This introduction to automorphic forms on adelic groups G(A) emphasises the role of representation theory. The exposition is driven by examples, and collects extends many results scattered throughout literature, in particular Langlands constant term formula for Eisenstein series as well Casselman–Shalika p-adic spherical Whittaker function. book also covers more advanced topics such Hecke algebras L-functions. Many these mathematical have natural interpretations string theory, so some basic...

We give a recursive method for computing all values of basis Whittaker functions unramified principal series invariant under an Iwahori or parahoric subgroup split reductive group $G$ over nonarchimedean local field $F$. Structures in the proof have surprising analogies to features certain solvable lattice models. In case $G=\mathrm{GL}_r$ we show that there exist models whose partition precisely these values. Here `solvable' means family Yang-Baxter equations which imply, among other...

Abstract In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G an adelic split simply-laced group. Let $\pi $ be minimal or next-to-minimal representation . We prove that any $\eta \in \pi is completely determined by its Whittaker with respect to (possibly degenerate) characters the unipotent radical fixed Borel subgroup, analogously Piatetski-Shapiro–Shalika formula for cusp $\operatorname {GL}_n$ also derive explicit formulas expressing form, as well all...

We provide an introduction to the theory of Eisenstein series and automorphic forms on real simple Lie groups G, emphasising role representation theory. It is useful take a slightly wider view define all objects over (rational) adeles A, thereby also paving way for connections number theory, Langlands program. Most results we present are already scattered throughout mathematics literature but our exposition collects them together driven by examples. Many interesting aspects these functions...

Type A Demazure atoms are pieces of Schur functions, or sets tableaux whose weights sum to such functions. Inspired by colored vertex models Borodin and Wheeler, we will construct solvable lattice partition functions atoms; the proof this makes use a Yang-Baxter equation for five-vertex model. As biproduct, on Kashiwara's $\mathcal{B}_\infty$ crystal give new algorithms computing Lascoux-Sch\"utzenberger keys.

The technique of generating new solutions to 4D gravity-matter systems by dimensional reduction a -model is extended supersymmetric configurations supergravity. conditions required for the preservation supersymmetry under isometry transformations in target space are found. Some examples illustrating given.

We study the question of Eulerianity (factorizability) for Fourier coefficients automorphic forms, and we prove a general transfer theorem that allows one to deduce certain from another coefficient. also establish ‘hidden’ invariance property coefficients. apply these results minimal next-to-minimal representations, large class Fourier–Jacobi In particular, parabolic with characters maximal rank Eisenstein series in representations groups ADE-type are interest string theory.

Abstract We construct a family of solvable lattice models whose partition functions include ‐adic Whittaker for general linear groups from two very different sources: Iwahori‐fixed vectors and metaplectic covers. Interpolating between them by Drinfeld twisting, we uncover unexpected relationships Iwahori functions. This leads to new Demazure operator recurrence relations spherical In prior work the authors it was shown that row transfer matrices certain could be represented as ‘half‐vertex...

In this paper we compute new values of Iwahori Whittaker functions on $n$-fold metaplectic covers $\widetilde{G}$ $\mathbf{G}(F)$ with $\mathbf{G}$ a split reductive group over non-archimedean local field $F$. For every function $\phi$, and for $g\in\widetilde{G}$, evaluate $\phi(g)$ by recurrence relations the Weyl using novel "vector Demazure-Whittaker operators." The general formula strategy proof are inspired ideas appearing in theory integrable systems. Specializing to case $\mathbf{G}...

Abstract We consider a special class of unipotent periods for automorphic forms on finite cover reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>K</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , which we refer to as Fourier coefficients associated the data ‘Whittaker pair’. describe quasi-order coefficients, and an...

We show that spherical Whittaker functions on an $n$-fold cover of the general linear group arise naturally from quantum Fock space representation $U_q(\widehat{\mathfrak{sl}}(n))$ introduced by Kashiwara, Miwa and Stern (KMS). arrive at this connection reconsidering solvable lattice models known as `metaplectic ice' whose partition are metaplectic functions. First, we a certain Hecke action coinvariants agrees (up to twisting) with Ginzburg, Reshetikhin, Vasserot. This allows us expand...