Let \( G \) denote an affine Kac--Moody group over the local field \( \mathbb{F}_q((s))\). We establish a local Birkhoff decomposition for a subset of \( G \) in terms of a pair of subgroups roughly of the form \( G(\mathbb{F}_q[[t]])\) and \( G(\mathbb{F}_q[t^{-1}]) \). Our techniques are global-to-local and use the reduction theory for loop groups due to H. Garland. Building on these ideas, we establish the finiteness of a set whose cardinality is related to the spherical \( R \)-polynomials in D. Muthiah's conjectural double-affine Kazhdan--Lusztig theory.

We describe the structure of the Whittaker or Gelfand-Graev module on a \( n \)-fold metaplectic cover of a \(p\)-adic group \(G\) at both the Iwahori and spherical level. We express our answer in terms of the representation theory of a quantum group at a root of unity attached to the Langlands dual group of \( G \). To do so, we introduce an algebro-combinatorial model for these modules and develop for it a Kazhdan-Lusztig theory involving new generic parameters. These parameters can either be specialized to Gauss sums to recover the \(p\)-adic theory or to the natural grading parameter in the representation theory of quantum groups. As an application of our results, we deduce geometric Casselman-Shalika type results for metaplectic covers, conjectured in a slightly different context by S. Lysenko, as well as prove a variant of G. Savin's local Shimura type correspondences at the Whittaker level.

In this paper, we associate a family of infinite-rank pro-Hermitian Euclidean lattices to elements of a formal loop group and a highest weight representation of the underlying affine Kac--Moody algebra. In the case that the element has a polynomial representative, we can prove our lattices are theta-finite in the sense of Bost, allowing us to attach to each of our lattices a well-defined theta-like function

Starting from some linear algebraic data (a Weyl-group invariant bilinear form) and some arithmetic data (a bilinear Steinberg symbol), we construct a central extension of a Kac-Moody group generalizing the work of Matsumoto. Specializing our construction over non-archimedean local fields, for each positive integer \(n\) we obtain the notion of \(n\)-fold metaplectic covers of Kac-Moody groups. In this setting, we prove a Casselman-Shalika type formula for Whittaker functions.

We relate Iwahori-Whittaker functions on metaplectic covers to certain Demazure-Lusztig operators, the latter of which are built from a Weyl group action previously considered by G. Chinta and P. Gunnells. Using a certain combinatorial identity for the sum of these Demazure-Lusztig operators, we obtain an analogue of the Casselman-Shalika formula for spherical Whittaker functions in this context.

We introduce unramified Whittaker functions and prove an analogue of the Casselman-Shalika formula for \(p\)-adic loop groups. Our formula differs from the naive generalization from the finite-dimensional case by a factor depending purely on the imaginary roots.

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 L-functions in their constant terms). Our result is the \( \mathbb{Q} \)-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.

In this paper we give an elementary proof of certain finiteness results about affine Kac-Moody groups over a local non-archimedian field \( \mathcal{K} \). Our results imply those proven earlier using either algebraic geometry or a Kac-Moody version of the Bruhat-Tits building. The above finiteness results allow one to formulate an affine version of the Gindikin-Karpelevich formula, which coincides with the one obtained earlier by Braverman-Finkelberg-Kazhdan when \( \mathcal{K} \) has positive characteristic. We deduce this formula from an affine version of the Macdonald formula for the spherical function, which will be proved in a subsequent publication.

In this paper we develop the theory of the Iwahori-Hecke algebra associated to an affine Kac-Moody group over a local non-archimedian field. The resulting algebra is shown to be closely related to Cherednik's double affine Hecke algebra. Furthermore, using these results, we give an explicit description of the affine Satake isomorphism, generalizing Macdonald's formula for the spherical function in the finite-dimensional case.

We give a geometric construction of Garland's Eisenstein series on loop groups defined over a function field of finite characteristic. This identifies the loop Eisenstein series with certain geometric generating functions considered by Kapranov involving moduli spaces of bundles (with additional data) on an algebraic surface. We also consider a related question of giving a categorical interpretation of a local Deligne-Riemann-Roch theorem.

Following Borcherds, we show a certain class of vertex algebras can be uniquely constructed from a bialgebra together with a twisted mul- tiplication by a bicharacter. We illustrate this construction in the case of Heisenberg and lattice vertex algebras. As a consequence, we see that these vertex algebras can be recovered from their \(2\)-point correlation functions and their underlying bialgebra structure.

We show that the isometry dimension of a finite group \( G \) is equal to the dimension of a minimal-dimensional faithful real representation of \( G \). Using this result, we answer several questions of Albertson and Boutin.