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).     

Macdonald polynomials and BGG reciprocity for current algebras

We study the category $\mathcal I _{\mathrm{gr }}$ of graded representations with finite-dimensional graded pieces for the current algebra $\mathfrak{g }\otimes \mathbf{C }[t]$ where $\mathfrak{g }$ is a simple Lie algebra. This category has many similarities with the category $\mathcal O $ of modules for $\mathfrak{g }$ , and in this paper, we prove an analog of the famous BGG duality in the case of $\mathfrak{sl }_{n+1}$ .     

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.     

The geometric meaning of Zhelobenko operators

Let $ \mathfrak{g} $ be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, $ \mathfrak{b} $ a Borel subalgebra of $ \mathfrak{g} $ , $ \mathfrak{h}\subset \mathfrak{b} $ the Cartan sublagebra, and N ? G the unipotent subgroup corresponding to the nilradical $ \mathfrak{n}\subset \mathfrak{b} $ . We show that the explicit formula for the extremal projection operator for $ \mathfrak{g} $ obtained by Asherova, Smirnov, and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence $ N\times \mathfrak{h}\to \mathfrak{b} $ given by the restriction of the adjoint action. Simple geometric proofs of formulas for the “classical” counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.     

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.     

Categories of bounded \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) - and \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) -modules

Let $ \mathfrak{g} $ be a reductive Lie algebra over $ \mathbb{C} $ and $ \mathfrak{k} \subset \mathfrak{g} $ be a reductive in $ \mathfrak{g} $ subalgebra. We call a $ \mathfrak{g} $ -module M a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module whenever M is a direct sum of finite-dimensional $ \mathfrak{k} $ -modules. We call a $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -module M bounded if there exists $ {C_M} \in {\mathbb{Z}_{{ \geqslant 0}}} $ such that for any simple finite-dimensional $ \mathfrak{k} $ -module E the dimension of the E-isotypic component is not greater than C _M dim E. Bounded $ \left( {\mathfrak{g}{\hbox{,}}\;\mathfrak{k}} \right) $ -modules form a subcategory of the category of $ \mathfrak{g} $ -modules. Let V be a finite-dimensional vector space. We prove that the categories of bounded $ \left( {\mathfrak{sp}\left( {{{\mathrm{S}}^2}V \oplus {{\mathrm{S}}^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ - and $ \left( {\mathfrak{sp}\left( {{\varLambda^2}V \oplus {\varLambda^2}{V^{*}}} \right),\;\mathfrak{gl}(V)} \right) $ -modules are isomorphic to the direct sum of countably many copies of the category of representations of some explicitly described quiver with relations under some mild assumptions on the dimension of V .     

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.     

A commutant realization of Odake’s algebra

The bcβγ-system $ \mathcal{W} $ of rank 3 has an action of the affine vertex algebra $ {V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ , and the commutant vertex algebra $ \mathcal{C}=\mathrm{Com}\left( {{V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right),\mathcal{W}} \right) $ contains copies of V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and Odake’s algebra $ \mathcal{O} $ . Odake’s algebra is an extension of the N = 2 super-conformal algebra with c = 9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and $ \mathcal{O} $ form a Howe pair (i.e., a pair of mutual commutants) inside $ \mathcal{C} $ . More generally, any finite-dimensional representation of a Lie algebra $ \mathfrak{g} $ gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of $ \mathfrak{s}{{\mathfrak{l}}_2} $ .     

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 ${J : \mathfrak g \to V}$ .     

MULTIPLICITY-FREE PRIMITIVE IDEALS ASSOCIATED WITH RIGID NILPOTENT ORBITS

Let G be a simple algebraic group defined over ?. Let e be a nilpotent element in $ \mathfrak{g} $ = Lie(G) and denote by U ( $ \mathfrak{g} $ , e) the finite W-algebra associated with the pair ( $ \mathfrak{g} $ , e). It is known that the component group Γ of the centraliser of e in G acts on the set ? of all one-dimensional representations of U ( $ \mathfrak{g} $ , e). In this paper we prove that the fixed point set ?^Γ is non-empty. As a corollary, all finite W-algebras associated with $ \mathfrak{g} $ admit one-dimensional representations. In the case of rigid nilpotent elements in exceptional Lie algebras we find irreducible highest weight $ \mathfrak{g} $ -modules whose annihilators in U ( $ \mathfrak{g} $ ) come from one-dimensional representations of U ( $ \mathfrak{g} $ , e) via Skryabin’s equivalence. As a consequence, we show that for any nilpotent orbit $ \mathcal{O} $ in $ \mathfrak{g} $ there exists a multiplicity-free (and hence completely prime) primitive ideal of U ( $ \mathfrak{g} $ ) whose associated variety equals the Zariski closure of $ \mathcal{O} $ in $ \mathfrak{g} $ .     

CR-quadrics with a symmetry property

For non-degenerate CR-quadrics ${Q \subset \mathbb{C}^{n}}$ it is well known that the real Lie algebra ${\mathfrak{g} = \mathfrak{hol}(Q)}$ of all infinitesimal CR-automorphisms has a canonical grading ${\mathfrak{g} = \mathfrak{g}^{-2} \oplus\mathfrak{g}^{-1} \oplus\mathfrak{g}^{0} \oplus\mathfrak{g}^{1} \oplus\mathfrak{g}^{2}}$ . While the first three spaces in this grading, responsible for the affine automorphisms of Q, are always easy to describe this is not the case for the last two. In general, it is even difficult to determine the dimensions of ${\mathfrak{g}^{1}}$ and ${\mathfrak{g}^{2}}$ . Here we consider a class of quadrics with a certain symmetry property for which ${\mathfrak{g}^{1}, \mathfrak{g}^{2}}$ can be determined explicitly. The task then is to verify that there exist enough interesting examples. By generalizing the ?ilov boundaries of irreducible bounded symmetric domains of non-tube type we get a collection of basic examples. Further examples are obtained by ‘tensoring’ any quadric having the symmetry property with an arbitrary commutative (associative) unital *-algebra A (of finite dimension). For certain quadrics this also works if A is not necessarily commutative.     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

Algebraic and geometric properties of quadratic Hamiltonians determined by sectional operators

Following the terminology introduced by V. V. Trofimov and A. T. Fomenko, we say that a self-adjoint operator $\varphi :\mathfrak{g}* \to \mathfrak{g}$ is sectional if it satisfies the identity ad _?x ^* a = ad _β ^* x, $x \in \mathfrak{g}*$ , where $\mathfrak{g}$ is a finite-dimensional Lie algebra and $a \in \mathfrak{g}*$ and $\beta \in \mathfrak{g}$ are fixed elements. In the case of a semisimple Lie algebra $\mathfrak{g}$ , the above identity takes the form [?x, a] = [β, x] and naturally arises in the theory of integrable systems and differential geometry (namely, in the dynamics of n-dimensional rigid bodies, the argument shift method, and the classification of projectively equivalent Riemannian metrics). This paper studies general properties of sectional operators, in particular, integrability and the bi-Hamiltonian property for the corresponding Euler equation $\dot x = ad_{\varphi x}^* x$ .     

Irreducible representations of the classical Lie superalgebra of type A(0,n) in prime characteristic

Let $\mathfrak{g }=\mathfrak{s }\mathfrak{l }(1|n+1)$ be the classical Lie superalgebra of type $A(0,n)$ over an algebraically closed field of prime characteristic $p>2$ . A sufficient condition is provided for baby Kac $\mathfrak{g }$ -modules to be simple. Moreover, simple $\mathfrak{g }$ -modules with (quasi) regular semisimple characters are classified. In particular, up to isomorphism, all the simple modules for $\mathfrak{s }\mathfrak{l }(1|2)$ are determined, and representatives and dimensions of simples are precisely given. As an application, simple modules for the general linear Lie superalgebra $\mathfrak{g }\mathfrak{l }(1|n+1)$ with certain $p$ -characters are classified. In particular, a complete classification of simple $\mathfrak{g }\mathfrak{l }(1|2)$ -modules is given.     

On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces

Let ${\mathfrak{g}=\mathfrak{g}^{\bar 0}\oplus \mathfrak{g}^{\bar 1}}$ be a ${\mathbb{Z}_2}$ -graded Lie algebra. We study the posets of abelian subalgebras of ${\mathfrak{g}^{\bar 1}}$ which are stable w.r.t. a Borel subalgebra of ${\mathfrak{g}^{\bar 0}}$ . In particular, we find a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, and Suter.     

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .     

Equivariant map superalgebras

Suppose a group $\Gamma $ acts on a scheme $X$ and a Lie superalgebra $\mathfrak {g}$ . The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from $X$ to $\mathfrak {g}$ . We classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of $X$ is finitely generated, $\Gamma $ is finite abelian and acts freely on the rational points of $X$ , and $\mathfrak {g}$ is a basic classical Lie superalgebra (or $\mathfrak {sl}\,(n,n)$ , $n \ge 1$ , if $\Gamma $ is trivial). We show that they are all (tensor products of) generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on $X$ . Furthermore, in the case that the even part of $\mathfrak {g}$ is semisimple, we show that all such modules are in fact (tensor products of) evaluation modules. On the other hand, if the even part of $\mathfrak {g}$ is not semisimple (more generally, if $\mathfrak {g}$ is of type I), we introduce a natural generalization of Kac modules and show that all irreducible finite dimensional modules are quotients of these. As a special case, our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras.     

Positive definite superfunctions and unitary representations of lie supergroups

For a broad class of Fréchet-Lie supergroups $ \mathcal{G} $ , we prove that there exists a correspondence between positive definite smooth (resp., analytic) superfunctions on $ \mathcal{G} $ and matrix coefficients of smooth (resp., analytic) unitary representations of the Harish-Chandra pair (G, $ \mathfrak{g} $ ) associated to $ \mathcal{G} $ . As an application, we prove that a smooth positive definite superfunction on $ \mathcal{G} $ is analytic if and only if it restricts to an analytic function on the underlying manifold of $ \mathcal{G} $ . When the underlying manifold of $ \mathcal{G} $ is 1-connected we obtain a necessary and sufficient condition for a linear functional on the universal enveloping algebra U( $ {{\mathfrak{g}}_{\mathbb{C}}} $ ) to correspond to a matrix coefficient of a unitary representation of (G, $ \mathfrak{g} $ ). The class of Lie supergroups for which the aforementioned results hold is characterised by a condition on the convergence of the Trotter product formula. This condition is strictly weaker than assuming that the underlying Lie group of $ \mathcal{G} $ is a locally exponential Fréchet-Lie group. In particular, our results apply to examples of interest in representation theory such as mapping supergroups and diffeomorphism supergroups.