Generalized Bessel series and multiplicity problem in complex semisimple Lie algebra theory

The use of differintegral operators $D^\nu $ with $D = {d / {dz}}$ and $\nu \in \mathbb{C}$ and hyperdifferential operators, together with some operational calculus manipulations leads to new formulae of representation of usual special functions. These formulae are useful for obtaining generating functions and differential recurrence relations of orthogonal polynomials.

Goldie rank in the enveloping algebra of a semisimple Lie algebra, I

Let \(\) be a complex semisimple Lie algebra and let \(\), resp. \(\), be its symmetric, resp. enveloping algebra. It is shown in [1] that finite \(\)-algebras can be realized, up to a central extension, as the algebra of invariants \(\), resp. \(\), for the adjoint action of a Lie subalgebra \(\) of \(\). For \(\) nilpotent, we use the Taylor lemma to give a description (generators and relations) for these algebras.

Invariant eigendistributions on a semisimple Lie algebra and homology classes on the conormal variety. I. An integral formula

We define and classify splints of root systems of complex semisimple Lie algebras. In a few instances, splints play a role in determining branching rules of a module over a complex semisimple Lie algebra when restricted to a subalgebra. In these particular cases, the set of submodules with respect to the subalgebra themselves may be regarded as the character of an auxiliary Lie algebra which may or may not be another Lie subalgebra.

It is well known that the Weyl groups play a key role in the theory of the structure of Lie algebras and their linear representations, as well as in the theory of Chevalley groups. Suppose g is a complex semisimple Lie algebra, h its Cartan subalgebra, Σ and Π denote its root system and the basis of the root system, respectively, and w α i indicates the reflection determined by αi ∈ Π. The so-called Weyl group, denoted by W,refers to the finite group generated by all w α i ( αi ∈ Π). There have been many works published on the structure and properties of W [6, 7]. The following Molien identity related to the global structure of W is well known [3]: $$\underset{w\in W}{\mathop{\Sigma}}\,\frac{1}{\det\left( 1-tw \right)}=\frac{1}{\prod{{{\left( 1-{{t}^{{{d}_{i}}}} \right)}^{,}}}}$$ (8.1) where d i signify the Poincare indices of g. In 1956, by topological methods based on Morse’s theory, Bott proved another identity [2]: $$\underset{w\in W}{\mathop \sum }\,{{t}^{l\left(w \right)}}=\prod{\frac{1-{{t}^{{{d}_{i}}}}}{1-{{t}^{,}}}}$$ (8.2) where l(w) refers to the Π-length of w;i.e.,wmay be expressed by the number of minimal products of w αi ( αi ∈ Π). This important formula may shed new light on our understanding the structure of W. In 1966, Solomon offered a purely algebraic proof of (8.2), thus solving the order problem of the Chevalley simple groups [5, 3].

A theory of special functions should derive as many of the numerous formulas and properties as possible from some guiding principles. One of these principles is the F-equation, which Truesdell has put forward, and which is satisfied by all simple special functions. Many results about addition theorems, generating functions, integral representations etc. follow immediately from the F-equation. One disadvantage, however, is that the F-equation does not constitute a specific property of the simple special functions, generating instead functions of a much more general character.

We associate to an arbitrary positive root [Formula: see text] of a complex semisimple finite-dimensional Lie algebra [Formula: see text] a twisting endofunctor [Formula: see text] of the category of [Formula: see text]-modules. We apply this functor to generalized Verma modules in the category [Formula: see text] and construct a family of [Formula: see text]-Gelfand–Tsetlin modules with finite [Formula: see text]-multiplicities, where [Formula: see text] is a commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] generated by a Cartan subalgebra of [Formula: see text] and by the Casimir element of the [Formula: see text]-subalgebra corresponding to the root [Formula: see text]. This covers classical results of Andersen and Stroppel when [Formula: see text] is a simple root and previous results of the authors in the case when [Formula: see text] is a complex simple Lie algebra and [Formula: see text] is the maximal root of [Formula: see text]. The significance of constructed modules is that they are Gelfand–Tsetlin modules with respect to any commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] containing [Formula: see text]. Using the Beilinson–Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of [Formula: see text]-Gelfand–Tsetlin modules which contains constructed modules as well as the category [Formula: see text].

We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra \mathfrak{g} is a Noetherian and finitely generated ring. If the surface has punctures, we also prove that it has no non-trivial zero divisors (i.e., it is a domain). Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin–Turaev functor for the quantum group U_{q}(\mathfrak{g}) , and which coincides with the Kauffman bracket skein algebra when \mathfrak{g}=\mathfrak{sl}_{2} . We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras.

Sur la méthode des orbites pour les orbites régulières d'une algèbre de Lie semi-simple

Let \(\mathfrak {g}\) be a complex semisimple Lie algebra and \(Y_{\hbar }(\mathfrak {g})\) its Yangian. Drinfeld proved that the universal R-matrix \(\mathcal {R}(s)\) of \(Y_{\hbar }(\mathfrak {g})\) gives rise to rational solutions of the \(\operatorname {QYBE}\) on irreducible, finite-dimensional representations of \(Y_{\hbar }(\mathfrak {g})\). This result was recently extended by Maulik–Okounkov to symmetric Kac–Moody algebras and representations arising from geometry. We show that rationality ceases to hold on arbitrary finite-dimensional representations, if one requires such solutions to be natural and compatible with tensor products. Equivalently, the tensor category of finite-dimensional representations of \(Y_{\hbar }(\mathfrak {g})\) does not admit rational commutativity constraints. We construct instead two meromorphic commutativity constraints, which are related by a unitarity condition. Each possesses an asymptotic expansion in s which has the same formal properties as \(\mathcal {R}(s)\), and therefore coincides with it by uniqueness. In particular, we give a constructive proof of the existence of \(\mathcal {R}(s)\). Our construction relies on the Gauss decomposition \(\mathcal {R}^+(s)\cdot \mathcal {R}^0(s)\cdot \mathcal {R}^-(s)\) of \(\mathcal {R}(s)\). The divergent abelian term \(\mathcal {R}^0\) was resummed on finite-dimensional representations by the first two authors in Gautam and Toledano Laredo (Publ Math Inst Hautes Études Sci 125:267–337, 2017). In the present paper, we construct \(\mathcal {R}^{\pm }(s)\), prove that they are rational on finite-dimensional representations, and that they intertwine the standard coproduct of \(Y_{\hbar }(\mathfrak {g})\) and the deformed Drinfeld coproduct introduced in loc. cit.

Let $$\mathfrak {g}$$ g be a complex semisimple Lie algebra and let $$\textbf{U}_q(\mathfrak g)$$ U q ( g ) denote the associated Drinfel’d Jimbo quantized enveloping algebra. In this paper we study spherical functions of $$\textbf{U}_q(\mathfrak g)$$ U q ( g ) related to characters. We show invariance under the Wang-Zhang braid group operators and show relative Weyl group invariance, when restricted to the quantum torus.

We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The cases of sl(3) and sl(4) are computed explicitly. This allows us to formulate the conjecture that, as a bigraded vector space, the center of a regular block of the small quantum sl(m) at a root of unity is isomorphic to Haiman's diagonal coinvariant algebra for the symmetric group S-m.

Let $\mathsf{u}_q(\mathfrak{g})$ be the small quantum group associated with a complex semisimple Lie algebra $\mathfrak{g}$ and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three different actions of $\mathfrak{g}$ on the center of $\mathsf{u}_q(\mathfrak{g})$ and on the higher derived center of $\mathsf{u}_q(\mathfrak{g})$. Based on the triviality of this action for $\mathfrak{g} = \mathfrak{sl}_2, \mathfrak{sl}_3, \mathfrak{sl}_4$, we conjecture that, in finite type A, central elements of the small quantum group $\mathsf{u}_q(\mathfrak{sl}_n)$ arise as the restriction of central elements in the big quantum group $\mathsf{U}_q(\mathfrak{sl}_n)$. We also study the role of an ideal $\mathsf{z}_{\mathrm{Hig}}$ known as the Higman ideal in the center of $\mathsf{u}_q(\mathfrak{g})$. We show that it coincides with the intersection of the Harish-Chandra center and its Fourier transform, and compute the dimension of $\mathsf{z}_{\mathrm{Hig}}$ in type A. As an illustration we provide a detailed explicit description of the derived center of $\mathsf{u}_q(\mathfrak{sl}_2)$ and its various symmetries.

Déformation par quantification et rigidité des algèbres enveloppantes