Langlands Dual Lie Algebra Research Articles

Representations of Shifted Quantum Affine Algebras

AbstractWe develop the representation theory of shifted quantum affine algebras $\mathcal {U}_\mu (\hat {\mathfrak {g}})$ and of their truncations, which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge theories. Our approach is based on novel techniques, which are new in the cases of shifted Yangians or ordinary quantum affine algebras as well: realization in terms of asymptotical subalgebras of the quantum affine algebra $\mathcal {U}_q(\hat {\mathfrak {g}})$, induction and restriction functors to the category $\mathcal {O}$ of representations of the Borel subalgebra $\mathcal {U}_q(\hat {\mathfrak {b}})$ of $\mathcal {U}_q(\hat {\mathfrak {g}})$, relations between truncations and Baxter polynomiality in quantum integrable models, and parametrization of simple modules via Langlands dual interpolation. We first introduce the category $\mathcal {O}_\mu $ of representations of $\mathcal {U}_\mu (\hat {\mathfrak {g}})$ and we classify its simple objects. Then we establish the existence of fusion products and we get a ring structure on the sum of the Grothendieck groups $K_0(\mathcal {O}_\mu )$. We classify simple finite-dimensional representations of $\mathcal {U}_\mu (\hat {\mathfrak {g}})$ and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We prove a truncation has only a finite number of simple representations and we introduce a related partial ordering on simple modules. Eventually, we state a conjecture on the parametrization of simple modules of a non-simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations.

Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and geometric Satake equivalence

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.

Feigin–Frenkel Image of Witten–Kontsevich Points

AbstractThe Witten–Kontsevich KdV tau function of topological gravity has a generalization to an arbitrary Drinfeld–Sokolov hierarchy associated to a simple complex Lie algebra. Using the Feigin–Frenkel isomorphism we describe the affine opers describing such generalized Witten–Kontsevich functions in terms of Segal–Sugawara operators associated to the Langlands dual Lie algebra. In the case where the Lie algebra is simply laced there is a second role these Segal–Sugawara operators play: their action, in the basic representation of the affine algebra associated to the Lie algebra, singles out the Witten–Kontsevich tau function within the phase space. We show that these two Langlands dual roles of Segal–Sugawara operators correspond to a duality between the first and last operator for a complete set of Segal–Sugawara operators.

Affine Gaudin models and hypergeometric functions on affine opers

We conjecture that quantum Gaudin models in affine types admit families of higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further conjecture that the Hamiltonians themselves are naturally expressed as such integrals.We go on to describe the space of meromorphic affine opers on an arbitrary Riemann surface. We prove that it fibres over the space of meromorphic connections on the canonical line bundle Ω. Each fibre is isomorphic to the direct product of the space of sections of the square of Ω with the direct product, over the exponents j not equal to 1, of the twisted cohomology of the jth tensor power of Ω.

Monodromy and Vinberg fusion for the principal degeneration of the space of G-bundles

We study the geometry and the singularities of the principal direction of the Drinfeld-Lafforgue-Vinberg degeneration of the moduli space of G-bundles Bun_G for an arbitrary reductive group G, and their relationship to the Langlands dual group of G. In the first part of the article we study the monodromy action on the nearby cycles sheaf along the principal degeneration of Bun_G. We describe the weight-monodromy filtration in terms of the combinatorics of the Langlands dual group of G and generalizations of the Picard-Lefschetz oscillators found in [Sch1]. Our proofs use certain local models for the principal degeneration whose geometry is studied in the second part. Our local models simultaneously provide two types of degenerations of the Zastava spaces, which together equip the Zastava spaces with the geometric analog of a Hopf algebra structure. The first degeneration corresponds to the usual Beilinson-Drinfeld fusion of divisors on the curve. The second degeneration is new and corresponds to what we call Vinberg fusion: It is obtained not by degenerating divisors on the curve, but by degenerating the group G via the Vinberg semigroup. On the level of cohomology the Vinberg fusion gives rise to an algebra structure, while the Beilinson-Drinfeld fusion gives rise to a coalgebra structure; the Hopf algebra axiom is a consequence of the underlying geometry. It is natural to conjecture that this Hopf algebra agrees with the universal enveloping algebra of the positive part of the Langlands dual Lie algebra. The above procedure would then yield a novel and highly geometric way to pass to the Langlands dual side: Elements of the Langlands dual Lie algebra are represented as cycles on the above moduli spaces, and the Lie bracket of two elements is obtained by deforming the cartesian product cycle along the Vinberg degeneration.

Nearby cycles of Whittaker sheaves on Drinfeld’s compactification

In this article we give a geometric construction of a tilting perverse sheaf on Drinfeld’s compactification, by applying the nearby cycles functor to a family of nondegenerate Whittaker sheaves. Its restrictions along the defect stratification are shown to be certain perverse sheaves attached to the nilpotent radical of the Langlands dual Lie algebra. We also describe the subquotients of the monodromy filtration using the Picard–Lefschetz oscillators introduced by Schieder. We give an argument that the subquotients are semisimple based on the action, constructed by Feigin, Finkelberg, Kuznetsov, and Mirković, of the Langlands dual Lie algebra on the global intersection cohomology of quasimaps into flag varieties.

Residual automorphic forms and spherical unitary representations of exceptional groups

Arthur has conjectured that the unitarity of a number of representations can be shown by nding appropriate automorphic realizations. This has been veried for classical groups by Mglin and for the exceptional Chevalley group G2 by Kim. In this paper we extend their results on spherical representations to the remaining exceptional groups E6, E7, E8, and F4. In particular, we prove Arthur’s conjecture that the spherical constituent of an unramied principal series of a Chevalley group over any local eld of characteristic zero is unitarizable if its Langlands parameter coincides with half the weighted marking of a coadjoint nilpotent orbit of the Langlands dual Lie algebra.

Algebras of twisted chiral differential operators and affine localization of $${\mathfrak {g}}$$ -modules

We propose a notion of algebra of twisted chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of “smallest” such modules are irreducible \({{\hat{{\mathfrak{g}}}}}\) -modules, and all irreducible \({{\mathfrak{g}}}\) -integrable \({{\hat{{\mathfrak{g}}}}}\) -modules at the critical level arise in this way.

Miura opers and critical points of master functions

Critical points of a master function associated to a simple Lie algebra\(\mathfrak{g}\) come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra\(^t \mathfrak{g}\). The proof is based on the correspondence between critical points and differential operators called the Miura opers.For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms of critical points composing the population.

Discrete Miura Opers and Solutions of the Bethe Ansatz Equations

Solutions of the Bethe ansatz equations associated to the XXX model of a simple Lie algebra come in families called the populations. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra. The proof is based on the correspondence between the solutions of the Bethe ansatz equations and special difference operators which we call the discrete Miura opers. The notion of a discrete Miura oper is one of the main results of the paper. For a discrete Miura oper D, associated to a point of a population, we show that all solutions of the difference equation DY=0 are rational functions, and the solutions can be written explicitly in terms of points composing the population.

Opers on the Projective Line, Flag Manifolds and Bethe Ansatz

We consider the problem of diagonalization of the hamiltonians of the Gaudin model, which is a quantum chain model associated to a simple Lie algebra. The hamiltonians of this model act on the tensor product of finite-dimensional representations of this Lie algebra. We show that the eigenvalues of the Gaudin hamiltonians are encoded by the so-called on the projective line, associated to the Langlands dual Lie algebra. These opers have regular singularities at the marked points with prescribed residues and trivial monodromy representation. The Bethe Ansatz is a procedure to construct explicitly the eigenvectors of the generalized Gaudin hamiltonians. We show that each solution of the Bethe Ansatz equations defines what we call a oper on the projective line. Moreover, we show that the space of Miura opers is a union of copies of the flag manifold (of the dual group), one for each oper. This allows us to prove that all solutions of the Bethe Ansatz equations, corresponding to a fixed oper, are in one-to-one correspondence with the points of an open dense subset of the flag manifold. The Bethe Ansatz equations can be written for an arbitrary Kac-Moody algebra, and we prove an analogue of the last result in this more general setting. For the Lie algebras of types A,B,C similar results were obtained by other methods by I.Scherbak and A.Varchenko and by E.Mukhin and A.Varchenko.

Long Abelian ideals

We study Abelian ideals of a Borel subalgebra consisting of long roots. It is shown that methods of Cellini and Papi can be extended to this situation. A uniform expression for the number of long Abelian ideals is given. We also show that there is a one-to-one correspondence between the long Abelian ideals and B-stable commutative subalgebras in the little adjoint representation of the Langlands dual Lie algebra.

Another realization of the category of modules over the small quantum group

Let g be a semi-simple simply-connected Lie algebra and let U ℓ be the corresponding quantum group with divided powers, where ℓ is an even order root of unity. Let in addition u ℓ⊂ U ℓ be the corresponding “small” quantum group. In this paper we establish the following relation between the categories of representations of U ℓ and u ℓ. We show that the category of u ℓ-modules is naturally equivalent to the category of U ℓ-modules, which have a Hecke eigen-property with respect to representations lifted by means of the quantum Frobenius map U ℓ→U( g ̌ ) , where g ̌ is the Langlands dual Lie algebra. This description allows to express the regular linkage class in the category u ℓ-mod in terms of perverse sheaves on the affine flag variety with a Hecke eigen-property. Moreover, it can serve as a basis to the program to understand the connection between the category u ℓ-mod and the category of representations of the corresponding affine algebra at the critical level.

Intersection cohomology of Drinfeld‚s compactifications

Let X be a smooth complete curve, G be a reductive group and $ P \subset G $ a parabolic. Following Drinfeld, one defines a (relative) compactification $ \widetilde{\hbox{\rm Bun}\,}_P $ of the moduli stack of P-bundles on X. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of $ \widetilde{\hbox{\rm Bun}\,}_P $ . The description is given in terms of the combinatorics of the Langlands dual Lie algebra $ \check{\mathfrak g} $ .

Lusztig's Canonical Quotient and Generalized Duality

We give a new characterization of Lusztig's canonical quotient, a finite group attached to each special nilpotent orbit of a complex semisimple Lie algebra. This group plays an important role in the classification of unipotent representations of finite groups of Lie type. We also define a duality map. To each pair of a nilpotent orbit and a conjugacy class in its fundamental group, the map assigns a nilpotent orbit in the Langlands dual Lie algebra. This map is surjective and is related to a map introduced by Lusztig (and studied by Spaltenstein). When the conjugacy class is trivial, our duality map is just the one studied by Spaltenstein and by Barbasch and Vogan which has an image of the set of special nilpotent orbits.

Crystals via the affine Grassmannian

Let G be a connected reductive group over ℂ, and let $\mathfrak {g}$∨ be the Langlands dual Lie algebra. Crystals for $\mathfrak {g}$∨ are combinatorial objects that were introduced by M. Kashiwara (cf., e.g., [6]) as certain "combinatorial skeletons" of finite-dimensional representations of $\mathfrak {g}$∨. For every dominant weight λ of $\mathfrak {g}$∨ Kashiwara constructed a crystal B(λ) by considering the corresponding finite-dimensional representation of the quantum group Uq($\mathfrak {g}$∨) and then specializing it to q=0. Other (independent) constructions of B(λ) were given by G. Lusztig (cf. [9]) using the combinatorics of root systems and by P. Littelmann (cf. [7]) using the "Littelmann path model." It was also shown in [5] that the family of crystals B(λ) is unique if certain reasonable conditions are imposed (cf. Theorem 1.1). The purpose of this paper is to give another (rather simple) construction of the crystals B(λ) using the geometry of the affine Grassmannian \mathscr {G}$G=G($\mathscr{K}$)/G($\mathscr{O}$) of the group G, where $\mathscr{K}$=ℂ((t)) is the field of Laurent power series and $\mathscr{O}$=ℂ[[t]] is the ring of Taylor series. We then check that the family B(λ) satisfies the conditions of the uniqueness theorem from [5], which shows that our crystals coincide with those constructed in the references above. It would be interesting to find these isomorphisms directly (cf., however, [10]).