We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. As an application, we construct Iwasawa decompositions for classical real or complex Banach-Lie groups associated with the Schatten ideals ${\mathfrak S}_p({\mathcal H})$ on a complex separable Hilbert space ${\mathcal H}$ if $1<p<\infty$.
The Cartan and Iwasawa decompositions of real reductive Lie groups play a fundamental role in the representation theory of the groups and their corresponding symmetric spaces. These decompositions are defined by an involution with a compact fixed-point group, called a Cartan involution. For an arbitrary involution, one can consider similar decompositions. We offer a generalization of the Cartan and Iwasawa decompositions for algebraic groups defined over an arbitrary field $k$ and a general involution.
The purpose of this paper is to prove Duflo's conjecture for $(G,π, AN)$ where $G$ is a simple Lie group of Hermitian type and $π$ is a discrete series of $G$ and $AN$ is the maximal exponential solvable subgroup for an Iwasawa decomposition $G=KAN$. This is essentially reduced from the following general theorem we prove in this paper: let $G$ be a connected semisimple Lie group . Then a strongly elliptic $G$-coadjoint orbit $\mathcal{O}$ is holomorphic if and only if $\text{p}(\mathcal{O})$ is an open $AN$-coadjoint orbit, where $\text{p} : \mathfrak{g}^* \longrightarrow (\mathfrak{a}\oplus\m…
We prove faithfulness of infinite-dimensional generalised Verma modules for Iwasawa algebras corresponding to split simple Lie algebras with a Chevalley basis. We use this to prove faithfulness of all infinite-dimensional highest-weight modules in the case of type $A_{2}$. In this case we also show that all prime ideals of the corresponding Iwasawa algebras are annihilators of finite-dimensional simple modules.
Let $G=GL_{n}(R)$, with the usual Cartan decomposition $G=Kexp(p_0)$ and the usual Iwasawa decomposition $G=NAK$. We determine the image of $exp(p_0)$ under the projection of $G$ to $K$ through the Iwasawa decomposition. As an application, we prove a positivity result about the matrix coefficients of principal series representations of $G$.
Let $H$ be a compact $p$-adic analytic group without torsion element, whose Lie algebra is split semisimple and $\mathfrak{N}_H(G)$ be the full subcategory of the category of finitely generated modules over the Iwasawa algebra $Λ_G$ that are also finitely generated as $Λ_H$-modules, where $G = \mathbb{Z}_{p} \times H$. We show that if the class of a module $N$ in the Grothendieck group of $\mathfrak{N}_H(G)$ equals to the class of a completely faithful module, then $q(N)$ is also completely faithful, where $q(N)$ denotes the image of $N$ via the quotient functor modulo the full subcategory of …
We construct generators of the center of the universal enveloping algebra of the complex orthogonal Lie algebra realized as the alternative matrices of size $n$. These elements are constructed in accordance with the Iwasawa decomposition of the real rank one indefinite orthogonal Lie algebra. We also discuss the Iwasawa decomposition of the Pfaffian.
Given a $p$-adic field $K$ and a nilpotent uniform pro-$p$ group $G$, we prove that all primitive ideals in the $K$-rational Iwasawa algebra $KG$ are maximal, and can be reduced to a particular standard form. Setting $\mathcal{L}$ as the associated $\mathbb{Z}_p$-Lie algebra of $G$, our approach is to study the action of $KG$ on a Dixmier module $\widehat{D(λ)}$ over the affinoid envelope $\widehat{U(\mathcal{L})}_K$, and to prove that all primitive ideals can be reduced to annihilators of modules of this form.
We prove a remarkable generalization of a convexity theorem for semisimple symmetric spaces G/H established earlier in 1986 by the second named author. The latter result generalized Kostant's non-linear convexity theorem for the Iwasawa decomposition of a real semisimple Lie group. The present generalization involves Iwasawa decompositions related to minimal parabolic subgroups of G of arbitrary type instead of the particular type relative to H considered in 1986.
We suggest a method of constructing special nonunitary representations of semisimple Lie groups using representations of Iwasawa subgroups. As a typical example, we study the group $U(2,2)$.
For a prime $p>2$, let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_p$, $G(1)$ be the first congruence kernel of $G$ and $Ω_{G(1)}$ be the mod-$p$ Iwasawa algebra defined over the finite field $\mathbb{F}_p$. Ardakov, Wei, Zhang have shown that if $p$ is a "nice prime " ($p \geq 5$ and $p \nmid n+1$ if the Lie algebra of $G(1)$ is of type $A_n$), then every non-zero normal element in $Ω_{G(1)}$ is a unit. Furthermore, they conjecture in their paper that their nice prime condition is superfluous. The main goal of this article is to provide an entirely new proof…
In this article we establish a version of Koszul duality for filtered rings arising from $p$-adic Lie groups. Our precise setup is the following. We let $G$ be a uniform pro-$p$ group and consider its completed group algebra $Ω=k[\![G]\!]$ with coefficients in a finite field $k$ of characteristic $p$. It is known that $Ω$ carries a natural filtration and $\text{gr} Ω=S(\frak{g})$ where $\frak{g}$ is the (abelian) Lie algebra of $G$ over $k$. One of our main results in this paper is that the Koszul dual $\text{gr} Ω^!=\bigwedge \frak{g}^{\vee}$ can be promoted to an $A_{\infty}$-algebra in such…
We prove the finiteness of the cohomology of torsion-free lattices in a semisimple Lie group of real rank one with coefficients in the distribution vector globalization of Harish-Chandra modules. The cohomology is expressed in terms of automorphic and cusp forms. We also consider the Lie-algebra cohomology of these globalizations for the nilpotent part of the Iwasawa decomposition of the group.
Let $ X = G/K $ be a rank one Riemannian symmetric space of noncompact type. In view of the Iwasawa decomposition $ G = NAK $ of the underlying semisimple Lie group, we can also view $ X $ as the solvable extension $ S = NA $ of the Iwasawa group $ N$ which is known to be a $H$-type group. In this work we study the holomorphic extendability of eigenfunctions of the Laplace-Beltrami operator $ Δ_S$ on $ S $ to certain domains in the complexification of the nilpotent group $ N$. We can also do the same for any $H$-type group $ N $ not necessarily an Iwasawa group. The results are accomplished by…
The lack of a $p$-adic Haar measure causes many methods of traditional representation theory to break down when applied to continuous representations of a compact $p$-adic Lie group $G$ in Banach spaces over a given $p$-adic field $K$. For example, Diarra showed that the abelian group $G=\dZ$ has an enormous wealth of infinite dimensional, topologically irreducible Banach space representations. We therefore address the problem of finding an additional ''finiteness'' condition on such representations that will lead to a reasonable theory. We introduce such a condition that we call ''admissibili…
We discuss Euler characteristics for finitely generated modules over Iwasawa algebras. We show that the Euler characteristic of a module is well-defined whenever the 0th homology group is finite if and only if the relevant compact p-adic Lie group is finite-by-nilpotent and that in this case all pseudo-null modules have trivial Euler characteristic. We also prove some other results relating to the triviality of Euler characteristics for pseudo-null modules. We also prove some analogous results for the Akashi series of Coates et al.
It was proved by Huckleberry that the Akhiezer-Gindikin domain is included in the ``Iwasawa domain'' using complex analysis. But we can see that we need no complex analysis to prove it. In this paper, we generalize the notions of the Akhiezer-Gindikin domain and the Iwasawa domain for two associated symmetric subgroups in real Lie groups and prove the inclusion. Moreover, by the symmetry of two associated symmetric subgroups, we also give a direct proof of the known fact that the Akhiezer-Gindikin domain is included in all cycle spaces.
We obtain minimal dimension matrix representations for each of the Lie algebras of dimension five, six, seven, and eight obtained by Turkowski that have a non-trivial Levi decomposition. The Key technique involves using subspace associated to a particular representation of semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition.
By using of Euler's difference table, we obtain simple explicit formula for the decomposition of $k$-th tensor power of adjoint representation of $A_n$ Lie algebra at $2 k \le{n+1}$.
We discuss Cauchy type decompositions of crystal graphs for general linear Lie superalgebras. More precisely, we consider bicrystal graph structures on various sets of matrices of non-negative integers, and obtain their decompositions with explicit combinatorial isomorphisms.
We determine approximations to the decomposition matrices for unipotent $\ell$-blocks of several series of finite reductive groups of classical and exceptional type over $\FF_q$ of low rank in non-defining good characteristic~$\ell$.
We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0, and $A\in\mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan-Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$) such that $A=S+N$.
There exist principal $\mathfrak{sl}_2$ subalgebras for hyperbolic Kac-Moody Lie algebras. In the case of rank 2 symmetric hyperbolic Kac-Moody Lie algebras, certain $\mathfrak{sl}_2$ subalgebras are constructed. These subalgebras are generalizations of principal $\mathfrak{sl}_2$ subalgebras. We show that the rank 2 symmetric hyperbolic Kac-Moody Lie algebras themselves are irreducibly decomposed under the action of this $\mathfrak{sl}_2$ subalgebras. Furthermore, we classify irreducible components of the decomposition. In particular, we obtain multiplicities of unitary principal series and c…
The author has previously shown that solvable Lie A-algebras and complemented solvable Lie algebras decompose as a vector space direct sum of abelian subalgebras, and their ideals relate nicely to this decomposition. However, neither of these classes is contained in the other. The object of this paper is to find a larger class of algebras, containing each of these classes, in which these same results hold.
