Related papers: Exceptional Siegel-Weil theorems for compact $\mat…

Let G be a simply connected simple algebraic group over an algebraically closed field K of characteristic p>0 with root system R, and let ${\mathfrak g}={\cal L}(G)$ be its restricted Lie algebra. Let V be a finite dimensional ${\mathfrak…

We study the structure of an algebraic supergroup $\mathbb{G}$ and establish the Borel-Weil theorem for $\mathbb{G}$ to give a systematic construction of all simple supermodules over an arbitrary field. Especially when $\mathbb{G}$ has a…

We develop a theory of modular forms on the groups $\mathrm{SO}(3,n+1)$, $n \geq 3$. This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and…

We know that any element A of the group SO(3) can be represented as A = A1 A2 A1', where A1, A1' are elements of SO1(2)={A is an element of SO(3) | Ae1=e1}, and SO2(2)={A is an element of SO(3) | Ae2=e2} . This fact is known as Euler's…

Let $U_\epsilon(\mathfrak g)$ be the simply connected quantized enveloping algebra associated to a finite-dimensional complex simple Lie algebra $\mathfrak g$ at the roots of unity. The De Concini-Kac-Procesi conjecture on the dimension of…

In this paper we consider some global constructions of liftings of automorphic representations attached to some commuting pairs in the exceptional group $F_4$. We consider two families of integrals. The first uses the minimal representation…

In this note we describe a seemingly new approach to the complex representation theory of the wreath product $G\wr S_d$ where $G$ is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We…

Cuspidal automorphic representations $\tau$ of $\mathrm{PGL}_2$ correspond to global long root $A$-parameters for $\mathsf{G}_2$. Using an exceptional theta lift between $\mathrm{PU}_3$ and $\mathsf{G}_2$, we construct the associated global…

Given the maximal compact subalgebra $\mathfrak{k}(A)$ of a split-real Kac-Moody algebra $\mathfrak{g}(A)$ of type $A$, we study certain finite-dimensional representations of $\mathfrak{k}(A)$, that do not lift to the maximal compact…

Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$ and Weyl group $W$. We build up a graded map $(\mathcal H\otimes \bigwedge\mathfrak h\otimes \mathfrak h)^W\to (\bigwedge \mathfrak g\otimes \mathfrak…

For $G$ a connected, reductive group over an algebraically closed field $k$ of large characteristic, we use the canonical Springer isomorphism between the nilpotent variety of $\mathfrak{g}:=\mathrm{Lie}(G)$ and the unipotent variety of $G$…

We consider a special theta lift $\theta(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $\theta(f)$ on $\mathrm{SO}(4,4)$. This lift can be considered an analogue of the Saito-Kurokawa lift, where now the…

We introduce the notion of a generalized spin representation of the maximal compact subalgebra of a symmetrizable Kac-Moody algebra in order to show that, if defined over a formally real field, every such subalgebra has a non-trivial…

Let G be the fundamental group of the complement of a K(G,1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group (as defined in the paper). The subgroup of elements in the complex…

We develop the theory of integrable representations for an arbitrary maximal parabolic subalgebra of an affine Lie algebra. We see that such subalgebras can be thought of as arising in a natural way from a Borel--de Siebenthal pair of…

We review the Kohno-Drinfeld theorem as well as a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection D on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes…

Let $F$ be a non-archimedean local field of characteristic different from $2$ and of residual characteristic $p$. We generalise the theory of the Weil representation over $F$ with complex coefficients to $\ell$-modular representations…

We construct two parametrizations of the non compact exceptional Lie group G=E7(-25), based on a fibration which has the maximal compact subgroup K=(E6 x U(1))/Z_3 as a fiber. It is well known that G plays an important role in the N=2 d=4…

A class of Z_2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form g = g_0 + g_1, with g_0 = so(V) + W_0 and g_1 = W_1,…

We present an asymptotic expansion for quaternionic self-adjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. The result exhibits a striking duality…