Conductors and newforms for U(1,1)
Let F be a non-Archimedean local field whose residue characteristic is odd. In this paper we develop a theory of newforms forU (1, 1)(F), building on previous work onSL2(F). This theory is analogous to the results of Casselman forGL2(F) and Jacquet, Piatetski-Shapiro, and Shalika forGLn(F). To a representation π ofU(1, 1)(F), we attach an integer c(π) called the conductor of π, which depends only on theL-packet π containing π. A newform is a vector in π which is essentially fixed by a congruence subgroup of level c(π). We show that our newforms are always test vectors for some standard Whittaker functionals, and, in doing so, we give various explicit formulae for newforms.
L-indistinguishability and R groups for the special linear group
In an earlier paper [7] we showed how to realize explicitly the irreducible constituents of unitary principal series representations of the special linear group &5,,(k) when k is a nondiscrete locally compact field of characteristic 0. We saw that the set of constituents has a canonical simply transitive group action on it by a finite abelian group, and we interpreted this finite abelian group as the Galois group of a canonical finite abelian Galois field extension of k. The various irreducible constituents of a single unitary principal series representation are L-indistinguishable in the sense of Langlands [2,24,25]. In fact, it is expected that they exhaust a whole equivalence class under Lindistinguishability. Our purpose in this paper is to generalize our parametrization of the members of these special L-packets, exhibiting for an arbitrary L-packet a simply transitive group action by a finite abelian group that is the Galois group of a canonical finite abelian Galois field extension of k. However, we can give only a relative proof of our results, because the Iinal definition of L-indistinguishability for SL,(k) awaits completion of results about the discrete series of GL,(k). We shall therefore cast our results as theorems in the presence of two widely believed “Working Hypotheses” about GL,(k). These hypotheses are stated in Section 4. Our main theorems are that these Working Hypotheses imply
An injectable and thermosensitive hydrogel: Promoting periodontal regeneration by controlled-release of aspirin and erythropoietin.
Periodontitis is an inflammatory disease induced by complex interactions between host immune system and plaque microorganism. Alveolar bone resorption caused by periodontitis is considered to be one of the main reasons for tooth loss in adults. To terminate the alveolar bone resorption, simultaneous anti-inflammation and periodontium regeneration is required, which has not appeared in the existing methods. In this study, chitosan (CS), β-sodium glycerophosphate (β-GP), and gelatin were used to prepare an injectable and thermosensitive hydrogel, which could continuously release aspirin and erythropoietin (EPO) to exert pharmacological effects of anti-inflammation and tissue regeneration, respectively. The releasing profile showed that aspirin and EPO could be continuously released from the hydrogels, which exhibited no toxicity both in vitro and in vivo, for at least 21 days. Immunohistochemistry staining and micro-CT analyses indicated that administration of CS/β-GP/gelatin hydrogels loaded with aspirin/EPO could terminate the inflammation and recover the height of the alveolar bone, which is further confirmed by histological observations. Our results suggested that CS/β-GP/gelatin hydrogels are easily prepared as drug-loading vectors with excellent biocompatibility, and the CS/β-GP/gelatin hydrogels loaded with aspirin/EPO are quite effective in anti-inflammation and periodontium regeneration, which provides a great potential candidate for periodontitis treatment in the dental clinic. Statement of Significance To terminate the alveolar bone resorption caused by periodontitis, simultaneous anti-inflammation and periodontium regeneration is required, which has not appeared in the existing methods. Here, (1) the chitosan (CS)/β-sodium glycerophosphate/gelatin hydrogels loaded with aspirin/erythropoietin (EPO) can form at body temperature in 5 min with excellent biocompatibility in vitro and in vivo; (2) The faster release of aspirin than EPO in the early stage is beneficial for anti-inflammation and provides a microenvironment for ensuring the regeneration function of EPO in the following step. In vivo experiments revealed that the hydrogels are effective in the control of inflammation and regeneration of the periodontium. These results indicate that our synthesized hydrogels have a great potential in the future clinical application.