Mathematics

For a general linear supergroup G=GL(m|n) , we consider a natural isomorphism ϕ:G→ U − × G ev × U + , where G ev is the even subsupergroup of G , and U − , U + are appropriate odd unipotent subsupergroups of G . We compute the action of odd superderivations on the images ϕ ∗ ( x ij ) of the generators of K[G] . We describe a specific ordering of the dominant weights X(T ) + of GL(m|n) for which there exists a Donkin-Koppinen filtration of the coordinate algebra K[G] . Let Γ be a finitely generated ideal Γ of X(T ) + and O Γ (K[G]) be the largest Γ -subsupermodule of K[G] having simple composition factors of highest weights λ∈Γ . We apply combinatorial techniques, using generalized bideterminants, to determine a basis of G -superbimodules appearing in Donkin-Koppinen filtration of O Γ (K[G]) .

We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type. It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see [Spä12,2.12]) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro--Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition.

We introduce a cup-cap duality in the Koszul calculus of N-homogeneous algebras. As an application, we prove that the graded symmetry of the Koszul cap product is a consequence of the graded commutativity of the Koszul cup product. We propose a conceptual approach that may lead to a proof of the graded commutativity, based on derived categories in the framework of DG algebras and DG bimodules. Various enriched structures are developed in a weaker situation corresponding to N>2.

We construct a class of non-weight modules over the twisted N=2 superconformal algebra $\T$. Let $\mathfrak{h}=\C L_0\oplus\C G_0$ be the Cartan subalgebra of $\T$, and let $\mathfrak{t}=\C L_0$ be the Cartan subalgebra of even part $\T_{\bar 0}$. These modules over $\T$ when restricted to the h are free of rank 1 or when restricted to the t are free of rank 2 . We provide the sufficient and necessary conditions for those modules being simple, as well as giving the sufficient and necessary conditions for two $\T$-modules being isomorphic. We also compute the action of an automorphism on them. Moreover, based on the weighting functor introduced in \cite{N2}, a class of intermediate series modules A ? are obtained. As a byproduct, we give a sufficient condition for two $\T$-modules are not isomorphic.

In this article, we generalize Duflo's conjecture to understand the branching laws of non-discrete series. We give a unified description on the geometric side about the restriction of an irreducible unitary representation ? of GL n (k) , k=R or C , to the mirabolic subgroup, where ? is attached to a certain kind of coadjoint orbit.

We introduce a class of equivalences, which we call generalized semi-infinite Hecke equivalences, between certain categories of representations of graded associative algebras which appear in the setting of semi-infinite cohomology for associative algebras and categories of representations of related algebras of Hecke type which we call semi-infinite Hecke algebras. As an application we obtain an equivalence between a category of representations of a non-twisted affine Lie algebra g ? of level ?? h ???�k , where h ??is the dual Coxeter number of the underlying semisimple Lie algebra g and k?�C , and the category of finitely generated representations of the W-algebra associated to g ? of level k . When k=??h ??this yields an equivalence between a category of representations of g ? of central charge ??h ??and the category Coh( Op L G ( D ? )) of coherent sheaves on the space Op L G ( D ? ) of L G -opers on the punctured disc D ? , where L G is the Langlands dual group to the algebraic group of adjoint type with Lie algebra g . This can be regarded as a version of the local geometric Langlands correspondence. The above mentioned equivalences generalize to the case of affine Lie algebras the Skryabin equivalence between the categories of generalized Gelfand-Graev representations of g and the categories of representations of the corresponding finitely generated W-algebras, and Kostant's results on the classification of Whittaker modules over g .

We consider branching laws for the restriction of some irreducible unitary representations Π of G=O(p,q) to its subgroup H=O(p−1,q) . In Kobayashi (arXiv:1907.07994), the irreducible subrepresentations of O(p−1,q) in the restriction of the unitary Π | O(p−1,q) are determined. By considering the restriction of packets of irreducible representations we obtain another very simple branching law, which was conjectured in Orsted-Speh (arXiv:1907.07544).

We give a short proof based on Lusztig's generalized Springer correspondence of some results of [BrCh,BaCr,P].

We formulate a Satake isomorphism for the integral spherical Hecke algebra of an unramified p -adic group G and generalize the formulation to give a description of the Hecke algebra H G (V) of weight V , where V is a lattice in an irreducible algebraic representation of G .

Let Λ be an algebra with a indecomposable projective-injective module. Adachi gave a method to construct the Hasse quiver of support τ -tilting Λ -modules. In this paper, we will show that it can be restricted to τ -tilting modules.