The Bershadsky-Polyakov algebras are the subregular quantum Hamiltonian reductions of the affine vertex operator algebras associated associated with [Formula: see text]. In (D. Adamović, K. Kawasetsu and D. Ridout, A realisation of the Bershadsky–Polyakov algebras and their relaxed modules, Lett. Math. Phys. 111 (2021) 38, arXiv:2007.00396 [math.QA]), we realized these algebras in terms of the regular reduction, Zamolodchikov’s W3-algebra, and an isotropic lattice vertex operator algebra. We also proved that a natural construction of relaxed highest-weight Bershadsky-Polyakov modules has the property that the result is generically irreducible. Here, we prove that this construction, when combined with spectral flow twists, gives a complete set of irreducible weight modules whose weight spaces are finite-dimensional. This gives a simple independent proof of the main classification theorem of (Z. Fehily, K. Kawasetsu and D. Ridout, Classifying relaxed highest-weight modules for admissible-level Bershadsky–Polyakov algebras, Comm. Math. Phys. 385 (2021) 859–904, arXiv:2007.03917 [math.RT]) for nondegenerate admissible levels and extends this classification to a category of weight modules. We also deduce the classification for the nonadmissible level k=-[Formula: see text], which is new.
Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let $$W=V {\oplus}Fe$$ with the form Q extending q with Q(e) = 1. Consider the standard embedding $$\mathrm{O}(V) \hookrightarrow \mathrm{O}(W)$$ and the two-sided action of $$\mathrm{O}(V)\times \mathrm{O}(V)$$ on $$\mathrm{O}(W)$$ . In this note we show that any $$\mathrm{O}(V) \times \mathrm{O}(V)$$ -invariant distribution on $$\mathrm{O}(W)$$ is invariant with respect to transposition. This result was earlier proven in a bit different form in van Dijk (Math Z 193:581–593, 1986) for $$F={\mathbb{R}}$$ , in Aparicio and van Dijk (Complex generalized Gelfand pairs. Tambov University, 2006) for $$F={\mathbb{C}}$$ and in Bosman and van Dijk (Geometriae Dedicata 50:261–282, 1994) for p-adic fields. Here we give a different proof. Using results from Aizenbud et al. (arXiv:0709.1273 (math.RT), submitted), we show that this result on invariant distributions implies that the pair (O(V), O(W)) is a Gelfand pair. In the archimedean setting this means that for any irreducible admissible smooth Fréchet representation (π, E) of $$\mathrm{O}(W)$$ we have $$ dim Hom_{\mathrm{O}(V)}(E,\mathbb{C}) \leq 1.$$ A stronger result for p-adic fields is obtained in Aizenbud et al. (arXiv:0709.4215 (math.RT), submitted).
This paper is a subsequent paper of math.RT/0607673. Here we consider the irreducible components of Springer fibres (or orbital varieties) for two-column case in GL}_n. We describe the intersection of two irreducible components, and specially give the necessary and sufficient condition for this intersection to be of codimension one. Since an orbital variety in two-column case is a finite union of the Borel orbits, we solve the initial question by determining orbits of codimension one in the closure of a given orbit. We show that they are parameterized by a specific set of involutions called descendants, already introduced by the first author in a previous work. Applying this result we show that the intersections of two components of codimension one are irreducible and provide the combinatorial description in terms of Young tableaux of the pairs of such components.
We study a 2-parametric family of probability measures on the space of countable point configurations on the punctured real line (the points of the random configuration are concentrated near zero). These measures (or, equivalently, point processes) have been introduced in Part II (A. Borodin, math.RT/9804087) in connection with the problem of harmonic analysis on the infinite symmetric group. The main result of the present paper is a determinantal formula for the correlation functions. The formula involves a kernel called the matrix Whittaker kernel. Each of its two diagonal blocks governs the projection of the process on one of the two half-lines; the corresponding kernel on the half-line was studied in Part III (A. Borodin and G. Olshanski, math/RT/9804088). While the diagonal blocks of the matrix Whitaker kernel are symmetric, the whole kernel turns out to be $J$-symmetric, i.e., symmetric with respect to a natural indefinite inner product. We also discuss a rather surprising connection of our processes with the recent work by B. Eynard and M. L. Mehta (cond-mat/9710230) on correlations of eigenvalues of coupled random matrices.
Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Satake compactification of a Hermitian locally symmetric space; Baily and Borel proved this is always geometrically rational. Satake compactifications for which all the real boundary components are equal-rank symmetric spaces are a natural generalization of the Baily-Borel-Satake compactification. Recent work (see math.RT/0112250, math.RT/0112251) indicates that this is the natural setting for many results about cohomology of compactifications of locally symmetric spaces. In this paper we prove any Satake compactification for which all the real boundary components are equal-rank symmetric spaces is geometrically rational aside from certain rational rank 1 or 2 exceptions; we completely analyze geometric rationality for these exceptional cases. The proof uses Casselman's criterion for geometric rationality. We also prove that a Satake compactification is geometrically rational if the representation is defined over the rational numbers.
Let X be a locally symmetric space associated to a reductive algebraic group G defined over Q. L-modules are a combinatorial analogue of constructible sheaves on the reductive Borel-Serre compactification of X; they were introduced in [math.RT/0112251]. That paper also introduced the micro-support of an L-module, a combinatorial invariant that to a great extent characterizes the cohomology of the associated sheaf. The theory has been successfully applied to solve a number of problems concerning the intersection cohomology and weighted cohomology of the reductive Borel-Serre compactification [math.RT/0112251], as well as the ordinary cohomology of X [math.RT/0112250]. In this paper we extend the theory so that it covers L^2-cohomology. In particular we construct an L-module whose cohomology is the L^2-cohomology of X and we calculate its micro-support. As an application we obtain a new proof of the conjectures of Borel and Zucker.
H. Hertz called any manifold M with a given nonintegrable distribution {\it nonholonomic}. Vershik and Gershkovich stated and R. Montgomery proved that the space of germs of any nonholonomic distribution on M with an open and dense orbit of the diffeomorphism group is either (1) of codimension one or (2) an Engel distribution. No analog of this statement for supermanifolds is formulated yet, we only have some examples: our list (an analog of E.Cartan's classification) of simple Lie superalgebras of vector fields with polynomial coefficients and a particular (Weisfeiler) grading contains 16 series similar to contact ones and 11 exceptional algebras preserving nonholonomic structures. Here we compute the cohomology corresponding to the analog of the Riemann tensor for the SUPERmanifolds corresponding to the 15 exceptional simple vectorial Lie superalgebras, 11 of which are nonholonomic. The cohomology for analogs of the Riemann tensor for the manifolds with an exceptional Engel manifolds are computed in math.RT/0202213.
We give a summary of the results from Parts I-V (math.RT/9804086, math.RT/9804087, math.RT/9804088, math.RT/9810013, math.RT/9810014). Our work originated from harmonic analysis on the infinite symmetric group. The problem of spectral decomposition for certain representations of this group leads to a family of probability measures on an infinite-dimensional simplex, which is a kind of dual object for the infinite symmetric group. To understand the nature of these measures we interpret them as stochastic point processes on the punctured real line and compute their correlation functions. The correlation functions are given by multidimensional integrals which can be expressed in terms of a multivariate hypergeometric series (the Lauricella function of type B). It turns out that after a slight modification (`lifting') of the processes the correlation functions take a common in Random Matrix Theory (RMT) determinantal form with a certain kernel. The kernel is expressed through the classical Whittaker functions. It depends on two parameters and admits a variety of degenerations. They include the well-known in RMT sine and Bessel kernels as well as some other Bessel-type kernels which, to our best knowledge, are new. The explicit knowledge of the correlation functions enables us to derive a number of conclusions about the initial probability measures. We also study the structure of our kernel; this finally leads to a constructive description of the initial measures. We believe that this work provides a new promising connection between RMT and Representation Theory.
If A is a finite dimensional nilpotent associative algebra over a finite field k, the set G=1+A of all formal expressions of the form 1+a, where a is an element of A, has a natural group structure, given by (1+a)(1+b)=1+(a+b+ab). A finite group arising in this way is called an algebra group. One can also consider G as a unipotent algebraic group over k. We study representations of G from the point of view of ``geometric character theory'' for algebraic groups over finite fields (cf. G. Lusztig, ``Character sheaves and generalizations'', math.RT/0309134). The main result of this paper is a construction of canonical injective ``base change maps'' between - the set of isomorphism classes of complex irreducible representations of G', and - the set of isomorphism classes of complex irreducible representations of G'', which commute with the natural action of the Galois group Gal(k''/k), where k' is a finite extension of k and k'' is a finite extension of k', and G', G'' are the finite algebra groups obtained from G by extension of scalars.
This paper is a sequel to math.RT/0601155. Let G be a complex symplectic group. In math.RT/0601155, we constructed a certain G-variety N = N_1, which we call the (1-) exotic nilpotent cone. In this paper, we study the set of G-orbits of the variety N. It turns out that the variety N gives a variant of the Springer correspondence for the Weyl group of type C, but shares a similar flavor with that of type A case. (I.e. there appears no non-trivial local system and the correspondence is bijective.) As an application, we present one sufficient condition for the bijectivity of our exotic Deligne-Langlands correspondence [K1].
We give a construction of the two-dimensional tame symbol as the commutator of a group-like monoidal groupoid which is obtained from some group of k-linear operators acting in a two-dimensional local field and corresponds to some third cohomology class of this group. We give also the hypothetical method for the proof of the two-dimensional Parshin reciprocity laws. This text was written in 2003 as preprint 03-13 of the Humboldt University of Berlin and was available at this http URL (only evident misprints are corrected now). Later E. Frenkel and X. Zhu obtained in arXiv:0810.1487 [math.RT] more general results concerning the third cohomology classes of groups acting on two-dimensional local fields, and the author and X. Zhu obtained in arXiv:1002.4848 [math.AG] the proof of the Parshin reciprocity laws on an algebraic surface similar to the Tate proof of the residue formula on an algebraic curve.
We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper (math.RT/0006004) we showed that these polynomials are eigenfunctions for commuting difference operators. Our central result now is the "transposition formula", a generalization of Okounkov's binomial theorem (q-alg/9608021) for shifted Jack polynomials. From this formula, we derive an interpolation formula, an evaluation formula, a scalar product, a binomial theorem, and properties of the algebra generated by the multiplication and difference operators.
Abstract An axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras was presented in [R.M. Adin, F. Brenti, Y. Roichman, A unified construction of Coxeter group representations (I), Adv. Appl. Math., in press, arXiv: math.RT/0309364 ]. Combinatorial aspects of this construction are studied in this paper. In particular, the symmetric group case is investigated in detail. The resulting representations are completely classified and include the irreducible ones.
First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to construct adjoint and coadjoint representations as usual. For a finite group G of nilpotency class 2 of odd order, I construct a basis in its group algebra C[G], parameterized by elements of g* so that the elements of coadjoint orbits form bases of simple two-side ideals of C[G]. That construction gives us a one-to-one correspondence between G-orbits in g* and classes of equivalence of irreducible unitary representations of G, implying a very simple character formula. The properties of that correspondence are similar to the properties of the analogous correspondence given by Kirillov's orbit method for nilpotent connected and simply connected Lie groups. The diagram method introduced in my article 'Diagrams of Representations, math.RT/9803079 (1998)' and my thesis 'A Combinatorial Approach to Representations of Lie Groups and Algebras, University of Pennsylvania (1998)' gives us a convenient way to study normal forms on the orbits and corresponding representations.
This text is a continuation to"Notes sur l'indice des alg\`ebres de Lie (I)", math.RT/0605499 ----- Ce texte est une suite \`a :"Notes sur l'indice des alg\`ebres de Lie (I)".