Let $k$ be the algebraic closure of a finite field, $G$ a Chevalley group over $k$, $U$ the maximal unipotent subgroup of $G$. To each orthogonal subset $D$ of the root system of the group $G$ and each set $ξ$ of $|D|$ non-zero scalars from $k$ one can assign the coadjoint orbit of the group $U$. We prove that the dimension of such an orbit does not depend on $ξ$. We also give an upper bound of the dimension in terms of the Weyl group.
Let $Φ$ be a classical root system and $k$ be a field of sufficiently large characteristic. Let $G$ be the classical group over $k$ with the root system $Φ$, $U$ be its maximal unipotent subgroup and $\mathfrak{u}$ be the Lie algebra of $U$. Let $D$ be an orthogonal subset of $Φ$ and $Ω$ be a coadjoint orbit of $U$ associated with $D$. We construct a polarization of $\mathfrak{u}$ at the canonical form on $Ω$. We also find the dimension of $Ω$ in terms of the Weyl group of $Φ$. As a corollary, we determine all possible dimensions of irreducible complex represenations of the group $U$ for the c…
We define in an axiomatic fashion a \emph{Coxeter datum} for an arbitrary Coxeter group $W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear paring without any integrality and non-degeneracy requirements. These representations are not required to be embeddings of $W$ in the orthogonal group of any vector space, and they give rise to a pair of inter-related root systems generalizing the classical root systems of Coxeter groups. We obtain comparison results between these non-orthogonal root systems and the classical root …
We show that an orthogonal root number of a tempered L-parameter decomposes as the product of two other numbers: the orthogonal root number of the principal parameter and the value on a certain involution of Langlands's central character for the parameter. The formula resolves a conjecture of Gross and Reeder and computes root numbers of Weil-Deligne representations arising in the work of Hiraga, Ichino, and Ikeda on the Plancherel measure.
Let $G$ be a semisimple Lie group and let $\g =\n_- +\hh +\n$ be a triangular decomposition of $\g= \hbox{Lie}\,G$. Let $\b =\hh +\n$ and let $H,N,B$ be Lie subgroups of $G$ corresponding respectively to $\hh,\n$ and $\b$. We may identify $\n_-$ with the dual space to $\n$. The coadjoint action of $N$ on $\n_-$ extends to an action of $B$ on $\n_-$. There exists a unique nonempty Zariski open orbit $X$ of $B$ on $\n_-$. Any $N$-orbit in $X$ is a maximal coadjoint orbit of $N$ in $\n_-$. The cascade of orthogonal roots defines a cross-section $\r_-^{\times}$ of the set of such orbits leading to…
Let $G$ be a complex simply-connected semisimple Lie group and let $\g=\hbox{\rm Lie}\,G$. Let $\g = \n_- +\hh + \n$ be a triangular decomposition of $\g$. One readily has that $\hbox{\rm Cent}\,U(\n)$ is isomorphic to the ring $S(\n)^{\n}$ of symmetric invariants. Using the cascade ${\cal B}$ of strongly orthogonal roots, some time ago we proved (see [K]) that $S(\n)^{\n}$ is a polynomial ring $\Bbb C[ξ_1,...,ξ_m]$ where $m$ is the cardinality of ${\cal B}$. The authors in [LW] introduce a very nice representation-theoretic method for the construction of certain elements in $S(\n)^{\n}$. A ke…
We investigate the local descents for special orthogonal groups over p-adic local fields of characteristic zero, and obtain an explicit spectral decomposition of the local descents at the first occurrence index in terms of the local Langlands data via the explicit local Langlands correspondence and explicit calculations of relevant local root numbers. The main result can be regarded as a refinement of the local Gan-Gross-Prasad conjecture.
We describe an algorithm for classifying the closed subsets of a root system, up to conjugation by the associated Weyl group. Such a classification of an irreducible root system is closely related to the classification of the regular subalgebras, up to inner automorphism, of the corresponding simple Lie algebra. We implement our algorithm to classify the closed subsets of the irreducible root systems of ranks 3 through 7. We present a complete description of the classification for the closed subsets of the rank 3 irreducible root system. We employ this root system classification to classify al…
Let $\mathfrak{g}$ be a finite or an affine type Lie algebra over $\mathbb{C}$ with root system $Δ$. We show a parabolic generalization of the partial sum property for $Δ$, which we term the parabolic partial sum property. It allows any root $β$ involving (any) fixed subset $S$ of simple roots, to be written as an ordered sum of roots, each involving exactly one simple root from $S$, with each partial sum also being a root. We show three applications of this property to weights of highest weight $\mathfrak{g}$-modules: (1)~We provide a minimal description for the weights of all non-integrable …
Université Paris-Saclay（巴黎萨克雷大学）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛数学研究所）；Universidade Estadual de Campinas（坎皮纳斯州立大学）
Finite Root Sums in Symmetrizable Kac-Moody Algebras
New York City College of Technology, City University of New York（纽约市立大学纽约城市学院）；CUNY Graduate Center, City University of New York（纽约市立大学研究生中心）；University of Toronto（多伦多大学）
Let $R$ be an indecomposable root system. It is well known that any root is part of a basis $B$ of $R$. But when can you extend a set of two or more roots to a basis $B$ of $R$? A $π$-system is a linearly independent set of roots, $C$, such that if $α$ and $β$ are in $C$, then $α- β$ is not a root. We will use results of Dynkin and Bourbaki to show that with two exceptions, $A_3 \subset B_n$ and $A_7 \subset E_8$, an indecomposable $π$-system whose Dynkin diagram is a subdiagram of the Dynkin diagram of $R$ can always be extended to a basis of $R$.
Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\mathfrak{t} \subset \mathfrak{g}$ be a toral subalgebra of $\mathfrak{g}$. As a $\mathfrak{t}$-module $\mathfrak{g}$ decomposes as \[\mathfrak{g} = \mathfrak{s} \oplus \big(\oplus_{ν\in \mathcal{R}} \mathfrak{g}^ν\big)\] where $\mathfrak{s} \subset \mathfrak{g}$ is the reductive part of a parabolic subalgebra of $\mathfrak{g}$ and $\mathcal{R}$ is the Kostant root system associated to $\mathfrak{t}$. When $\mathfrak{t}$ is a Cartan subalgebra of $\mathfrak{g}$ the decomposition above is nothing but the root decomposition of $\mathfra…
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra and $M$ be a $\mathfrak{g}$-module. The Fernando-Kac subalgebra of $\mathfrak{g}$ associated to $M$ is the subset $\mathfrak{g}[M]\subset\mathfrak{g}$ of all elements $g\in\mathfrak{g}$ which act locally finitely on $M$. A subalgebra $\mathfrak{l}\subset\mathfrak{g}$ for which there exists an irreducible module $M$ with $\mathfrak{g}[M]=\mathfrak{l}$ is called a Fernando-Kac subalgebra of $\mathfrak{g}$. A Fernando-Kac subalgebra of $\mathfrak{g}$ is of finite type if in addition $M$ can be chosen to have finite Jordan-Hölder $\mathfrak{l}…
We count the number of strictly positive $B$-stable ideals in the nilradical of a Borel subalgebra and prove that the minimal roots of any $B$-stable ideal are conjugate by an element of the Weyl group to a subset of the simple roots. We also compute the number of ideals whose minimal roots are conjugate to a fixed subset of simple roots.
Chari and Greenstein [Adv. Math. 2009] introduced combinatorial subsets of the roots of a finite-dimensional simple Lie algebra $\mathfrak{g}$ which were important in studying Kirillov-Reshetikhin modules over $U_q(\widehat{\mathfrak{g}})$ and their specializations. Later, Khare [J. Algebra. 2016] studied these subsets for many highest weight $\mathfrak{g}$-modules (in finite type), under the name of weak-$\mathbb{A}$-faces (for a subgroup $\mathbb{A}$ of $(\mathbb{R},+)$), and more generally, $(\{2\};\{1,2\})$-closed subsets. These notions extend and unify the faces of Weyl polytopes as well …
Let $Q$ be a finite quiver and $G\subseteq\Aut(\mathbbm{k}Q)$ a finite abelian group. Assume that $\hat{Q}$ and $Γ$ is the generalized Mckay quiver and the valued graph corresponding to $(Q, G)$ respectively. In this paper we discuss the relationship between indecomposable $\hat{Q}$-representations and the root system of Kac-Moody algebra $\mathfrak{g}(Γ)$. Moreover, we may lift $G$ to $\bar{G}\subseteq\Aut(\mathfrak{g}(\hat{Q}))$ such that $\mathfrak{g}(Γ)$ embeds into the fixed point algebra $\mathfrak{g}(\hat{Q})^{\bar{G}}$ and $\mathfrak{g}(\hat{Q})^{\bar{G}}$ as $\mathfrak{g}(Γ)$-module i…
We define and study supercharacters of the classical finite unipotent groups of symplectic and orthogonal types (over any finite field of odd characteristic). We show how supercharacters for groups of those types can be obtained by restricting the supercharacter theory of the finite unitriangular group, and prove that supercharacters are orthogonal and provide a partition of the set of all irreducible characters. We also describe all irreducible characters of maximum degree in terms of the root system, and show how they can be obtained as constituents of particular supercharacters.
We investigate the finite-dimensional representation theory of two-parameter quantum orthogonal and symplectic groups that we found in [BGH] under the assumption that $rs^{-1}$ is not a root of unity and extend some results [BW1, BW2] obtained for type $A$ to types $B$, $C$ and $D$. We construct the corresponding $R$-matrices and the quantum Casimir operators, by which we prove that the complete reducibility Theorem also holds for the categories of finite-dimensional weight modules for types $B$, $C$, $D$.
Let Q be a Dynkin quiver, and let P(Q) be the corresponding preprojective algebra. Let I be a set of pairwise different indecomposable irreducible components of varieties of P(Q)-modules such that generically there are no extensions between these components. We show that the number of elements in I is at most the number of positive roots of Q. Furthermore, we give a module theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.
Let $G=SO(8n+4,\mathbb{C})$ ($n\ge 1$). Let $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ Let $P (\supset B)$ denote the maximal parabolic subgroup of $G$ corresponding to the simple root $α_{4n+2}$. In this article, we prove projective normality of the GIT quotients of certain Schubert varieties in the orthogonal Grassmannian $G/P$ with respect to the descent of a suitable $T$-linearized very ample line bundle.
