. One more ingredient of this article besides the geometric results above is (a combinatorial version of) standard monomial theory for level-zero extremal weight modules over quantum affine algebras, which is described in terms of semi-infinite Lakshmibai–Seshadri paths. In fact, in our Pieri–Chevalley formula, the positivity of structure coefficients is proved by giving an explicit representation-theoretic meaning through semi-infinite Lakshmibai–Seshadri paths.
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TL;DR: In this article, the Pontryagin product structure on the equivariant group of an affine Grassmannian was shown to coincide with the tensor structure of a semi-infinite flag manifold.
Abstract: We explain that the Pontryagin product structure on the equivariant $K$-group of an affine Grassmannian considered in [Lam-Schilling-Shimozono, Compos. Math. {\bf 146} (2010)] coincides with the tensor structure on the equivariant $K$-group of a semi-infinite flag manifold considered in [K-Naito-Sagaki, Duke Math. {\it to appear}]. Then, we construct an explicit isomorphism between the equivariant $K$-group of a semi-infinite flag manifold with a suitably localized equivariant quantum $K$-group of the corresponding flag manifold. These exhibit a new framework to understand the ring structure of equivariant quantum $K$-theory and the Peterson isomorphism.
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TL;DR: In this article, the authors show that the Schubert varieties of a semi-infinite flag variety admit a nice (ind)scheme structure, its projective coordinate ring has a unique scheme, and it admits Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic.
Abstract: We exhibit basic algebro-geometric results on the formal model of semi-infinite flag varieties and its Schubert varieties over an algebraically closed field $\bK$ of characteristic $
eq 2$ from scratch. We show that the formal model of a semi-infinite flag variety admits a unique nice (ind)scheme structure, its projective coordinate ring has a $\Z$-model, and it admits a Frobenius splitting compatible with the boundaries and opposite cells in positive characteristic. This establishes the normality of the Schubert varieties of the quasi-map space with a fixed degree (instead of their limits proved in [K, Math. Ann. {\bf 371} no.2 (2018)]) when $\mathsf{char} \, \bK =0$ or $\gg 0$, and the higher cohomology vanishing of their nef line bundles in arbitrary characteristic $
eq 2$. Some particular cases of these results play crucial roles in our proof [K, arXiv:1805.01718] of a conjecture by Lam-Li-Mihalcea-Shimozono [J. Algebra {\bf 513} (2018)] that describes an isomorphism between affine and quantum $K$-groups of a flag manifold.
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1,110 citations
TL;DR: In this article, the basic relationship between G and G is discussed, and a canonical construction of G, starting from G, is presented, which leads to a rather explicit construction of a Hopf algebra by Tannakian formalism.
Abstract: As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the co-roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a reductive group, we write G for its companion and call it the dual group G. The notion of the dual group itself does not appear in Satake's paper, but was introduced by Langlands, together with its various elaborations, in [LI], [L2] and is a cornerstone of the Langlands program. It also appeared later in physics [MO], [GNO]. In this paper we discuss the basic relationship between G and G. We begin with a reductive G and consider the affine Grassmannian Qx, the Grassmannian for the loop group of G. For technical reasons we work with formal algebraic loops. The affine Grassmannian is an infinite dimen sional complex space. We consider a certain category of sheaves, the spherical perverse sheaves, on ?r. These sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a Hopf algebra, by what has come to be known as Tannakian formalism. The resulting Hopf algebra turns out to be the ring of functions on G. In this interpretation, the spherical perverse sheaves on the affine Grassman nian correspond to finite dimensional complex representations of G. Thus, instead of defining G in terms of the classification of reductive groups, we pro vide a canonical construction of G, starting from G. We can carry out our construction over the integers. The spherical perverse sheaves are then those with integral coefficients, but the Grassmannian remains a complex algebraic object.
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TL;DR: In this article, a theory of affine flag varieties and Schubert varieties for reductive groups over a Laurent power series local field k((t)) with k a perfect field was developed.
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TL;DR: In this paper, the authors consider the problem of constructing moduli for curves using tools from algebraic stacks and give general conditions under which the fixed points and the quotient of an algebraic stack are algebraic.
Abstract: The motivation at the origin of this article is to investigate some ways of constructing moduli for curves, and covers above them, using tools from stack theory. This idea arose from reading Bertin–Mezard [BeM, esp. Sec. 5] and Abramovich– Corti–Vistoli [ACV]. Our approach is in the spirit of most recent works, where one uses the flexibility of the language of algebraic stacks. This language has two (twin) aspects: category-theoretic on one side and geometric on the other. Some of our arguments, especially in Section 8, are formal arguments involving general constructions concerning group actions on algebraic stacks (this is more on the categoric side). They are, intrinsically, natural enough to preserve the “modular” aspect. In trying to isolate these arguments, we were led to write results of independent interest. It seemed therefore more adequate to present them in a separate, self-contained part. Thus the article is split into two parts of comparable size. More specifically, groups are ubiquitous in algebraic geometry (when one focuses on curves and maps between them, examples include the automorphism group, fundamental group, monodromy group, permutation group of the ramification points, . . .). It is natural to ask whether we can handle group actions on stacks in the same fashion as we do on schemes. For example, we expect: that the quotient of the stack of curves with ordered marked points Mg,n by the symmetric group should classify curves with unordered marked points; that if G acts on a scheme X then the fixed points of the stack Pic(X) under G should be related to G-linearized line bundles on X; and that the quotient of the modular stack curve X1(N ) by (Z/NZ)× should be X0(N ) (the notation is, we hope, well known to the reader). Other important examples appear in the literature: action of tori on stacks of stable maps in Gromov–Witten theory [Ko; GrPa], and action of the symmetric group Sd on a stack of multisections in [L-MB, (6.6)]. Our aim is to provide the material necessary to handle the questions raised here and then answer them, as well as to give other applications. Let us now explain in more detail the structure and results of this paper. In Part A we discuss the notion of a group action on a stack. We are mainly interested in giving general conditions under which the fixed points and the quotient of an algebraic stack are algebraic. In Sections 1 and 2 we give definitions and basics on actions. For simplicity let us now consider a flat group scheme G and an algebraic stack M, both of finite presentation (abbreviated fp) over some
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