Mathematics

Let L be the space of spinors on the 3-sphere that are the restrictions of the Laurent polynomial type harmonic spinors on C^2. L becomes an associative algebra. For a simple Lie algebra g, the real Lie algebra Lg generated by the tensor product of L and g is called the g-current algebra. The real part K of L becomes a commutative subalgebra of L. For a Cartan subalgebra h of g, h tensored by K becomes a Cartan subalgebra Kh of Lg. The set of non-zero weights of the adjoint representation of Kh corresponds bijectively to the root space of g. Let g=h+e+ f be the standard triangular decomposition of g, and let Lh, Le and Lf respectively be the Lie subalgebras of Lg generated by the tensor products of L with h, e and f respectively . Then we have the triangular decomposition: Lg=Lh+Le+Lf, that is also associated with the weight space decomposition of Lg. With the aid of the basic vector fields on the 3-shpere that arise from the infinitesimal representation of SO(3) we introduce a triple of 2-cocycles {c_k; k=0,1,2} on Lg. Then we have the central extension: Lg+ \sum Ca_k associated to the 2-cocycles {c_k; k=0,1,2}. Adjoining a derivation coming from the radial vector field on S^3 we obtain the second central extension g^=Lg+ \sum Ca_k + Cn. The root space decomposition of g^ as welll as the Chevalley generators of g^ will be given.

Let π 1 ,…, π k be smooth irreducible representations of p -adic general linear groups. We prove that the parabolic induction product π 1 ×⋯× π k has a unique irreducible quotient whose Langlands parameter is the sum of the parameters of all factors (cyclicity property), assuming that the same property holds for each of the products π i × π j ( i<j ), and that for all but at most two representations π i × π i remains irreducible (square-irreducibility property). Our technique applies the recently devised Kashiwara-Kim notion of a normal sequence of modules for quiver Hecke algebras. Thus, a general cyclicity problem is reduced to the recent Lapid-Mínguez conjectures on the maximal parabolic case.

In this paper we construct a new basis for the cyclotomic completion of the center of the quantum gl N in terms of the interpolation Macdonald polynomials. Then we use a result of Okounkov to provide a dual basis with respect to the quantum Killing form (or Hopf pairing). The main applications are: 1) cyclotomic expansions for the gl N Reshetikhin--Turaev link invariants and the universal gl N knot invariant; 2) an explicit construction of the unified gl N invariants for integral homology 3-spheres using universal Kirby colors. These results generalize those of Habiro for sl 2 . In addition, we give a simple proof of the fact that the universal gl N invariant of any evenly framed link and the universal sl N invariant of any 0 -framed algebraically split link are ? -invariant, where ?=Y/2Y with the root lattice Y .

Let G be a finite group and k an algebraically closed field of characteristic p>0 . We define the notion of a Dade kG -module as a generalization of endo-permutation modules for p -groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade kG -modules forms a group under tensor product, and the group obtained this way is isomorphic to the Dade group D(G) defined by Lassueur. We also consider the subgroup D Ω (G) of D(G) generated by relative syzygies Ω X , where X is a finite G -set. If C(G,p) denotes the group of superclass functions defined on the p -subgroups of G , there are natural generators ω X of C(G,p) , and we prove the existence of a well-defined group homomorphism Ψ G :C(G,p)→ D Ω (G) that sends ω X to Ω X . The main theorem of the paper is the verification that the subgroup of C(G,p) consisting of the dimension functions of k -orientable real representations of G lies in the kernel of Ψ G .

Bezrukavnikov-Finkelberg-Mirković [Compos. Math. {\bf 141} (2005)] identified the equivariant K -group of an affine Grassmannian, that we refer as (the coordinate ring of) a BFM space á là Teleman [Proc. ICM Seoul (2014)], with a version of Toda lattice. We give a new system of generators and relations of a certain localization of this space, that can be seen as a version of its Darboux coordinate. This establishes a conjecture in Finkelberg-Tymbaliuk [Progress in Math. {\bf 300} (2019)] that relates the BFM space of a connected reductive algebraic group with those of Levi subgroups.

Previously, the last two authors found large families of decomposable Specht modules labelled by bihooks, over the Iwahori--Hecke algebra of type B . In most cases we conjectured that these were the only decomposable Specht modules labelled by bihooks, proving it in some instances. Inspired by a recent semisimplicity result of Bowman, Bessenrodt and the third author, we look back at our decomposable Specht modules and show that they are often either semisimple, or very close to being so. We obtain their exact structure and composition factors in these cases. In the process, we determine the graded decomposition numbers for almost all of the decomposable Specht modules indexed by bihooks.

We compute the decomposition numbers of the unipotent characters lying in the principal ??-block of a finite group of Lie type B 2n (q) or C 2n (q) when q is an odd prime power and ??is an odd prime number such that the order of q mod ??is 2n . Along the way, we extend to these finite groups the results of \cite{DVV19} on the branching graph for Harish-Chandra induction and restriction.

We prove that pointwise finite-dimensional S^1 persistence modules over an arbitrary field decompose uniquely, up to isomorphism, into the direct sum of a bar code and finitely-many Jordan cells. These persistence modules have also been called angle-valued or circular persistence modules. We allow either a cyclic order or partial order on S^1 and do not have additional finiteness requirements on the modules. We also show that a pointwise finite-dimensional S^1 persistence module is indecomposable if and only if it is a bar or Jordan cell (a string or a band module, respectively, in representation theory). Along the way we classify the isomorphism classes of such indecomposable modules.

Let V be a finite-dimensional vector space over a field of characteristic two. As the main result of this paper, for every nilpotent element e?�sl(V) , we describe the Jordan normal form of e on the sl(V) -modules ??2 (V) and S 2 (V) . In the case where e is a regular nilpotent element, we are able to give a closed formula. We also consider the closely related problem of describing, for every unipotent element u?�SL(V) , the Jordan normal form of u on ??2 (V) and S 2 (V) . A recursive formula for the Jordan block sizes of u on ??2 (V) was given by Gow and Laffey (J. Group Theory 9 (2006), 659-672). We show that their proof can be adapted to give a similar formula for the Jordan block sizes of u on S 2 (V) .

In this paper we obtain a decomposition formula of the uniform projection of the Weil character of a finite reductive dual pair consisting of a symplectic group and an even orthogonal group. This is the first and major step to give an explicit description of the Howe correspondence on unipotent characters confirming a conjecture by Aubert-Michel-Rouquier.