Quotients, automorphisms and differential operators

Let $V$ be a $G$-module where $G$ is a complex reductive group. Let Z:=\quot VG denote the categorical quotient and let $\pi\colon V\to Z$ be the morphism dual to the inclusion \O(V)^G\subset\O(V). Let $\phi\colon Z\to Z$ be an algebraic automorphism. Then one can ask if there is an algebraic map $\Phi\colon V\to V$ which lifts $\phi$, i.e., $\pi(\Phi(v))=\phi(\pi(v))$ for all $v\in V$. In \cite{Kuttler} the case is treated where V=r\lieg is a multiple of the adjoint representation of $G$. It is shown that, for $r$ sufficiently large (often $r\geq 2$ will do), any $\phi$ has a lift. We consider the case of general representations (satisfying some mild assumptions). It turns out that it is natural to consider holomorphic lifting of holomorphic automorphisms of $Z$, and we show that if a holomorphic $\phi$ and its inverse lift holomorphically, then $\phi$ has a lift $\Phi$ which is an automorphism such that $\Phi(gv)=\sigma(g)\Phi(v)$, $v\in V$, $g\in G$ where $\sigma$ is an automorphism of $G$. We reduce the lifting problem to the group of automorphisms of $Z$ which preserve the natural grading of \O(Z)\simeq\O(V)^G. Lifting does not always hold, but we show that it always does for representations of tori in which case algebraic automorphisms lift to algebraic automorphisms. We extend Kuttler's methods to show lifting in case $V$ contains a copy of \lieg.Comment: 23 pages, minor revisions. To appear in J. London Math. Societ

Markup meets middleware

We describe a distributed system architecture that supports the integration of different front-office trading systems with middle and back-office systems, each of which have been procured from different vendors. The architecture uses a judicious combination of object-oriented middleware and markup languages. In this combination an object request broker implements reliable trade data transport. Markup languages, particularly XML, are used to address data integration problems. We show that the strengths of middleware and markup languages are complementary and discuss the benefits of deploying middleware and markup languages in a synergistic manner

Growth Pattern of Untreated Boys with Simple Virilizing Congenital Adrenal Hyperplasia Indicates Relative Androgen Insensitivity during the First Six Months of Life

Context: Mild forms of simple virilizing congenital adrenal hyperplasia (CAH) may be missed in newborn screening. In the pre-newborn-screening era, missed diagnosis of simple virilizing CAH was not infrequent in boys. Elevated adrenal androgens lead to accelerated growth and bone maturation. Traditional treatment of CA H consists of the suppression of ACTH through glucocorticoid replacement, in an attempt to reduce excessive androgen production. Objective: To retrospectively analyze early growth pattern and bone maturation in untreated boys with simple virilizing CAH. Patients: In the pre-newborn screening era, 13 boys had a late diagnosis of simple virilizing classical CAH. Diagnosis of 21-hydroxylase deficiency was confirmed by mutation analysis of the CYP21A2 gene in all patients. Growth data were retrospectively collected from standarized preventive medical checkups at the regular pediatrician until the time of diagnosis of CAH. Results: Length was 0.1 +/- 0.8 SDS (mean +/- SD) at birth, 0.2 +/- 1 SDS at 3 months, 0.2 +/- 0.9 SDS at 6 months, 0.7 +/- 1 SDS at 1 year, +1.1 +/- 0.9 SDS at 2 years and +1.8 +/- 1.2 SDS at 4 years. At diagnosis, mean chronological age was 4.4 +/- 1.6 years and height SDS was 2 +/- 1.7. Bone age was accelerated (9.4 +/- 4 years) at diagnosis. Signs that had led to diagnosis were pubic hair (n = 11), accelerated growth rate (n = 6) and birth of an affected sister (n = 3). Despite late start of hydrocortisone treatment, mean final height was 1 +/- 0.9 SDS. Seven of 18 patients had a final height within 1 SD of target height. Conclusion: Height velocity is not markedly increased in untreated boys with simple virilizing CAH in the first 6 months of life, indicating that infants are relatively androgen insensitive during that period. After the first 6 months of life, growth velocity increases significantly and elevated androgens lead to advanced skeletal maturation. This observation has implications for lower hydrocortisone doses to be used in CAH children during the first 6 months of life. In addition, staying alert for clinical symptoms and signs of simple virilizing CAH is still warranted, since mild forms may be missed in newborn screening. Copyright (C) 20105. Karger AG, Base