arXiv subjects
Publications and source records attributed to Alexander Braverman.
We deduce the Kazhdan-Lusztig conjecture on the multiplicities of simple modules over a simple complex Lie algebra in Verma modules in category O from the equivariant geometric Satake correspondence and the analysis of torus fixed points in zastava spaces. We make similar speculations for the affine Lie algebras and W-algebras.
We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup with the category of $GL(N-1,{\mathbb C}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $GL_N$. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.
Let G be an algebraic reductive group over a an algebraically closed field of positive characteristic. Choose a parabolic subgroup $P$ in $G$ and denote by $U$ its unipotent radical. Let $X$ be a $G$-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of l-adic sheaves on $X$ with respect to a generic character of $U$ commutes with Verdier duality. In the first example we take $χ$ to be an arbitrary $G$-variety and we prove the above property for all $\overline{P}$-equivariant sheaves on $X$ where $\overline{P}$ is an opposite parabolic subgroup assuming $χ$ satisfies a strong nondegeneracy condition (such a $χ$ exists for some but not all choices of $P$). In the case when $P$ is a Borel subgroup it is enough to require that the sheaf in question is $\overline{U}$ equivariant where $\overline{U}$ is the unipotent radical of $\overline{P}$. In the second example we take $X = G$ where $G$ acts by left translations and we prove the corresponding result when $P$ is a Borel subgroup for sheaves equivariant under the adjoint action of $G$ (the latter result was conjectured by B. C. Ngo who proved it for $G = GL(n)$). As an application we reprove a theorem of N. Katz and G. Laumon about local acyclicity of the kernel of the Fourier-Deligne transform.
We show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differential-reflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, we define a new $q$-deformation of this algebra, which we call cyclotomic DAHA. Namely, we give a $q$-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA with DAHA, and homology with K-theory, and show that they give the same algebra. In addition, we show that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, we apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of $q$-deformed quasiinvariants. In the appendix by H. Nakajima and D. Yamakawa (added in version 2), the authors explain the relations between multiplicative bow varieties and (various versions of) multiplicative quiver varieties for a cyclic quiver.
This is the third companion paper of arXiv:1601.03586. When a gauge theory has a flavor symmetry group, we construct a partial resolution of the Coulomb branch as a variant of the definition. We identify the partial resolution with a partial resolution of a generalized slice in the affine Grassmannian, Hilbert scheme of points, and resolved Cherkis bow variety for a quiver gauge theory of type ADE or affine type A.
These are (somewhat informal) lectures notes for the CIME summer school "Geometric Representation Theory and Gauge Theory" in June 2018. In these notes we review the results and constructions of a series of our joint papers with H.Nakajima where a mathematical definition of Coulomb branches of 3d N=4 quantum gauge theories (of cotangent type) is given. We also discuss some further constructions (such as categories of line operators in the corresponding topologically twisted theories) and briefly discuss a connection to local geometric Langlands correspondence.
Let $G$ be a split reductive $p$-adic group. Let ${\mathcal H}(G)$ be its Hecke algebra and let ${\mathcal C}(G)\supset {\mathcal H}(G)$ be the Harish-Chandra Schwartz algebra. The purpose of this note is to give a spectral interpretation of Lusztig's asymptotic Hecke algebra $J$ (which contains the Iwahori part of ${\mathcal H}(G)$ as a subalgebra), which shows that $J$ is a subalgebra of ${\mathcal C} (G)$. This spectral description also allows to define a version of $J$ beyond the Iwahori component - i.e. we define certain subalgebra ${\mathcal J}(G)$ of ${\mathcal C}(G)$ which contains ${\mathcal H}(G)$. We explain a relation between ${\mathcal J}(G)$ and the Schwartz space of the basic affine space studied by us about 20 years ago.
Let $F$ be a local non-archimedian field and $G$ be the group of $F$-points of a split connected reductive group over $F$. In a previous aricle we defined an algebra $\mathcal J(G)$ of functions on $G$ which contains the Hecke algebra $\mathcal H(G)$ and is contained in the Harish-Chandra Schwartz algebra $\mathcal C(G)$. We consider $\mathcal J(G)$ as an algebraic analog the algebra $\mathcal C(G)$. Given a parabolic subgroup $P$ of $G$ with a Levi subgroup $M$ and the unipotent radical $U_P$ we write $X_P:=G/U_P$. In this paper we study two versions of the Schwartz space of $X_P$. The first is $\mathcal S(X_P):=\mathcal J({\mathcal S} _c(X_P))$ and the 2nd is the space spanned by functions of the form $Φ_{Q,P}(ϕ)$ where $Q$ is another parabolic with the same Levi subgroup, $ϕ\in \mathcal S_c(X_Q)$ and $Φ_{Q,P}$ is a normalized intertwining operator from $L^2(X_Q)$ to $L^2(X_P)$. We formulate a series of conjectures about these spaces, for example, we conjecture that $\mathcal S'(X_P)\subset \mathcal S(X_P)$ and that this embedding is an isomorphism on $M$-cuspidal part. We give a proof of some of our conjectures.
Let G be a reductive p-adic group. Let $Φ$ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that $Φ$ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modiification of the arguments of J.F.Dat who proved our result in one direction.
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.
In this note, we extend the results of arxiv:1111.2266 and arxiv:1203.1583 to the non simply laced case. To this end we introduce and study the twisted zastava spaces.
Let g be a semi-simple Lie algebra. In this paper we study the spaces of based quasi-maps from the projective line P^1 to the flag variety of g (it is well-known that their singularities are supposed to model the singularities of the so called semi-infinite Schubert varieties which are hard to define directly). In the first part of the paper we show that the above spaces are normal and in the case when g is simply laced they are also Gorenstein and have rational singularities. In the second part of the paper we compute the character of the ring of functions on the above spaces; in view of the above results this computation can be thought of as a computation of the (equivariant) K-theoretic J-function of the flag variety of g. We show that when g is simply laced the above characters satisfy the "fermionic recursion" version of the difference quantum Toda lattice (due to B.Feigin, E.Feigin, M.Jimbo, T.Miwa and E.Mukhin). As a byproduct we show that the equivariant K-theoretic J-function of the flag variety of a simply laced Lie algebra g is the universal eigen-function of the difference quantum Toda lattice, thus proving a conjecture of Givental and Lee. Some modification of this result is also shown to hold for non-simply laced g. We also discuss an extension of the above results to the case when g is an affine Lie algebra (this extension is conjectural except when g=sl(N)).
Let G be a semi-simple simply connected group over complex numbers. In this paper we give a geometric definition of the (dual) Weyl modules over the group G[t] and show that their characters form an eigen-function of the lattice version of the q-Toda integrable integrable system (defined by means of the quantum group version of Kostant-Whittaker reduction due to Etingof and Sevostyanov). All the proofs are algebro-geometric and rely on our previous work which interprets the universal eigen-function of the q-Toda system in terms of rings of functions on the spaces of based quasi-maps from P^1 to the flag variety of G. We discuss in detail the relation between the current work and the works of Cherednik, Ion, Sanderson and Gerasimov-Lebedev-Oblezin.
We describe the (equivariant) intersection cohomology of certain moduli spaces ("framed Uhlenbeck spaces") together with some structures on them (such as e.g.\ the Poincaré pairing) in terms of representation theory of some vertex operator algebras ("$\mathscr W$-algebras").
Let $G$ be an almost simple simply connected group over complex numbers. For a positive element $α$ of the coroot lattice of $G$ let $Z^α$ denote the space of based maps from the projective line to the flag variety of $G$ of degree $α$. This space is known to be isomorphic to the space of framed euclidean $G$-monopoles with maximal symmetry breaking at infinity of charge $α$. In [Finkelberg-Kuznetsov-Markarian-Mirković] a system of (étale, rational) coordinates on $Z^α$ is introduced. In this note we compute various known structures on $Z^α$ in terms of the above coordinates. As a byproduct we give a natural interpretation of the Gaiotto-Witten superpotential and relate it to the theory of Whittaker D-modules introduced by D.Gaitsgory.
We construct the action of the quantum group U_v(sl_n) by the natural correspondences in the equivariant localized $K$-theory of the Laumon based Quasiflags' moduli spaces. The resulting module is the universal Verma module. We construct geometrically the Shapovalov scalar product and the Whittaker vectors. It follows that a certain generating function of the characters of the global sections of the structure sheaves of the Laumon moduli spaces satisfies a $v$-difference analogue of the quantum Toda lattice system, reproving the main theorem of Givental-Lee. The similar constructions are performed for the affine Lie agebra \hat{sl}_n.
This paper is a continuation of a previous paper in which the first two authors have introduced the spherical Hecke algebra and the Satake isomorphism for an untwisted affine Kac-Moody group over a non-archimedian local field. In this paper we develop the theory of the Iwahori-Hecke algebra associated to these same groups. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dimensional case.
In this note we formulate and prove a version of Cartan decomposition for holomorphic loop groups, similar to Cartan decomposition for $p$-adic loop groups, discussed proved by Garland (and later by the authors by geometric mathods). The main technical tool that we use is the (well-known) interpretation of twisted conjugacy classes in the holomorphic loop group in terms of principal holomorphic bundles on an elliptic curve.