Motivated by recent developments on visible actions on complex manifolds, we raise a question whether or not the multiplication of three subgroups $L$, $G'$ and $H$ surjects a Lie group $G$ in the setting that $G/H$ carries a complex structure and contains $G'/G' \cap H$ as a totally real submanifold.
Particularly important cases are when $G/L$ and $G/H$ are generalized flag varieties, and we classify pairs of Levi subgroups $(L, H)$ such that $L G' H = G$, or equivalently, the real generalized flag variety $G'/H \cap G'$ meets every $L$-orbit on the complex generalized flag variety $G/H$ in the setting that $(G, G') = (U(n), O(n))$.
For such pairs $(L, H)$, we introduce a \textit{herringbone stitch} method to find a generalized Cartan decomposition for the double coset space $L \backslash G/H$, for which there has been no general theory in the non-symmetric case.
Our geometric results provides a unified proof of various multiplicity-free theorems in representation theory of general linear groups.
Let $G$ be a real-reductive Lie group and let $G_1$ and $G_2$ be two subgroups given by involutions. We show how the technique of gradient maps can be used in order to obtain a new proof of Matsuki's parametrization of the closed double cosets $G_1\backslash G/G_2$ by Cartan subsets. We also describe the elements sitting in non-closed double cosets.
We study a class of double coset spaces R_A \backslash G_1 \times G_2 /R_C, where G_1 and G_2 are connected reductive algebraic groups, and R_A and R_C are certain spherical subgroups of G_1 \times G_2 obtained by ``identifying'' Levi factors of parabolic subgroups in G_1 and G_2. Such double cosets naturally appear in the symplectic leaf decompositions of Poisson homogeneous spaces of complex reductive groups with the Belavin-Drinfeld Poisson structures. They also appear in orbit decompositions of the De Concini-Procesi compactifications of semi-simple groups of adjoint type. We find explicit…
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.
We make a fine study of the SO(2,C)-double coset decomposition in SL(2,C) and give a full description of the intersection of the various cells with the complex crown Xi of SL(2,R)/SO(2). A non-linear convexity theorem is proved and applications to analytically continued spherical functions are given.
A Cartan decomposition for symmetric pairs plays an important role to study not only orbit geometry of the symmetric spaces but also harmonic analysis on them. For non-symmetric reductive pairs, there are examples of generalizations of Cartan decompositions for some spherical complex homogeneous spaces such as complex line bundles over the complexified Hermitian symmetric spaces and triple spaces. This paper provides new examples of a Cartan decomposition for non-symmetric reductive pairs, namely, reductive non-symmetric spherical pairs of rank-one type. We also show that the action of some co…
In this note we formulate and prove a version of Cartan decomposition for holomorphic loop groups, similar to Cartan decomposition for $p$-adic loop groups, discussed proved by Garland (and later by the authors by geometric mathods). The main technical tool that we use is the (well-known) interpretation of twisted conjugacy classes in the holomorphic loop group in terms of principal holomorphic bundles on an elliptic curve.
Let A be a local commutative principal ideal ring. We study the double coset space of GL(n,A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A^n. If k is the length of the ring, we determine for which of the pairs (n,k) the double coset space depend on the ring in question. For n=3, we give a complete parametrisation of the double coset space and provide estimates on the rate of growth of the number of double cosets.
Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Let $J=J(F[N])$ denote the Jacboson radical of $F[N]$ and $I={\rm Ann}(J)=\{α\in F[G]|Jα=0\}$. We have another algebra $F[G]/I$. We study the decomposition of Cartan matrix of $F[G]$ according to $F[G/N]$ and $F[G]/I$. This decomposition establishs some connections between Cartan invariants and chief composition factors of $G$. We find that existing zero-defect $p$-block in $N$ depends on the properties of $I$ in $F[G]$ or Cartan invariants. When we consider the Cartan invariants for a block algebra $B$ of $G$, the decomposition is…
Let $G$ be a non-compact group, $K$ the compact subgroup fixed by a Cartan involution and assume $G/K$ is an exceptional, symmetric space, one of Cartan type $E,F $ or $G$. We find the minimal integer, $L(G),$ such that any convolution product of $L(G)$ continuous, $K$-bi-invariant measures on $G$ is absolutely continuous with respect to Haar measure. Further, any product of $L(G)$ double cosets has non-empty interior. The number $L(G)$ is either $2$ or $3$% , depending on the Cartan type, and in most cases is strictly less than the rank of $G$.
For G = SL(3,R) and G = SO(2,n), we give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan decomposition G = KAK of G, and then carry out an approximate calculation of the intersection of KHK with A, for each closed, connected subgroup H of G.
We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna's slice theorem as well as a version of the Hilbert-Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.
We give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G = SU(2,n) has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan decomposition G = K A K of G, and then carry out an approximate calculation of the intersection of KHK with A, for each closed, connected subgroup H of G. This generalizes the work of Hee Oh and Dave Witte for G = SO(2,n).
Let F be a non Archimedean locally compact field of residue characteristic different from 2, let G be a connected reductive group defined over F, let s be an involutive F-automorphism of G and H an open F-subgroup of the fixed points group of s. We denote by G(F) (resp. H(F)) the group of F-points of G (resp. H). In this paper, we obtain an analogue of the Cartan decomposition for the reductive symmetric space G(F)/H(F). More precisely, we obtain a decomposition of G(F) as a union of H(F)-cosets which is related to the H(F)-conjugacy classes of maximal s-anti-invariant F-split tori in G. When …
The Iwasawa decomposition $\mathfrak{g}=\mathfrak{k}_0 \oplus \hat{\mathfrak{a}}_0 \oplus \mathfrak{n}_0$ of the real semisimple Lie algebra $\mathfrak{g}_0$ comes from its Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{p}_0$. Then we get $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{b}_0$ where $\mathfrak{b}_0=\hat{\mathfrak{a}}_0 \oplus \mathfrak{n}_0$. The question of knowing if the index were additive in the decomposition $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{b}_0$ goes back M. Raïs \cite{Rais}. In \cite{Moreau3}, I wrote that the index always is additive …
Let $G$ be a finite group. Let $H, K$ be subgroups of $G$ and $H \backslash G / K$ the double coset space. Let $Q$ be a probability on $G$ which is constant on conjugacy classes ($Q(s^{-1} t s) = Q(t)$). The random walk driven by $Q$ on $G$ projects to a Markov chain on $H \backslash G /K$. This allows analysis of the lumped chain using the representation theory of $G$. Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on $GL_n(q)$ onto a Markov chain on $S_n$ via the Bruhat decompositio…
Let H be a subgroup of a finite group G. We use Markov chains to quantify how large r should be so that the decomposition of the r tensor power of the representation of G on cosets on H behaves (after renormalization) like the regular representation of G. For the case where G is a symmetric group and H a parabolic subgroup, we find that this question is precisely equivalent to the question of how large r should be so that r iterations of a shuffling method randomize the Robinson-Schensted-Knuth shape of a permutation. This equivalence is rather remarkable, if only because the representation th…
Külshammer, Olsson and Robinson conjectured that a certain set of numbers determined the invariant factors of the $\ell$-Cartan matrix for $S_n$ (equivalently, the invariant factors of the Cartan matrix for the Iwahori-Hecke algebra $\mathcal{H}_n(q)$, where $q$ is a primitive $\ell$th root of unity). We call these invariant factors Cartan invariants. In a previous paper, the second author calculated these Cartan invariants when $\ell=p^r$, $p$ prime, and $r\leq p$ and went on to conjecture that the formulae should hold for all $r$. Another result was obtained, which is surprising and counteri…
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 $G$.
Cartan motion groups: regularity of K-finite matrix coefficients
Universite Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208（里昂第一大学、法国国家科学研究中心、里昂中央理工学院、里昂国立应用科学学院、让·莫内大学、ICJ联合研究实验室）
