Non-compact groups of inner type and factorization

Compact symmetric spaces, triangular factorization, and Cayley coordinates

Let U/K represent a connected, compact symmetric space, where theta is an involution of U that fixes K, phi: U/K to U is the geodesic Cartan embedding, and G is the complexification of U. We investigate the intersection of phi(U/K) with the Bruhat decomposition of G corresponding to a theta-stable triangular, or LDU, factorization of the Lie algebra of G. When g in phi(U/K) is generic, the corresponding factorization g=ld(g)u is unique, where l in N^-, d(g) in H, and u in N^+. In this paper we present an explicit formula for d in Cayley coordinates, compute it in several types of symmetric spa…

Unitary representations of totally disconnected locally compact groups satisfying Ol'shanskii's factorization

Inspired by Ol'shanskii's work, we provide an axiomatic framework to describe certain irreducible unitary representations of non-discrete unimodular totally disconnected locally compact groups. We then look at the applications to certain groups of automorphisms of locally finite trees and semi-regular right-angled buildings.

超光滑向量的强分解与紧李群表示相关

Strong factorization of ultradifferentiable vectors associated with compact Lie group representations

本文针对紧李群表示的超光滑向量，证明了强分解定理并解决了Gimperlein等人的猜想。

Loops in SU(2), Riemann Surfaces, and Factorization, I

In previous work we showed that a loop $g\colon S^1 \to {\rm SU}(2)$ has a triangular factorization if and only if the loop $g$ has a root subgroup factorization. In this paper we present generalizations in which the unit disk and its double, the sphere, are replaced by a based compact Riemann surface with boundary, and its double. One ingredient is the theory of generalized Fourier-Laurent expansions developed by Krichever and Novikov. We show that a ${\rm SU}(2)$ valued multiloop having an analogue of a root subgroup factorization satisfies the condition that the multiloop, viewed as a trans…

Loops in noncompact groups and factorization

In [11] we showed that a loop in a simply connected compact Lie group $\dot{U}$ has a unique Birkhoff (or triangular) factorization if and only if the loop has a unique root subgroup factorization (relative to a choice of a reduced sequence of simple reflections in the affine Weyl group). In this paper our main purpose is to investigate Birkhoff and root subgroup factorization for loops in a noncompact type semisimple Lie group $\dot G_0$ of inner type. In [4] we showed that for an element of $\dot G_0$, i.e. a constant loop, there is a unique Birkhoff factorization if and only if there is a r…

Character Formulas from Matrix Factorizations

In this paper I present a new and unified method of proving character formulas for discrete series representations of connected Lie groups by applying a Chern character-type construction to the matrix factorizations of [FT] and [FHT3]. In the case of a compact group I recover the Kirillov formula, thereby exhibiting the work of [FT] as a categorification of the Kirillov correspondence. In the case of a real semisimple group I recover the Rossman character formula with only a minimal amount of analysis. The appeal of this method is that it relies almost entirely on highest-weight theory, which …