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Developing an idea of Kapranov and Voevodsky, we introduce a model of weak omega-categories based on directed complexes, combinatorial presentations of pasting diagrams. We propose this as a convenient framework for higher-dimensional…

This is Part IV of a thematic series currently consisting of a monograph and four essays. This essay examines the form of induced representations of locally p-adic Lie groups G which is appropriate for the abelian category of ${\mathcal…

We prove the following result in relative representation theory of a reductive p-adic group $G$: Let $U$ be the unipotent radical of a minimal parabolic subgroup of $G$, and let $\psi$ be an arbitrary smooth character of $U$. Let $S \subset…

In this paper, we study the class of parabolically geometrically finite (PGF) subgroups of mapping class groups, introduced by Dowdall-Durham-Leininger-Sisto. We prove a combination theorem for graphs of PGF groups (and other…

We study the representation theory of the symmetric group $S_n$ in positive characteristic $p$. Using features of the LLT-algorithm we give a conjectural description of the projective cover $P(\lambda)$ of the simple module $D(\lambda)$…

A crucial role in representation theory of loop groups of reductive Lie groups and their Lie algebras is played by their non-trivial second cohomology classes which give rise to their central extensions (the affine Kac-Moody groups and Lie…

By restricting to a special class of smooth functions, the local action of the symmetry group is globalized. This special class of functions is constructed using parabolic induction.

A W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. This paper concentrates on the study of 1-dimensional representations of these algebras. Under some conditions on a nilpotent element…

We construct the polynomial induction functor, which is the right adjoint to the restriction functor from the category of polynomial representations of a general linear group to the category of representations of its Weyl group. This…

We show that the reduction mod p of an orthogonal linear representation is orthogonal, and we generalize this fact to representations of algebras with involution.The proofs make an essential use of the notion of " middle lattices ".

For $SU(2)$ (or $SO(3)$) Donaldson theory on a 4-manifold $X$, we construct a simple geometric representative for $\mu$ of a point. Let $p$ be a generic point in $X$. Then the set $\{ [A] | F_A^-(p) $ is reducible $\}$, with coefficient…

Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect…

This is the second paper in a series where we study arithmetic applications of the multiple elliptic Gamma functions originated in mathematical physics. In the first article in this series we defined geometric families of these functions…

A fundamental problem from invariant theory is to describe the endomorphism algebra of multilinear functions on a representation V invariant under the action of a group G. According to Weyl's classic, a first main (later: fundamental)…

Let $p$ be an odd prime number, and $F$ a nonarchimedean local field of residual characteristic $p$. We classify the smooth, irreducible, admissible genuine mod-$p$ representations of the twofold metaplectic cover…

Let G be a quasi-split reductive group over a non-archimedean local field. We establish a local Langlands correspondence for all irreducible smooth complex G-representations in the principal series. The parametrization map is injective, and…

We construct a canonical irreducible representation for the orthofermion algebra of arbitrary order, and show that every representation decomposes into irreducible representations that are isomorphic to either the canonical representation…

We classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities. For the natural…

Many relevant applications of group theoretical methods to physical problems are related, in some manner, to classification schemes by means of symmetry groups. In these schemes, irreducible representations of a Lie group have to be…

We study the theta lifting for real unitary groups and completely determine the theta lifts of discrete series representations. In particular, we show that these theta lifts can be expressed as cohomologically induced representations in the…