Transfer of K-types on local theta lifts of characters and unitary lowest weight modules

In this paper we study representations of the indefinite orthogonal group O(n,m) which are local theta lifts of one dimensional characters or unitary lowest weight modules of the double covers of the symplectic groups. We apply the transfer of K-types on these representations of O(n,m), and we study their effects on the dual pair correspondences. These results provide examples that the theta lifting is compatible with the transfer of K-types. Finally we will use these results to study subquotients of some cohomologically induced modules

Derivatives for smooth representations of GL(n,R) and GL(n,C)

The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS12].Comment: First version of this preprint was split into 2. The proofs of two theorems which are technically involved in analytic difficulties were separated into "Twisted homology for the mirabolic nilradical" preprint. All the rest stayed in v2 of this preprint. v3: version to appear in the Israel Journal of Mathematic

Optimizing the monomer structure of polyhedral oligomeric silsesquioxane for ion transport in hybrid organic–inorganic block copolymers

Poly(ethylene oxide)-b-polyhedral oligomeric silsesquioxane (PEO–POSS) mixed with lithium bis(trifluoromethanesulfonyl)imide salt is a nanostructured hybrid organic–inorganic block copolymer electrolyte that may enable lithium metal batteries. The synthesis and characteristics of three PEO–POSS block copolymer electrolytes which only differ by their POSS silica cage substituents (ethyl, isobutyl, and isooctyl) is reported. Changing the POSS monomer structure results in differences in both thermodynamics and ion transport. All three neat polymers exhibit lamellar morphologies. Adding salt results in the formation of a disordered window which closes and gives way to lamellae at higher salt concentrations. The width of disordered window decreases with increasing length of the POSS alkyl chain substituent from ethyl to isobutyl and is absent in the isooctyl sample. Rheological measurements demonstrate good mechanical rigidity when compared with similar all-organic block copolymers. While salt diffusion coefficient and current ratio are unaffected by substituent length, ionic conductivity increases as the length of the alkyl chain substituent decreases: the ethyl substituent is optimal for ion transport. This is surprising because conventional wisdom suggests that ion transport occurs primarily in the PEO-rich domains, that is, ion transport should be unaffected by substituent length after accounting for the minor change in conducting phase volume fraction. © 2020 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2020 © 2020 Wiley Periodicals, Inc. J. Polym. Sci. 2020, 58, 363–371