Hengfei Lu
Let F be a local field of character zero. Let E be a quadratic field extension of F. We show that any P-invariant linear functional on a GL(n,E)-distinguished irreducible smooth admissible representation of GL(2n,F) is also GL(n,E)-invariant where P is a mirabolic subgroup of GL(n,E).
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We study the role of the mirabolic subgroup $P$ of $G=\mathbf{GL}_n(F)$ ($F$ a $p$-adic field) in smooth irreducible representations of $G$ that possess a non-zero invariant functional relative to a subgroup of the form $H_{k} = \mathbf{GL}_k(F)\times \mathbf{GL}_{n-k}(F)$. We show that if a non-zero $H_1$-invariant functional exists on a representation, then every $P\cap H_1$-invariant functional must equal to a scalar multiple of it. When $k>1$, we give a reduction of the same problem to a question about invariant distributions on the nilpotent cone of the tangent space of the symmetric spac…
Using linear periods on the mirabolic subgroup of $GL(n,F)$, for $F$ a non archimedean local field, we give a list of the maximal Levi subgroups of $GL(n,F)$ which can distinguish a discrete series, and a generic representation. We also obtain the functional equation of the local exterior-square $L$-function of a generic representation of $GL(n,F)$ when $n=2m$ is even. Then we discuss the relation between Shalika models and local models for representations of $GL(2m,F)$. Finally we give a necessary and sufficient condition on the cuspidal representation $ρ$ for a generalised Steinberg represen…
Let $V$ be a $GL_n(\mathbb{R})$-distinguished, irreducible, admissible representation of $GL_n(\mathbb{C})$. We prove that any continuous linear functional on $V$, which is invariant under the action of the real mirabolic subgroup, is automatically $GL_n(\mathbb{R})$-invariant.
Twisted Jacquet modules associated to maximal parabolic subgroups and cuspidal representations of $GL(n, q)$
本文研究有限域上一般线性群尖表示关于极大抛物子群的扭Jacquet模的结构，通过建立扭Jacquet函子的范畴等价并构造具有泛性质的表示，直接解决了所有参数情形下的模结构问题。
Let F be a p-adic field, and Gn one of the groups GL(n, F), GSO(2n-1, F), GSp(2n, F), or GSO(2(n - 1), F). Using the mirabolic subgroup or analogues of it, and related "derivative" functors, we give an asymptotic expansion of functions in the Whittaker model of generic representations of Gn, with respect to a minimal set of characters of subgroups of the maximal torus. Denoting by Zn the center of Gn, and by Nn the unipotent radical of its standard Borel subgroup, we characterize generic representations occurring in L2(ZnNn\Gn) in terms of these characters. This is related to a conjecture of L…
Let $F$ be a non-archimedean local field of characteristic zero. We study the linear period problem for the pair $(G,H_{p,p+1})=(GL_{2p+1}(F), GL_{p}(F)\times GL_{p+1}(F))$ and we prove that any bi-$(H_{p,p+1},μ)$-invariant generalized function on $G$ is invariant under the matrix transpose when μis a good character. We also show that any $P\cap H_{p,p+1}$-invariant linear functional on an $H_{p,p+1}$-distinguished irreducible smooth representation of $G$ is also $H_{p,p+1}$-invariant when F is nonarchimedean, where $P$ is a standard mirabolic subgroup of $G$ with last row vector $(0,\cdots,0,…
Let $F$ be a $p$-adic field, and $G_n$ one of the groups $GL(n,F)$, $GSO(2n-1,F)$, $GSp(2n,F)$, or $GSO(2(n-1),F)$. Using the mirabolic subgroup or analogues of it, and related "derivative" functors, we give an asymptotic expansion of functions in the Whittaker model of generic representations of $G_n$, with respect to a minimal set of characters of subgroups of the maximal torus. Denoting by $Z_n$ the center of $G_n$, and by $N_n$ the unipotent radical of its standard Borel subgroup, we characterize generic representations occurring in $L^2(Z_nN_n\backslash G_n)$ in terms of these characters.…
Let $G=GL_{n}(\mathbb{C})$ and $1\neψ:\mathbb{C}\to\mathbb{C}^{\times}$ be an additive character. Let $U$ be the subgroup of upper triangular unipotent matrices in $G$. Denote by $θ$ the character $θ:U\to\mathbb{C}$ given by \[ θ(u):=ψ(u_{1,2}+u_{2,3}+...+u_{n-1,n}). \] Let $P$ be the mirabolic subgroup of $G$ consisting of all matrices in $G$ with the last row equal to $(0,0,...,0,1)$. We prove that if $π$ is an irreducible generic representation of $GL_{n}(\mathbb{C})$ and $\mathcal{W}(π,ψ)$ is its Whittaker model, then the space $\{f|_{P}:P\to \mathbb{C}:\, f\in \mathcal{W}(π,ψ)\}$ contains…
A Note On the Orbits of a Symmetric Subgroup in the Flag Variety
本文研究连通约化代数群旗簇上对称子群的 2^r 个轨道子集，证明其闭包沿集合内其他轨道光滑，并由此导出分裂实群表示理论的推论。
Let G be the split special orthogonal group of degree 2n+1 over a field F of char F \ne 2. Then we describe G-orbits on the triple flag varieties G/P\times G/P\times G/P and G/P\times G/P\times G/B with respect to the diagonal action of G where P is a maximal parabolic subgroup of G of the shape (n,1,n) and B is a Borel subgroup. As by-products, we also describe GL_n-orbits on G/B, Q_{2n}-orbits on the full flag variety of GL_{2n} where Q_{2n} is the fixed-point subgroup in Sp_{2n} of a nonzero vector in F^{2n} and 1\times Sp_{2n}-orbits on the full flag variety of GL_{2n+1}. In the same way, …
We use the variety of one-parameter subgroups to define a numerical invariant for a representation of an infinitesimal group scheme. For an indecomposable module M of complexity 1, this number is related to the period of M.
Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic zero. An irreducible admissible representation $π$ of $GL(n,E)$ is said to be distinguished with respect to $GL(n,F)$ if it admits a non-trivial linear form that is invariant under the action of $GL(n,F)$. It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the $F$-points of the mirabolic subgroup when $π$ is unitary and generic. In this paper, we prove that the essential vector of [JPSS81…
Let $H$ be a connected spherical subgroup of a semisimple algebraic group $G$. In this paper, we give a criterion for $H$-orbit closures in the flag variety of $G$ to have nice geometric and cohomological properties. Our main tool is the method of Frobenius splitting and of global F-regularity.
Real double flag variety for the symmetric pair $(U(p,p),GL_{p}(\mathbb{C}))$ and Galois cohomology
本文针对对称对$(U(p,p),GL_{p}(\text{C}))$，在$p=2$时利用伽罗瓦上同调对实双旗簇的轨道分解$H\ackslash (H/B_H\ imes G/P_S)$完成了分类。
We show that each orbit of a Borel subgroup $B_{n-1}$ of GL(n-1) (respectively SO(n-1)) on the flag variety of GL(n) (respectively of SO(n)) is a bundle over a $B_{n-1}$-orbit on a generalized flag variety of GL(n-1) (respectively SO(n-1)), with fiber isomorphic to an orbit of an analogous subgroup on a smaller flag variety. As a consequence, we develop an inductive procedure to classify $B_{n-1}$-orbits on the flag variety. Our method is essentially uniform in the two cases. As further consequences, in the sequel to this paper we give an explicit combinatorial classification of orbits and det…
Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup H of B acting with finitely many orbits on the flag variety G/B, and we classify the H-orbits in G/B in terms of suitable combinatorial invariants. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G/B, and we give a combinatorial model for this action in terms of weight polytopes.
Let $G$ be a connected reductive algebraic group over an algebraically closed field ${\bf k}$ of characteristic not equal to 2, let $\B$ be the variety of all Borel subgroups of $G$, and let $K$ be a symmetric subgroup of $G$. Fixing a closed $K$-orbit in $\B$, we associate to every $K$-orbit on $\B$ some subsets of the Weyl group of $G$, and we study them as invariants of the $K$-orbits. When ${\bf k} = {\mathbb C}$, these invariants are used to determine when an orbit of a real form of $G$ and an orbit of a Borel subgroup of $G$ have non-empty intersection in $\B$. We also characterize the i…
We give a pattern avoidance criterion to classify the orbits of Sp(p,C) x Sp(q,C) (resp. GL(n,C)) on the flag variety of type C_{p+q} (resp. D_n) with rationally smooth closure. We show that all such orbit closures fiber (with smooth fiber) over a smaller flag variety, and hence are in fact smooth. In addition we prove that the classification is insensitive to isogeny.
The main result of this paper is a new parameterization of the orbits of a symmetric subgroup on a partial flag variety. The parameterization is in terms of certain Spaltenstein varieties, on one hand, and certain nilpotent orbits, on the other. One of the motivations is related to enumerating special unipotent representations of real reductive groups. Another motivation is understanding (a portion of) the closure order on the set of nilpotent coadjoint orbits.
Symmetric $k$-varieties are a natural generalization of symmetric spaces to general fields $k$. We study the action of minimal parabolic $k$-subgroups on symmetric $k$-varieties and define a map that embeds these orbits within the orbits corresponding to algebraically closed fields. We develop a condition for the surjectivity of this map in the case of $k$-split groups that depends only on the dimension of a maximal $k$-split torus contained within the fixed point group of the involution defining the symmetric $k$-variety.
Action of Hecke algebra on the double flag variety of type AIII
本文针对AIII型双旗簇，通过组合方法结合图确定了Hecke代数在其轨道空间上的显式作用，并得到Weyl群在该空间上的表示为诱导表示直和的结论。
Let G be a simple algebraic group over C with the Weyl group W. For a unipotent element u of G, let B_u be the variety of Borel subgroups of G containing u. Let L be a Levi subgroup of a parabolic subgroup of G with the Weyl subgroup W_L. Assume that u is in L and let B_u^L be the corresponding variety for L. The induction theorem of Springer representations describes the W-module structure of H*(B_u) in terms of the W_L-module structure of H*(B_u^L). In this paper, we prove a certain refinement of the induction theorem by considering the action of a cyclic group of order e on H^*(B_u). As a c…
We consider the action of the Levi subgroup of a parabolic subgroup that stabilizes a Schubert variety. We show that a smooth Schubert variety is a homogeneous space for a parabolic subgroup, or it has a smooth Schubert divisor. Further, we show that all smooth Schubert varieties in a (partial) flag variety of a rank two simple algebraic group are spherical.
Double flag varieties of Levi type with a finite number of orbits
本文研究 Levi 型双旗簇上有限轨道问题，利用箭图表示分类一般线性群中使作用轨道有限的抛物子群对。
In this paper we study a key example of a Hermitian symmetric space and a natural associated double flag variety, namely for the real symplectic group $G$ and the symmetric subgroup $L$, the Levi part of the Siegel parabolic $P_S$. We give a detailed treatment of the case of the maximal parabolic subgroups $Q$ of $L$ corresponding to Grassmannians and the product variety of $G/P_S$ and $L/Q$; in particular we classify the $L$-orbits here, and find natural explicit integral transforms between degenerate principal series of $L$ and $G$.
We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GL_n(C) on the variety of x-nilpotent complex matrices. We obtain a criterion as to whether the action admits a finite number of orbits and specify a system of representatives for the orbits in the finite case of 2-nilpotent matrices. Furthermore, we give a set-theoretic description of their closures and specify the minimal degenerations in detail for the action of the Borel subgroup. Concerning the action on the nilpotent cone, we obtain a generic normal form of the orbits which yields a U-normal form as well…
Let G be a connected and reductive algebraic group over an algebraically closed field of characteristic p > 0. An interesting class of representations of G consists of those G-modules having a good filtration -- i.e. a filtration whose layers are the standard highest weight modules obtained as the space of global sections of G-linearized line bundles on the flag variety of G. Let H be a connected and reductive subgroup of G. One says that (G,H) is a Donkin pair, or that H is a good filtration subgroup of G, if every G-module with good filtration also has a good filtration as an H-module. In th…
The fixed point locus of the smooth Jordan quiver variety under the action of the finite subgroups of $\mathrm{SL}_2(\mathbb{C})$
本文研究了 $\mathrm{SL}_2(\mathbb{C})$ 的有限子群作用于光滑 Jordan 箭图簇时不动点轨迹的不可约分解，利用 McKay 箭图簇和仿射李代数根格给出组合模型，并证明花环积奇点的每个射影辛消解都对应点 Hilbert 概型不动点轨迹的一个不可约分量。
Let G be a semisimple complex Lie group. In this article, we study Geometric Invariant Theory on a flag variety G/B with respect to the action of a principal 3-dimensional simple subgroup S of G. We determine explicitly the GIT-equivalence classes of S-ample line bundles on G/B. We show that, under mild assumptions, among the GIT-classes there are chambers, in the sense of Dolgachev-Hu. The Hilbert quotients Y=X//S with respect to various chambers form a family of Mori dream spaces, canonically associated with G. We are able to determine three important cones in the Picard group of any of thes…
Let $L_w$ be the Levi part of the stabilizer $Q_w$ in $GL_N$ (for left multiplication) of a Schubert variety $X(w)$ in the Grassmannian $G_{d,N}$. For the natural action of $L_w$ on $\mathbb{C}[X(w)]$, the homogeneous coordinate ring of $X(w)$ (for the Plücker embedding), we give a combinatorial description of the decomposition of $\mathbb{C}[X(w)]$ into irreducible $L_w$-modules; in fact, our description holds more generally for the action of the Levi part $L$ of any parabolic subgroup $Q$ that is contained in $Q_w$. This decomposition is then used to show that all smooth Schubert varieties, …
Let $G$ be a reductive linear algebraic group over an algebraically closed field $\mathbb{K}$ of characteristic $2$. Fix a parabolic subgroup $P$ such that the corresponding parabolic subgroup over $\mathbb{C}$ has abelian unipotent radical and fix a Levi subgroup $L\subseteq P$. We parametrize the orbits of a Borel $B\subseteq P$ over the Hermitian symmetric variety $G/L$ supposing the root system $Φ$ is irreducible. For $Φ$ simply laced we prove a combinatorial characterization of the Bruhat order over these orbits. We also prove a formula to compute the dimension of the orbits from combinat…
Levi subgroup actions on Schubert varieties are studied. In the case of partial flag varieties, the horospherical actions are determined. This leads to a characterization of the toroidal and horospherical partial flag varieties with Picard number 1. In the more general case, we provide a set of necessary conditions for the action of a Levi subgroup on a Schubert variety to be toroidal. The singular locus of a (co)minuscule Schubert variety is shown to contain all the $L_{max}$-stable Schubert subvarieties, where $L_{max}$ is the standard Levi subgroup of the maximal parabolic which acts on the…
Orthogonal and symplectic orbits in the affine flag variety of type A
本文研究A型仿射旗簇中正交群或辛群的轨道，通过建立双陪集与扭仿射对合之间的显式双射，首次给出这些轨道的组合描述。
In this paper, for any simple, simply connected algebraic group $G$ of type $B_n,C_n$ or $D_n$ and for any maximal parabolic subgroup $P$ of $G$, we describe all minimal dimensional Schubert varieties in $G/P$ admitting semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$. In this paper, we also describe, for any semi-simple simply connected algebraic group $G$ and for any Borel subgroup $B$ of $G$, all Coxeter elements $τ$ for which the Schubert variety $X(τ)$ admits a semistable point for the action of the torus $T$ with respect to a non-trivi…
We introduce the notion of ideal triangle in the Bruhat-Tits building associated to a split group -- it is analogous to the usual notion of triangle, but one vertex is "at infinity" in a certain direction. We prove that the algebraic variety of based ideal triangles with prescribed side-lengths is naturally isomorphic to a suitable variety of genuine triangles. From theorems pertaining to genuine triangles, we deduce saturation theorems related to branching to Levi subgroups and to the constant term homomorphisms.
We observe that the expansion in the basis of Schubert cycles for $H^*(G/B)$ of the class of a Richardson variety stable under a spherical Levi subgroup is described by a theorem of Brion. Using this observation, along with a combinatorial model of the poset of certain symmetric subgroup orbit closures, we give positive combinatorial descriptions of certain Schubert structure constants on the full flag variety in type $A$. Namely, we describe $c_{u,v}^w$ when $u$ and $v$ are inverse to Grassmannian permutations with unique descents at $p$ and $q$, respectively. We offer some conjectures for si…
For a reductive Lie algbera over an algbraically closed field of charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its orbit under $B$ in the Grassmannian.The variety $X$ plays an important role in the study of the commuting variety. In thisnote, we prove that $X$ is Gorenstein with rational singularities.
We describe the centralizer of irreducible representations from a finitely generated group $Γ$ to $PSL(p,\mathbb{C})$ where $p$ is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety $χ^i(Γ,PSL(p,\mathbb{C}))$. When $Γ$ is a free group of rank $l\geq 2$ or the fundamental group of a closed Riemann surface of genus $g\geq 2$, we give a complete description of this locus and prove that this locus is exactly the set of…
Let L be k((ε)), where k is an algebraic closure of a finite field with q elements and εis an indeterminate, and let σbe the Frobenius automorphism. Let G be a split connected reductive group over the fixed field of σin L, and let I be the Iwahori subgroup of G(L) associated to a given Borel subgroup of G. Let W be the extended affine Weyl group of G. Given x in W and b in G(L), we have some subgroup of G(L) that acts on the affine Deligne-Lusztig variety X_x(b) = {gI in G(L)/I : g^{-1}bσ(g) is in IxI} and hence a representation of this subgroup on the Borel-Moore homology of the variety. We i…
Let $G$ be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ In this article, we show that $α$ is a co-minuscule root if and only if for any parabolic subgroup $Q$ containing $B$ properly, there is no Schubert variety $X_{Q}(w)$ in $G/Q$ such that the minimal parabolic subgroup $P_α$ of $G$ is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of $X_{Q}(w).$
Let $G$ be a simple simply-laced algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, $B$ be a Borel subgroup of $G$ containing a maximal torus $T$ of $G.$ In this article, we show that $ω_α$ is a minuscule fundamental weight if and only if for any parabolic subgroup $Q$ containing $B$ properly, there is no Schubert variety $X_{Q}(w)$ in $G/Q$ such that the minimal parabolic subgroup $P_α$ of $G$ is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of $X_{Q}(w).$
We describe the effective and the big cones of a projective symmetric variety. Moreover, we give a necessary and sufficient combinatorial criterion for the bigness of a nef divisor on a projective symmetric variety. When the variety is toroidal and the divisor is $G$-stable, such criterion has an explicit geometric interpretation. Finally, we describe the spherical closure of a symmetric subgroup.
Let $P$ be a parabolic subgroup in $SL_n(\mathbb C)$. We show that there is a $SL_n(\mathbb C)$-stable closed subvariety of an affine Schubert variety in an infinite dimensional partial Flag variety (associated to the Kac-Moody group ${\widehat{SL_n}(\mathbb C)}$) which is a natural compactification of the cotangent bundle to $SL_n(\mathbb C)/P$. As a consequence, we recover the Springer resolution for any orbit closure inside the variety of nilpotent matrices.
Uniform decomposition of the flag scheme by a symmetric subgroup action
该研究在局部常值假设下，建立了旗簇对称子群轨道分解经典结果的概形论版本，证明了轨道概形的仿射嵌入性，并通过经典例子给出相关概形论推论。
Let $ G $ be a connected reductive algebraic group over $ \C $. We denote by $ K = (G^θ)_{0} $ the identity component of the fixed points of an involutive automorphism $ θ$ of $ G $. The pair $ (G, K) $ is called a symmetric pair. Let $Q$ be a parabolic subgroup of $K$. We want to find a pair of parabolic subgroups $P_{1}$, $P_{2}$ of $G$ such that (i) $P_{1} \cap P_{2} = Q$ and (ii) $P_{1} P_{2}$ is dense in $G$. The main result of this article states that, for a simple group $G$, we can find such a pair if and only if $(G, K)$ is a Hermitian symmetric pair. The conditions (i) and (ii) yield …
We study the ring of sections A(X) of a complete symmetric variety X, that is of the wonderful completion of G/H where G is an adjoint semi-simple group and H is the fixed subgroup for an involutorial automorphism of G. We find generators for Pic(X), we generalize the PRV conjecture to complete symmetric varieties and construct a standard monomial theory for A(X) that is compatible with G orbit closures in X. This gives a degeneration result and the rational singularityness for A(X).
A normal variety $X$ is called $H$-spherical for the action of the complex reductive group $H$ if it contains a dense orbit of some Borel subgroup of $H$. We resolve a conjecture of Hodges--Yong by showing that their spherical permutations are characterized by permutation pattern avoidance. Together with results of Gao--Hodges--Yong this implies that the sphericality of a Schubert variety $X_w$ with respect to the largest possible Levi subgroup is characterized by this same pattern avoidance condition.
Let us consider the group $G = < x,y \mid x^m = y^n>$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ and the character variety $X(G)$ in $SL(2,\C)$ of the group $G$,obtaining by elementary methods an explicit primary decomposition of the ideal corresponding to $X(G)$ in the coordinates $X=t_x$, $Y=t_y$ and $Z=t_{xy}$. As an easy consequence, a formula for computing the number of irreducible components of $X(G)$ as a function of $m$ and $n$ is given. We provide a combinatorial description of $X(G)$ and we prove that in most cases it is possible t…
The affine closure of cotangent bundles of horospherical spaces
本文通过新的几何方法证明了对于任何horospherical子群H，G/H为拟仿射时，其余切丛的仿射闭包是辛概形，并验证了Ginzburg和Kazhdan的猜想。
In this paper we determine the stringy motivic volume of log terminal horospherical $G$-varieties of complexity one, where $G$ is a connected reductive linear algebraic group. The stringy motivic volume of a log terminal variety is an invariant of singularities which was introduced by Batyrev and plays an important role in mirror symmetry for Calabi--Yau varieties. A horospherical $G$-variety of complexity one is a normal $G$-variety which is equivariantly birational to a product $C \times G/H$, where $C$ is a smooth projective curve and the closed subgroup $H$ contains a maximal unipotent sub…
Blow-up of a generalized flag variety
本文研究广义旗簇沿光滑 Schubert 簇爆破后的线丛正性，给出其丰富或 nef 的充要条件，并据此刻画爆破成为弱 Fano 的情形。
The subgroup K=GL_p x GL_q of GL_{p+q} acts on the (complex) flag variety GL_{p+q}/B with finitely many orbits. We introduce a family of polynomials that specializes to representatives for cohomology classes of the orbit closures in the Borel model. We define and study K-orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the H-polynomials and the Kazhdan-Lusztig-Vogan polynomials.
We give explicit formulas for torus-equivariant fundamental classes of closed $K$-orbits on the flag variety $G/B$ when $G$ is one of the classical groups $SL(n,\C)$, $SO(n,\C)$, or $Sp(2n,\C)$, and $K$ is a symmetric subgroup of $G$. We describe parametrizations of each orbit set and the combinatorics of its weak order, allowing us to compute formulas for the equivariant classes of all remaining orbit closures using divided difference operators. In each of the cases in type A, we realize the $K$-orbit closures as universal degeneracy loci of a certain type, involving a vector bundle $V$ over …
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