Related papers: On the Siegel-Weil Theorem for Loop Groups (I)

We formulate and prove the Siegel-Weil formula for loop groups.

In this paper, we propose a formula relating certain residues of Eisenstein series on symplectic groups. These Eisenstein series are attached to parabolic data coming from Speh representations. The proposed formula bears a strong similarity…

We study the Siegel-Weil formula in the second term range ($n+1\leq m\leq n+r$) for unitary groups of hermitian forms over a skew-field $D$ with involution of the second kind.

We survey a number of Weyl type laws that have recently been established in low-dimensional symplectic geometry. These have had a number of applications, which we also introduce. We sketch a number of proofs so that the reader can get a…

In this paper, we introduce a new kind of Siegel upper half space and consider the symplectic geometry on it explicitly under the action of the group of all holomorphic transformations of it. The results and methods will form a basis for…

We develop the structure theory of symplectic Lie groups based on the study of their isotropic normal subgroups. The article consists of three main parts. In the first part we show that every symplectic Lie group admits a sequence of…

We establish a Siegel-Weil formula for classical groups over a function field with odd characteristic, which asserts in many cases that the Siegel Eisenstein series is equal to an integral of a theta function. This is a function-field…

We give a new proof of Quillen's conjecture for solvable groups via a geometric and explicit method. For p-solvable groups, we provide both a new proof using the Classification of Finite Simple Groups and an asymptotic version without…

This note outlines the Borel-Weil-Bott theory for (arbitrary-genus) flag varieties of loop groups, following ideas of G. Segal. Recent revivalist interest in the "WZW fusion rules" suggests that my original (CMP 1995) treatment of the…

We will give the graded ring of Siegel modular forms of degree two with respect to a non-split symplectic group explicitly.

We prove the higher Siegel--Weil formula for \emph{corank one} terms, relating (1) the $r^{\rm th}$ central derivatives of corank one Fourier coefficients of Siegel--Eisenstein series, and (2) the degrees of special cycles of virtual…

We extend the Weil representation of infinite-dimensional symplectic group to a representation a certain category of linear relations.

We generalize the Uhlenbeck-Segal theory for harmonic maps into compact semi-simple Lie groups to general Lie groups equipped with torsion free bi-invariant connection.

We study equivariant Hermitian K-theory for representations of symplectic groups, especially $\mathrm{SL}_2$. The results are used to establish an Atiyah-Segal completion theorem for Hermitian $K$-theory and symplectic groups.

The representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the question of whether a given representation is symplectic or…

We present a new proof of the transformation law of $\vartheta_1$ under the action of the generator of the full modular group $\Gamma$ using Siegel's method.

In this note a combinatorial formula related to the symmetric group is generalized to an arbitrary finite Weyl group.

We study the structure of a family of algebras which encodes a generalization of the Pieri Rule for the complex orthogonal group. In particular, we show that each of these algebras has a standard monomial basis and has a flat deformation to…

A theory of cyclic elements in semisimple Lie algebras is developed. It is applied to an explicit construction of regular elements in Weyl groups.

We prove a general structure theorem for finitely presented torsion modules over a class of commutative rings that need not be Noetherian. As a first application, we then use this result to study the Weil- \'etale cohomology groups of…