Howard Garland

The arithmetic theory of loop algebras

We let Z, Q, R, and @ denote the rational integers, rational numbers, real numbers, and complex numbers, respectively. Let gc denote a complex simple Lie algebra and fix a Cartan subalgebra ljc C gc . We fix a choice of simple roots ffl ,..*, 01~ (1= dim ljc) in lj”, , the dual space of ljc . We let iyrVl denote the negative of the corresponding highest root OL,, and we define an 1 x I matrix A, and an (2 + 1) x (I + 1) matrix a by

Entirety of cuspidal Eisenstein series on loop groups

In this paper, we prove the entirety of loop group Eisenstein series induced from cusp forms on the underlying finite dimensional group, by demonstrating their absolute convergence on the full complex plane. This is quite in contrast to the finite-dimensional setting, where such series only converge absolutely in a right half plane (and have poles elsewhere coming from

Eisenstein series on loop groups: Maass-Selberg relations 3

L

The Macdonald-Kac formulas as a consequence of the Euler-Poincaré principle

-functions in their constant terms). Our result is the

Lie algebra homology and the Macdonald-Kac formulas

{\Bbb Q}

The arithmetic theory of loop groups

-analog of a theorem of A.~Braverman and D.~Kazhdan from the function field setting, who previously showed the analogous Eisenstein series there are finite sums.

EXISTENCE OF LATTICES IN KAC–MOODY GROUPS OVER FINITE FIELDS

In this paper we prove the Maass-Selberg relations for the inner product of truncated, pseudo Eisenstein series. One can then extract the Maass-Selberg relations for truncated Eisenstein series themselves from those for truncated pseudo-Eisenstein series—a task which we shall complete in a subsequent paper.

Absolute convergence of Eisenstein series on loop groups

Publisher Summary This chapter presents a summary of a previous paper, where the authors computed the homology of certain infinite-dimensional Lie algebras and used these computations, together with the Euler-Poincare principle to obtain the identities of I. G. Macdonald and V. G. Kac, Kac. The chapter overviews certain Lie algebras associated with symmetrizable Cartan matrices and introduces the related notions of the Weyl groups and root systems of these Lie algebras. These Lie algebras and related notions were first defined and studied by Kac and Moody. The chapter introduces certain derivations D i of a Kac-Moody algebra g , and then defines an extended algebra g e , obtained by adjoining such derivations to g . The notion of a quasisimple module is presented where a certain subalgebra u - of g , associated with a certain subset of simple roots is defined. Subsequently, an overview of the combinatorial identities of Kac and Macdonald is presented in the chapter.