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We consider the KZ differential equations over C \mathbb {C} in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field F p \mathbb {F}_p . We study the polynomial solutions of these differential equations over F p \mathbb {F}_p , constructed in a previous work joint with V. Schechtman and called the F p \mathbb {F}_p -hypergeometric solutions. The dimension of the space of F p \mathbb {F}_p -hypergeometric solutions depends on the prime number p p . We say that the KZ equations have ample reduction for a prime p p , if the dimension of the space of F p \mathbb {F}_p -hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over C \mathbb {C} . Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis F p \mathbb {F}_p -hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials ( z i − z j ) M i + M j (z_i-z_j)^{M_i+M_j} are replaced with ( z i − z j ) M i + M j − p (z_i-z_j)^{M_i+M_j-p} and the Euler gamma function Γ ( x ) \Gamma (x) is replaced with a suitable F p \mathbb {F}_p -analog Γ F p ( x ) \Gamma _{\mathbb {F}_p}(x) defined on F p \mathbb {F}_p .

We consider the differential KZ equations over C \mathbb C in the case, when the hypergeometric solutions are one-dimensional hyperelliptic integrals of genus g g . In this case the space of solutions of the differential KZ equations is a 2 g 2g -dimensional complex vector space. We also consider the same differential equations modulo p s p^s , where p p is an odd prime number and s s is a positive integer, and over the field Q p \mathbb Q_p of p p -adic numbers. We describe a construction of polynomial solutions of the differential KZ equations modulo p s p^s . These polynomial solutions have integer coefficients and are p s p^s -analogs of the hyperelliptic integrals. We call them the p s p^s -hypergeometric solutions. We consider the space M p s \mathcal M_{p^s} of all p s p^s -hypergeometric solutions, which is a module over the ring of polynomial quasi-constants modulo p s p^s . We study basic properties of M p s \mathcal M_{p^s} , in particular its natural filtration, and the dependence of M p s \mathcal M_{p^s} on s s . We show that the p p -adic limit of M p s \mathcal M_{p^s} as s → ∞ s\to \infty gives us a g g -dimensional vector space of solutions of the differential KZ equations over the field Q p \mathbb Q_p . The solutions over Q p \mathbb Q_p are power series at a certain asymptotic zone of the KZ equations. In the appendix written jointly with Steven Sperber we consider all asymptotic zones of the KZ equations in the special case g = 1 g=1 of elliptic integrals. It turns out that in this case the p p -adic limit of M p s \mathcal M_{p^s} as s → ∞ s\to \infty gives us a one-dimensional space of solutions over Q p \mathbb Q_p at every asymptotic zone. We apply Dwork’s theory of the classical hypergeometric function over Q p \mathbb Q_p and show that our germs of solutions over Q p \mathbb Q_p defined at different asymptotic zones analytically continue into a single global invariant line subbundle of the associated KZ connection. Notice that the corresponding KZ connection over C \mathbb C does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over Q p \mathbb Q_p . Also in the appendix we follow Dwork and describe the Frobenius transformations of solutions of the KZ equations for g = 1 g=1 . Using these Frobenius transformations we recover the unit roots of the zeta functions of the elliptic curves defined by the affine equations y 2 = β x ( x − 1 ) ( x − α ) y^2= \beta \,x(x-1)(x-\alpha ) over the finite field F p \mathbb F_p . Here α , β ∈ F p × , α ≠ 1 \alpha ,\beta \in \mathbb F_p^\times , \alpha \ne 1 . Notice that the same elliptic curves considered over C \mathbb {C} are used to construct the complex holomorphic solutions of the KZ equations for g = 1 g=1 .

We consider the equivariant quantum differential equation for the projective space $$P^{n-1}$$ and introduce a compatible system of difference equations. We prove an equivariant gamma theorem for $$P^{n-1}$$ , which describes the asymptotics of the differential equation at its regular singular point in terms of the equivariant characteristic gamma class of the tangent bundle of $$P^{n-1}$$ . We describe the Stokes bases of the differential equation at its irregular singular point in terms of the exceptional bases of the equivariant K-theory algebra of $$P^{n-1}$$ and a suitable braid group action on the set of exceptional bases. Our results are an equivariant version of the well-known results of Dubrovin and Guzzetti..

We consider the KZ differential equations over $$\mathbb C$$ in the case where its multidimensional hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $$\mathbb{F}_p$$ . We study the space of polynomial solutions of these differential equations over $$\mathbb{F}_p$$ , constructed in a previous work by V. Schechtman and the author. The module of these polynomial solutions defines an invariant subbundle of the associated KZ connection modulo $$p$$ . We describe the algebraic equations for that subbundle and argue that the equations correspond to highest weight vectors of the associated $$\widehat{{\mathfrak{sl}_2}}$$ Verma modules over the field $$\mathbb{F}_p$$ .

We introduce a Frobenius-like structure for the $\frak{sl}_2$ Gaudin model. Namely, we introduce potential functions of the first and second kind. We describe the Shapovalov form in terms of derivatives of the potential of the first kind and the action of Gaudin Hamiltonians in terms of derivatives of the potential of the second kind., Comment: Latex, 14 pages

We consider a pair of quiver varieties $(X;X^{\prime})$ related by 3D mirror symmetry, where $X =T^*{Gr}(k,n)$ is the cotangent bundle of the Grassmannian of $k$-planes of $n$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $X \times X^{\prime} $ (the mother function) whose restrictions to $X$ and $X^{\prime} $ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $X$ and $X^{\prime}$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.

We consider the space of solutions of the Bethe ansatz equations of the s l N ^ \widehat {\frak {sl}_N} XXX quantum integrable model, associated with the trivial representation of s l N ^ \widehat {\frak {sl}_N} . We construct a family of commuting flows on this space and identify the flows with the flows of coherent rational Ruijesenaars-Schneider systems. For that we develop in full generality the spectral transform for the rational Ruijesenaars-Schneider system.

In the previous paper by Tarasov and Varchenko the equivariant quantum differential equation ( q D E qDE ) for a projective space was considered and a compatible system of difference q K Z qKZ equations was introduced; the space of solutions to the joint system of the q D E qDE and q K Z qKZ equations was identified with the space of the equivariant K K -theory algebra of the projective space; Stokes bases in the space of solutions were identified with exceptional bases in the equivariant K K -theory algebra. This paper is a continuation of the paper by Tarasov and Varchenko. We describe the relation between solutions to the joint system of the q D E qDE and q K Z qKZ equations and the topological-enumerative solution to the q D E qDE only, definitionned as a generating function of equivariant descendant Gromov-Witten invariants. The relation is in terms of the equivariant graded Chern character on the equivariant K K -theory algebra, the equivariant Gamma class of the projective space, and the equivariant first Chern class of the tangent bundle of the projective space. We consider a Stokes basis, the associated exceptional basis in the equivariant K-theory algebra, and the associated Stokes matrix. We show that the Stokes matrix equals the Gram matrix of the equivariant Grothendieck-Euler-Poincaré pairing wrt to the basis, which is the left dual to the associated exceptional basis. We identify the Stokes bases in the space of solutions with explicit full exceptional collections in the equivariant derived category of coherent sheaves on the projective space, where the elements of those exceptional collections are just line bundles on the projective space and exterior powers of the tangent bundle of the projective space. These statements are equivariant analogs of results of G. Cotti, B. Dubro-vin, D. Guzzetti, and S. Galkin, V. Golyshev, H. Iritani.

We consider the KZ differential equations over $${\mathbb {C}}$$ in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field $${\mathbb {F}}_p$$ . We study the space of polynomial solutions of these differential equations over $${\mathbb {F}}_p$$ , constructed in a previous work by Schechtman and the second author. Using Hasse–Witt matrices, we identify the space of these polynomial solutions over $${\mathbb {F}}_p$$ with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over $${\mathbb {F}}_p$$ and the hypergeometric solutions over $${\mathbb {C}}$$ .

It is known that solutions of the KZ equations can be written in the form of multidimensional hypergeometric integrals. In 2017 in a joint paper of the author with V. Schechtman the construction of hypergeometric solutions was modified, and solutions of the KZ equations modulo a prime number $p$ were constructed. These solutions modulo $p$, called the $p$-hypergeometric solutions, are polynomials with integer coefficients. A general problem is to determine the number of independent $p$-hypergeometric solutions and understand the meaning of that number. In this paper we consider the KZ equations associated with the space of singular vectors of weight $n-2r$ in the tensor power $W^{\otimes n}$ of the vector representation of $\frak{sl}_2$. In this case, the hypergeometric solutions of the KZ equations are given by $r$-dimensional hypergeometric integrals. We consider the module of the corresponding $p$-hypergeometric solutions, determine its rank, and show that the rank equals the dimension of the space of suitable square integrable differential $r$-forms., Comment: Latex, 19 pages

The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field $F_p$ with a prime number $p$ of elements were constructed recently. In this paper we consider the example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus $g$. It is known that in this case the total $2g$-dimensional space of holomorphic solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field $F_p$ in this case gives only a $g$-dimensional space of solutions, that is, a "half" of what the complex analytic construction gives. We also show that all the constructed polynomial solutions over the field $F_p$ can be obtained by reduction modulo $p$ of a single distinguished hypergeometric solution. The corresponding formulas involve the entries of the Cartier-Manin matrix of the hyperelliptic curve. That situation is analogous to the example of the elliptic integral considered in the classical Y.I. Manin's paper in 1961., Latex, 16 pages

We prove an $\mathbb F_p$-Selberg integral formula of type $A_n$, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$. For the type $A_1$ the formula was proved in a previous paper by the authors., Comment: Latex, 21 pages, v2: misprints corrected, remarks, references added. arXiv admin note: text overlap with arXiv:2011.14248

We consider the Frobenius algebra of functions on the critical set of the master function of a weighted arrangement of hyperplanes in $\C^k$ with normal crossings. We construct two potential functions (of first and second kind) of variables labeled by hyperplanes of the arrangement and prove that the matrix coefficients of the Grothendieck residue bilinear form on the algebra are given by the $2k$-th derivatives of the potential function of first kind and the matrix coefficients of the multiplication operators on the algebra are given by the $(2k+1)$-st derivatives of the potential function of second kind. Thus the two potentials completely determine the Frobenius algebra. The presence of these potentials is a manifestation of a Frobenius like structure similar to the Frobenius manifold structure. We introduce the notion of an elementary subarrangement of an arrangement with normal crossings. It turns out that our potential functions are local in the sense that the potential functions are sums of contributions from elementary subarrangements of the given arrangement. This is a new phenomenon of locality of the Grothendieck residue bilinear form and multiplication on the algebra. It is known that this Frobenius algebra of functions on the critical set is isomorphic to the Bethe algebra of this arrangement. (That Bethe algebra is an analog of the Bethe algebras in the theory of quantum integrable models.) Thus our potential functions describe that Bethe algebra too., Latex , 26 pages, a misprint is corrected. arXiv admin note: text overlap with arXiv:1410.2438

We construct polynomial solutions of the KZ differential equations over a finite field $F_p$ as analogs of hypergeometric solutions., Comment: Latex 24 pages, two references added

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame, which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.

We prove an $\mathbb F_p$-Selberg integral formula, in which the $\mathbb F_p$-Selberg integral is an element of the finite field $\mathbb F_p$ with odd prime number $p$ of elements. The formula is motivated by analogy between multidimensional hypergeometric solutions of the KZ equations and polynomial solutions of the same equations reduced modulo $p$., Comment: Latex, 19 pages, v2: references, remarks, formula (4.7) added

In this paper we strengthen the results of [SV] by presenting their derived version. Namely, we define a "derived Knizhnik - Zamolodchikov connection"\ and identify it with a "derived Gauss - Manin connection"., Comment: 30 pages

We consider two complexes. The first complex is the twisted de Rham complex of scalar meromorphic differential forms on projective line, holomorphic on the complement to a finite set of points. The second complex is the chain complex of the Lie algebra of $\mathfrak{sl}_2$-valued algebraic functions on the same complement, with coefficients in a tensor product of contragradient Verma modules over the affine Lie algebra $\hat{{\mathfrak{sl}_2}}$. In [Schechtman V., Varchenko A., Mosc. Math. J. 17 (2017), 787-802] a construction of a monomorphism of the first complex to the second was suggested and it was indicated that under this monomorphism the existence of singular vectors in the Verma modules (the Malikov-Feigin-Fuchs singular vectors) is reflected in the relations between the cohomology classes of the de Rham complex. In this paper we prove these results.

Let $X$ be a holomorphic symplectic variety with a torus $\mathsf{T}$ action and a finite fixed point set of cardinality $k$. We assume that elliptic stable envelope exists for $X$. Let $A_{I,J}= \operatorname{Stab}(J)|_{I}$ be the $k\times k$ matrix of restrictions of the elliptic stable envelopes of $X$ to the fixed points. The entries of this matrix are theta-functions of two groups of variables: the K\"ahler parameters and equivariant parameters of $X$. We say that two such varieties $X$ and $X'$ are related by the 3d mirror symmetry if the fixed point sets of $X$ and $X'$ have the same cardinality and can be identified so that the restriction matrix of $X$ becomes equal to the restriction matrix of $X'$ after transposition and interchanging the equivariant and K\"ahler parameters of $X$, respectively, with the K\"ahler and equivariant parameters of $X'$. The first examples of pairs of 3d symmetric varieties were constructed in [Rim\'anyi R., Smirnov A., Varchenko A., Zhou Z., arXiv:1902.03677], where the cotangent bundle $T^*\operatorname{Gr}(k,n)$ to a Grassmannian is proved to be a 3d mirror to a Nakajima quiver variety of $A_{n-1}$-type. In this paper we prove that the cotangent bundle of the full flag variety is 3d mirror self-symmetric. That statement in particular leads to nontrivial theta-function identities.

We consider the $\mathfrak{gl}_N$ Gaudin model of a tensor power of the standard vector representation. The geometric Langlands correspondence in the Gaudin model relates the Bethe algebra of the commuting Gaudin Hamiltonians and the algebra of functions on a suitable space of $N$-th order differential operators. In this paper we introduce a third side of the correspondence: the algebra of functions on the critical set of a master function. We construct isomorphisms of the third algebra and the first two. Our main technical tool is the Bethe vector averaging maps, which is a new object.

Let \A be an affine hyperplane arrangement in $\C^\ell$ with complement $U$. Let $f_1, \..., f_n$ be linear polynomials defining the hyperplanes of \A, and $A^\cdot$ the algebra of differential forms generated by the 1-forms $d \log f_1, \..., d \log f_n$. To each $l \in \C^n$ we associate the master function $\Phi=\Phi_l = \prod_{i=1}^n f_i^{l_i}$ on $U$ and the closed logarithmic 1-form $\omega= d \log \Phi$. We assume $\omega$ is an element of a rational linear subspace $D$ of $A^1$ of dimension $q>1$ such that the multiplication map $\bigwedge^k(D) \to A^k$ is zero for $p, Comment: v2: Major revision. Example 3.17 added to illustrate effects of singularities, thanks to an anonymous referee. To appear in "Arrangements of Hyperplanes - Sapporo 2009," Adv. Studies in Pure Math., in press. v3: final version; minor corrections

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $K$-theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the $K$-theoretic stable envelopes and our elliptic stable envelopes. We show that the $K$-theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the $\frak{gl}_2$ case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the $K$-theoretic stable envelopes., Latex, 37 pages; v.2: Appendix and Figure 1 added; v.3: missing shift in Theorem 2.9 added and a proof of Theorem 2.9 added

We refine the statement of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr. and prove the first non-trivial cases of these conjectures. Our results provide a q-deformation of the computation of genus 1 conformal blocks via elliptic Selberg integrals by Felder-Stevens-Varchenko. They allow us to give precise formulations for the affine Macdonald conjectures in the general case which are consistent with computer computations. Our method applies recent work of the second named author to relate these conjectures in the case of $U_q(\widehat{\mathfrak{sl}}_2)$ to evaluations of certain theta hypergeometric integrals defined by Felder-Varchenko. We then evaluate the resulting integrals, which may be of independent interest, by well-chosen applications of the elliptic beta integral introduced by Spiridonov., 26 pages. v3: minor edits for published version

We consider a weighted family of $n$ parallelly transported hyperplanes in a $k$-dimensioinal affine space and describe the characteristic variety of the Gauss-Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the associated point in the Grassmannian Gr$(k,n)$. The Laurent polynomials are in involution. An intermediate object between the differential equations and the characteristic variety is the algebra of functions on the critical set of the associated master function. We construct a linear isomorphism between the vector space of the Gauss-Manin differential equations and the algebra of functions. The isomorphism allows us to describe the characteristic variety. It also allowed us to define an integral structure on the vector space of the algebra and the associated (combinatorial) connection on the family of such algebras., Comment: Latex, 24 pages, v2: references added, misprints corrected; v3: misprint correct

We consider a family of generic weighted arrangements of $n$ hyperplanes in $\C^k$ and show that the Gauss-Manin connection for the associated hypergeometric integrals, the contravariant form on the space of singular vectors, and the algebra of functions on the critical set of the master function define a Frobenius like structure on the base of the family. As a result of this construction we show that the matrix elements of the linear operators of the Gauss-Manin connection are given by the 2k+1-st derivatives of a single function on the base of the family, the function called the potential of second kind, see formula (6.46)., Comment: AmsLaTeX, 55 pages, misprints corrected, a references added, abstract and introduction extended

We discuss the Bethe ansatz in the Gaudin model on the tensor product of finite-dimensional $sl_2$-modules over the field $F_p$ with $p$ elements, where $p$ is a prime number. We define the Bethe ansatz equations and show that if $(t^0_1,\dots,t^0_k)$ is a solution of the Bethe ansatz equations, then the corresponding Bethe vector is an eigenvector of the Gaudin Hamiltonians. We characterize solutions $(t^0_1,\dots,t^0_k)$ of the Bethe ansatz equations as certain two-dimensional subspaces of the space of polynomials $F_p[x]$. We consider the case when the number of parameters $k$ equals 1. In that case we show that the Bethe algebra, generated by the Gaudin Hamiltonians, is isomorphic to the algebra of functions on the scheme defined by the Bethe ansatz equation. If $k=1$ and in addition the tensor product is the product of vector representations, then the Bethe algebra is also isomorphic to the algebra of functions on the fiber of a suitable Wronski map., Comment: Latex, v2 and v3: misprints corrected, v4: misprints corrected, a reference added

We consider the cotangent bundle T^*F_\lambda of a GL_n partial flag variety, \lambda = (\lambda_1,...,\lambda_N), |\lambda|=\sum_i\lambda_i=n, and the torus T=(C^*)^{n+1} equivariant cohomology H^*_T(T^*F_\lambda). In [MO], a Yangian module structure was introduced on \oplus_{|\lambda|=n} H^*_T(T^*F_\lambda). We identify this Yangian module structure with the Yangian module structure introduced in [GRTV]. This identifies the operators of quantum multiplication by divisors on H^*_T(T^*F_\lambda), described in [MO], with the action of the dynamical Hamiltonians from [TV2, MTV1, GRTV]. To construct these identifications we provide a formula for the stable envelope maps, associated with the partial flag varieties and introduced in [MO]. The formula is in terms of the Yangian weight functions introduced in [TV1], c.f. [TV3, TV4], in order to construct q-hypergeometric solutions of qKZ equations.

We define a $\mathfrak{gl}_N$-stratification of the Grassmannian of $N$ planes $\mathrm{Gr}(N,d)$. The $\mathfrak{gl}_N$-stratification consists of strata $\Omega_{\mathbf{\Lambda}}$ labeled by unordered sets $\mathbf{\Lambda}=(\lambda^{(1)},\dots,\lambda^{(n)})$ of nonzero partitions with at most $N$ parts, satisfying a condition depending on $d$, and such that $(\otimes_{i=1}^n V_{\lambda^{(i)}})^{\mathfrak{sl}_N}\ne 0$. Here $V_{\lambda^{(i)}}$ is the irreducible $\mathfrak{gl}_N$-module with highest weight $\lambda^{(i)}$. We show that the closure of a stratum $\Omega_{\mathbf{\Lambda}}$ is the union of the strata $\Omega_{\mathbf\Xi}$, $\mathbf{\Xi}=(\xi^{(1)},\dots,\xi^{(m)})$, such that there is a partition $\{I_1,\dots,I_m\}$ of $\{1,2,\dots,n\}$ with $ {\rm {Hom}}_{\mathfrak{gl}_N} (V_{\xi^{(i)}}, \otimes_{j\in I_i}V_{\lambda^{(j)}}\big)\neq 0$ for $i=1,\dots,m$. The $\mathfrak{gl}_N$-stratification of the Grassmannian agrees with the Wronski map. We introduce and study the new object: the self-dual Grassmannian $\mathrm{sGr}(N,d)\subset \mathrm{Gr}(N,d)$. Our main result is a similar $\mathfrak{g}_N$-stratification of the self-dual Grassmannian governed by representation theory of the Lie algebra $\mathfrak {g}_{2r+1}:=\mathfrak{sp}_{2r}$ if $N=2r+1$ and of the Lie algebra $\mathfrak g_{2r}:=\mathfrak{so}_{2r+1}$ if $N=2r$., Comment: LaTeX, 30 pages, 2 figures

We consider a tensor product \( V(b)=\otimes_{i=1}^n{{\mathbb{C}}^N}\left( {{b_i}} \right) \) of the Yangian \( Y\left( {\mathfrak{g}{{\mathfrak{l}}_N}} \right) \) evaluation vector representations. We consider the action of the commutative Bethe subalgebra \( {{\mathcal{B}}^q}\subset Y\left( {\mathfrak{g}{{\mathfrak{l}}_N}} \right) \) on a \( \mathfrak{g}{{\mathfrak{l}}_N} \)-weight subspace \( V{(b)_{\uplambda}}\subset V(b) \) of weight λ. Here the Bethe algebra depends on the parameters q = (q1, . . . , qN ). We identify the \( {{\mathcal{B}}^q} \) -module V (b)λ with the regular representation of the algebra of functions on a fiber of a suitable discrete Wronski map. For q = (1, . . . , 1), we study the action of \( {{\mathcal{B}}^q} \) on the space \( V(b)_{\lambda}^{\mathrm{sing}} \) of singular vectors of a certain weight and identify the \( {{\mathcal{B}}^q} \) -module \( V(b)_{\lambda}^{\mathrm{sing}} \) with the regular representation of the algebra of functions on a fiber of another suitable discrete Wronski map.

We consider the conormal bundle of a Schubert variety \(S_I\) in the cotangent bundle \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) of the Grassmannian \({{\mathrm{\mathrm {Gr}}}}\) of \(k\)-planes in \({{\mathrm{\mathbb {C}}}}^n\). This conormal bundle has a fundamental class \({\kappa _I}\) in the equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\). Here \({{\mathrm{\mathbb T}}}=({{\mathrm{\mathbb {C}}}}^*)^n\times {{\mathrm{\mathbb {C}}}}^*\). The torus \(({{\mathrm{\mathbb {C}}}}^*)^n\) acts on \(T^*\!{{\mathrm{\mathrm {Gr}}}}\) in the standard way and the last factor \({{\mathrm{\mathbb {C}}}}^*\) acts by multiplication on fibers of the bundle. We express this fundamental class as a sum \(Y_I\) of the Yangian \(Y(\mathfrak {gl}_2)\) weight functions \((W_J)_J\). We describe a relation of \(Y_I\) with the double Schur polynomial \([S_I]\). A modified version of the \(\kappa _I\) classes, named \(\kappa '_I\), satisfy an orthogonality relation with respect to an inner product induced by integration on the non-compact manifold \(T^*\!{{\mathrm{\mathrm {Gr}}}}\). This orthogonality is analogous to the well known orthogonality satisfied by the classes of Schubert varieties with respect to integration on \({{\mathrm{\mathrm {Gr}}}}\). The classes \((\kappa '_I)_I\) form a basis in the suitably localized equivariant cohomology \(H^*_{{{\mathrm{\mathbb T}}}}(T^*\!\!{{\mathrm{\mathrm {Gr}}}})\). This basis depends on the choice of the coordinate flag in \({{\mathrm{\mathbb {C}}}}^n\). We show that the bases corresponding to different coordinate flags are related by the Yangian R-matrix.

We introduce families \( \mathcal{B}_n^S\left( {{z_1},\ldots,{z_n}} \right) \) and \( \mathcal{B}_{{n,\hbar}}^S\left( {{z_1},\ldots,{z_n}} \right) \) of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra \( \mathbb{C}\left[ {\mathfrak{S}n} \right] \) of the symmetric group. Bethe subalgebras are deformations of the Gelfand−Zetlin subalgebra of \( \mathbb{C}\left[ {\mathfrak{S}n} \right] \). We describe various properties of Bethe subalgebras.

Using the vanishing cycles of simple singularities, we study the eigenvectors of Cartan matrices of finite root systems, and of q-deformations of these matrices., Comment: 27 pages; a physical section describing an experiment related to the PF vector of E8 is added

We define an elliptic version of the stable envelope of Maulik and Okounkov for the equivariant elliptic cohomology of cotangent bundles of Grassmannians. It is a version of the construction proposed by Aganagic and Okounkov and is based on weight functions and shuffle products. We construct an action of the dynamical elliptic quantum group associated with gl2 on the equivariant elliptic cohomology of the union of cotangent bundles of Grassmannians. The generators of the elliptic quantum groups act as difference operators on sections of admissible bundles, a notion introduced in this paper., Symmetry Integrability and Geometry: Methods and Applications, 14, ISSN:1815-0659

We describe \,$q$-hypergeometric solutions of the equivariant quantum differential equations and associated qKZ difference equations for the cotangent bundle $T^*F_\lambda$ of a partial flag variety \,$F_\lambda$\,. These \,$q$-hypergeometric solutions manifest a Landau-Ginzburg mirror symmetry for the cotangent bundle. We formulate and prove Pieri rules for quantum equivariant cohomology of the cotangent bundle. Our Gamma theorem for \,$T^*F_\lambda$ \,says that the leading term of the asymptotics of the \,$q$-hypergeometric solutions can be written as the equivariant Gamma class of the tangent bundle of $T^*F_\lambda$ multiplied by the exponentials of the equivariant first Chern classes of the associated vector bundles. That statement is analogous to the statement of the gamma conjecture by B.\,Dubrovin and by S.\,Galkin, V.\,Golyshev, and H.\,Iritani, see also the Gamma theorem for \,$F_\lambda$ \,in Appendix B., Comment: Latex, 47 pages; v3: title extended, appendix on the gamma theorem for $T^*F_\lambda$ added; v4: new Section 11 and new Appendix A added, equivariant Gamma theorem for $F_\bla$ added

We identify the Bethe algebra of the Gaudin model associated to gl(N) acting on a suitable representation with the center of the rational Cherednik algebra and with the algebra of regular functions on the Calogero-Moser space.

We interpret the equivariant cohomology \(H_{GL_n }^* \)(ℱλ,ℂ) of a partial flag variety ℱλ parametrizing chains of subspaces 0 = F0 ⊂ F1 ⊂ … ⊂ FN = ℂn, dimFi/Fi−1 = λi, as the Bethe algebra Open image in new window of the Open image in new window-weight subspace Open image in new window of a Open image in new window[t]-module Open image in new window.

For a set of quasi-exponentials with real exponents, we consider the discrete Wronskian (also known as Casorati determinant) with pure imaginary step 2h. We prove that if the coefficients of the discrete Wronskian are real and for every its roots the imaginary part is at most |h|, then the complex span of this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This result is a generalization of the statement of the B. and M. Shapiro conjecture on spaces of polynomials. The proof is based on the Bethe ansatz for the XXX model., Comment: Latex, 9 pages

Using equivariant localization formulas we give a formula for conformal blocks at level one on the sphere as suitable polynomials. Using this presentation we give a generating set in the space of conformal blocks at any level if the marked points on the sphere are generic.

We prove the B. and M. Shapiro conjecture that says that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This in particular implies the following result: If all ramification points of a parametrized rational curve $ f : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset CP^r $. The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linear operator on a Euclidean space has a real spectrum. In Appendix we discuss properties of differential operators associated with Bethe vectors in the Gaudin model and, in particular, prove a conditional statement: we deduce the transversality of certain Schubert cycles in a Grassmannian from the simplicity of the spectrum of the Gaudin Hamiltonians., Comment: Latex, 18 pages, revised version