Restricted Root Systems and Spin Representations

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic approach to studying both of them. GRSs inherit many of the properties of root systems. In particular, every GRS defines a crystallographic hyperplane arrangement. We believe that GRSs provide an intrinsic counterpart to finite Weyl groupoids and crystallographic hyperplane arrangements, extending the relationship between finite Weyl groupoids and crystallographic hyperplane arrangements established by Cuntz. An important difference between GRSs and root systems is that GRSs may lack a (large enough) Weyl group. In order to compensate for this, we introduce the notion of a virtual reflection, building on a construction of Penkov and Serganova in the context of root systems of Lie superalgebras. The most significant new feature of GRSs is that, along with subsystems, one can define quotient GRSs. Both Kostant root systems and root systems of Lie superalgebras are equivalent to quotients of root systems and all root systems are isomorphic to quotients of simply-laced root systems. We classify all rank 2 GRSs and show that they are equivalent to quotients of root systems. Finally, we discuss in detail quotients of root systems. In particular we provide all isomorphisms and equivalences among them. Our results on quotient of root systems provide a different point of view on flag manifolds, reproving results of Alekseevsky and Graev.

In the present paper we continue the project of systematic classification and construction of invariant differential operators for non-compact semisimple Lie groups. This time we make the stress on one of the main building blocks, namely the Verma modules and the corresponding parabolic subalgebras. In particular, we start the study of the relation between the parabolic subalgebras of real semisimple Lie algebras and of their complexification. Two cases are given in more detail: the conformal algebra of 4D Minkowski space-time and the minimal parabolics of classical real semisimple Lie algebras.

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet series are fundamental objects. The goal of this paper is to define these series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet series are multiplicative, but the multiplicativity is twisted , so the Dirichlet series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet series resembling our An+1 multiple Dirichlet series. There is no a priori reason that we are

It is well known that the Weyl groups play a key role in the theory of the structure of Lie algebras and their linear representations, as well as in the theory of Chevalley groups. Suppose g is a complex semisimple Lie algebra, h its Cartan subalgebra, Σ and Π denote its root system and the basis of the root system, respectively, and w α i indicates the reflection determined by αi ∈ Π. The so-called Weyl group, denoted by W,refers to the finite group generated by all w α i ( αi ∈ Π). There have been many works published on the structure and properties of W [6, 7]. The following Molien identity related to the global structure of W is well known [3]: $$\underset{w\in W}{\mathop{\Sigma}}\,\frac{1}{\det\left( 1-tw \right)}=\frac{1}{\prod{{{\left( 1-{{t}^{{{d}_{i}}}} \right)}^{,}}}}$$ (8.1) where d i signify the Poincare indices of g. In 1956, by topological methods based on Morse’s theory, Bott proved another identity [2]: $$\underset{w\in W}{\mathop \sum }\,{{t}^{l\left(w \right)}}=\prod{\frac{1-{{t}^{{{d}_{i}}}}}{1-{{t}^{,}}}}$$ (8.2) where l(w) refers to the Π-length of w;i.e.,wmay be expressed by the number of minimal products of w αi ( αi ∈ Π). This important formula may shed new light on our understanding the structure of W. In 1966, Solomon offered a purely algebraic proof of (8.2), thus solving the order problem of the Chevalley simple groups [5, 3].

The main purpose of this paper is to lay the foundations of generalizing the AdS/CFT (holography) idea beyond the conformal setting. The main tool is to find suitable realizations of the bulk and boundary via group theory. We use all ten families of classical real semisimple Lie groups [Formula: see text] and Lie algebras [Formula: see text]. For this are used several group and algebra decompositions: the global Iwasawa decomposition and the local Bruhat and Sekiguchi-like decompositions. The same analysis is applied to the exceptional real semisimple Lie algebras.

Let O 1 \mathcal {O}_1 and O 2 \mathcal {O}_2 be adjoint nilpotent orbits in a real semisimple Lie algebra. Write O 1 ≥ O 2 \mathcal {O}_1\geq \mathcal {O}_2 if O 2 \mathcal {O}_2 is contained in the closure of O 1 . \mathcal {O}_1. This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form E V of E 7 . E_7.

Let O 1 \mathcal {O}_1 and O 2 \mathcal {O}_2 be adjoint nilpotent orbits in a real semisimple Lie algebra. Write O 1 ≥ O 2 \mathcal {O}_1\geq \mathcal {O}_2 if O 2 \mathcal {O}_2 is contained in the closure of O 1 . \mathcal {O}_1. This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the two noncompact nonsplit real forms of the simple complex Lie algebra E 7 . E_7.

Let S(g ) be the symmetric algebra over the complexification 9 of the real semisimple Lie algebra g.For u £ S(g ), d(u) is the corresponding differential operator on g. 3)(g) denotes the algebra generated by 1. Preliminaries.Our aim here is to give alternative proofs to some results of Harish-Chandra on radial components of invariant differential operators on a real semisimple Lie algebra [l], [2].Let 9 be a semisimple Lie algebra over R, g^ its complexif¡cation, r] a Cartan subalgebra of g, and \)c complexification of \).Denote by g' the set of regular elements of g and £)' = i> n g'.We have H e i,' ¡f and only if zr(/7) ¿ 0,

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p. For each p, the number of nonzero such coefficients is equal to the order of the Weyl group, and each nonzero coefficient is a product of n-th order Gauss sums. The root system plays a basic role in the combinatorics underlying the proof of the functional equations.

Let g be a complex simple Lie algebra, h its Cartan subalgebra, and Σ and \( \prod =\left\{ {{\beta }_{1}},\ldots ,{{\beta }_{i}} \right\} \)the root system of g and a basis of the root system, respectively. It is known that the Weyl group W(g) is a finite group generated by reflections [3] $${w_{{\rm B}i}}:\xi \in h,\xi \to \xi - \frac{{2\left( {\xi ,} \right.}}{{\left( {{\beta _{i,}}} \right.}}\frac{{\left. {{\beta _i}} \right)}}{{\left. {{\beta _i}} \right)}}{\beta _{i \bullet }}$$

The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by simple reflections and Coxeter relations. In a broad sense this answers a question of Serganova. Our weak version of the exchange condition allows us to prove Matsumoto’s theorem. Therefore the word problem is solved for the groupoid.

Suppose be a simple involution of a semisimple algebraic group and suppose H is the subgroup of G of points fixed by . If the restricted root system is of type or and G is simply connected, or if the restricted root system is of type and G is of adjoint type, then we describe a standard monomial theory and the equations for the coordinate ring using the standard monomial theory and the Plucker relations of an appropriate (maybe infinite-dimensional) Grassmann variety.

The effects of frequent and shallow soil wetting by surface drip irrigation on root growth, morphology, and location, and their impact on plant sensitivity to irrigation management were studied in cotton (Gossypium hirsutum L.). Daily drip irrigation, which wetted the 0 to 40-cm soil depth, encouraged root development mainly around the drippers. Water extraction took place mostly from 0 to 20 cm below the drippers, where the roots were concentrated. Shallowness of root growth was not altered by the expansion and deepening of the wetted soil zone which resulted from an increase in amount of irrigation water. The shallow and restricted root system was characterized by a high fraction of thin roots (less than 1 mm dia.) which comprised almost 90% of the root dry matter. Root proximity to the drippers and the limited amount of water in the rooted soil led to a sensitive and quick response of the plants to small amounts of irrigation. A supply of 1.0 mm H2O given at midday to 70 day-old plants resulted in a leaf water potential (Lψw) increase from −1.64 to −1.32 MPa over a 20-min period. This amount of irrigation comprised 15% of the average daily quantity. A 24 h delay in irrigation to 80 dayold plants was enough to decrease Lψw from −1.41 to −2.42 MPa. This decrease was caused by a soil water deficit of less than 6 mm H2O. Extending the irrigation delay to 72 h affected yield and earliness, although the deficient amount of water was supplied over the several days after the treatment. A strong response to minor, but continuous, differences in the daily irrigation amount was detected. Differences in irrigation of less than 1 mm H2O per day applied during the whole growth season substantially affected Lωw, yield and earliness. It was concluded that the establishment of a shallow and restricted root system resulted in strong dependence of the plants on frequent and sufficient supply of water, and temporary minor changes in irrigation affected plant water status and productivity.

In this paper, we describe a connection that exists among (a) the number of singular points along the trajectory of Toda flow, (b) the cohomology of a compact subgroup $K$, and (c) the number of points of a Chevalley group $K({\mathbb F}_q)$ related to $K$ over a finite field ${\mathbb F}_q$. The Toda lattice is defined for a real split semisimple Lie algebra $\mathfrak g$, and $K$ is a maximal compact Lie subgroup of $G$ associated to $\mathfrak g$. Relations are also obtained between the singularities of the Toda flow and the integral cohomology of the real flag manifold $G/B$ with $B$ the Borel subgroup of $G$ (here we have $G/B=K/T$ with a finite group $T$). We also compute the maximal number of singularities of the Toda flow for any real split semisimple algebra, and find that this number gives the multiplicity of the singularity at the intersection of the varieties defined by the zero set of Schur polynomials.