pinecone/core-2020-05-10-deduplication huggingface dataset:
Finite difference quantum Toda lattice via equivariant K-theory

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.Comment: Some corrections are made in Sections 2,
Can one factor the classical adjoint of a generic matrix?

Let k be a field, n a positive integer, X a generic nxn matrix over k (i.e.,
a matrix (x_{ij}) of n^2 independent indeterminates over the polynomial ring
k[x_{ij}]), and adj(X) its classical adjoint. It is shown that if char k=0 and
n is odd, then adj(X) is not the product of two noninvertible nxn matrices over
k[x_{ij}]. If n is even and >2, a restricted class of nontrivial factorizations
occur. The nonzero-characteristic case remains open.
  The operation adj on matrices arises from the (n-1)st exterior power functor
on modules; the same question can be posed for matrix operations arising from
other functors.Comment: Revised version contains answer to "even n" question left open in
  original version. (Answer due to Buchweitz & Leuschke; simple proof in this
  note.) Copy at http://math.berkeley.edu/~gbergman/papers will always have
  latest version; revisions sent to arXiv only for major change
Algebraic and hamiltonian approaches to isostokes deformations

We study a generalization of the isomonodromic deformation to the case of
connections with irregular singularities. We call this generalization Isostokes
Deformation. A new deformation parameter arises: one can deform the formal
normal forms of connections at irregular points. We study this part of the
deformation, giving an algebraic description. Then we show how to use loop
groups and hypercohomology to write explicit hamiltonians. We work on an
arbitrary complete algebraic curve, the structure group is an arbitrary
semisimiple group.Comment: 23 pages, minor corrections in the introduction, references expande
Classification of smooth affine spherical varieties

Let G be a complex reductive group. A normal G-variety X is called spherical
if a Borel subgroup of G has a dense orbit in X. Of particular interest are
spherical varieties which are smooth and affine since they form local models
for multiplicity free Hamiltonian K-manifolds, K a maximal compact subgroup of
G. In this paper, we classify all smooth affine spherical varieties up to
coverings, central tori, and C*-fibrations.Comment: v1: 23 pages, uses texdraw; v2: 25 pages, introduction updated, Lemma
  7.2 fixed, references added, typos correcte
Homogeneous quaternionic Kaehler structures and quaternionic hyperbolic space

34 pages.-- MSC codes: 53C30; 53C26.-- Printed version published Dec 2006.-- ArXiv pre-print available at: http://arxiv.org/abs/math/0504550An explicit classification of homogeneous quaternionic Kaehler structures by real tensors is derived and we relate this to the\ud
representation-theoretic description found by Fino. We then show how the quaternionic hyperbolic space HH(n) is characterised by admitting homogeneous structures of a particularly simple type. In the process we\ud
study the properties of different homogeneous models for HH(n).Partially supported by DGICYT, Spain, under Grant MTM2005-00173. Partially supported by the EDGE, Research Training Network HPRN-CT-2000-0010, of the European Human Potential Programme.Peer reviewe
Yangians and Mickelsson Algebras I

We study the composition of the functor from the category of modules over the
Lie algebra gl_m to the category of modules over the degenerate affine Hecke
algebra of GL_N introduced by I. Cherednik, with the functor from the latter
category to the category of modules over the Yangian Y(gl_n) due to V.
Drinfeld. We propose a representation theoretic explanation of a link between
the intertwining operators on the tensor products of Y(gl_n)-modules, and the
`extremal cocycle' on the Weyl group of gl_m defined by D. Zhelobenko. We also
establish a connection between the composition of two functors, and the
`centralizer construction' of the Yangian Y(gl_n) discovered by G. Olshanski.Comment: publication details added. arXiv admin note: substantial text overlap
  with arXiv:math/060627
Exact cohomology sequences with integral coefficients for torus actions

Using methods applied by Atiyah in equivariant K-theory, Bredon obtained
exact sequences for the relative cohomologies (with rational coefficients) of
the equivariant skeletons of (sufficiently nice) T-spaces, T=(S^1)^n, with free
equivariant cohomology over the cohomology of BT. Here we characterise those
finite T-CW complexes with connected isotropy groups for which an analogous
result holds with integral coefficients.Comment: similar main result as in our preprint math.AT/0307112, but the proof
  is more elementary; 10 pages; final versio
Groups of outer type E6 with trivial Tits algebras

In two 1966 papers, Jacques Tits gave a construction of exceptional Lie
algebras (hence implicitly exceptional algebraic groups) and a classification
of possible indexes of simple algebraic groups. For the special case of his
construction that gives groups of type E6, we connect the two papers by
answering the question: Given an Albert algebra A and a separable quadratic
field extension K, what is the index of the resulting algebraic group?Comment: 29 pages. For possibly newer versions, see
  http://www.mathcs.emory.edu/~skip/preprints.htm
Visible actions on symmetric spaces

A visible action on a complex manifold is a holomorphic action that admits a
$J$-transversal totally real submanifold $S$.
  It is said to be strongly visible if there exists an orbit-preserving
anti-holomorphic diffeomorphism $\sigma$ such that $\sigma |_S = \mathrm{id}$.
  In this paper, we prove that for any Hermitian symmetric space $D = G/K$ the
action of any symmetric subgroup $H$ is strongly visible.
  The proof is carried out by finding explicitly an orbit-preserving
anti-holomorphic involution and a totally real submanifold $S$.
  Our geometric results provide a uniform proof of various multiplicity-free
theorems of irreducible highest weight modules when restricted to reductive
symmetric pairs, for both classical and exceptional cases, for both finite and
infinite dimensional cases, and for both discrete and continuous spectra
A Deodhar type stratification on the double flag variety

We describe a stratification on the double flag variety $G/B^+\times G/B^-$
of a complex semisimple algebraic group $G$ analogous to the Deodhar
stratification on the flag variety $G/B^+$, which is a refinement of the
stratification into orbits both for $B^+\times B^-$ and for the diagonal action
of $G$, just as Deodhar's stratification refines the orbits of $B^+$ and $B^-$.
  We give a coordinate system on each stratum, and show that all strata are
coisotropic subvarieties. Also, we discuss possible connections to the positive
and cluster geometry of $G/B^+\times G/B^-$, which would generalize results of
Fomin and Zelevinsky on double Bruhat cells and Marsh and Rietsch on double
Schubert cells.Comment: 21 page
Isometric group actions on Banach spaces and representations vanishing
  at infinity

Our main result is that the simple Lie group $G=Sp(n,1)$ acts properly
isometrically on $L^p(G)$ if $p>4n+2$. To prove this, we introduce property
$({\BP}_0^V)$, for $V$ be a Banach space: a locally compact group $G$ has
property $({\BP}_0^V)$ if every affine isometric action of $G$ on $V$, such
that the linear part is a $C_0$-representation of $G$, either has a fixed point
or is metrically proper. We prove that solvable groups, connected Lie groups,
and linear algebraic groups over a local field of characteristic zero, have
property $({\BP}_0^V)$. As a consequence for unitary representations, we
characterize those groups in the latter classes for which the first cohomology
with respect to the left regular representation on $L^2(G)$ is non-zero; and we
characterize uniform lattices in those groups for which the first $L^2$-Betti
number is non-zero.Comment: 28 page
Cohomology of the minimal nilpotent orbit

We compute the integral cohomology of the minimal non-trivial nilpotent orbit
in a complex simple (or quasi-simple) Lie algebra. We find by a uniform
approach that the middle cohomology group is isomorphic to the fundamental
group of the sub-root system generated by the long simple roots. The modulo
$\ell$ reduction of the Springer correspondent representation involves the sign
representation exactly when $\ell$ divides the order of this cohomology group.
The primes dividing the torsion of the rest of the cohomology are bad primes.Comment: 29 pages, v2 : Leray-Serre spectral sequence replaced by Gysin
  sequence only, corrected typo
Cohomogeneity one manifolds and selfmaps of nontrivial degree

We construct natural selfmaps of compact cohomgeneity one manifolds with
finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds
with simple cohomology rings this yields in certain cases relations between the
order of the Weyl group and the Euler characteristic of a principal orbit. We
apply our construction to the compact Lie group SU(3) where we extend identity
and transposition to an infinite family of selfmaps of every odd degree. The
compositions of these selfmaps with the power maps realize all possible degrees
of selfmaps of SU(3).Comment: v2, v3: minor improvement
Affine T-varieties of complexity one and locally nilpotent derivations

Let X=spec A be a normal affine variety over an algebraically closed field k
of characteristic 0 endowed with an effective action of a torus T of dimension
n. Let also D be a homogeneous locally nilpotent derivation on the normal
affine Z^n-graded domain A, so that D generates a k_+-action on X that is
normalized by the T-action. We provide a complete classification of pairs (X,D)
in two cases: for toric varieties (n=\dim X) and in the case where n=\dim X-1.
This generalizes previously known results for surfaces due to Flenner and
Zaidenberg. As an application we compute the homogeneous Makar-Limanov
invariant of such varieties. In particular we exhibit a family of non-rational
varieties with trivial Makar-Limanov invariant.Comment: 31 pages. Minor changes in the structure. Fixed some typo
Traces on finite W-algebras

We compute the space of Poisson traces on a classical W-algebra, i.e., linear functionals invariant under Hamiltonian derivations. Modulo any central character, this space identifies with the top cohomology of the corresponding Springer fiber. As a consequence, we deduce that the zeroth Hochschild homology of the corresponding quantum W-algebra modulo a central character identifies with the top cohomology of the corresponding Springer fiber. This implies that the number of irreducible finite-dimensional representations of this algebra is bounded by the dimension of this top cohomology, which was established earlier by C. Dodd using reduction to positive characteristic. Finally, we prove that the entire cohomology of the Springer fiber identifies with the so-called Poisson-de Rham homology (defined previously by the authors) of the centrally reduced classical W-algebra.National Science Foundation (U.S.) (grant DMS-0504847)National Science Foundation (U.S.) (grant DMS-0900233
Spherical orbit closures in simple projective spaces and their
  normalizations

Let G be a simply connected semisimple algebraic group over an algebraically
closed field k of characteristic 0 and let V be a rational simple G-module of
finite dimension. If G/H \subset P(V) is a spherical orbit and if X is its
closure, then we describe the orbits of X and those of its normalization. If
moreover the wonderful completion of G/H is strict, then we give necessary and
sufficient combinatorial conditions so that the normalization morphism is a
homeomorphism. Such conditions are trivially fulfilled if G is simply laced or
if H is a symmetric subgroup.Comment: 24 pages, LaTeX. v4: Final version, to appear in Transformation
  Groups. Simplified some proofs and corrected minor mistakes, added
  references. v3: major changes due to a mistake in previous version
Solvable Lie algebras are not that hypo

We study a type of left-invariant structure on Lie groups, or equivalently on
Lie algebras. We introduce obstructions to the existence of a hypo structure,
namely the 5-dimensional geometry of hypersurfaces in manifolds with holonomy
SU(3). The choice of a splitting g^*=V_1 + V_2, and the vanishing of certain
associated cohomology groups, determine a first obstruction. We also construct
necessary conditions for the existence of a hypo structure with a fixed
almost-contact form. For non-unimodular Lie algebras, we derive an obstruction
to the existence of a hypo structure, with no choice involved. We apply these
methods to classify solvable Lie algebras that admit a hypo structure.Comment: 21 pages; v2: presentation improved, typos corrected, notational
  conflicts eliminated. To appear in Transformation Group
Simple locally compact groups acting on trees and their germs of
  automorphisms

Automorphism groups of locally finite trees provide a large class of examples
of simple totally disconnected locally compact groups. It is desirable to
understand the connections between the global and local structure of such a
group. Topologically, the local structure is given by the commensurability
class of a vertex stabiliser; on the other hand, the action on the tree
suggests that the local structure should correspond to the local action of a
stabiliser of a vertex on its neighbours.
  We study the interplay between these different aspects for the special class
of groups satisfying Tits' independence property. We show that such a group has
few open subgroups if and only if it acts locally primitively. Moreover, we
show that it always admits many germs of automorphisms which do not extend to
automorphisms, from which we deduce a negative answer to a question by George
Willis. Finally, under suitable assumptions, we compute the full group of germs
of automorphisms; in some specific cases, these turn out to be simple and
compactly generated, thereby providing a new infinite family of examples which
generalise Neretin's group of spheromorphisms. Our methods describe more
generally the abstract commensurator group for a large family of
self-replicating profinite branch groups
On hyperbolicity of SU(2)-equivariant, punctured disc bundles over the
  complex affine quadric

Given a holomorphic line bundle over the complex affine quadric $Q^2$, we
investigate its Stein, SU(2)-equivariant disc bundles. Up to equivariant
biholomorphism, these are all contained in a maximal one, say $\Omega_{max}$.
By removing the zero section to $\Omega_{max}$ one obtains the unique Stein,
SU(2)-equivariant, punctured disc bundle over $Q^2$ which contains entire
curves. All other such punctured disc bundles are shown to be Kobayashi
hyperbolic.Comment: 15 pages, v2: minor changes, to appear in Transformation Group
Embeddings of SL(2,Z) into the Cremona group

Geometric and dynamic properties of embeddings of SL(2,Z) into the Cremona
group are studied. Infinitely many non-conjugate embeddings which preserve the
type (i.e. which send elliptic, parabolic and hyperbolic elements onto elements
of the same type) are provided. The existence of infinitely many non-conjugate
elliptic, parabolic and hyperbolic embeddings is also shown.
  In particular, a group G of automorphisms of a smooth surface S obtained by
blowing-up 10 points of the complex projective plane is given. The group G is
isomorphic to SL(2,Z), preserves an elliptic curve and all its elements of
infinite order are hyperbolic.Comment: to appear in Transformation Group
Braided racks, Hurwitz actions and Nichols algebras with many cubic
  relations

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over
groups such that the underlying rack is braided and the homogeneous component
of degree three of the Nichols algebra satisfies a given inequality. This
assumption turns out to be equivalent to a factorization assumption on the
Hilbert series. Besides the known Nichols algebras we obtain a new example. Our
method is based on a combinatorial invariant of the Hurwitz orbits with respect
to the action of the braid group on three strands.Comment: v2: 35 pages, 6 tables, 14 figure
Equivariant pretheories and invariants of torsors

In the present paper we introduce and study the notion of an equivariant
pretheory: basic examples include equivariant Chow groups, equivariant K-theory
and equivariant algebraic cobordism. To extend this set of examples we define
an equivariant (co)homology theory with coefficients in a Rost cycle module and
provide a version of Merkurjev's (equivariant K-theory) spectral sequence for
such a theory. As an application we generalize the theorem of
Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a
G-equivariant pretheory we associate a graded ring which serves as an invariant
of E. In the case of Chow groups this ring encodes the information concerning
the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes
of the respective Tits algebras.Comment: 23 pages; this is an essentially extended version of the previous
  preprint: the construction of an equivariant cycle (co)homology and the
  spectral sequence (generalizing the long exact localization sequence) are
  adde
Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius
kernels of unipotent and parabolic algebraic group schemes and of their quantum
analogs. Given a simple algebraic group $G$, a parabolic subgroup $P_J$, and
its unipotent radical $U_J$, we determine the ring structure of the cohomology
ring $H^\bullet((U_J)_1,k)$. We also obtain new results on computing
$H^\bullet((P_J)_1,L(\lambda))$ as an $L_J$-module where $L(\lambda)$ is a
simple $G$-module with high weight $\lambda$ in the closure of the bottom
$p$-alcove. Finally, we provide generalizations of all our results to the
quantum situation.Comment: 18 pages. Some proofs streamlined over previous version. Additional
  details added to some proofs in Section
On spherical twisted conjugacy classes

Let G be a simple algebraic group over an algebraically closed field of good
odd characteristic, and let theta be an automorphism of G arising from an
involution of its Dynkin diagram. We show that the spherical theta-twisted
conjugacy classes are precisely those intersecting only Bruhat cells
corresponding to twisted involutions in the Weyl group. We show how the
analogue of this statement fails in the triality case. We generalize to good
odd characteristic J-H. Lu's dimension formula for spherical twisted conjugacy
classes.Comment: proof of Lemma 6.4 polished. The journal version is available at
  http://www.springerlink.com/content/k573l88256753640
On Non-Abelian Symplectic Cutting

We discuss symplectic cutting for Hamiltonian actions of non-Abelian compact
groups. By using a degeneration based on the Vinberg monoid we give, in good
cases, a global quotient description of a surgery construction introduced by
Woodward and Meinrenken, and show it can be interpreted in algebro-geometric
terms. A key ingredient is the `universal cut' of the cotangent bundle of the
group itself, which is identified with a moduli space of framed bundles on
chains of projective lines recently introduced by the authors.Comment: Various edits made, to appear in Transformation Groups. 28 pages, 8
  figure
K-orbit closures on G/B as universal degeneracy loci for flagged vector
  bundles with symmetric or skew-symmetric bilinear form

We use equivariant localization and divided difference operators to determine
formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit
closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one
of the symmetric subgroups $O(n,\C)$ or $Sp(n,\C)$. We realize these orbit
closures as universal degeneracy loci for a vector bundle over a variety
equipped with a single flag of subbundles and a nondegenerate symmetric or
skew-symmetric bilinear form taking values in the trivial bundle. We describe
how our equivariant formulas can be interpreted as giving formulas for the
classes of such loci in terms of the Chern classes of the various bundles.Comment: Minor revisions and corrections suggested by referees. Final version,
  to appear in Transformation Group
The algebra of $SL_3(\mathbb{C})$ conformal blocks

We construct and study a family of toric degenerations of the algebra of
conformal blocks for a stable marked curve $(C, \vec{p})$ with structure group
$SL_3(\mathbb{C}).$ We find that this algebra is Gorenstein. For the genus $0,
1$ cases we find the level of conformal blocks necessary to generate the
algebra. In the genus 0 case we also find bounds on the degrees of relations
required to present the algebra. Along the way we recover polyhedral rules for
counting conformal blocks originally due to Senechal, Mathieu, Kirillov, and
Walton.Comment: 22 pages, 13 figure
Invariant Hilbert schemes and desingularizations of quotients by classical groups

International audienceLet $W$ be a finite-dimensional representation of a reductive algebraic group $G$. The invariant Hilbert scheme $\mathcal{H}$ is a moduli space that classifies the $G$-stable closed subschemes $Z$ of $W$ such that the affine algebra $k[Z]$ is the direct sum of simple $G$-modules with prescribed multiplicities. In this article, we consider the case where $G$ is a classical group acting on a classical representation $W$ and $k[Z]$ is isomorphic to the regular representation of $G$ as a $G$-module. We obtain families of examples where $\mathcal{H}$ is a smooth variety, and thus for which the Hilbert-Chow morphism $\gamma: \mathcal{H} \rightarrow W//G$ is a canonical desingularization of the categorical quotient
Representation Theory in Complex Rank, I

P. Deligne defined interpolations of the tensor category of representations of the symmetric group S [subscript n] to complex values of n. Namely, he defined tensor categories Rep(S [subscript t]) for any complex t. This construction was generalized by F. Knop to the case of wreath products of S[subscript n] with a finite group. Generalizing these results, we propose a method of interpolating representation categories of various algebras containing S [subscript n] (such as degenerate affine Hecke algebras, symplectic reflection algebras, rational Cherednik algebras, etc.) to complex values of n. We also define the group algebra of S [subscript n] for complex n, study its properties, and propose a Schur-Weyl duality for Rep(S [subscript t]).National Science Foundation (U.S.) (Grant DMS-0504847)National Science Foundation (U.S.) (Grant DMS-1000113
REPRESENTATIONS OF AFFINE SUPERALGEBRAS AND MOCK THETA FUNCTIONS

We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra  sℓˆ[subscript 2|1] (resp.  psℓˆ[subscript 2|2]) can be modified, using Zwegers’ real analytic corrections, to form a modular (resp. S-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N = 2 (resp.N = 4) superconformal algebras with central charge 3(1 − (2 m + 2)/M), where m ∈ ℤ[subscript ≥0], M ∈ ℤ[subscript ≥2], gcd(2 m + 2, M) = 1 if m > 0 (resp. 6 (m/M − 1), where m ∈ ℤ[subscript ≥1], M ∈ ℤ[subscript ≥2], gcd(2 m, M) = 1 if m > 1), whose modified characters and supercharacters form a modular invariant family