Let $k$ be an algebraically closed field. Let $B$ be the Borel subgroup of $\mGL_n(k)$ consisting of nonsingular upper triangular matrices. Let $\frb = \mLie B$ be the Lie algebra of upper triangular $n \times n$ matrices and $\fru$ the Lie subalgebra of $\frb$ consisting of strictly upper triangular matrices. We classify all Lie ideals $\frn$ of $\frb$, satisfying $\fru' \subseteq \frn \subseteq \fru$, such that $B$ acts (by conjugation) on $\frn$ with a dense orbit. Further, in case $B$ does not act with a dense orbit, we give the minimal codimension of a $B$--orbit in $\frn$. This can be viewed as a first step towards the difficult open problem of classifying of all ideals $\frn \subseteq \fru$ such that $B$ acts on $\frn$ with a dense orbit.
The proofs of our main results require a translation into the representation theory of a certain quasi-hereditary algebra $\cA_{t,1}$. In this setting we find the minimal dimension of $\mExt^1_{\cA_{t,1}}(M,M)$ for a $Δ$-good $\cA_{t,1}$--module of certain fixed $Δ$-dimension vectors.
We describe an approach to express the geometric side of the Arthur-Selberg trace formula in terms of zeta integrals attached to prehomogeneous vector spaces. This will provide explicit formulas for weighted orbital integrals and for the coefficients by which they are multiplied in the trace formula. We implement this programme for the principal unipotent conjugacy class. The method relies on certain convergence results and uses the notions of induced conjugacy classes and canonical parabolic subgroups. So far, it works for certain types of conjugacy classes, which covers all classes appearing…
We give a branching law for subgroups fixed by an involution. As an application we give a generalization of the Cartan-Helgason theorem and a noncompact analogue of the Borel-Weil theorem.
Let V be a finite dimensional representation of the connected complex reductive group H. Denote by G the derived subgroup of H and assume that the categorical quotient of V by G is one dimensional. In this situation there exists a homomorphism, denoted by rad, from the algebra A of G-invariant differential operators on V to the first Weyl algebra. We show that the image of rad is isomorphic to the spherical subalgebra of a Cherednik algebra, whose parameters are determined by the b-function of the relative invariant associated to the prehomogeneous vector space (H : V). If (H : V) is furthemor…
We construct an action of the Hecke monoid on the category of rational modules for the quantum negative Borel subgroup of the quantum general linear group. We also show that this action restricts to the category of polynomial modules for this quantum subgroup and induces an action on the category of modules for quantised Borel-Schur algebras.
Colmez has given a recipe to associate a smooth modular representation Omega(W) of the Borel subgroup of GL_2(Q_p) to a F_p^bar-representation W of Gal(Qp^bar/Qp) by using Fontaine's theory of (phi,Gamma)-modules. We compute Omega(W) explicitly and we prove that if W is irreducible and dim(W)=2, then Omega(W) is the restriction to the Borel subgroup of GL_2(Q_p) of the supersingular representation associated to W in Breuil's correspondence.
Let G be the group of symplectic (unimodular) automorphisms of the free associative algebra on two generators. A theorem of G.Wilson and the first author asserts that G acts transitively the Calogero-Moser spaces C_n for all n. We generalize this theorem in two ways: first, we prove that the action of G on C_n is doubly transitive, meaning that G acts transitively on the configuration space of (ordered) pairs of points in C_n; second, we prove that the diagonal action of G on the product of (any number of) copies of C_n is transitive provided the corresponding n's are pairwise distinct. In the…
When does Borel's theorem on free subgroups of semisimple groups generalize to other groups? We initiate a systematic study of this question and find positive and negative answers for it. In particular, we fully classify fundamental groups of surfaces and von Dyck groups that satisfy Borel's theorem. Further, as a byproduct of this theory, we make headway on a question of Breuillard, Green, Guralnick, and Tao concerning double word maps.
This paper studies Emerton's Jacquet module functor for locally analytic representations of p-adic reductive groups. When P is a parabolic subgroup whose Levi factor M is not commutative, we show that passing to an isotypical subspace for the derived subgroup of M gives rise to essentially admissible locally analytic representations of the torus Z(M), which have a natural interpretation in terms of rigid geometry. We use this to extend Emerton's representation-theoretic construction of eigenvarieties by constructing eigenvarieties interpolating automorphic representations whose local component…
Let $F$ be a non-Archimedean local field and let $p$ be the residual characteristic of $F$. Let $G=GL_2(F)$ and let $P$ be a Borel subgroup of $G$. In this paper we study the restriction of irreducible representations of $G$ on $E$-vector spaces to $P$, where $E$ is an algebraically closed field of characteristic $p$. We show that in a certain sense $P$ controls the representation theory of $G$. We then extend our results to smooth $\oK[G]$- modules of finite length and unitary $K$-Banach space representations of $G$, where $\oK$ is the ring of integers of a complete discretely valued field $K…
Let $G$ be a quasi-simple algebraic group defined over an algebraically closed field $k$ and $B$ a Borel subgroup of $G$ acting on the nilradical $\mathfrak{n}$ of its Lie algebra $\mathfrak{b}$ via the Adjoint representation. It is known that $B$ has only finitely many orbits in only five cases: when $G$ is of type $A_{n}$ for $n \leq 4$, and when $G$ is type $B_{2}$. In this paper, we elaborate on this work in the case when $G =SL_{n +1}(k)$ (type $A_{n})$, for $n \leq 4$, by finding the polynomial defining equations of each orbit. Consequences of these equations include the dimension of the…
We give an explicit description of the restriction of discrete series representations of SU(2,1) to a Borel subgroup and to a maximal exponential solvable subgroup and we interpret our results in the framework of orbit method, Hamiltonian geometry and "$\text{Spin}_{c}$" quantization. Our results give a positive answer to a Duflo's conjecture for SU(2,1).
Let G be a unipotent algebraic subgroup of some GL_m(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G \cap GL_m(Z). This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.
Let E be a finite extension of Fp. Using Fontaine's theory of (phi,Gamma)-modules, Colmez has shown how to attach to any irreducible E-linear representation of Gal(Qpbar/Qp) an infinite dimensional smooth irreducible E-linear representation of B_2(Qp) that has a central character. We prove that every such representation of B_2(Qp) arises in this way. Our proof extends to algebraically closed fields E of characteristic p. In this case, infinite dimensional smooth irreducible E-linear representations of B_2(Qp) having a central character arise in a similar way from irreducible E-linear represent…
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 $\frak{g}= Lie G$. Let $\frak{g} = \frak{n}_- +\frak{h} + \frak{n}$ be a triangular decomposition of $\frak{g}$. One readily has that $Cent\,U({\frak n})$ is isomorphic to the ring $S({\frak n})^{\frak\n}$ of symmetric invariants. Using the cascade ${\cal B}$ of strongly orthogonal roots, some time ago we proved that $S({\frak n})^{{\frak n}$ is a polynomial ring $\Bbb C [ξ_1,...,ξ_m]$ where $m$ is the cardinality of ${\cal B}$. Using this result we establish that the maximal coadjoint of $N = exp \frak {n}$ has codimension $m$…
Let $B$ be a Borel subgroup of a semisimple algebraic group $G$, and let $\mathfrak a$ be an abelian ideal of $\mathfrak b=Lie(B)$. The ideal $\mathfrak a$ is determined by certain subset $Δ_{\mathfrak a}$ of positive roots, and using $Δ_{\mathfrak a}$ we give an explicit classification of the $B$-orbits in $\mathfrak a$ and $\mathfrak a^*$. Our description visibly demonstrates that there are finitely many $B$-orbits in both cases. We also describe the Pyasetskii correspondence between the $B$-orbits in $\mathfrak a$ and $\mathfrak a^*$ and the invariant algebras $\Bbbk[\mathfrak a]^U$ and $\B…
Let $G$ be a complex simply-connected semisimple Lie group and let $\frak{g}= Lie G$. Let $\frak{g} = \frak{n}_- +\frak{h} + \frak{n}$ be a triangular decomposition of $\frak{g}$. One readily has that $Cent\,U({\frak n})$ is isomorphic to the ring $S({\frak n})^{\frak\n}$ of symmetric invariants. Using the cascade ${\cal B}$ of strongly orthogonal roots, some time ago we proved that $S({\frak n})^{{\frak n}$ is a polynomial ring $\Bbb C [ξ_1,...,ξ_m]$ where $m$ is the cardinality of ${\cal B}$. Using this result we establish that the maximal coadjoint of $N = exp \frak {n}$ has codimension $m$…
Let F be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on F with finitely many orbits, and let V be an H-orbit closure in F. Expanding the cohomology class of V in the basis of Schubert classes defines a union V_0 of Schubert varieties in F with positive multiplicities. If G is simply-laced, we show that these multiplicites are equal to the same power of 2. For arbitrary G, we show that V_0 is connected in codimension 1. If moreover all multiplicities are 1, we show that the singularities of V are rational, and we construct a flat degeneration of…
The flag variety of a complex reductive linear algebraic group G is by definition the quotient G/B by a Borel subgroup. It can be regarded as the set of Borel subalgebras of Lie(G). Given a nilpotent element e in Lie(G), one calls Springer fiber the subvariety formed by the Borel subalgebras which contain e. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by C. De Concini, G. Lusztig, and C. Procesi asserts that, when G is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In th…
Let $p$ be a prime number and $\Bbbk=\bar{\mathbb{F}}_p$, the algebraic closure of the finite field $\mathbb{F}_p$ of $p$ elements. Let ${\bf G}$ be a connected reductive group defined over $\mathbb{F}_p$ and ${\bf B}$ be a Borel subgroup of ${\bf G}$ (not necessarily defined over $\mathbb{F}_p$). We show that for each (one-dimensional) character $θ$ of ${\bf B}$ (not necessarily rational), there is a unique (up to isomorphism) irreducible $\Bbbk{\bf G}$-module $\mathbb{L}(θ)$ containing $θ$ as a $\Bbbk{\bf B}$-submodule, and moreover, $\mathbb{L}(θ)$ is isomorphic to a parabolic induction fro…
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…
Let G_0 be a simply connected noncompact real simple Lie group with maximal compact subgroup K_0. Assume that rank(G_0) = rank(K_0) so that G_0 has discrete series representations. If G_0/K_0 is Hermitian symmetric, there exists a relatively simple discrete series of G_0, called holomorphic discrete series. Now assume that G_0/K_0 is not Hermitian symmetric. In this case, we can define Borel-de Siebenthal discrete series of G_0 analogous to holomorphic discrete series. We consider a certain circle subgroup of K_0 whose centralizer L_0 is such that K_0/L_0 is an irreducible compact Hermitian sy…
We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$, especially of the Borel subgroup $B$ and of the standard unipotent subgroup $U$ of the latter on the nilpotent cone of complex nilpotent matrices. We obtain generic normal forms of the orbits and describe generating (semi-) invariants for the Borel semi-invariant ring as well as for the $U$-invariant ring. The latter is described in more detail in terms of algebraic quotients by a special toric variety closely related. The study of a GIT-quotient for the Borel-action is initiated.
This paper studies nilpotent orbits in complex simple Lie algebras from the viewpoint of strongly visible actions in the sense of T. Kobayashi. We prove that the action of a maximal compact group consisting of inner automorphisms on a nilpotent orbit is strongly visible if and only if it is spherical, namely, admitting an open orbit of a Borel subgroup. Further, we find a concrete description of a slice in the strongly visible action. As a corollary, we clarify a relationship among different notions of complex nilpotent orbits: actions of Borel subgroups (sphericity); multiplicity-free represe…
A closed subgroup $H$ of a connected reductive group $G$ is called $\textit{spherical}$ if a Borel subgroup in $G$ has an open orbit on $G/H$. We give a combinatorial characterization for a spherical subgroup to be contained in another one which generalizes previous work by Knop. As an application, we compute the Luna datum of the identity component of a spherical subgroup which yields a characterization of connectedness for spherical subgroups.
Let $G$ be a higher rank semisimple linear algebraic group over a non-Archimedean local field. The simplicial complexes corresponding to any sequence of pairwise non-conjugate irreducible lattices in $G$ are Benjamini-Schramm convergent to the Bruhat-Tits building. Convergence of the relative Plancherel measures and normalized Betti numbers follows. This extends the work of Abert, Bergeron, Biringer, Gelander, Nokolov, Raimbault and Samet from real Lie groups to linear groups over arbitrary local fields. Along the way, various results concerning Invariant Random Subgroups and in particular a v…
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 reductive algebraic group defined over an algebraically closed field $\mathbbm k$ of characteristic zero. We consider the commuting variety $\mathcal C(\mathfrak u)$ of the nilradical $\mathfrak u$ of the Lie algebra $\mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $\mathfrak u$ with only a finite number of orbits, we verify that $\mathcal C(\mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {\em distinguished} $B$-orbits in $\mathfrak u$. We observe that in general $\mathcal C(\mathfrak u)$ is not equidim…
Let $G$ be a complex reductive group and $H=G^θ$ be its fixed point subgroup under a Galois involution $θ$. We show that any $H$-distinguished representation $π$ (i.e $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\neq0$) satisfies: 1) $π^θ\simeq\tildeπ$, where $\tildeπ$ is the contragredient representation and $π^θ$ is the twist of $π$ under $θ$. 2) $\mathrm{dim}_{\mathbb{C}}\left(π^{*}\right)^{H}\leq\left|B\backslash G/H\right|$, where $B$ is a Borel subgroup of $G$. By proving Statement 1), we give a partial answer to a conjecture by Lapid.
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…
We give three interpretations of the number $b$ of orbits of the Borel subgroup of upper triangular matrices on the variety $\ms{X}$ of complete quadrics. First, we show that $b$ is equal to the number of standard Young tableaux on skew-diagrams. Then, we relate $b$ to certain values of a modified Hermite polynomial. Third, we relate $b$ to a certain cell decomposition on $\ms{X}$ previously studied by De Concini, Springer, and Strickland. Using these, we give asymptotic estimates for $b$ as the dimension of the quadrics increases.
A subgroup H of an algebraic group G is said to be strongly solvable if H is contained in a Borel subgroup of G. This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in reductive complex algebraic groups: Luna's general classification of arbitrary spherical subgroups restricted to the strongly solvable case, Luna's 1993 classification of strongly solvable wonderful subgroups, and the author's 2011 classification of strongly solvable spherical subgroups. We give a detailed presentation of all the t…
Let $\cA$ be a commutative unital Banach algebra, $\g$ be a semisimple complex Lie algebra and $G(\cA)$ be the 1-connected Banach--Lie group with Lie algebra $\g \otimes \cA$. Then there is a natural concept of a parabolic subgroup $P(\cA)$ of $G(\cA)$ and we obtain generalizations $X(\cA) := G(\cA)/P(\cA)$ of the generalized flag manifolds. In this note we provide an explicit description of all homogeneous holomorphic line bundles over $X(\cA)$ with non-zero holomorphic sections. In particular, we show that all these line bundles are tensor products of pullbacks of line bundles over $X(\C)$ b…
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 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 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 $X_{Q}(w).$
Let ${\mathfrak g}$ be a complex simple Lie algebra with Borel subalgebra ${\mathfrak b}$. Consider the semidirect product $I{\mathfrak b}={\mathfrak b}\ltimes{\mathfrak b}^*$, where the dual ${\mathfrak b}^*$ of ${\mathfrak b}$, is equipped with the coadjoint action of ${\mathfrak b}$ and is considered as an abelian ideal of $I{\mathfrak b}$. We describe the automorphism group ${\operatorname{Aut}}(I{\mathfrak b})$ of the Lie algebra $I{\mathfrak b}$. In particular we prove that it contains the automorphism group of the extended Dynkin diagram of ${\mathfrak b}$. In type $A_n$, the dihedral s…
Let G be a connected reductive algebraic group and let G'=[G,G] be its derived subgroup. Let (G,V) be a multiplicity free representation with a one dimensional quotient (see definition below). We prove that the algebra D(V)^{G'} of G'-invariant differential operators with polynomial coefficients on V, is a quotient of a so-called Smith algebra over its center. Over C this class of algebras was introduced by S.P. Smith as a class of algebras similar to the enveloping algebra U(sl(2)) of sl(2). Our result generalizes the case of the Weil representation, where the associative algebra generated by…
Let G be a complex reductive group acting on a finite-dimensional complex vector space H. Let B be a Borel subgroup of G and let T be the associated torus. The Mumford cone is the polyhedral cone generated by the T-weights of the polynomial functions on H which are semi-invariant under the Borel subgroup. In this article, we determine the inequalities of the Mumford cone in the case of the linear representation associated to a quiver and a dimension vector n=(n_x). We give these inequalities in terms of filtered dimension vectors, and we directly adapt Schofield's argument to inductively deter…
One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps in the standard development, including the self-normalizing property of Borel subgroups.
Let k be an algebraically closed field of characteristic p>0 and let G be a connected reductive group over k. Let B be a Borel subgroup of G and let g and b be the Lie algebras of G and B. Denote the first Frobenius kernels of G and B by G_1 and B_1. Furthermore, denote the algebras of polynomial functions on G and g by k[G] and k[g], and similar for B and b. The group G acts on k[G] via the conjugation action and on k[g] via the adjoint action. Similarly, B acts on k[B] via the conjugation action and on k[b] via the adjoint action. We show that, under certain mild assumptions, the cohomology …
Let $G$ be a reductive algebraic group and $H$ its reductive subgroup. Fix a Borel subgroup $B\subset G$ and a maximal torus $T\subset B$. The Cartan space $\a_{G,G/H}$ is, by definition, the subspace of $\Lie(T)^*$ generated by the weights of $B$-semiinvariant rational functions on $G/H$. We compute the spaces $\a_{G,G/H}$.
In this paper we introduce (weakly) root graded Banach--Lie algebras and corresponding Lie groups as natural generalizations of group like $\GL_n(A)$ for a Banach algebra $A$ or groups like $C(X,K)$ of continuous maps of a compact space $X$ into a complex semisimple Lie group $K$. We study holomorphic induction from holomorphic Banach representations of so-called parabolic subgroups $P$ to representations of $G$ on holomorphic sections of homogeneous vector bundles over $G/P$. One of our main results is an algebraic characterization of the space of sections which is used to show that this spac…
In this paper, we show that, under certain assumptions, a parabolic induction $Ind_B^Gλ$ from the Borel subgroup $B$ of a (real or $p$-adic) reductive group $G$ decomposes into a direct sum of the form: \[ Ind_B^Gλ= \left(Ind_P^G St_M\otimes χ_0\right) \oplus \left(Ind_P^G \mathbf{1}_M\otimes χ_0\right), \] where $P$ is a parabolic subgroup of $G$ with Levi subgroup $M$ of semi-simple rank $1$, $\mathbf{1}_M$ is the trivial representation of $M$, $St_M$ is the Steinberg representation of $M$ and $χ_0$ is a certain character of $M$. We construct examples of this phenomenon for all simply-connec…
Let G be an algebraic reductive group over a an algebraically closed field of positive characteristic. Choose a parabolic subgroup $P$ in $G$ and denote by $U$ its unipotent radical. Let $X$ be a $G$-variety. The purpose of this paper is to give two examples of a situation in which the functor of averaging of l-adic sheaves on $X$ with respect to a generic character of $U$ commutes with Verdier duality. In the first example we take $χ$ to be an arbitrary $G$-variety and we prove the above property for all $\overline{P}$-equivariant sheaves on $X$ where $\overline{P}$ is an opposite parabolic s…
Let $G$ be a semisimple simply connected complex algebraic group. Let $U$ be the unipotent radical of a Borel subgroup in $G$. We describe the coordinate rings of $U$ (resp., $G/U$, $G$) in terms of two (resp., four, eight) birational charts introduced in [L94, L19] in connection with the study of total positivity.
We prove the Grothendieck-Serre conjecture for quasi-split reductive groups schemes. Our method involves reducing to the Borel subgroup in order to conclude the result from purity for tori and the structure theorem for unipotent radicals of parabolic subgroups in a reductive group.
Let G be a connected reductive complex affine algebraic group, and let X denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in X, extending a theorem of Richardson and proving a Mumford-type result about topological singularities; this resolves conjectures of Florentino-Lawton. In particular, we compute the codimension of the orbifold singular locus using facts about Borel-de Siebenthal subgroups. We then use the codimension bound to calculate higher homotopy groups of the smooth locus of X, proving conjectures of Florentino-Lawt…
We show that the categories of smooth ${\rm SL}_2({\mathbb Q}_p)$-representations (resp. ${\rm GL}_2({\mathbb Q}_p)$-representations) of level $1$ on $p$-torsion modules are equivalent with certain explicitly described equivariant coefficient systems on the Bruhat-Tits tree; the coefficient system assigned to a representation $V$ assigns to an edge $τ$ the invariants in $V$ under the pro-$p$-Iwahori subgroup corresponding to $τ$. The proof relies on computations of the group cohomology of a compact open subgroup group $N_0$ of the unipotent radical of a Borel subgroup.
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…
Suppose $L$ is a semisimple Levi subgroup of a connected Lie group~$G$, $X$ is a Borel $G$-space with finite invariant measure, and $α\colon X \times G \to \GL_n(\real)$ is a Borel cocycle. Assume $L$ has finite center, and that the real rank of every simple factor of~$L$ is at least two. We show that if $L$ is ergodic on~$X$, and the restriction of~$α$ to~$X \times L$ is cohomologous to a homomorphism (modulo a compact group), then, after passing to a finite cover of~$X$, the cocycle $α$ itself is cohomologous to a homomorphism (modulo a compact group).
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.
