A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.     

Parabolic contractions of semisimple Lie algebras and their invariants

Let $G$ be a connected semisimple algebraic group with Lie algebra $\mathfrak{g }$ and $P$ a parabolic subgroup of $G$ with $\mathrm{Lie\, }P=\mathfrak{p }$ . The parabolic contraction $\mathfrak{q }$ of $\mathfrak{g }$ is the semi-direct product of $\mathfrak{p }$ and a $\mathfrak{p }$ -module $\mathfrak{g }/\mathfrak{p }$ regarded as an abelian ideal. We are interested in the polynomial invariants of the adjoint and coadjoint representations of $\mathfrak{q }$ . In the adjoint case, the algebra of invariants is easily described and it turns out to be a graded polynomial algebra. The coadjoint case is more complicated. Here we found a connection between symmetric invariants of $\mathfrak{q }$ and symmetric invariants of centralisers $\mathfrak{g }_e\subset \mathfrak{g }$ , where $e\in \mathfrak{g }$ is a Richardson element with polarisation $\mathfrak{p }$ . Using this connection and results of Panyushev et al. (J Algebra 313:343–391, 2007), we prove that the algebra of symmetric invariants of $\mathfrak{q }$ is free for all parabolic subalgebras in types $\mathbf A$ and $\mathbf C$ and some parabolics in type $\mathbf B$ . This technique also applies to the minimal parabolic subalgebras in all types. For $\mathfrak{p }=\mathfrak{b }$ , a Borel subalgebra of $\mathfrak{g }$ , one gets a contraction of $\mathfrak{g }$ recently introduced by Feigin (Selecta Math 18:513–537, 2012) and studied from invariant-theoretic point of view in our previous paper (Panyushev and Yakimova in Ann Inst Fourier 62(6):2053–2068, 2012).     

On the Left and Right Brylinski-Kostant Filtrations

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ a Borel subalgebra, and $\mathfrak{h}\subset\mathfrak{b}$ a Cartan subalgebra. Let V be a finite dimensional simple $U(\mathfrak{g})$ module. Based on a principal s-triple (e,h,f) and following work of Kostant, Brylinski (J Amer Math Soc 2(3):517–533, 1989) defined a filtration $\mathcal{F}_e$ for all weight subspaces V _μ of V and calculated the dimensions of the graded subspaces for μ dominant. In Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) these dimensions were calculated for all μ. Let δM(0) be the finite dual of the Verma module of highest weight 0. It identifies with the global functions on a Weyl group translate of the open Bruhat cell and so inherits a natural degree filtration. On the other hand, up to an appropriate shift of weights, there is a unique $U(\mathfrak{b})$ module embedding of V into δM(0) and so the degree filtration induces a further filtration $\mathcal{F}$ on each weight subspace V _μ . A casual reading of Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) might lead one to believe that $\mathcal{F}$ and $\mathcal{F}_e$ coincide. However this is quite false. Rather one should view $\mathcal{F}_e$ as coming from a left action of $U(\mathfrak{n})$ and then there is a second (Brylinski-Kostant) filtration $\mathcal{F}'_e$ coming from a right action. It is $\mathcal{F}'_e$ which may coincide with $\mathcal{F}$ . In this paper the above claim is made precise. First it is noted that $\mathcal{F}$ is itself not canonical, but depends on a choice of variables. Then it is shown that a particular choice can be made to ensure that $\mathcal{F}=\mathcal{F}'_e$ . An explicit form for the unique left $U(\mathfrak{b})$ module embedding $V\hookrightarrow\delta M(0)$ is given using the right action of $U(\mathfrak{n})$ . This is used to give a purely algebraic proof of Brylinski’s main result in Brylinski (J Amer Math Soc 2(3):517–533, 1989) which is much simpler than Joseph et al. (J Amer Math Soc 13(4):945–970, 2000). It is noted that the dimensions of the graded subspaces of $\rm{gr}_{\mathcal{F}_e} V_{\!\mu}$ and $\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}$ can be very different. Their interrelation may involve the Kashiwara involution. Indeed a combinatorial formula for multiplicities in tensor products involving crystal bases and the Kashiwara involution is recovered. Though the dimensions for the graded subspaces of $\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}$ are determined by polynomial degree, their values remain unknown.     

Complement of forms

The notions of the parallel sum, the parallel difference, and the complement of two nonnegative sesquilinear forms were introduced and studied by Hassi, Sebestyé and de Snoo in Hassi et al. (Oper Theory Adv Appl 198:211–227, 2010) and Hassi et al. (J Funct Anal 257(12):3858–3894, 2009). In this paper we continue these investigations. The Galois correspondence induced by the map ${\mathfrak{m} \mapsto \mathfrak{m}_\mathfrak{t}}$ (where ${\mathfrak{m}_\mathfrak{t}}$ denotes the ${\mathfrak{t}}$ -complement of ${\mathfrak{m}}$ ) is also studied. Inspired by the work of Eriksson and Leutwiler Eriksson and Leutwiler (Math Ann 274:301–317, 1986), we introduce the notion of quasi-unit for nonnegative sesquilinear forms. The quasi-units are characterized by means of the complement and the disjoint part. It is also shown that the ${{\mathfrak{t}}}$ -quasi-units coincide with the extreme points of the convex set ${\mathfrak{z}: 0 \leq \mathfrak{z} \leq \mathfrak{t}\}}$ .     

Supports of (\mathfrak{g},\mathfrak{k})-modules of finite type

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{k}$ be a reductive subalgebra in $\mathfrak{g}$ . We say that a $\mathfrak{g}$ -module M is a $(\mathfrak{g},\mathfrak{k})$ -module if M, considered as a $\mathfrak{k}$ -module, is a direct sum of finite-dimensional $\mathfrak{k}$ -modules. We say that a $(\mathfrak{g},\mathfrak{k})$ -module M is of finite type if all $\mathfrak{k}$ -isotopic components of M are finite-dimensional. In this paper we prove that any simple $(\mathfrak{g},\mathfrak{k})$ -module of finite type is holonomic. A simple $\mathfrak{g}$ -module M is associated with the invariants V(M), V(LocM), and L(M) reflecting the ??directions of growth of M.?? We also prove that for a given pair $(\mathfrak{g},\mathfrak{k})$ the set of possible invariants is finite.     

Cohomology and Support Varieties for Restricted Lie Superalgebras

Let $(\mathfrak{g}, [p]) $ be a restricted Lie superalgebra over an algebraically closed field k of characteristic p?>?2. Let $\mathfrak{u}(\mathfrak{g})$ denote the restricted enveloping algebra of $\mathfrak{g}$ . In this paper we prove that the cohomology ring $\operatorname{H}^\bullet(\mathfrak{u}(\mathfrak{g}), k)$ is finitely generated. This allows one to define support varieties for finite dimensional $\mathfrak{u}(\mathfrak{g})$ -supermodules. We also show that support varieties for finite dimensional $\mathfrak{u}(\mathfrak{g})$ - supermodules satisfy the desirable properties of a support variety theory.     

Yangians and quantum loop algebras

Let $\mathfrak{g }$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $U_\hbar (L\mathfrak g )$ of $\mathfrak{g }$ degenerates to the Yangian ${Y_\hbar (\mathfrak g )}$ . We strengthen this result by constructing an explicit algebra homomorphism $\Phi $ from $U_\hbar (L\mathfrak g )$ to the completion of ${Y_\hbar (\mathfrak g )}$ with respect to its grading. We show moreover that $\Phi $ becomes an isomorphism when ${U_\hbar (L\mathfrak g )}$ is completed with respect to its evaluation ideal. We construct a similar homomorphism for $\mathfrak{g }=\mathfrak{gl }_n$ and show that it intertwines the actions of $U_\hbar (L\mathfrak gl _{n})$ and $Y_\hbar (\mathfrak gl _{n})$ on the equivariant $K$ -theory and cohomology of the variety of $n$ -step flags in ${\mathbb{C }}^d$ constructed by Ginzburg–Vasserot.     

Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster Reeb parallel shape operator

In a paper due to Jeong et al. (Kodai Math J 34(3):352–366, 2011) we have shown that there does not exist a hypersurface in $G_{2}({\mathbb{C }}^{m+2})$ with parallel shape operator in the generalized Tanaka–Webster connection (see Tanaka in Jpn J Math 20:131–190, 1976; Tanno in Trans Am Math Soc 314(1):349–379, 1989). In this paper, we introduce the notion of the Reeb parallel in the sense of generalized Tanaka–Webster connection for a hypersurface $M$ in $G_{2}({\mathbb{C }}^{m+2})$ and prove that $M$ is an open part of a tube around a totally geodesic $G_2(\mathbb{C }^{m+1})$ in $G_2(\mathbb{C }^{m+2})$ .     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

Local behavior and hitting probabilities of the \text{ Airy}_1 process

We obtain a formula for the $n$ -dimensional distributions of the $\text{ Airy}_1$ process in terms of a Fredholm determinant on $L^2(\mathbb{R })$ , as opposed to the standard formula which involves extended kernels, on $L^2(\{1,\dots ,n\}\times \mathbb{R })$ . The formula is analogous to an earlier formula of Prähofer and Spohn (J Stat Phys 108(5–6):1071–1106, 2002) for the $\text{ Airy}_2$ process. Using this formula we are able to prove that the $\text{ Airy}_1$ process is Hölder continuous with exponent $\frac{1}{2}$ —and that it fluctuates locally like a Brownian motion. We also explain how the same methods can be used to obtain the analogous results for the $\text{ Airy}_2$ process. As a consequence of these two results, we derive a formula for the continuum statistics of the $\text{ Airy}_1$ process, analogous to that obtained in Corwin et al. (Commun Math Phys 2011, to appear) for the $\text{ Airy}_2$ process.     

An abstract setting for hamiltonian actions

In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain ω on a Lie algebra ${\mathfrak h}$ with values in an ${\mathfrak h}$ -module V, we associate subalgebras ${\mathfrak {sp}(\mathfrak h,\omega) \supseteq \mathfrak {ham}(\mathfrak h,\omega)}$ of symplectic, resp., hamiltonian elements. Then ${\mathfrak {ham}(\mathfrak h,\omega)}$ has a natural central extension which in turn is contained in a larger abelian extension of ${\mathfrak {sp}(\mathfrak h,\omega)}$ . In this setting, we study linear actions of a Lie group G on V which are compatible with a homomorphism ${\mathfrak g \to \mathfrak {ham}(\mathfrak h,\omega)}$ , i.e., abstract hamiltonian actions, corresponding central and abelian extensions of G and momentum maps ${J : \mathfrak g \to V}$ .     

Morrey-type regularity of solutions to parabolic problems with discontinuous data

We consider regular oblique derivative problem in cylinder Q _T ?=????× (0, T), ${\Omega\subset {\mathbb R}^n}$ for uniformly parabolic operator ${{{\mathfrak P}}=D_t- \sum_{i,j=1}^n a^{ij}(x)D_{ij}}$ with VMO principal coefficients. Its unique strong solvability is proved in Manuscr. Math. 203?C220 (2000), when ${{{\mathfrak P}}u\in L^p(Q_T)}$ , ${p\in(1,\infty)}$ . Our aim is to show that the solution belongs to the generalized Sobolev?CMorrey space ${W^{2,1}_{p,\omega}(Q_T)}$ , when ${{{\mathfrak P}}u\in L^{p,\omega} (Q_T)}$ , ${p\in (1, \infty)}$ , ${\omega(x,r):\,{\mathbb R}^{n+1}_+\to {\mathbb R}_+}$ . For this goal an a priori estimate is obtained relying on explicit representation formula for the solution. Analogous result holds also for the Cauchy?CDirichlet problem.     

Vector partition functions and generalized dahmen and micchelli spaces

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory. Here we introduce a space of functions on a lattice which generalizes the space of quasipolynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X, introduced by Dahmen and Micchelli [8]. This space $ \mathcal{F}(X) $ contains the partition function $ {\mathcal{P}_{(X)}} $ . We prove a “localization formula” for any f in $ \mathcal{F}(X) $ , inspired by Paradan's decomposition formula [12]. In particular, this implies a simple proof that the partition function $ {\mathcal{P}_{(X)}} $ is a quasi-polynomial on the Minkowski differences $ \mathfrak{c} - B(X) $ , where c is a big cell and B(X) is the zonotope generated by the vectors in X, a result due essentially to Dahmen and Micchelli.     

Extending structures for Lie algebras

Let $\mathfrak{g }$ be a Lie algebra, $E$ a vector space containing $\mathfrak{g }$ as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on $E$ such that $\mathfrak{g }$ is a Lie subalgebra of $E$ . A general product, called the unified product, is introduced as a tool for our approach. Let $V$ be a complement of $\mathfrak{g }$ in $E$ : the unified product $\mathfrak{g } \,\natural \, V$ is associated to a system $(\triangleleft , \, \triangleright , \, f, \{-, \, -\})$ consisting of two actions $\triangleleft $ and $\triangleright $ , a generalized cocycle $f$ and a twisted Jacobi bracket $\{-, \, -\}$ on $V$ . There exists a Lie algebra structure $[-,-]$ on $E$ containing $\mathfrak{g }$ as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras $(E, [-,-]) \cong \mathfrak{g } \,\natural \, V$ . All such Lie algebra structures on $E$ are classified by two cohomological type objects which are explicitly constructed. The first one $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ will classify all Lie algebra structures on $E$ up to an isomorphism that stabilizes $\mathfrak{g }$ while the second object $\mathcal{H }^{2} (V, \mathfrak{g })$ provides the classification from the view point of the extension problem. Several examples that compute both classifying objects $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ and $\mathcal{H }^{2} (V, \mathfrak{g })$ are worked out in detail in the case of flag extending structures.     

The Quantized Walled Brauer Algebra and Mixed Tensor Space

In this paper we investigate a multi-parameter deformation $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ of the walled Brauer algebra which was previously introduced by Leduc (1994). We construct an integral basis of $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $\mathfrak{B}_{r,s}^n(q)= \mathfrak{B}_{r,s}^n(q^{-1}-q,q^n,[n]_q)$ on mixed tensor space and prove that the kernel is free over the ground ring R of rank independent of R. As an application, we prove one side of Schur–Weyl duality for mixed tensor space: the image of $\mathfrak{B}_{r,s}^n(q)$ in the R-endomorphism ring of mixed tensor space is, for all choices of R and the parameter q, the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra U of the general linear Lie algebra $\mathfrak{gl}_n$ on mixed tensor space. Thus, the U-invariants in the ring of R-linear endomorphisms of mixed tensor space are generated by the action of $\mathfrak{B}_{r,s}^n(q)$ .     

Tilting Theory and Functor Categories II. Generalized Tilting

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $\mathrm{Mod}(\mathcal{C})$ , from a skeletally small preadditive category $\mathcal{C}$ to the category of abelian groups, initiated in [15]. We introduce the notion of a generalized tilting category $\mathcal{T}$ , and we concentrate here on extending Happel’s theorem to $\mathrm{Mod}(\mathcal{C})$ ; more specifically, we prove that there is an equivalence of triangulated categories $\mathcal{D}^{b}( \mathrm{Mod}(\mathcal{C}))\cong \mathcal{D}^{b}(\mathrm{Mod}(\mathcal{T}))$ . We then add some restrictions on our category $\mathcal{C}$ , in order to obtain a version of Happel’s theorem for the categories of finitely presented functors. We end the paper proving that some of the theorems for artin algebras, relating tilting with contravariantly finite categories proved in Auslander and Reiten (Adv Math 12(3):306–366, 1974; Adv Math 86(1):111–151, 1991), can be extended to the category of finitely presented functors $\mathrm{mod}(\mathcal{C})$ , with $\mathcal{C}$ a dualizing variety.     

\mathfrak D -parallelism of normal and structure Jacobi operators for hypersurfaces in complex two-plane Grassmannians

In this paper, we give non-existence theorems for Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C }^{m+2})$ with $\mathfrak D $ -parallel normal Jacobi operator ${\bar{R}}_N$ and $\mathfrak D $ -parallel structure Jacobi operator $R_{\xi }$ if the distribution $\mathfrak D $ or $\mathfrak D ^{\bot }$ component of the Reeb vector field is invariant by the shape operator, respectively.     

Erratum to: Algèbres de lie quasi-réductives

In Corollary 12(ii) and Theorem 13(v) of [1] we omitted the hypothesis dim $ \mathfrak{z}\leq 1 $ . Moreover, in some places the symbol $ \mathbb{K} $ must be replaced by the symbol $ {{\mathbb{K}}^{\times }} $ .