Let B be a reductive Lie subalgebra of a semi-simple Lie algebra of the same rank both over the complex numbers. To each finite dimensional irreducible representation $V_λ$ of F we assign a multiplet of irreducible representations of B with m elements in each multiplet, where m is the index of the Weyl group of B in the Weyl group of F. We obtain a generalization of the Weyl character formula; our formula gives the character of $V_λ$ as a quotient whose numerator is an alternating sum of the characters in the multiplet associated to $V_λ$ and whose denominator is an alternating sum of the characters of the multiplet associated to the trivial representation of F.
Let $X$ be a locally symmetric space defined by a simple Chevalley group $G$ and a congruence subgroup of $G(\mathbb Q)$. In this generality, the Weyl law for $X$ was proved by Lindenstrauss--Venkatesh. In the case where $G$ is simply connected, we sharpen their result by giving a power saving estimate for the remainder term.
Let $G$ be a reductive algebraic group over $\mathbb{Q}$ and $Γ\subset G(\mathbb{Q})$ an arithmetic subgroup. Let $K_\infty\subset G(\mathbb{R})$ be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of $G(\mathbb{R})$ in $L^2(Γ\backslash G(\mathbb{R}))$ of a fixed $K_\infty$-type $σ$. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type $σ$ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Ar…
Root subgroup factorization is a refinement of triangular (or LDU) factorization. For a complex reductive Lie group, and a choice of reduced factorization of the longest Weyl group element, the forward map from root subgroup coordinates to triangular coordinates is polynomial. We show that the inverse is rational. There is an algorithm for the inverse (involving LDU factorization), and a related explicit formula for Haar measure in root subgroup coordinates. In classical cases there are preferred reduced factorizations of the longest Weyl group elements, and conjecturally in these cases there …
Let $G$ be a connected reductive group over a non-archimedean local field $F$ and $I$ be an Iwahori subgroup of $G(F)$. Let $I_n$ is the $n$-th Moy-Prasad filtration subgroup of $I$. The purpose of this paper is two-fold: to give some nice presentations of the Hecke algebra of connected, reductive groups with $I_n$-level structure; and to introduce the Tits group of the Iwahori-Weyl group of groups $G$ that split over an unramified extension of $F$. The first main result of this paper is a presentation of the Hecke algebra $\mathcal H(G(F),I_n)$, generalizing the previous work of Iwahori-Matsu…
We obtain an irreducibility criterion for generalized principal series, extending known and frequently employed results for principal series. Our approach rests on a newly observed semi-direct product decomposition of the relative Weyl group and its action on generalized principal series, in conjunction with the theory of Jacquet module. We thus circumvent the obstacle that the Weyl group of a general Levi subgroup is not a Coxeter group. Our statements on irreducibility are formulated in terms of a subgroup of the Knapp--Stein R-group, which arises naturally from our decomposition of the Weyl…
Let G be a simple reductive group over the complex numbers. Let W be the Weyl group of G. We propose a description of the Springer representations of W associated to various unipotent classes of G by a purely algebraic method involving the families of various reflection subgroups of W and which is suitable to computer calculations.
We propose and prove an identity relating the Poincaré polynomials of stabilizer subgroups of the affine Weyl group and of the corresponding stabilizer subgroups of the Weyl group.
This paper is motivated by several combinatorial actions of the affine Weyl group of type $C_n$. Addressing a question of David Vogan, it was shown in an earlier paper that these permutation representations are essentialy multiplicity-free~\cite{arXiv:2009.13880}. However, the Gelfand trick, which was indispensable in~\cite{arXiv:2009.13880} to prove this property for types $C_n$ and $B_n$, is not applicable for other classical types. Here we present a unified approach to fully answer the analogous question for all irreducible affine Weyl groups. Given a finite Weyl group $W$ with maximal para…
We classify (*)-subgroups of compact Lie groups of adjoint type, and associate a twisted root system to every (*)-subgroup. We study the structure of twisted root system in several aspects: properties of the small Weyl group W_{small} and its normal subgroups W_{tiny} and W_{f}; properties of finite root datum; structure of strips of infinite roots.
The Peter-Weyl idempotent $e_{\mathcal{P}}$ of a parahoric subgroup ${\mathcal{P}}$ is the sum of the idempotents of irreducible representations of $\mathcal{P}$ which have a nonzero Iwahori fixed vector. The convolution algebra associated to $e_{\mathcal{P}}$ is called a Peter-Weyl Iwahori algebra. We show any Peter-Weyl Iwahori algebra is Morita equivalent to the Iwahori-Hecke algebra. Both the Iwahori-Hecke algebra and a Peter-Weyl Iwahori algbera have a natural $\mathbb{C}^\star$-algebra structure, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebra…
We show that the centraliser of the maximal compact subgroup of the real orthogonal or symplectic groups acting on tensors of their standard representation are isomorphic to cyclotomic Brauer algebras. We also show that for the symplectic group this cyclotomic Brauer algebra splits into summands of walled Brauer algebras.
This paper is part of the program to classify Kazhdan-Lusztig cells for Weyl groups of type $D_n$. We prove analogous results to those of section 4 of Kazhdan-Lusztig's original paper, this time related to a parabolic subgroup of type $D_4$. We also show how this is used in the definition of the generalized $τ$-invariant.
In a previous paper I have defined a new basis for the representation ring of a Weyl group. In this paper we show that the new basis is related to the standard basis by an upper triangular unipotent matrix. We also give a new parametrization of representations in a fixed family by certain pairs of subgroups of a finite group attached to the family. We outline an extension to the case of unipotent representations of a finite Chevalley group.
In joint work with J.-S. Koskivirta, we had previously introduced the notion of "quasi-constant" character (of a maximal torus of a connected reductive group over a field); we showed that over an algebraically closed field it naturally unifies the notions "minuscule" and "co-minuscule". In this note we describe a characterization of quasi-constant fundamental weights in terms of the Weyl group of a maximal Levi subgroup. Equivalently, purely in the language of root systems, the result characterizes special and co-special vertices of Dynkin diagrams in terms of Weyl groups of maximal sub-root s…
In the present paper we further the study of the compression cone of a real spherical homogeneous space $Z=G/H$. In particular we provide a geometric construction of the little Weyl group of $Z$ introduced recently by Knop and Krötz. Our technique is based on a fine analysis of limits of conjugates of the subalgebra $\mathrm{Lie}(H)$ along one-parameter subgroups in the Grassmannian of subspaces of $\mathrm{Lie}(G)$. The little Weyl group is obtained as a finite reflection group generated by the reflections in the walls of the compression cone.
Given a regular supercuspidal representation $ρ$ of the Levi subgroup $M$ of a standard parabolic subgroup $P=MN$ in a connected reductive group $G$ defined over a non-archimedean local field $F$, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set $JH(Ind^G_P(ρ))$ of Jordan--H{ö}lder constituents of the Harish-Chandra parabolic induction representation $Ind^G_P(ρ)$, vastly generalizing Rodier structure theorem for $P=B=TU$ Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essent…
We classify and explicitly construct the irreducible graded representations of anti-spherical Hecke categories which are concentrated in one degree. Each of these homogeneous representations is one-dimensional and can be cohomologically constructed via a BGG resolution involving every (infinite dimensional) standard representation of the category. We hence determine the complete first row of the inverse parabolic $p$-Kazhdan--Lusztig matrix for an arbitrary Coxeter group and an arbitrary parabolic subgroup. This generalises the Weyl--Kac character formula to all Coxeter systems (and their para…
In this paper, we determine the modular invariants of finite modular pseudo-reflection subgroups of the finite general linear group $ \text{GL}_n(q) $ acting on the tensor product of the symmetric algebra $ S^{\bullet}(V) $ and the exterior algebra $ \wedge^{\bullet}(V) $ of the natural $ \text{GL}_n(q) $-module $ V $. We are particularly interested in the case where $ G $ is a subgroup of the parabolic subgroups of $ \text{GL}_n(q) $ which is a generalization of Weyl group of Cartan type Lie algebra.
We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application…
As a sequel of \cite{Ou}, in this shot note, we investigate block basis for coinvariants of finite modular pseudo-reflection groups. We are particularly interested in the case where $ G $ is a subgroup of the parabolic subgroups of $ GL_n(q) $ which generalizes the Weyl groups of restricted Cartan type Lie algebra.
We study coadjoint $B$-orbits on $\mathfrak{n}^*$, where $B$ is a Borel subgroup of a complex orthogonal group $G$, and $\mathfrak{n}$ is the Lie algebra of the unipotent radical of $B$. To each basis involution $w$ in the Weyl group $W$ of $G$ one can assign the associated $B$-orbit $Ω_w$. We prove that, given basis involutions $σ$, $τ$ in $W$, if the orbit $Ω_σ$ is contained in the closure of the orbit $Ω_τ$ then $σ$ is less than or equal to $τ$ with respect to the Bruhat order on $W$. For a basis involution $w$, we also compute the dimension of $Ω_w$ and present a conjectural description of…
The space spanned by the characters of twisted affine Lie algebras admit the action of certain congruence subgroups of $SL(2,\mathbb{Z})$. By embedding the characters in the space spanned by theta functions, we study an $SL(2,\mathbb{Z})$-closure of the space of characters. Analogous to the untwisted affine Lie algebra case, we construct a commutative associative algebra (fusion algebra) structure on this space through the use of the Verlinde formula and study important quotients. Unlike the untwisted cases, some of these algebras and their quotients, which relate to the trace of diagram autom…
我们获得了一个公式，它对某些群体的代表性理论的基础上的一个faisceau caractère的基本特征的价值，liés au groupe de Weyl。这一公式是MÜglin和Waldspurger的一般结果，它取决于对某些特定区域的认识，这些区域是尖形的。在第二个时间点，在一个类简单的情况下，我们确定了这些子组的共轭关系。利用与外尔群有关的某些有限群的表示论，得到了特征层的特征函数值的一个公式.这个公式，由于MÜglin和Waldspurger以前的结果的推广，依赖于某些约化子群的知识，承认尖点特征层。对于拟单群，我们通过确定直至共轭的所有此类子群，使公式真正明确。
From a certain induced representation $\mathcal{P}_\ell$ of a double affine Weyl group, we construct a ring $\mathcal{F}_\ell$ that is isomorphic to the fusion ring, or Verlinde algebra, associated to affine Lie algebras at fixed positive integer level for both twisted and untwisted type. The induced representation, which also has a natural commutative associative algebra structure and is modular invariant with respect to certain congruence subgroups, contains $\mathcal{F}_\ell$ as an ideal and we show how it naturally inherits the modular invariance property from $\mathcal{P}_\ell.$ This cons…
